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 Studies of Lightcone World Sheet Dynamics in Perturbation Theory and with Monte Carlo Simulations
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 GUDMUNDSSON, SKULI
 Copyright Date:
 2008
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 Conceptual lattices ( jstor )
Feynman diagrams ( jstor ) Gluons ( jstor ) Mathematics ( jstor ) Momentum ( jstor ) Monte Carlo methods ( jstor ) Quantum field theory ( jstor ) Scalars ( jstor ) Simulations ( jstor ) Vertices ( jstor ) City of Gainesville ( local )
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 University of Florida
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 3/1/2007
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STUDIr7. OF LIGHTCO'' WORLD I' T DY" AMICS
IN PERTURBATION THEORY
AND WITH MONTE CARLO '.,' IULATIONS
By
A Di: STATION PF \ .* TED TO THE GRADUATE SC 001
OF THE UNIV:i X I OF FLORIDA IN PARTIAL FULFI I E
OF THE WR.'UIT U I TS FOR, T;TT, DEGREE OF
DO( 'I OR (C: Pi i i,:OSOFIH .
UNIVERSITY OF FLORIDA
: ULI GUDMU[')
'Skuli (}udmiuiidssor
I dedicate this work to . children Darnfel, Lsa and Isar.
AC:T'OWLEDCTS
I would like to thank B. : : rmy supervisor, for his r .': ons and
Sent ':.ort and guidance during this research. I also thank him for his frien, 1 and
understanding when times were hard but ( y for his ability to keep me interested
and motivated.
T.. IFT Fellowship, which was awarded to me for the period '::' : was crucial
to the successful :::i)letion of this project.
I should also mention that rnmy graduate studies in the USA were supported in part
by the Icelandic Student Loan Institute as well as the Fulbright Institute which s G orted
me as a Fellow and finally the AmericanScandinavian Association ..orted me with its
S1... :i ..ors Grant." I thank these institutions for their financial support.
Mostly I am grateful for beloved children Daniel, Lisa and Isar, whom I miss more
than words can : .. : all the difficult times and darkest moments my children
were a1 : anchor, : .ic and : : life. To : : real family, those whom I could
a] fall back on: Mamma, pabbi, Inga, Kalli, Tobbi and their families I am forever
grateful. T1: stood by me through difficulties at times when it. was the most difficult.
Finally, I wish to thank the following: Alberto and Bobby (for a true friend ',, that
will never be forgotten); I!' 1 (for his calm and sensitive advice, but more importantly,
for his 1. .y as a friend), ( : (for his un . promising 1.. 1) and ' ort) and ( : (for
a] being my lifelong best friend).
TABLE OF CONTENTS
page
ACKNOW LEDG 11 1 NTS .................................
LIST O F TABLES . . . . . . . . . .
LIST OF FIGURES . . . . . . . . .
A B ST R A C T . . . . . . . . . .
CHAPTER
1 INTRODUCTION ..................................
1.1 Lightcone Variables and Other Conventions .................
1.2 't Hooft's Large Nc Lim it ............................
1.3 W ork on Planar Diagrams ...........................
2 LIGHTCONE WORLD SHEET FORMA II. I .. ................
2.1 Introduction to the Lightcone World Sheet Formalism ............
2.2 Supersymmetric Gauge Tli. i ..........................
2.2.1 SUSY YangMills Quantum Field Theory ..............
2.2.2 SUSY YangMills as a Lightcone World Sheet . . .
3 PERTURBATION THEORY ON THE WORLD SHEET . . . .
3.1 Gluon Self Energy . . . . . .
3.2 OneLoop Gluon Cubic Vertex: Internal Gluons .....
3.2.1 A Feynman Diagram Calculation ..........
3.2.2 Simplification: The World Sheet Picture .....
3.3 Adding SUSY Particle Content: Fermions and Scalars .
3.4 Discussion of Results .....................
3.5 Details of the Loop Calculation . . .....
3.5.1 Feynmandiagram Calculation: Evaluation of F' .
3.5.2 Feynmandiagram Calculation: Evaluation of FV .
3.5.3 Feynmandiagram Calculation: Evaluation of FV^ .
3.5.4
3.5.5
Feynmandiagram Calculation: Divergent Parts of Ini
Details of SUSY Particle Calculation . ...
. . . 4 1
. . . 44
. . . 45
. . . 53
. . 53
. . . 55
. . . 58
. . . 58
. . . 59
. . . 6 1
tegrals and Sums 63
. . . 68
4 THE MONTE CARLO APPROACH . . . . .
4.1 Introduction to Monte Carlo Techniques . . . .
. . 72
. . 72
4.1.1 Mathematics: Markov Chains . . .
4.1.2 Expectation Values of Operators . . .
4.1.3 A Simple Example: Bosonic Chain . . .
4.1.4 Another Simple Example: ID Ising Spins . .
4.1.5 Statistical Errors and Data Analysis . ...
4.2 Application to 2D Tr 3 . . . . .
4.2.1 Generating the Lattice Configurations . ..
4.2.2 Using the Lattice Configurations . . .
4.2.3 Comparison of small M results: Exact Numerical vs.
4.3 A Real Test of 2D Tr3 . . . . .
APPENDIX
A COMPUTER SIMULATION: DESIGN . . . .
A.1 Object Oriented Approach, Software Design . . .
A .1.1 Basic Ideas . . . . . .
A.1.2 Object Oriented Programming: General Concepts and
A.1.3 Organization of the Simulation Code . . .
A.2 Description of the Computer Functions . . .
Monte Carlo .
Nomenclature
B COMPUTER SIMULATION: EXAMPLE HEADER FILE . . . 127
REFERENCES . . . . . . . . . 130
BIOGRAPHICAL SKETCH . . . . . . . . 132
73
77
78
79
82
86
87
92
97
101
LIST OF TABLES
Table
21 Lightcone Feynman rules . . . .
41 Monte Carlo concepts in mathematics and physics .
42 Test results for ID Ising system. . . .
43 Basic spinflip probabilities . . . .
44 World Sheet spin pictures. . . . .
45 Monte Carlo vs. Exact results for M = 2 . .
A1 Software file structure. . . . . .
page
. . . . 27
. . . . 77
. . . . 81
. . . . 88
. . . . 91
. . . . 101
. . . . 117
LIST OF FIGURES
Figure
11 Example of 't Hooft's doubleline notation. . . .
12 Various ways to draw Feynman diagrams. . . .
21 World Sheet picture of free scalar field theory. . . .
22 Grassmann field snaking around the World Sheet. . ...
23 Quartic from cubics for a simple case. . . . .
31 One loop gluon self energy. . . . . .
32 Cubic vertex kinematics. . . . . . .
33 One loop diagrams for fixed I in the Lightcone World Sheet. .
41 Test results for the simple example of a bosonic chain .
42 Test results for ID Ising system. . . . .
43 Examples of allowed and disallowed spin configurations. .
44 Basic double spinflip. . . . . . .
45 Double spinflip, with distant vertex modification. . ...
46 Fitting MonteCarlo data to exp. . . . .
47 Energy levels of a typical QFT. . . . . .
48 Monte Carlo vs. Exact results for M = 2. . . .
9 Monte Carlo vs. Exact results for M
10 Monte Carlo vs. Exact results for M
11 M 12 D&K and MC comparison. .
12 Data analysis and data fitting for M
13 AE as a function of 1/a. . ...
14 AE as a function of x. . ....
2, cyclicly symmetric observable.
12 simulations..................
12 simulations ..............
page
17
20
29
36
39
42
45
55
80
82
89
90
92
96
98
102
. 103
. 104
. 107
. 108
. 109
. 109
415 AE as a function of M . . . . . . . 110
A1 Organization of the computer code. . . . . . . 119
A2 Operations of the Monte Carlo layer. . . . . . . 123
Abstract of Dissertation Presented to the Graduate School
of the University of F i rida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
i OF L iC( i V'.)l! i) C': i i DYNA HiCS
IN PERTURBATION THEORY
AND WITH MONTE CARLO : IULATIONS
By
Skuli Gudmundsson
December
CI ... ": C es rn
Ma'. Department: F:
In this thesis we l :y < uter based simulation techniques to tackle large Nc
.. ium field thec: on the lightcone. The basis for the investigation was set by Bardakci
and T.. in :"'* when they showed how planar diagrarns of such theories could be
mapped into the dynamics of a worldsheet. We call such a reformulated field theory a
S7 .'.. world sheet.
7: T Lightcone World Sheet formalism offers a new view on lightcone field theories
which allows for .: lications of stringtheory techniques, the greatly augmenting the
machinery available for investigating such theories. As an investigation of the Lightcone
World Sheet itself the current work .! stochastic modelling techni m ., as a means of
cla: ; : : the formalism and developing intuition for further work. Seen as an investigation
of the und um field thec the : ch taken here is to capitalize on the
only two dimensions of the worldsheet as opposed to the four dimensions of the field
theory.
CHAPTER 1
INTRODUCTION
In this dissertation we will present a connection between field theories and string
theory which has enjoyed an increasing interest by researchers in the field of theoretical
high energy physics. This is true especially since Maldacena proposed that IIB supersting
theory on an AdS5 x S5 background is equivalent to supersymmetric YangMills field theory
with extended A = 4 supersymmetry (a (.. iii t i'e called the AdS/CFT correspondence
[1]). This is the view that although string theory has so far been unsuccessful in its task of
predicting experimental results as an all inclusive th., .ii vof i. iii,.: in its own right, its
techniques are potentially applicable in describing mathematically limits or views of older
field theories which themselves have a strong experimental foundation.
The string techniques in field theory have a myriad of possible applications. Wherever
the standard perturbative advances fail, some other technique is needed, the confinement
problem in QCD being an obvious example. Mechanisms which have been proposed
to confine the quarks of QCD include color flux tubes and gluon chains. A complete
description of the mechanism would of course eventually involve a stringy limit of
the underlying field theory. It may be that a string description of particular physical
mechanisms, and not of the theory as a whole, is what will prevail. The idea dates back
to 1970 [2] and [3] and was further established by t'Hooft in 1974 [4]. t'Hooft's approach
was to construct a systematic expansion in 1/NA for a general field theory, where the
meaning of N, could be made definite for "nonGauge" theories by means of a global
SU(Nc) symmetry group acting on a matrix of fields. The 1/NA expansion turned out to
single out planar Feynman diagrams in expressions for observables of the theory, planar
in the sense that in his doubleline notation .r v could be drawn on a plane i.e., with no
lines intersecting. When the planar diagrams in t'Hooft's original work were selected, the
approach i.., . 1 a world sheet description which was nonlocal on the sheet. A similar,
but local, world sheet was much later explicitly constructed by Bardakci and Thorn [5]
and their formalism is essentially the backdrop and foundation to the work presented here.
We will in this dissertation present work that has been done in the framework of
this explicit local world sheet description of wellknown field theories. In Chapter 1 we
discuss this setting in general, the planar diagram approach and the lightcone. In Chapter
2 we present in some detail the Lightcone World Sheet established by Bardakci and
Thorn and later developed by them and others. The approach is to maintain clarity and
leave details and technicality in the original work. Next we turn to the two themes of
the dissertation. In Chapter 3 we discuss standard field theoretic perturbation theory in
the Lightcone World Sheet setting; since the world sheet mapping is done diagram per
diagram this should yield entirely known results but now in the context of a different
regulator. This chapter is almost entirely a recap from articles which I coauthored
[6] and [7]. It serves both to familiarize the reader with the setting and to test the
formalism in the perturbative limit. Chapter 4 deals with the formalism in a different
and complementary way, namely by use of numerical methods which are most effective
in the strong coupling regime. It was outside the scope of the current thesis to complete
the numerical investigation for other theories than the simplest ones that the Lightcone
World Sheet formalism had been developed for, namely two dimensional f3 scalar field
theory. We employ Monte Carlo methods similar to those of lattice gauge theory but on
the world sheet system describing the field theory. Although the general methodology of
Monte Carlo simulations has been well established, not only in lattice gauge theory and
statistical physics but in a wide range of applied mathematical settings, the application
here is completely novel and required the development of specialized computer programs.
It was partly for this reason that time did not permit a numerical investigation of more
interesting field theories, but mainly the reason was that at the time when this work was
done, the renormalization procedure for the more complicated theories had not yet been
developed. Such a renormalization is of course necessary for a direct numerical .:.: iech
since infinities cannot be handled by floating point numbers in a computer. i : program
could still be ,.. )loyed to the two dimensional (d scalar field t.. .. , since it is finite to
begin with. I : dissertation concludes with appendices that describe the implementation
of the : iuter simulation. T do not contain the i.licit source code, since this would
make for countless but rather explains and describes the .... uter code and its
development and organization. T:. actual source code can be obtained from me via email
should the reader be interested.
1.1 Lightcone Variables and Other Conventions
One starts out with so called r '..' variables which for D dimensional Minkowski
vectors X5P are defined as
xK + ( xD1)
2
x (1 1)
Stransverse components X k are often denoted by vector bold face type, so that the
Minkowski vector is presented by coordinates. (X+ X, X). T: Lorenz invariant scalar
product of vectors has the form X PY XI Y" X Y X+Y XY+ in the
Lightcone variables.
When D = 4, a case we find ourselves dealing with from time to time. then it is
sometimes convenient to use polarization defined by
xA (XI + 1x)
Xv (XI iX2). 12)
T.. the coordinates are (X+, X, XA, XV) but we do not use a special index because
this choice of coordinates can be treated in the same fashion as Cartesian ones, so that '/k
can mean k = 1, 2 but also k = A, V. Here the Lorentz invariant product can be written
XpYj, X^ + XvY^ X+Y XY+.
To clarify the meaning of these Lightcone coordinates we look a little closer at a
simple example. Consider a scalar field theory with Lagrangian density:
haa 1 (
= 02m2 V(0). (13)
We wish to study this theory in the canonical formalism, with x+ representing time, i.e.,
the evolution parameter. We write 40 =0+) so that
9,ap = (Vr)2 2 Q a9_ and then
In H _1 (14)
where II is the conjugate momentum to 0. We shall sometimes want to work with the
Lightcone Hamiltonian density, written as
P = (VO)2 + 2 + v(W) (1 5)
2 2
This procedure goes through analogously for Gauge theory, although the algebra is a
bit more cumbersome, as we shall see in section 2.2.1. Let us consider a pure U(NA) Gauge
Theory with Lagrangian:
1
S= Tr (GG) (16)
where
G/,, ,A, ,A, + i g [A, A,] (17)
A Z A T2 (18)
and the rs are the generators of U(NA) (the quantum operator nature of A. resides in the
coefficients A2). Now the index a labels the matrices in the chosen representation of the
Lie algebra U(NA) which is determined by the structure constants fabc by the relation:
[r, b] I fabcTc (19)
Since the matrices Ta form a basis for the U(Nc) algebra the indices must run from 1 to
NA the dimension of the algebra. A representation of U(Nc) is a set of matrices which
satisfy relation (19), and the so called fundamental representation is the set of Nc x Nc
unitary matrices (hence the label U(Nc)). Another such set (which satisfies (19)) are
the structure constants themselves, namely fabc thought of as a matrix, with the index a
labelling which generator and indices b and c labelling the rows and columns of the matrix.
The structure constants are in this way said to form the adjoint representation of the
algebra U(Nc). Notice that in this representation the matrices are Nc x Nc but there are
still just N) degrees of freedom, i.e. U(Nc) is found here as a small subalgebra among all
matrices of this size.
1.2 't Hooft's Large Nc Limit
In 1974 a paper [4] was published by 't Hooft explaining how a systematic expansion
in 1/Nc for field theories with U(Nc) symmetries could be achieved. The sheets on which
Feynman diagrams are drawn were classified into planar, onehole, twohole, etc. surfaces
and it was shown that this topological classification corresponds to the order of 1/Nc
in the expansion. In his article 't Hooft pointed out that this was also the topology of
the classes of string diagrams in the quantized dual string models with quarks at its
ends. He further pursued the string analogy by going to Lightcone frame and proposed
a worldsheet nonlocal Hamiltonian theory which would sum all Feynman diagrams
perturbatively with Nc  +oc and g2Nc fixed. The coupling of g2Nc has since been
called the 'tHooftcoupling and his worldsheet model was the beginning of extensive
considerations along the same lines by theoretical physicists. Since this work by 't Hooft
most certainly was the foundation on which the Lightcone WorldSheet picture later built,
we present here the basis of 't Hooft's results and notions which became standard in this
field.
't Hooft's arguments work equally well with pure U(NA) gauge theories as 'l. v do
with U(NA) gauge theories coupled to quarks or even scalar matrix field theories with
global U(NA) symmetry. 't Hooft presented his arguments for a theory coupled to quarks
with Lagrangian given by
L = Tr ( GP" 7 (,,D, + Tm(r,)) 4r with (110)
D rY = r + gA rY (111)
where the index r = 1,2, 3 labelled the quark family:
I1 p; 2 n; 3 A, (112)
each one being a vector which the fundamental representation of U(NA) can act on; i.e.
the boldface type 4, indicates a columnvector (and the 4, a linevector) which the matrix
indices of A. can act on. The covariant derivative in Eqn. (111) for example contains the
matrix multiplication of A. with 4r. The G are as in Eqn. (17).
The Feynmanrules are obtained in the usual manner and l,. v are nicely assembled
with figures in 't Hooft's original article. To illustrate his so called doubleline notation
we consider the pure Gauge propagator and cubic vertex, as shown in top two pictures in
figure 11. The lower two pictures in that same figure show the same obi. t in the double
line notation. This notation is essentially a convenient way of organizing and keeping track
of the color group U(NA). Since A, is in the fundamental representation of U(NA), then
there are N2 degrees of freedom associated with it, represented by the matrix elements of
the A. with A. = At, the d .., r both transposing the matrix and taking the conjugate
of all elements. The vector and matrix indices i, j of the fundamental representation,
going from 1 to NA, are denoted by an arrowed single line, incoming arrow for row vectors
and outgoing for column vectors. It is then clear that fermions being NAvectors will be
represented by single lines only but gauge bosons being Nc x N, matrices, by double
lines, incoming and outgoing. The Kronecker delta's identifying the matrix indices in the
traditional Feynman rules are directly and manifestly implemented by connecting the
single lines at the vertices, as shown in figure 11 below.
(k,) (i,) gpk6jl/(k2 1)
(n,j)
S (gav(k q)p/ +gap(P k), +g,(q P)a
P q
(1, k) P (m, i)
i i
SGauge Propagator as above without 6s.
SJ Cubic vertex as above without 6s.
Figure 11. Example of 't Hooft's doubleline notation. Each line hold a single matrix
index i or j resulting in the Gauge propagator to be represented by two
such lines. This manifestly implements the Kronecker delta's present in the
propagators as shown. An incoming arrow on a single line denotes that the
index it holds refers to a row and an outgoing arrow to a column index. This
means that fermion propagators (not shown) would be represented by single
lines.
The double line notation can also be implemented for FadeevPopov ghosts in the
Feynman gauge. In this notation, consider a large Feynman diagram with the Kronecker
delta's manifestly implemented graphically. A general such diagram would often require
the double lines to twist, i.e., one going over the other. If we imagine that linecrossing is
forbidden, then the twist can still be achieved by inserting a "wormhole" in the sheet on
which the diagram is drawn. It is therefore clear that with the Feynman rules according
to the doubleline notation, and with linecrossing forbidden, the sum over all diagrams
would have to include the sum over all topologies of the sheet on which the diagrams
are drawn. Notice also that if a single line closes in a contractible loop, the resulting
amplitude, however complicated, will have a factor of i ii = Nc. If the loop, however,
is not contractible, meaning that the surface on which it is drawn is not planar, then the
index sum does not decouple from the expressions to yield the factor of N,. A rigorous
proof that amplitudes drawn on Hhole surfaces will be suppressed from the planar ones
by 1/NH, is given in 't Hooft's paper [4] using Euler's formula: FP+V 22H relating
the number of faces (F), lines (P), vertices (V) and holes (H) of a planar shape. The limit
Nc +oc with the 't Hooft coupling held constant therefore singles out planar diagram.
Furthermore, the topology of the sheets on which the diagrams are drawn constitutes a
means of systematically improving the zeroth order Nc  +oc limit.
The concept of "the sheet on which Feynman diagrams are drawn" is not perhaps
fully useful until the Lightcone world sheet concept introduced in the next chapter
identifies this sheet with the world sheet itself. Then the topology is not that of "the
surface on which to draw the Feynman diagrams" but rather the topology of a world sheet
which itself directly holds the dynamical variables. The Lightcone parametrization enables
us to identify the p+ component of propagators with a space coordinate on this sheet and
x+ with the time coordinate. This means that first order 1/Nc corrections to the work
presented here this have a prescription in terms of the topology of the Lightcone World
Sheet.
1.3 Work on Planar Diagrams
The approach due to 't Hooft, considering quantum field theories with U(Nc)
symmetry perturbatively in 1/Nc, is particularly intriguing because it organizes the order
in perturbation theory according to the topology of a sheet which immediately calls for
an analogy with a string theory world sheet. Lightcone variables offer a parametrization
of the sheet on which one would draw Feynman diagrams, using x+ and p+ and further
promoting x+ to an evolution parameter. Each Feynman diagram can in such a way be
seen as a world sheet as shown in Figure 12.
The above way of identifying each Feynman diagram with a world sheet was further
implemented by Bardakci and Thorn and is the basis for the Lightcone World Sheet
formalism, which is the framework on which the work in this thesis is based. As was
discussed above, the stringfield theory connection has been an active subject of research
ever since the Maldacena AdS/CFT correspondence emerged. The development of the
Lightcone World Sheet idea is an rare example of the theoretical attempt at constructing
a string world sheet which sums planar diagrams of field theory. More emphasis has
been put into the opposite view, i.e., to recover field theoretic physics from the string
formulation. As was pointed out in Thorn and Tran [8] the next task is to develop a
tractable framework for extracting the physics from this new worldsheet formalism. The
first steps of organizing the renormalization in this picture were taken alongside developing
the supersymmetric Lightcone World Sheet [7] and are further repeated here as part of the
graduate work of the author. The necessary conclusion was later published in an article
which summarized the renormalized f3 worldsheet [9].
Previous work on summing planar diagrams should not be forgotten. Ideas and
methods of considering special cases of planar diagrams, those which would describe
simple models for forces between quarks, have been pursued. Among those, fishnet
diagrams are particularly interesting [10] because of how 'l. v, or at least fishnetlike
diagrams, seem to be emerging from the Monte Carlo studies presented later in this
dissertation.
The conclusion is that fishnets come about as particular subsums in the sum over
all planar diagrams if viewed in the Lightcone World Sheet picture; in other words, ],l v
3
4 4 4
.... .. 5 5
 6 
10 8 99
10 8 10 10 8
11 9 11 11
12 12 12
lure 12. Various to draw F : :... diagrams. T, figure shows various
a certain perturbative expression for an amplitude can be graphically
represented, a tcchrniU : invented and named after nnman. To the right the
traditional F. nman diagram is shown with pro .' ."s, i.e. field correlations,
depicted '( <: and straight lines. In the figure on the left the same graph
is drawn using 't Li:.oft's doubleline notation. ::: the middle is the scheme,
also by 't Hooft, which draws the Feynman diagram as a world sheet with each
numbered s ., e area of the gr.. '. ., I:: to the ]: : '.! rs.
are contained in that description. In contrast to the work presented in this thesis. the
Lightcone ''. ld Sheet has been investigated with an entirely different approximation
scheme, namely the socalled ,, :... 1 ', '. constitutes the next step
towards ':: ; the ::: erturbative from the new worldsheet formalism.
We will briefly discuss this method and the results for the Lightcone World Sheet
obtained from it. Although some knowledge of the Lightcone World Sheet is necessary
to understand this discussion, it still belongs here because it is really not part of the main
themes of the thesis.
T: mineainfield approximation effort to analyzing the Lightcone World Sheet model
for field theories (for simplicity, the scalar Q3 field theory) started soon after the original
formalism was put forth in [11]. In their paper, the authors review the Lightcone World
Sheet formalism as a means of realizing the sum over planar graphs by coupling the world
sheet fields to a two dimensional Ising spin system. They then regard the resulting two
dimensional system as a noninteracting string moving in a background described by the
Ising spin system. The meanfield approximation is therefore applied to the spin system
so that a qualitative understanding of the physics of the sums of planar graphs could be
achieved. This meanfield work was done in stages, where refinements and improvements
were constantly being published, and it is of course an ongoing effort since the more
complicated theories, such as QCD and Supersymmetric Gaugetheory, have yet to be
tackled. There are more recent views of this project [12, 13]. We will concentrate on the
early developments for sake of simplicity; the refinements and especially the string theory
implications are beyond the scope of this thesis.
As will be clear after the introduction to the Lightcone World Sheet formalism,
the Ising spin system describes the Feynman diagram topology of each graph. For
now it is sufficient to note that the Ising system represents the lines making up the
smaller rectangles in the center graph of Figure 12. Notice that as we go to higher
order perturbative Feynman diagrams, the solid lines become more numerous and in the
asymptotic regime we can imagine a limit where the lines acquire a finite density on the
world sheet. The authors of the meanfield papers [11, 14] refer to this mechanism as the
condensation of boundaries, and depending on the dynamics investigations on whether
string formation occurs through this condensation was considered in some limiting cases.
Further work on string formation and the implication is also found in later articles [12, 13].
To give the reader a taste of the meanfield idea, we will elaborate briefly on the
original attempts [11]. To keep track of the solid lines which in turn keep track of the
splitting of momentum between propagators, i.e., the topology of Feynman diagrams, a
scalar field Q on the world sheet is introduced which takes the value 1 on lines and 0
away from them. The authors then build a Lagrangian density through the Lightcone
World Sheet formalism on a continuous worldsheet with a simple cutoff (see chapter 2 for
a thorough discussion of the formalism). A continuous field Q replaces the Ising like spin
system introducing a cutoff so that the worldsheet boundary conditions only exactly hold
when the cutoff is removed while maintaining it amounts to imposing an infrared cutoff.
The constraints Q = 0 and = 1 are implemented by a Lagrange multiplier 7r(, 7), so
that '],. v end up with two fields 0(j, 7) and 7r(a7,7) on the worldsheet. These two fields
are treated as a background on which the quantum fields live, the authors then compute
the ground state of the quantum system in the presence of the background and then solve
the classical equations of motion for the background fields thereby minimizing the total
energy. In later refinements [14] the meanfields were taken to be scalar bilinears of the
target space worldsheet fields, which makes the meanfield approximation more clearly
applicable but obscures the interpretation of the meanfield as background, representing
the solid lines. A further extension to the meanfield approach was later published [8].
Here, instead of a uniform field 0(j, r) = on the worldsheet and representing the
"smearedout" solid lines (condensation), two fields 0(a, 7r) and 0'(a, ,r) are introduced at
alternatingt sites on a discretized worldsheet lattice. These fields, although each is taken
to be homogenous on the worldsheet, allow for inhomogeneity in the distribution of solid
lines. Note in particular that = Q' corresponds to the previous work but = 1 and
' = 0 corresponds to an ordering of solid lines reminiscent of an antiFerromagnetic Ising
spin arrangement. Such an arrangement of solid lines yields a socalled fishnet diagram
Not to be confused with the quantum matrix field Q in the original scalar f3
Lagrangian density.
t Not exactly alternating sites is required to reproduce the fishnet diagrams, but rather
spin configurations of the type 1, T, T, 1, T, T, T, I, ....
if written as a traditional Feynman diagram according to figure 12. Such diagrams
have been selectively summed before, but the novelty offered in this picture is that
configurations away from Q 1, t' = 0 take all other planar diagrams into account, in
an average way. In other words, the treatment of solid lines as in the meanfield approach
allows for a different way of organizing Feynman diagrams and approximating differently,
namely across all Feynman loop orders.
The Monte Carlo approach, presented in chapter 4, offers yet another approach. It
treats the solid lines mentioned above in a stochastic way. The terms selectivesummation
or importancesampling, often seen in Monte Carlo applications of statistical physics,
describe quite well how this approach works. A closer look, and a more meaningful one, is
reserved for chapter 4.
CHAPTER 2
LIGHTCONE WORLD SHEET FORMALISI.
In this chapter we give an overview of the construction we call the Lightcone World
Sheet. The formalism was originally invented by Bardakci and Thorn [5] and was
presented as a concrete mechanism to see the world sheet behavior of large N, matrix
quantum field theory. In that paper the authors establish the first Lightcone World Sheet
from a scalar matrix quantum field theory with an interaction term of gTro3/V N and
indicate how the the approach might be extended to more general field theories. This
extension is carried out in a number of later papers by Thorn and collaborators. The
formalism is the fundamental backdrop to the work presented in this thesis, since the
world sheet picture allows for new interpretations and methodology for investigating the
field theory, among others the Monte Carlo techniques of chapter 4. The Lightcone World
Sheet describes a wide collection of field theories in terms of a path integral over fields
that live in a two dimensional space referred to as the world sheet. The mechanics of the
procedure is described in this chapter.
2.1 Introduction to the Lightcone World Sheet Formalism
Consider the simplest case of a Lightcone World Sheet, namely the one presented by
Bardakci and Thorn in their paper from 2002 [5], describing a scalar matrix field theory
with cubic interaction. In order to shed light on the main concepts of the mechanism it
is in order to sketch in some detail the steps presented therein, since as a field theory it
is the simplest possible case, and because the matrix scalar field theory will be applied
later when we look at Monte Carlo studies. Consider therefore the dynamics of the planar
diagrams of a large Nc matrix quantum field theory with action given by
S = Tr { pa v+! m2& } (21)
Here 0 is a NAbyNA matrix of scalar fields and the derivative is taken elementwise in the
matrix. The propagator is given by
where a, 3, 7, 6 E {1, 2,..., N}. Using the double line notation due to t'Hooft as explained
earlier the Greek indices correspond to the "color" of the lines. As t'Hooft prescribed, we
deal with the colors diagrammatically and since we will be taking the large N, limit, thus
ignoring all but planar diagrams, we suppress the color factors S6 &3 altogether in what
follows. Introducing next Lightcone coordinates defined for a Ddimensional Minkowski
vector xP as
x (X= 0 Dl) /2. (22)
There is no transformation of the remaining components, and we distinguish them
instead by Latin indices, or as vector boldface type. The coordinates are (x+, x, xk) or
(x+, x, x) and the Lorentz invariant scalar product becomes x y x= x y x+y xy+
. By now choosing x+ to be the quantum evolution operator, or "time", its Hamiltonian
(, ,iii: p = p2/2p+ becomes the massless onshell "energy" of a particle. We choose
the variables (x+,p+,p) to represent the Feynman rules and arrive at the following
expression for the propagator:
This is called "mixed representation", i.e., Fourier transforming back the p variable
but retaining the momentum representation in the other components.
D(x+,p+,p) = eA(p)
J 271
0(x+) _ix+p /2p
2p+
where we have assumed p+ > 0. Next, discretize "imaginary time" and "momentum"
(p+ = Im, ix = ka with I 1,2,..., M and k = 1,2,..., N) and define To = m/a. The
expression becomes
D(x+,p+,p) (k) kp2/2lT
21mrn
So far the steps may admittedly seem adhoc and reminiscent of a cookbook recipe, but
notice that this propagator can be associated with a rectangular grid with width M and
length N. Furthermore, the imaginary time transcription has been shown to be analogous
to the analytic extension of the Schwinger representation to a real exponential [15],
which corresponds to the normal Wickrotation. We wish to associate the mathematical
expression for the propagator with a path integral of local variables on this grid. To do so,
notice that a propagator always connects two vertices so as long as some final expression
for an amplitude or other Feynman diagram calculation contains all terms and factors,
we are free to redefine the rules for the diagrammatic construction of the expressions. In
this line, we can assign the factor of 1/1 present in the propagator, to one of the vertices it
connects. Since Lightcone parametrization only allows for propagation forward in time, it
is meaningful to assign the factor to the earlier vertex connected by the propagator. This
creates an ..i,iin. Iry between fission and fusion vertices which now require independent
treatment. Instead, propagation becomes a simple exponential which was the goal. The
resulting represcribed Feynman rules are summarized in Table 21.
Table 21: Lightcone Feynman rules. The arrows denote the flow of "time" ix+.
9 1
87T3/2To MI+M2
1 2
8\3/2TO MIM2
1 2
S kp2/21To
The local world sheet variables alluded to above, are now within reach. Write the
total momentum as a difference p = qM q and define:
S = Sg + Sq
M1
Sq 20 (qI l q)2 (23)
j i=0
M2
S, T I c + M b+E I bj) (1 I c() (24)
f 1 i= 1_
Then
ex1{ /2 / ( 1 7 77
S(qM 2 )2 1 H I dq eSgSq (25)
SN j=1 i= 1
where bk, c are a pair of Grassmann fields for each point (i, k) on the grid (or lattice).
We implement qkM qM and q k q0 for all k by putting in Jfunctions. Proof of the
above identity and discussion of useful intermediate results are found in the original paper
[5], but neither is necessary to appreciate the fact that we have here a world sheet local
representation of the free field theory. To put it less dramatically, we have spread out the
very simple exponential momentum propagation over the width of a world sheet. We can
even write the above mentioned 6functions as discretized path integrals of an exponential
emphasizing the interpretation of (25) as a path integral over an N x M discretized world
sheet grid. Even though the construction is a bit cumbersome considering that we still
only have the free field theory, the important point to notice is that here is a completely
rigorous mechanism by which the dynamics of particle is described in the language of a
string. By using the same language, comparison between the actual dynamics of theories is
possible. We shall see, in the case of Gauge theory, that dramatic simplification occurs and
that the world sheet locality is truly a strong condition.
The formal continuum limit of expression (25) for the propagator is
Tee Dq DbDceso (26)
where
N M1 j
DqDbDc < Hdq'
j= 1 i= 1
So = dr da blc' (q )2 < S
The expression (26) is precisely the infinite tension limit of the bosonic Lightcone string.
Let us pause and summarize this Lightcone World Sheet picture of a propagator in
free scalar field theory. Figure (21) shows how easily this description lends itself to a
graphical representation: We have a world sheet with length T = Na and width p+ = Mm
organized as a lattice. On each site (i,j) E [1, M] x [1, N] lives a momentum variable q'
and a pair of ghost fields bi, c(. The discretization serves as a regulator of the theory.
We continue now to explain the scalar field theory but in less detail and refer the
interested reader to the original paper. In the Lightcone World Sheet interpretation, cubic
vertices of the field theory are places where the world sheet splits into two world sheets.
This is implemented by drawing solid lines on the grid where boundary conditions for b, c
and q are supplied as for the original world sheet boundaries. Vertex factors that must
be present according to the Feynman rules should somehow be inserted at beginnings and
ends of solid lines. It turns out that this can be done by locally altering the action. For
.v  4:4+4::+4 :4 :
44444
I I I I I I I
I I I I I I I
I I I I I I I
N  .  ...........
I I I I
I I I I
2 ,,<i ,i,
I I I I I I I I
1 2 i M
,ure 21. World Sheet picture of free scalar field theory. Ti.. figure shows the
diagrammatic setup for free scalar field theory in the Lightcone World Sheet
formalism. Ti.. picture describes a pro,. .*...' in Lightcone variables with
p = mM and p according to thie boundary values of thle fields q. T
r evolves in .... 7 ix+ aj and the ghost fields b and c
make sure that the world sheet interacts properly with other world sheets once
interactions are added.
example, to obtain the fusion vertex above we subtract one b, c link at a timeslice k just
after the end of the solid line giving rise to a factor of 1/MI
MM1
1_ p/2MTb '; I
exp b + ,_c(_+ (b b)(c c)  (,11 )2
I ex t11 2 ] i( o
T.. detailed manipulation of altering the action and inserting &functions to
implement bound: conditions can be done e namically by introducing an Ising spin
S.'m on the world sheet grid. Consider '. another set of world sheet (I namical
variables 4 which are equal to I1 if there is a solid line (a bound at site (i.j) and 1
if there is none. T .. local action manipulation terms, creating boundaries and inserting
1/M and couplings for vertices, are then multiplied with appropriate combinations of six's
so that they are present where ',., v should be. For example, at endpoints of solid lines we
have s = sj+1 so a factor of (1 + s')( + s )(1 s)/8 is 1 at the beginning of a solid
line but 0 otherwise, whereas the factor (1 s+)(1 + st)(1 + s~ )/8 is 1 at the end of a
solid line but 0 otherwise. We then put an overall sum over all s' configurations in front of
the whole path integral and the resulting expression then represents the sum of all planar
Feynman diagrams:
/N M1 d7 d d 7
Tf, N d exTp S e+ bi [v' + vP']}
SexP{2q 4 })
s 1 j 1 i 1 i 
exp iY (q qi Pi + (  PbP ) In 2 bPI[P1[P j bc
I i3
+ (bic b _) (1 P ,i_ I
where P7 = (1 + s')/2 and P (P) is a combination of s's which is 1 at beginnings (ends)
of solid lines that are at least 2 time steps long, and 0 otherwise.
The Isinglike spin system sj is clearly introduced simply to manipulate the presence
or absence of terms which have to do with boundaries of the world sheet. The spin system
is completely new and has no counterpart in the original field theory. In a way it is
precisely what makes the strong coupling regime reachable by this world sheet approach
as compared with Feynman diagram perturbation theory or lattice quantum field theory.
The various schemes to tackle the world sheet, by Monte Carlo simulation as will be done
here, or by introducing a meanfield [11, 14] or the antiferromagneticlike configuration
considered by Thorn and Tran [8], all involve a particular choice for the treatment of the
spin system, and of course, the treatment of renormalization. What happens to the spin
system in the continuum limit is along with renormalization the most interesting question
to be asking the Lightcone World Sheet.
2.2 Supersymmetric Gauge Theories
Casting pure YangMills theory into Lightcone World Sheet form has been done by
Thorn [16] shortly after the appearance of the first article. Since a unified and slightly
refined representation for general Gauge theories later appeared [7] we skip over the
otherwise crucial step in the development of the Lightcone World Sheet constructions, and
turn right to Supersymmetric Gauge Theories.
2.2.1 SUSY YangMills Quantum Field Theory
In their paper, Gudmundsson et. al. [7] build the extended = 2 and = 4
supersymmetric Gauge theory by means of dimensionalreduction. This method starts off
with an = 1 supersymmetric Gauge theory in higher dimensions and then reduce the
theory to D = 4 by making the fields independent on the extra D 4 dimensions. This
automatically creates the correct number of fields for the extended supersymmetry. The
Gauge bosons associated with the extra dimensions become just the scalars when their
Gauge symmetry in the extra dimensions becomes a global symmetry and similarly the
higher dimensional representation of the Chfluid algebra generates just the right number
of fermions. This method is particularly useful on the Lightcone World Sheet because
making the fields independent upon the extra dimensions can easily be implemented by
setting the extra q components equal to zero on the boundaries of the sheet. They are still
allowed to fluctuate in the bulk, which allows them to participate in the crucial generation
of quartics from cubics as will be described in the next section. In order to carry out the
mapping of theories
(AN,D) (1, 6) (N, D)= (2,4) and
(, D)= (1,10) (A, D)= (4,4)
we need to formulate the A = 1 supersymmetric Gauge theory in Ddimensions. The
Lagrangian density we start off with is given by
L = TrFFl' + iTrF0F(,,' ig [A,, ]) (27)
Fl = ,A A ,A, ig [A, A,], (28)
where FP are the D dimensional Dirac gamma matrices. Lightcone gauge dictates A_ = 0
and A+ is eliminated using Gauss' law. The "time" evolution operator P is obtained in
order to read off the Lightcone Feynman rules:
P dxdx (T + C +Q) (29)
where the individual terms which give the various vertices of the Feynman rules are given
by the expressions below. Using the Lightcone Dirac equation and Gauss' law in the
A_ = 0 Gauge one arrives at
T TrOAjAj i Tr bt a
C = igTrOAk A A Ak A A Ak + Tr )cb [ Ak ']
g + k Ok }b
!Tr{ [ctAl(k nk)cb b b V. A, b)
Q = Tr 2[Ak, OAk + AAj[Ai,Aj] + + Tr {,', t} b bt (210)
ig2 Tr ( [_Ak, Ak]{ t}) + Tr{ [CA](fk + )nkcb [Ak ,,
where
X Y = X Y X Y. (211)
Although we do intend to refer the reader to the original work [7, 16] for the details of the
dimensional reduction and formulation of the individual vertices of the theory, we present
this expression for the Lightcone Gauge Hamiltonian in order for the reader to be able
to read off the Feynman rules and vertex factors. In the next section we indicate (albeit
by the same means of qualitative handwaving as up until now) how the complete set of
vertices are generated by local insertions on the world sheet.
2.2.2 SUSY YangMills as a Lightcone World Sheet
The construction of the world sheet local action is analogous to the scalar matrix field
case. The path integral reproduces first the mixed representation propagator and then
adds interaction vertices by means of the spin system si as before. Recall that the qs
now have D 2 components with all but two identically zero on all boundaries. Clearly,
if the rich particle structure of other field theories is to fit into the picture, then all field
theoretic propagators must (and do) have the same simple exponential form as the scalar
propagator, times, at most, Kronecker deltas that describe the flow of spin and other
internal quantum numbers. T]. r, Fore the expression (26) is universal for the Lightcone
World Sheet form of a field theory. The general construction of the SUSY Lightcone World
Sheet follows very much the same procedure as the scalar case with a few fundamental
differences. In the Lightcone World Sheet picture the fundamental propagation is that of
momentum, represented by the width of the strip. Such things as the field theories' rich
collection of particles must be propagated through the sheet by means of "flavoring" the
strip (lattice sites) with dynamical variables. Furthermore, the vertices must come out of
local alterations on the world sheet, if the picture is to retain its elegance and its relation
to string theory. Instead of repeating the systematic construction of the world sheet theory
from the original work [7, 16], and as was done for the scalar theory above, we comment
on the main issues briefly. The discussion here is in the form of an existence argument,
we show how a Lightcone World Sheet description of supersymmetric YangMills theory
could be constructed. The interested reader is referred to the original work for a more
thorough and full picture.
After seeing the construction of the Lightcone World Sheet for the scalar matrix
theory, the first questions one needs to address when turning to SUSY YangMills are the
following:
1. Propagation of the gauge boson polarization on the strip.
2. Propagation of the fermion spinor information on the strip.
3. Creating the correct vertices from the information propagating mechanism.
4. Local representation of all the cubic vertices.
5. Quartic vertices, how does one join four strips in one point? Must we abandon world
sheet locality?
It turns out that issues 1), 2) and 3) are solved simultaneously in a rather elegant way
by introduction of world sheet Grassmann variables and in treating issue 4) the problem
with quartic vertices, issue 5), is solved automatically.
Consider for the moment, the world sheet as in the last section, with i e [1, Mj
and j e [1, N] labelling the sites as before and with the system of the q scalars and
for each such component a b, c ghost pair. With each site, we furthermore associate four
Grassmann field pairs S', S'. The ps are location labelling indices explained in a moment,
and a is an O(D 2) spinor index of the Chluid algebra of the fermions. The spinor
index allows the Grassmann field to carry all the fermionic information and by creation
of bilinears such as J" = 2 (D2)'/47ab the vector and scalar quantum numbers are
mitigated. To show how let us first explain how the four Grassmann fields are placed and
linked between sites. Referring to Figure 22 we draw the four Grassmann pairs around
each vertex as shown. The Grassmann action is of the form:
2K1 2K1
A = Sp l + > Sp (212)
p=1 p=1
and with (213)
DS = [dSKdSK1 ... dS1] [dSKdS2Kl ... dS1]. (214)
in other words, it is a sum of link terms between the pairs. This way, we can dynamically
remove "p links over boundaries and add "time" links to the action, simply by
removing the correct terms. This is done by imposing conditions on the Sas and Sas.
With the correct linking structure, we use the fact that
SDSe IS SK J DSe ^S SK tab (215)
J DSe SI SK = DSeASI SK =0. (216)
This ensures that the correct fermionic information travels from vertex to vertex.
Similarly
JDSeaS JK = JDSeASi JK = DSetJi S2K D)SeAJi SK = 0(217)
J DSeAJ{ J2 (218)
takes care of the vector and scalar particle information. At the vertices, along with other
insertions, Dirac gamma matrices connect the fermionic indices a, b with the vector indices
k. The details are carefully presented in the above mentioned article [7]. It is rather
interesting to notice that, as with the transverse momentum fields q on the world sheet,
the time derivative of the Grassmann's comes up only along boundaries. The mean field
approach, where the boundaries reach a finite density, should therefore exhibit 5 and S
dependence.
Let us next turn to items 4) and 5) from the list of issues we expect from the
Lightcone World Sheet description of Gauge theory. In the scalar theory the plain
Feynman vertices contained no momentum dependent factors but the treatment of the
propagators created factors of 1/M in the vertices. In the Gauge theory however, the
vertices are somewhat more complex. Recalling now the long expression (210) for the
Lightcone Hamiltonian, the Feynman rules contain a number of A3 and itAQ cubics as
well as A4, QtA2Q and (QtQ)2 quartics and the derivation of their coupling is found in the
t The case of A = 4 extended supersymmetry requires a slightly different treatment
since then the spinors are simultaneously Majorana and Weyl making Sa and S the same.
This different treatment still produces equivalent equations
1 2 i M
Figure 22.
Mi
I
I
r
i
M
Grassmann field snaking around the World Sheet. The figure shows how,
by imposing simple rules for connecting the four Grassmann pairs at each
site together, one can "sew" the Grassmann chain into the world sheet strip
and thereby propagating the information carried by the Grassmann's from
one vertex to the next. At the vertex (or rather, just below the vertex) the
initial chain (solid colored chain number 1) terminates and two new chains
are started (grey colored and white colored chains 2 and 3). It is possible via
local insertions at the vertex to break the required links and at boundaries
the Grassmann's connect in time rather than space. The Kronecker delta
identities in the text show that the Grassmann path integrals guarantees
that the same spinor or vector indices appears on both ends of the chains, in
the figure from 1, 2, 3 to 1, 2, 3 respectively. On the right, the corresponding
Feynman diagram is shown, with each leg labelled by its p+ momentum.
original paper [7] and for pure Gauge theory in papers by Thorn and others [10, 16]. In
short ']., v are in general, rational functions of the p+ entering into the vertex. Just as the
simple rational function 1/p+ was generated by use of the b, c ghost pairs these rational
functions must be created by local insertions on the world sheet. Consider the pure A3
cubic vertex, shown in the paper on Supersymmetric Lightcone World Sheet construction
[7] to be given by
V123 4 g ( K"3 K"2 Knl
V"4n3 3/ 2 +g +k+ 3 + 2k3+ where (219)
P3 P2 Pi
K = Pi 1 P1P2 PP2 P2P3 PP3 PPl
Note that here the labels 1,2 and 3 denote the three particles coming into the vertex, and
nk particle's k polarization. The moment qk are those of world sheet strips meeting at
the vertex and each is therefore a difference Pk A qB where A, B are boundaries of a
strip. The following identities hold irrespective of 1, i.e., irrespective of where on the strip
between the boundaries the Aq1 insertion is made
I Dqesq
qM q0 / Dq (q, q1_1) esq J DqAqlesq (220)
where Sq is the qaction unchanged from the scalar case Eq. (23) and the measure Dq
is just as in that case: Dq = d2q, ... d2qM_1. Comparing with Eq. (219) we see that
these identities are sufficient for constructing the couplings for the all As vertices. With
an addition of more ghost variables like the b, cs 'l., v are the basis for constructing all the
rational functions required for supersymmetric gauge theory cubic vertices in Lightcone
gauge.
Now the only thing left to explain is how the quartic vertices are handled. A'priori,
this would seem the 1 t4 obstacle to finding a world sheet local description of the
theory. The reason is that it seems not possible to recreate such vertices, as four strips of
arbitrary width and height cannot in general be joined in a point. It seems that locality
would have to be abandoned and the strips joined along a whole line. One hope, is that
one could construct local cubiclike insertions which would reproduce the quartic vertices
when occurring on the same timeslice, but even this can a'priori not be guaranteed. It
is therefore truly remarkable that the quartic vertices drop out of the very expressions
for the cubic vertices already present in the theory, when these are taken to occur on the
same timeslice. This is not only true for the Gauge quartics, but also for the fermionic
quartics of equation (210). As was briefly touched on before, this occurs because of the
fluctuations of the qs just when two Aq insertions are made on the same timeslice.
Recall that the lefthand side of Eq. 220 was independent upon where on the timeslice
the insertion was made (independent upon 1). Two insertion at k and I give
Dq (qk k1) ( q 1q_) e S (q I+ ii I (2 21)
where notation from Eq. 220 has been borrowed. The second term is the "quantumfluctuation"
term and is an addition to the simple concatenation of two cubics. Notice that M above
is the total width of the strips which have the two insertions. Consider a situation as in
Figure 23, with i., labelling the widths of the various strips. The double insertions at
il and i2 produce a quantum fluctuation of 1/(M1 + M4). Taking into account the _.M4
prefactor of the two cubic vertices and the 1/(M1 + M4) of the intermediate propagator
gives the combination [_M4/(M1 + M4)2. Adding the contribution for where the arrow
of Lightcone time on the intermediate particle goes the opposite way, we have the total
expression:
MM4 + MOM. 1 (M1 M4)( .) (222)
(Mi + M4)2 2 (Mi + M4)2
which is precisely the momentum dependence in expression (31) in Thorn [16] for one of
the quartic interactions. This very much simplified example serves to show how the
quartic seems to just miraculously fall out from the algebra. The above argument goes
In the paper [16] the author derives two Lightcone quartic vertices for the pure
YangMills theory, the above one which contains the "Coulomb" exchange and
contribution from the commutator squared, and another one slightly more complex. The
vertices defined there depend on the polarization of the incoming particles.
through in terms of all prefactors and for all particles and configurations. As before in
this section, the reader is referred to the original text for details.
I III I I
I I I I I I
II4 1111 
I I 1 1 1 1 1
I I I I11
, 4,,4
I I I I 1 1
  4 I I I
,4*+, 
: I I
1 t i 
Figure 23. Quartic from cubics for a simple case. The figure
shows two world sheet
shows two world sheet
strips (boundaries at 1, il and M) breaking into another set of two strips
(boundaries at 1, i2 and M) at timeslice j. The incoming world sheet strips
have M = il and I[_. = M il units of p+ momentum respectively whereas
the outgoing strips have = M i2 and M4 i2 units. To the right
is shown the corresponding Feynman diagram as a concatenation of two
cubics with the Lightcone time difference between them equal to zero. The
legs of the Feynman diagram are labelled by their respective fourmomenta.
For simplicity the example shown shows how the concatenation of two pure
bosonic cubics become a quartic, but the same applies to all other quartics of
the theory.
CHAPTER 3
PERTURBATION THEORY ON THE WORLD SHEET
As we have seen the Lightcone World Sheet formulation is set up in the framework of
the Lightcone. The = ix+ and p+ lattice as constructed by Bering, Rozowsky and Thorn
[10] is of course the starting point for the world sheet and the perturbative issues faced in
that work, of course remain present in the world sheet picture.
When using the discretized world sheet to calculate processes to a given order in
perturbation theory the insertions have been designed to exactly reproduce the cubic
vertices of the Lightcone Feynman rules in the continuum limit. The precise meaning of
this limit is that every solid line in the diagram is many lattice steps long and also is many
lattice steps away from every other solid line. Clearly a diagram in which one of these
criteria is not met is sensitive to the details of our discretization choice. In tree diagrams
one can always avoid these dangerous situations by restricting the external legs so that
, carry p+ so that the differences p+ p+ are several units of m for any pair i,j, and
so that the time of evolution, T, between initial and final states are also several units of
a. However, a diagram containing one or more loops will involve sums over intermediate
states that violate these inequalities, and because of field theoretic divergences the
dangerous regions of these sums can produce significant effects in the continuum limit.
In particular we should expect these effects to include a violation of Lorentz invariance,
in addition to the usual harmless effects that are absorbed into renormalized couplings.
Indeed, when a solid line is of order a few lattice steps in length, it produces a gap in
the gluon energy spectrum that is forbidden by Lorentz invariance. This effect can be
cancelled by a counterterm that represents a local modification of the world sheet action.
The hope is that all counterterms needed for a consistent renormalization program can
be implemented by local modifications of the world sheet dynamics. A slight weakening of
the last statement, that all counterterms will be consistent with world sheet locality, is in
fact a corii.ture put forward by Thorn in many of his papers on the subject. If Thorn's
(, ,ii. hi e were not true, it would make the formalism considerably less interesting.
We consider in this chapter renormalization to one loop in the context of the
Lightcone World Sheet, for a general class of Supersymmetric gauge theories formulated
earlier. We will see that with the rather unusual regulators, with discrete and p+ are
(equivalently, cutoffs in p and ix), we still obtain the correct well known result for
gauge coupling renormalization. In the calculations that follow we shall first revisit older
work done on the subject where close contact with the Feynman diagram picture has been
kept, because, even though the world sheet diagrams are completely equivalent, organizing
diagram according to loop order is only natural and apparent in the Feynman diagrams.
Following this rather thorough treatment we turn to a world sheet organization which
allows for some insights the Feynman picture does not. We shall also later see that the
world sheet diagrams organize themselves in a way much better suited for Monte Carlo
methods which are the second subject of this thesis.
We present here with permission, the work which was published before in a paper
titled "One loop calculations in gauge theories regulated on an x+p+ lattice" [6]. This
work constitutes in part the graduate work which was done towards the completion of
this thesis and is therefore presented here without additions or major modification. It also
relates very strongly to the second theme of the thesis, namely the Monte Carlo approach
of the Lightcone World Sheet formalism.
3.1 Gluon Self Energy
In addition to setting up the basic formulation of the x+, p+ lattice, Bering et. al.
[10] also calculated the oneloop gluon selfenergy diagram as a check of the faithfulness of
the lattice as a regulator of divergences. We recap their results but with the world sheet
conventions used here. In addition we consider a general particle content with Nf number
of fermions and N, number of scalars. The gluon self energy to one loop can be extracted
from the lowest order correction to a gluon propagator represented by a single solid line
segment on a world sheet strip as in Fig. 31.
ki
/ M
Figure 31. One loop gluon self energy. Because of time translation invariance only the
difference k = k2 k, is important.
With the conventions used here, the result analogous to Eqs. (52) and (53) of that
article, for fixed kl, k2, I with k = k2 k, > 1 and 0 < 1 < M, reads
92 k{2 2 t[ I (2+N8Nf)r t )]} (31)
87 2k2 I M1 M 2 M M
where u = ep2a/2Mm. This must be summed over I and k, and k2. For brevity in the
discussion of these sums, denote by A(l, M) the contents of the curly braces in the last
equation. Now consider the one loop correction to the gluon propagator, propagating
K T/a time steps. The loop starts at time k1a and ends at k2a and is positioned at
p+ = ml. Before introducing the counterterm we have the following expression for the
pro. I or correction:
K3 KI M1
1)r(p, A. K) U (A2 Ak)2 r 4(/, A1) (3 2)
ki1 k2ki+2 11
87 =1 )2t2 Il
M1
V 1 K InK + O(Ko) A(1 M). (3 4)
term linear in K comes from terms where the Ic is short (kA k1
sum is over the possible locations of it. It is clear that when n short loops arc summed
over their locations we get factors proportional to C"K"I/n! where C is the coefHcient of
K in the above linear term. Ti: : short loop behavior therefore exponentiates and causes a
shift of the 2 r;y". /2Mm, in the exponent of the free p r. T.. shift causes
a gap in the gluon en  spectrum that is forbidden in perturbation theory by Lorentz
invariance. We must therefore attempt to cancel this linear term in K order by order in
perturbation theory with a suitable choice of counterterm. One simple choice is a two
time step short loop of exactly the structure that went into the "'. :. selfenergy. i :. at
one loop order it will be proportional to the k 2 term and will have the form:
l 1
where we .."ust ( to cancel the term proportional to k in the propagator correction.
( : :::; = 4(1 /6) does the job and we are left with a logarithmic divergence which
will contribute to the wave function contribution to coupling renormalization. \We have:
In k = ln(1/a) + ln(T), with T = ka, the total evolution time. We can therefore absorb the
divergence in the wave function renormalization factor:
SM1 2 2 F(I/M)}
Z(M) I Inr(I/a) E I + 1 (36)
82 liAi I Aj6
T1,
where
F(x) = 2f,(x) + Nfff(x) + Nf, (x) (37)
x(1 x) 2 for i = g gluonss)
fi(x) = < x(1 x) for i = s (scalars) (38)
1/2 x(1 x) for i = f fermionss).
The first two terms in the I sum produce a ln(1/m) divergence and we notice the familiar
entanglement of ultraviolet (a  0) and infrared (m  0) divergences [17]. It has been
explained how these divergences disentangle [6] and we will discuss this further in the
next section. In (36) Nf counts the total number of fermionic states, so, for example, a
single Dirac fermion in 4 spacetime dimensions has Nf = 4. We see that Supersymmetry,
Nf = Nb = 2 + Ns, kills the I dependent term in the summand. If Nf = 8 as well, the wave
function contribution to coupling renormalization (apart from the entangled divergences)
vanishes. This is the particle content of A = 4 SUSY YangMills theory.
3.2 OneLoop Gluon Cubic Vertex: Internal Gluons
Now we turn to the contribution of the proper vertex to coupling renormalization.
The proper one loop correction to the cubic vertex is represented by a Feynman triangle
graph appearing in the worldsheet as shown in Fig. 32. With the external particles of
Fig. 32 restricted to be gluons (vector bosons) the one loop renormalization of the gauge
coupling requires calculating the triangle graph for the different particles of the theory
running around the loop. In the following it will be useful to employ the "complex i i
x^ = x1 + ix2 and x' = x' ix2 for the first two components of any transverse vector x,
and as the name of the section ii. 1 we consider first only gluons running around the
loop.
k = k2
;ure 32
3.2.1 A
1 M7 i Pm, P i P2
SCubic vertex kinematics. Basic kinematic setup for the one loop correction to
the cubic vertex. T..: moment and are taken to point into the vertex
whereas points out, so that momentum conservation reads pl p+ = .. By
time translation invariance we take one of the vertices to be at 0. We take
the external gluon lines to have polarizations ni,
Feynman Diagram Calculation
For 4licity in presentation we consider at first the case ni = n2 = A, ns = V.
Omitting terms which are convergent in the continuum limit, i.e., retaining only those
terms that contribute to the charge renormalization, we have
S)AAV
IfT ( 1'11 .)T ..2 4
1AS2
iB+1 12B2+ 13B3
(TiT I T T) 3
S39)
where F1 denotes the j:: :: e" contribution to charge renormalization, at one 1 from
gluons internally. In this section we will omit this descriptive notation and refer to F' as
simply F. We have used the following definitions:
1 ,7
I k2 ki
2 21 ; : I
Ti T2 I T3
ZT (TIT :. TITT
T' ,
1 k2
1 1
(3 10)
I T T .: .)
(311)
(312)
(313)
T) f
and
M1 M2 2 (1 12
A + + + (3 14)
12 1 ) 2( 4( + )2 1 ,+ 21 )
B, 12( +2 ( 2 (315)
M 2 + 1)2f
1)2 ( 1)2 M( 2 (
,M, 1)2 M:1 1(f 1)2 11"( I )2
Note the constraint IT, I (1" ) )T2 (Mi)T = 0, which implies that for fixed 1, o::1 two
of the T's are independent. Also, momentum conservation implies that K,j is cyclically
mmetric and we therefore use K K112 Ks2 Al31. ' introduce the following
notation that will help streamline some of the formulae: P1* = /' P* /M P*
For example,
K2 = Mi 'P1 PP + ) = M1' M(P P P*). (318)
We are now dealing with potentially ultraviolet divergent diagrams. To reveal the
ultraviolet structure we consider the continuum limit in the order a + 0 followed .
m  0. Recall that a :/ 0 serves as our ultraviolet cutoff. In the a  0 limit we can
attempt to replace the sums over kA, k' (k'A, k2) for kA > 0 (kA < 0) '. integral over T, and
T2 (Ts). Since we wish to keep 1 fixed in this first step, for the case kA > 0 we express eT
in terms of T\ and T1: T3 (1\ + ( i: + 1)7')/(Mi 1). For the case kf < 0, it is more
convenient to express T2 in terms of T, and 3: T ( (I'j +1'))/( + 1'). find
ET (MfT2 I '. T) / ( 1: 1) = (MTs I Ir T,)I/( 1. 1'). For the A term, this ccdurc
encounters no obstacle, and we obtain (displaying licitlyy the contribution for k, > 0)
IgAAV K^ M All (A1 1,1 iT2(1Ti I (/2 + l)T2)K2 A _TT2)
,q,, 9 rl2 11j, 1T1I VT T2) 4
( 1 ++ 2)
)4 + ( 1. . 2 ) (3 19)
It will be useful to note that H can be written in the alternative forms
H I= + 1) T22 [ )T, 2 320)
A1 (l 1,2 A] 1 1
Kp2 (,T T )'}'
(i + 1')T P* + I'T1 p* + (3 21)
where the first is useful when k ,l > 0 and the second for kA,l < 0.
However the B terms produce logarithmically divergent integral with this procedure,
so 1.. must be handled differently. To deal with these logarithmically divergent terms,
we first note the identities:
11 3 (MI 1) 7
SM ( 13 (3 22)
(Ti7+ T1 + T1)3 12 2M (1 2 +, T)2
S ((3 )
il 2M (MTf+ T) f ')
T2 A ( MM 3) Mi, i 2 ,
2 324)
( T I T % I I )3 0T ( 2 I M 1)2
(Ti I T I T3s)3 T (AT3 1T1i)2
where the partial derivatives are taken with T7 fixed.
Because of the divergences we can't immediately write the continuum limit of the B
terms as an integral. However we can make the substitution e11 (eHf e10) +
where H0 is chosen to be an appropriate simplified version of H, which coincides with 7H
at T = 0. For ki, I > 0, it is convenient to choose Ho (IT1 + (' 1) 2) P1*, whereas for
ki, < 0 ( (] + :' + '1i ) T) is more convenient. ii :: the factor (c eHII
regulates the integrand at small TI so that the sums then safely be .1 ',
integrals. We shall denote the contributions from these terms by F A. Then using the
above identities, an integration by parts (for which the surface term vanishes) makes the
integrand similar to that in Fj^" and simplifications can be achieved. For details see Sec
3.5.
pAAV g3 K M J _.9l3MI1 0 TK2(MI 1)2A'
AV B1 4x72To MI 1 Jo LH(M + MIT)3
lT(Ho H)(M1 l)MA' IT(Ho H)(M1 l)MIA]
1/ H(IT + + 1)(M + MIT) / 'H(M + MIT)2
+ (1 2 ) (326)
where
m 2M + 1)2 [ (M. 1)3
A' M /2].2 (327)
/IMI(3[M. + 1)2 MIM(M1 1) MIM(. + 1)2 ( 27)
Since the integrand of (326) is a rational function of T the last integral can also be done.
The evaluation is sketched in Sec 3.5.
There remains the contribution of the term eHo which would give a divergent
integral. However, because the TI, T2 (T1, T3) dependence in the exponential is disentangled
by our choice of Ho, the sums can be directly analyzed in the a + 0 limit, giving an
explicit expression for the divergent part in terms of the lattice cutoff. We denote this
contribution, containing the ultraviolet divergence of the triangles, by AAV. Referring to
Sec 3.5 for details we obtain
AAV g3K^A M [M1 I (N/l N2([. +1) ( 2p+1 a
pV + In + f
B2 20 M81M.1 1 MM, M M ap
]My1 1 Nil N2(. +l) I\
M M/M, M, M ) 3K 3 ( f
g3K A M '1 B/' 2p(+ a A.V[ (M 1)2a fa]
472ToMl. l [
+ ( 1 2) (328)
where we have defined
S ) 1)
" "+'
S B (MI 1)
In x
aC
(M[ 1)
l(A1)
( )3
A [M,D 1)
B2(' 1)
?V2 3
te, xt ex(t
(1 e )2r
e )
I I I
In 1B2 we can further simplify the term ortional to ln('i
the ultraviolet divergence of the triangle diagrams. We obtain
.K A M
\ ;' 0 t ,' ,, ; l' /
1 (M, 1 +)
X11 L I+omI 0
Al Al~
U' I)
. ), which contains
(' +1)3 1
+ ( 1 2)
4 ,T ^ M, L
Mi (I 11 ,2
A3 3
( .'+ )
7 ::1
1)
(1 + 2)
(3 )
where z/t is the digamma, function:
(x) d in(F(x)).
dx
(3 )
(330)
(3 31)
(34. )
In
ap1
f(
+
( + 1) + +(::') +37
(334)
Writing out the terms from interchanging I +> 2 in this expression. and simplifying we
obtain
F^ .3KA In + In 4 ( (A)+ (AI)+ ( 37)
1 8 M 12 1
+3 2 "
In ( 2( (M ( t 2 + 1)
3 K A2 3 1 n 4 ( 1n V 2
ln ___ _(__(_.'_)__,(_._))_+__ (11 '''+ F,,IL)I ,
i2 l
In " I II n P2 4 (nM r 4 ) I 4 in In
I pn+ 9M) I In
In~ 8In 11 ( 3')j}3
where the final expression, valid at large A, I, i has been arranged so that the auv
divergence appears symmetrically among the three legs of the vertex.
Putting everything together, the l:: :litude for the vertex function to one loop is
given in the continuum limit by
p ^AAV A {i S (1 2)}
16 i21 1 M 9'
_In + In 4 (In A] + + 
pp pi 9
In 41 n +( +1 1(3 .)
where
ST 2 Mi(M1 1)2TK2 A' IT(Ho H)(M1 1 )M1A'
'M /:H(T,1)(M1T + M)3 / '.H(T + 1. + 1)(M+ MIT)
IT(Ho H)(M1 l)MA (3 37)
3 /H(M + MIT)2
S1 = S )M[(, ) a l )2 ff () (338)
I2 I TP2,P1, S2= S 1, M1, (339)
and where we r
call, for convenience, our definitions (appropriate to the case kl, I > 0)
M2 f2 M2 V2 (IL. + 1)2 (1 1)2
A M+ M + +
12(1 1)2 l 2( + 1)2 (Mi 1)2 (22 + 1)2
A' M2 22 2 (3[ + 1)2 (M1 13
[M (3[. +1)2 1MiM(MV 1) MAM{(3[. +1)2
B (3 )3 (M1 1)3 MMI
B ( + +
1/(Mi1) /(.V+1) 1(Mi1)
(M1 1)T K2
H = H(T, 1) = (. + l)P* + lTP* + ( )T K2
M + MIT M1M
Ho Ho(T, 1) (1[. + )PI* + TP*
a M1 (FMp + 1)
l3 1IM
To complete the continuum limit we assume M, M1, 31 f large and attempt to replace
the sums over 1 by integrals over a continuous variable ( A l/MI, with 0 < ( < 1. This
procedure is obstructed by singular behavior of the integrand for near 0 or 1. When this
occurs, we introduce a cutoff c << 1, and only do the replacement for c < < 1 e,
dealing with the sums directly in the singular regions. The detailed analysis is presented
in the appendices of the paper on which this chapter is mostly based [6]. Referring to
40)
Eq. B.56 of that paper, we see that we can write
16w2T0 M1 [. 12 2
( np2p+ \_ M M 22 p2p f M 22
+ In 4 1n + +in [4in + } 1 +
p+p M, 9 p+p 1. 9
g3A^ 2 2 (M M p2 + Mp Mip p\ Mip2 2
+42T L 3 M M1p2 Mp Mp + MIj Mp 6
_I 1.' + Mp2 Mip2 1]2 _2_11
+(In *. + 7) 2 M In 3 + (341)
M_,2 Mp Mip2 + M1,2 Mp 6
Comparing the zeroth order vertex, 2gKAM/MIM_.To, to Eq. 341, we see that the
ultraviolet divergence of the triangle is contained in the multiplicative factor
1 + 9ln 4 (lnM + nMi + ln. + 37) (342)
t672 a 3
Note the entanglement of ultraviolet (In(1/a)) and infrared (In .[) divergences, typical of
Lightcone gauge. The In M's multiplying In(1/a) must cancel to give the correct charge
renormalization. To see how this happens, do the I sum in the gluon wave function
renormalization factor from before
Z(Q) 1 2 8(In M + 7) n2Q 4 (343)
t672 3 aQ2 3
Thus the appropriate wave function renormalization factor for the triangle, //Z(pi)Z(p2)Z(p),
contains the ultraviolet divergent factor
g2N 2
1 [4(lnMMi..+37) 11] In , (344)
167w2 a
so the divergence for the renormalized triangle is contained in the multiplicative factor
+ g 2 ln (345)
3 167 2 a
implying the correct relation of renormalized to bare charge
9R =g + 1 asNc in (346)
2471 a
where a8 = g2/27.
3.2.2 Simplification: The World Sheet Picture
It is however more interesting to do the calculation above in a slightly different
manner. It is ''. 1. .1 by the world sheet picture, that one combine self energy and
vertex diagrams at each value of the discrete p+ of the loop. If we back up the above
calculation and consider the contribution to charge renormalization before the p+ sum is
done we have
(g3 n(/) K^M1 2 1 1 2 fg(1/11.)
(1gluOns)AAV 42 ln(1/a)3_Y(A { K + . + _1 + } .
(1 2), (347)
The first three logarithmically divergent terms in the I summands again represent the
entanglement of infrared and ultraviolet divergences. These terms will cancel against terms
from the selfenergy, so that the entangled divergences never arise. We shall see this better
in the next section, when the full particle content is taken into account.
3.3 Adding SUSY Particle Content: Fermions and Scalars
The proper one loop correction to the cubic vertex is represented by a Feynman
triangle graph appearing in the world sheet as shown in Fig. 32.
With the external particles of Fig. 32 restricted to be gluons (vector bosons) the one
loop renormalization of the gauge coupling requires calculating the triangle graph for the
different particles of the theory running around the loop. In the following subsections it
will be useful to employ the "complex x^ = x1 + ix2 and xv = x1 ix2 for the first
two components of any transverse vector x.
Fermions
Referring to Section 3.5 for details of the calculation the result for the diagram depicted in
Fig. 32 with fermions on the internal lines is given by
fermionss AAV Nfag3A ln(1/a) 1 ) (1 2), (348)
for polarizations nl = n2 A, n3 = V.
Gluons
This calculation has been done for n, = n2 A, n3 = V in the paper [6] and it is very
similar to the fermion calculation. The contribution to charge renormalization is given by:
g3 I K^ [ 2 1 1 2 fg(I
(1pgluons)AAV 4 92a ln(1/a) K_ A 1 t 2 1 1 2__ + 9)
4 2 m MM I K 1i I M, I M. 1
(1 2), (350)
The first three logarithmically divergent terms in the I summands again represent the
entanglement of infrared and ultraviolet divergences and we will see in section 3.4 how
'!. v cancel against similar terms from the self energy contribution.
Scalars
Now consider scalars on internal lines and the same external polarizations as before.
Recall that the indices ni in Eq. (219) run from 1 to D 2. Let us use indices a, b for
directions 3 to D 2. Then dimensional reduction is implemented by taking p' = 0 for
all i and a. Using these conventions we will be interested in the special case of Eq. (219)
with n1 = a, n2 = b and n3 = V
FabV 1 ab 2K (351)
0 873/2 m M1 + (351)
and similarly for Fpb^. The evaluation of the diagram is analogous to the previous
calculations and the result corresponding to (348) is
(icalars)AAV N nag3 KA Mi1
(palarsAAV ln(l/a) ^ f (1/ .) (1 2). (352)
47r2M M I
1
3.4 Discussion of Results
The physical coupling can be measured by i V/ZF, the renormalized vertex function,
where F is the proper vertex and Zi is the wave function renormalization for leg i. To one
loop we write this in terms of our quantities as:
Y + Fo (Zi 1), (353)
i
where F0o is the tree level vertex and F, is our one loop result for the vertex:
p^AAV fiermions + pgluons + pscalars (3 54)
L /3 P3+ M1 2 1 1 F(1/1f.) (t 025)
4 w nM + 1 + + +
411 1 m. i1 1
Because of how loops are treated in the Lightcone World Sheet formalism we are
motivated to combine the one loop vertex result and the wave function renormalization
for a fixed position of the solid line representing the loop. In other words we renormalize
,Ju,"flu on the world sheet. To clarify this, note the three different ways to insert a one loop
correction to the cubic vertex at fixed I on the world sheet, as in Fig. 33. Notice that
k k k2
k k2
kO0
k k
k = k i k k. k k2
k = 0
k 0
I M1 if. 1 M1 if. 1 M1 if.
Figure 33: One loop diagrams for fixed I in the Lightcone World Sheet.
the first and last figures correspond to self energy diagrams for the legs with moment
(p ,, ) and (p,1 ) respectively. 1:: ever, the middle figure corresponds to a triangle
diagram with time ordering ki > 0. So combining our us results can calculate the Y
corresponding to this polarization for a fixed I <
^^ = 2 ^
Sa 2 1 1 F (1/ )
S.In( + /a) K+ (3 57)
2( 2 In(1/a) (358)
I ln( /1a) (3 l 59)
In(11/a) (3 A F(i r (3
WVe see that the terms of the form 1/1, 1/(MI 1) and t/(. 1) cancel in the
final expression for Y. TI.. terms multiply the ln(1/a) factor and would result in
In(1/m) factors if the sum is taken before F and v/7 are combined. ': :: represent the
entanglement of mn  0 with a + 0 divergences and we have seen how this entanglement
of divergences disappears locally on the worldsheet.
For (....:.1. :,, we : the result of the triangle diagram for general
polarization, i : entangled divergence does not depend on the polarization of the external
gluons. T. local disentanglement discussed above therefore goes through unchanged for
all polarizations.
We write out the results for the renormalized vertex Y where a subscript refers to
the two different time orderings, k1 > 0, (1 < MI) or ki < 0, (1 > Mi) respectively.
y^AA^ Y=VVV (361)
yAAV g KA M1 (1/.) F(/M) (3 62)
YAV 7"2aln(l/a) (362)
82 P 1P2 1 (1 MJ I
g3 1p+ ^ MI1 F(1/M1.) F(/ )63)
YAVA aln(1/a) (363)
8712 P lP 3 M
g3 p7+ K^A MI1 F(1/1) F(1/.)
YV a1n(1/a) Ey (364)
872 P2 P3 1 1 11. ) I
g3 K^ M31
V aln(l/a)PK F( /) F( (365)
l MI+1
yDAVA h 3 c g aln(t/a)PK M3' (1 A) F (_ __
YAaA I n(1 A 1 F(a[.) /[.) F(3[. (366)
1 P3 Ihl MI+1
P2 P3 / M1+1
Y ^ a In(1/a)Uln3 ) (367)
The expressions for the Y's with A 4 V are the same with K^A K. We stress that the
summands in the above expressions for Y are exactly contribution of the three diagrams in
Fig. 3 3 with the loop fixed at 1.
Define the coupling constant renormalization A(Nf, NI) by:
Yn'n2n3 Y41n2n3 + Y'1n2n3 1Fl2n3A(Nf, NS). (368)
We then have in the limit 3[ + +oo:
A(Nf, Ns) 8,2 ln(l/a) 3 3 6 (369)
which is the well known result. In particular we have asymptotic freedom when A > 0,
and A vanishes for the particle content of A/ 4 Supersymmetric YangMills theory,
Nf 8 and N 6.
For some cases such as the Supersymmetric (Nf = 2 + Ns) or pure YangMills
(Nf = N, = 0) the summands in the expressions for the Y's do not change sign. When
Nf < 8, so that these cases are asymptotically free, the summands on the right sides
of (364) and (366) have a sign which works against asymptotic freedom. Since the
full sum exhibits asymptotic freedom for each polarization, this means that that the
complementary time orderings, (363) and (367), must contribute more than their share
to asymptotic freedom. This fact may be useful for approximations involving selective
summation.
3.5 Details of the Loop Calculation
We present here some of the calculational details that were omitted from the above
sections for clarity.
3.5.1 Feynmandiagram Calculation: Evaluation of p^"V
In the calculation of F"' we start by integrating by parts. This transfers the
derivative to the factor (eH eHo). For definiteness take the case I > 0. Then we
compute
a H Ho) H H TIIP Ho Ho TIP* HK (M )TIT2
9T2 T T2 M, (MT2 + MITI) 2
(370)
The first two terms on the r.h.s. partly cancel after integration over T1, T2. This is because
the integrals are separately finite, so one can change variables T = T2T in each term
separately. For the first term we find
dT dT2(T1, T2) H(TI, T2) TIlI P cH(T1,T2
dTdT2I(T, 1) [H(T, 1) TIP*] H(T) dTI(T, 1) ,(371)
and the second term yields the same expression with H(T, 1)  Ho(T, 1), so the two terms
combine to
Jo/
.0
(3 72)
~((T, 1)TI ) 
P (T, 1) (T 1+ + 1)
r the contribution to the T integrand from these terms leads to the continuum
,AAV g, K^ A Ia 
T 'T 2 Alf
SB B2
SK,1 M 1
4 72To 7,[" ..1
S IT(Ho H) (MIf I)T 2
o ( L H(IT I ' ) HM1{(M M A1 T)2
,'.T(Mf ) I 2MIT 2 I B
f0 T IT(Ho H) ( )TK2 ]
o (IT 1' 1) HM1(M f ,_T) j
M 1' A A + 11 1 (1 +
(M 1)2
(M1k[ MIT)2
I ( 1 2 )
(M 1)2
(M MJiT)2
2)1 (3 73)
where we have defined
A
IM^\(1)2
Notice that this result combines neatly with I
^ I^ {i dT
IlT(1[0 H)(V 1) ': IT(h
H(IT i 1)(M MT) ~
to give
TK [(M 1)2 A'
H (( 1M Mi T)3
H) (MI 1)M
H(M IMIT)2
A
II ( 1 4 2 .75)
3.5.2 Feynmandiagram Calculation: Evaluation of F`
We analyze the continuum limit of the A'Av contribution to the B terms, which will
be retained as discrete sums over the k's. Again for definiteness we the case 1 > 0
Simple
limit
. 1 )
:.
'~ '\[(~ ~
21)
(3 74)
^
1 ,
in detail:
pAAV 3 K^i M
B2 I. F.
1 T, B I T2B2 T3 B3 r 1o 
ki _k I23
[+
il) 0 t,
kl 7 I 1 k' k ki ] A3, '
_(k k',3)3, u 2
2k, k 9
+ ( 1 2 )
(3 76)
where
B (A11 1)
/
SB2(M 1)
* + B3
_ + 1'.
Ui C e P2
S,  2 ap/2p
p 2I) L CO2
(3 77)
where u2 is to be used in the case k, l < 0 instead of u. C y the continuum limit
entails u1, .2
1, causing the k sums to diverge logarithmically. To make this explicit, we
first note the integral representation
kl +k'9
S(1 q + 1)2
kik1
J tdt
i tdt
S("at
UI (3 78)
u l _8)
1 '_
1)(e" 1) (1
1,1 + At)(1
I t
ul + 13t)
79)
where the approximate form is valid for I u\ < e < a, /3. Doing the integral in the
second term leads to
U( l 
(kiar + VI 3)2
 n F +
.'3 l p
au p1
0 In a
13
( In 3
a
+ 1 In c +
03, ic "
I td at
( ,at
, sums we require can be obtained from this identity '. differentiation with respect to
a or 'f. To present the results it is convenient to define a function f(x')
S1 +x
I In X + lirn
21 x e
In 1
X l(
7 (et t l)(etl/ 1 )
1 Xt eX (
( C"xl)2
K /^ MA
To .7Mi "
11
ill'
(1
/
M
/3 1
,, ,
kk',
kA1 A'
I
1) Ic It 1)
f ()
(3 81)
(382)
where the second form is obtained by integration by parts. It is evident from the first form
that f(x) f= (1/x). Also one can easily calculate f(1) 72/6. From the second form
one easily sees that f(x) ~ In x for x + 0, whence from the symmetry, f(x) ~ xIn for
x + oc. Exploiting the function f and its symmetries, we deduce
ki +k/2 2p a
t In f+ f (383)
kyk la+kc)2 a ap + \3
k1 kl+k2 1 2p () 1 (
2U1 In + f + f 
(kia + k' 0)2 2a 3
kjk 4122 1 F" 2p1 ( 1+tl]
SIn 2p+ + f )f' f (385)
2a p2 ap a2 20 /
Inserting these results into Eq. 376 produces
F g3A^ M M1M 1 Nl N2(M.+ 1)( 2pt (alf
MI 1 Nil N2( + l)\a,
4Ef'P + 1 2)
Sg3K^ M Mj B'1 2p A+ (M 1)2 f,
4rol.,_ 1 IIn ap, + f + f' All
47 2T0 M, 1 _M ap M3M
+ ( 1 2) (386)
where we have defined
B' =(M1 3 + MM1+ ( + )3 (387)
3/(. +1) l(MA 1) 3/(Mi 1)
3.5.3 Feynmandiagram Calculation: Evaluation of F.^
For large .3 the sum over I can be approximated by an integral over = l/M1 from
e < 1 to 1 c, plus sums for 1 < 1 < cM1 and MI(1 c) < I < M l 1 which contain
the divergences. These divergences are only present in the first sum on the r.h.s. of Eq. ??
for I < M1 and in the last sum for M1 I < MI. The middle sum contains no divergence
and can be replaced by an integral from 0 to 1 with no e cutoff. To extract these divergent
contributions, we can use the large argument expansion of "'
(3 )
u'(z + + Of( )
z 2z2 0< 3)
to isolate them. It is thus evident, that their coefficients will be )ortional to the
moments E f//ck"' Y ~ /' for k 0, 1, which are ecisely the moments con strained 1
the requirement that the gluon remain massless at one lo(.
For the endpoint near 1 0, we put z kM,( + l')/lM and write
k, M 1
1 + 1) ( i 2 )+ (
I All M + 1) '2k : .;, i I ')2
so the summand for small I becomes
A4(2M 1) (2 1 ) M
4A1! 2 iA2 2 M2
1, A/j M r 3 1 M1L
fk, 2M 1) (2MA2 1) hk
1) 2k I f 1)2 2 M1 (f ,)2
1 M2(2M 1) +( 1) 1 1) I12
12 ',: + 1)2 + 1)2
(3 
Summing I up to c.Mi gives
4M 2M2 MA2 4 4AMr2 4A d ._
1 I .1 3 1' 1. 6 M 'A
Inserting these results into Eq. ?? and writing out . licitlyy the 1
divergent part gives
2 t7)n for t 91)
<> 2 terms for the
 )(2 1 + ) )
2 + hk
+ 27) (2 + (392)
+ 2/) 2 + 3 92)
AAV
C1 '
*dKA kv^ I + M) 1 (2
I +^/ k ++ ( j I k
( 1 ++ 2 : 6 .f (" :'
k6 I J,
Only the third sum contributes near I = We again use the large argument
expansion of u'Y. But this time one only gets a logarithmic divergence, because the
difference of "/'s is of order (Md 1)2 as is the i.licit rational term. Putting z .
kMi(1. + 1)/1(Mi 1) and z2 = klM/M_(M1
'(z1 + 1) '(z2 + 1)
zi z2) 2z,
2z2
M (MI 1)2
~ MM1 ( +)(1 + (M1
kl M Ml(M[ + 1)
Thus the I ~ M1 endpoint divergence is just
g3 K^ 4M2I fk
872 M M1M. k 1
k 1
SM1 1
fk M
l M4(1e)
g3 4MK^ (2
87&2 MI 1. [ 6
1) (Iln Mi+ ; 'l 1)
Putting ('I together we obtain for the continuum limit of the triangle with internal
longitudinal gluons
k 1
AAV g3 K^
81 2To M
S <1 k(1 _ )2M
k 1 o 1. + (M
_+ t + k _M
+ jEck^iIT1
^r'^3
( 1+
[. + WMi))
M )
( + k [{M) fk
11. + (Ml )
[ M (1t _ 2
k 2 ([. + i)2
+ (1
fk (2 )(2. +M1 ) + hk
/fk I
M1i)(M + Mi(1
(3.+ Mi)2
+ k +
3f(1 )}
(1+ 0(M + +Mi)
Mi(10)2
2MTr2
3
+hk]
2M2
M 11
Sm2 (In e2MiJ_. + 27) 3
M I. 
3.5.4 Feynmandiagram Calculation: Divergent Parts of Integrals and Sums
The T integral in Eq. 326 can be evaluated by expanding the integrand
MITK2(M )2A' lT(Ho H)(MI 1)M2A'
31 /H(M +MIT)3 3 /H(IT+.1+l)(M+MIT)
IT(Ho H)(Mi 1)M2A
31 H(M +MIT)2
(396)
in partial fractions. First note that since (MIT + M)H is a quadratic polynomial, it may
be factored as
p l(T T)(T T_)
+h k]
g3 K^A
472To M
 2) }
(395)
1), we have
22 (MI 1) 2
1)) ~ 393)
k1y3
k '
(MIT + M)H
(397)
where T_ ~ (K2 M : )/!1: I and T+ ~ M .. /(K2 ) when I < M1.
i : the partial fraction expansion reads
R + 2 3 + 1, + ? .,(3
T 1T* 1 ( +T .)2 +1 (
with the RH i: 1 cendent, of T. Of course the RH are such that, I falls off at least as 1/T2
for large T, i.e., RI1 R2 R:,3/l R5/1 = 0. identity is helpful for determining R5.
:: we have
dT '. In(T) R2 n(2 ) 3 n In (3 99)
oI l. 1 + 1 MMAl MI l
`'i.. I are given explicitly by
M,M (,i )T (M, 1) 2 A A_
(T+ T) (MIT+ + )2 )
I (MM 1)1' (M l 1 A'
R2 j)j' r II ) 
( :~_~ _1 ) (:A T_ M ) '
(3 100)
2( +I 11 4]) (3 101)
(3102)
(311 :)
1[ (M.,i( _1) lM,( C ) 1 ,
RT =  ) 
. O M _
MiT++M Mi'
(3104)
T ].!
T_M
"hen the /'s are large, the sum over I can be replaced an integral over ( 1/'' as
long as i is kept away from the endpoints 0, 1. We can isolate the terms that give rise
to singular end point contributions and simplify them considerably. We shall then ate
the divergent contributions and ( :.1 them in detail.
First note that the worst enI. o:int divergence is c1 n I near c 0 or 1/(1 C)
near 1. T::: we can drop all terms down by a factor of 1/ for small I or by
(MI 1)/ for I near Mi. I. for I < we note that ITM +A Mi ) ~ K2 /. and
T+ /(M I I[1T+) ~ / K 2 and obtain
p2M JT 11 2 1 1 ~ p2, ,1
2 (2 + (3 10 o5)
1 lK 2 ( 2 i ') I 2 ,
Mi/ 2M Mi Mi.' ; ,
12 I 1(",," '",,) I M ,"+ .... ( '
R3 ~ 2I1
R4 r [
R,5 I K2
Combining the I s 0 en, Ioint contributions gives
2 2
R In R2 In I
K 2 + f.,2
in
A M + K2 K2 + ,1
I i M. M l
ln 11 ^
I iK2 h K2 2
. in (K2 I A
M. 1 Ai
3 4 I
In
2 +
K2 1}
+ 11 Ki
In
IlK
In 
Ii i
A'
1 '. 'r
in
I} for < : 3110)
For the other endpoint, 1M < 1 T, the roots of the p< :. :..' 1 (A I MT) H : roach
T A/l/A/i and Ta "/. Which of these roots is ...... 1 d by T depends on the
S: values, but since the formulae are syrrmmetric under their interchange, we can
choose to use the first in place of T, and the second in place of T_. Since the denominator
M 1 1T = 0 in this limit, we need to carefi:1 evaluate
1 T1 MT K 2
dTI i
(3107)
(3108)
(3109)
r1.. we obtain for small Il 1,
frif(M Mip2) 1p92
(,M 1)K2 (Mi 0(A)(M 
( I) ( K M..2)
1
R3 I 
(Mi )
Combining the I p !.': endpoint contributions gives
R3 + R5 M, p2 14
M1,I Mn
2Mi My MI ]
2( U 4 A In 1
(M! 1)1 Ai[p2 A M
for M I < "
(3 116)
In writing the I sum as an integral these endpoint divergences can be separated 1
picking e
1 < e(A and IMi(fI
e) <1< M 1. For
these parts of the sum the above .. roximations can be made and the sum evaluated:
S Mi I
/Mi(1e)
7( (1 I cM)
Sdin00  et
Jo 1 ('
1" I U/ in + _
+ J d
( )( e t 1 )
12( + ((2) 
2 '2
In C) + C
2
M111 )
(3 111)
(3112)
(3 113)
(3114)
(3 115)
, Af ; I MI )
(MI 1) 2
SdT I
eMl i
l l
Ii
11
(3 117)
The rest of the sum is replaced by an integral over e < _< 1 e
IdT /~ l
< J o M
(In M + )( 2Mip2
n 17) L Mip 2 Mp2
Mp + )_2 Mlp2
Mp 1 MpI
M1 1 M ( [_1,2
Mip + p K2 + Mi2j
] l2 + /1 )1 L] / /12
M] 2M1p2
In (In cMl + 7) + In2( CMI) C 2 2 (3118)
l Finally,2 we2 must extract the divergent contributions that arise from replacing the sums
Finally, we must extract the divergent contributions that arise from replacing the sums
MI1
MIB
M f 
[M ^
All (M 1)2
M3+ A
in Eq. 336 by an integral. First, for I n Mi, a/f3 0 1 and only the first term gives a
singular endpoint contribution,
MI1 2
S ~ 2Mf (1)[ (1 + Mi) + 7] ~ M[ln M + 7].
l Mi(le)
(3120)
On the other hand, for 1 0, we have
(M1
f IM)
M13 f
IM
(M13[.
iM
i Mil fo 1 t et
In 1 dte'int n t
IM Jo (1 et)2
1M [T12 Mi1 I].
1 + 1 + In 1 1 .
M, 11. t12 1MIM
IM
n M (3121)
(3122)
The integral in the first line is zero because the integrand is a derivative of a function
vanishing at the endpoints. Inserting these approximations, we obtain
M2 M 1 2 eM, 1
A 12 + M( I
inM
l2)
[ (1 + CMI) + 7] 62
SM1 I
^Y
Mj11
Zs,
(3119)
eMl
S ~
l=1
(3123)
+ M MIpK22 _'
K2
Putting Eqs.3118,3120,3123 together, some simplification occurs and we obtain
 S(1 /M1 + IdT d IdT + dS(
M1d S !Jo/ Jo dJ(l
Mijl.T2 2._[ 2Mip2 Mlp2 1/ i 2
+ M 211 + (in M, + 7) 2ML I + + In
MMp2 Mp Mp2 + __I 1, Mp_
+ M 2 In3 
K2 MiP + 3_ in K2 + MI[ 31,2
[ M 21 6Mlp 3_
n (n M +7)+ In2( ) + C (3124)
When we add the contribution with 1 ++ 2, the antisymmetry of some of the coefficients
leads to further simplification as well as a reduction in the degree of divergence of some of
the terms
t1 1S(I/Mi) + IdT + (1 2 )
j1 do
1e o 1 2, pp+2 p+ 2 +
d IdT + d/S() + (1 2 ) In 2 2PI P12I
(1JO J c P1P2+P2P1 P2
2p+ 2 + 2 +2 p p2 + 2 2p+2 + 2 p2
+ In In + p I 2 in + In 6+
L+P2 + 2 pi p pi 2p + p2p pIp2 p+p2 p+p p+p 3]
Sp_ \ P 1_ p2 1 P2 P2 P22P 12
2+ 2 +2 +2 2 + + + 2 + 2
p+ K2 +2 + p+pp K2 2 31 + +p 2
(inP P2 2 1)3 ,\12 1 p2 P2P1
27 2M1 2Mlp2 Mp 31 Mnp2 2
3M + (In M1 + ) Mp + 1 In 3 +
3M2 Mip +31 2 Mp 6
( 2 _2 M 2 ? 22 1
2_(I, MIPM in 3 + (3 125)
+ L1,2 Mp M p + 2 Mp 6
As c  0 the first two lines on the r.h.s. approach a finite c independent answer. The
third line is explicitly finite. All divergences are shown in the last two lines. As .3 oo
there is a leading linear divergence as well as a single logarithmic subleading divergence.
3.5.5 Details of SUSY Particle Calculation
We consider the diagram depicted in Fig. 32 with fermions on internal lines and
gluons with polarizations nl, n2 and n3 on external lines. In section 2.2 the method of
constructing the world sheet vertices is outlined and referring to the paper itself [7] for the
detailed expressions we get
dq exp a kl(Pi p)2+ P? (k2 ki)(p3 )2_)
1 1 kl,k2
{ r / \ r / \1 11
Tr (7"7) P P P +ag P3 _P P 2
2 KM 1 / /1 M (16i
ag(7 P3 P + P P3
2m M. I I m 1 M.( \
a ( 7 2 7 ) P i P rP 3 i ) + a g P 3 P P 2] ( 3 1 2 6 )
Notice that this is the expression associated with fermion arrows running counterclockwise
around the loop. The other diagram contributes the same amount as this one. Also,
this expression is for k, > 0, the other time ordering k, < 0 is obtained by making the
substitution pl + P2 as in the gluon calculation of the paper [6]. We now proceed much as
in that calculation by completing the square in the exponent of Eq. (3126) and shifting
momentum
Ml 'f q tIr + t2+ a6 H
11 klk2 ( 2 q H _
Tr 2 ___ X i (7 X ) + g 3
Tr 1 (M 1) M (Mi1) H I(. 1) m
t n2
(X2 (2[t) 2_/+g 2 (3127)
2 A (M1)(pf. 1) (M. 1) M
with
3 tlKT t2K+ tlKT (
X1 + t2 + 6 Mlq", X' 6 + t +, 6 = + M 3128)
S kl k2 k2 kl (19
1 k2T' k3 (3129)
a tlt3P + 1t 2 2 3 2
H a tIt3p= + t2p + W3P3 (3130)
2m tI + t2 + 3
In the .. '.,egral only the terms "oportional to q2 times the Caussian will exhibit a # 0
divergences so we retain only those, i ::' general loop integral is given by
f ,q ktI 2 + t2 + t3\
2Jdqx1 3 \2 2" ,/a
( /a)2 (M {MI r Kr i 3 1 2 'k k 3 k
2(1, 1 12 1 13)3 yk r e K "
(3131)
(T. arrow means that a  0 finite terms have been 1 .: ..1 ) Some simplification
can be done right away, for example the term : :tional to Tr (7"' y" "" 2 t) after
contracting with the momentum integral is :tional to
Z Tr (' /q' rY r ( K)) =(4 Do) Tr (n /f77
u2( K))
(3132)
where D0 is the .... ..... dimensionality of the loop momentum integral, that is the
reduced dimension DIo 4 so this term vanishes. Further simplifications can be seen when
a particular external polarization is chosen.
S: detailed a  0 behavior becomes 'ent when the sum over A and A is done.
With a little work it can be shown that
1 i H Iln(1 /a)
'; 21 3 ) 2
(e11 hn1 1 2 11
,C (tI ( /a)
13 e In)2 ( a) 1
k (11 1 12 1 )2
13 )2
kjk (11 1 1,2 1 s "
Carrying this through for the polarization na =
Nfag K" ln(I /a) 1f 2
32712m MiM"' r
2A, 1)( 1f 1)
2M, 1
= = n yields
 )Mj I 2 (
(3 133)
M[ 1 1 ( 1
+ (3 134)
(3 1
i)]}
(3
(3 1 ..)
In the continuum limit we have Z7 f1(1/M) [ K. "/ dx f(x) for any continuous
function f. Ti. :'clore, after adding the ki < 0 contribution and mun 1 by 2 for the
_
other orientation of the fermion loop we obtain
Nf ag3KT M,2 + M22 + M32 ln(1/)_2
lI n(1/a) (3 137)
1672m MM,[_.. 3
In contrast, the calculation and result for the n n2 A, n3 = V polarization is
quite a lot simpler. The expression analogous to (3136) is
Nfag3K^ln(1/a) M I 1/
l 16r1 2 1 L (3138)
The result shown in (348) is obtained from this one by adding the k, < 0 contribution
and multiplying by two which accounts for the other orientation of the fermion arrows in
the loop.
CHAPTER 4
THE MONTE CARLO APPROACH
This chapter presents the second theme of the thesis, namely the numerically
implemented Monte Carlo approach to studying the Tr(o3) field theory in two dimensions.
We will see why, in this case, such a stochastic approach is preferred over a deterministic
numerical investigation. The mathematical framework for Markov chains is well
established and is discussed in detail in basic textbooks on the subject [18].
4.1 Introduction to Monte Carlo Techniques
Since the advent of easy, ubiquitous access to powerful computers, numerical methods
have come to be widely used in physics. Quantum field theory is no exception to this.
Numerical methods can, in principle, be categorized as deterministic or stochastic. Both
types yield approximate answers but in the case of deterministic methods the error can
be traced back to the finiteprecision representation of real numbers used in computers.
As the name implies, deterministic methods operate in a predefined manner. Stochastic
methods, on the other hand, rely heavily on statistics and errors originate not only from
floating point representation of numbers, but also from the statistical interpretation of the
results, since random numbers are used as inputs of the simulation. Monte Carlo methods
are examples of stochastic numerical methods and ],,. v are so widely used that the term
" Ionte Carlo" is sometimes taken to mean any stochastic numerical method.
4.1.1 Mathematics: Markov Chains
The underlying mathematical construction for Monte Carlo techniques is that of the
Markov chain. Given a probability space (Q, A, P) a Markov chain is a sequence of
random variables (Xk)k>o distributed in some way over a statespace S with the property
that:
P(Xk XkXki = Xk1,Xk2 Xk2,...,Xo = xo) = P(Xk XkXki = Xk1i) = T1,x _.
(41)
for some i,.n., r mapping T. Notice that the random variable Xk depends on Xk1 only,
and not the earlier random variables in the sequence. This is sometimes referred to as the
Markov chain's lack of memory. A distribution p : S > [0, 1] for an element Xk of the
Markov chain is defined to be the probability that Xk takes the value x or:
pk(x) P(Xk x). (4 2)
Clearly xrs p(x) = 1 for any distribution (hence the name). It is useful to think of the
Markov index k in (Xk) as time and investigate the evolution of the distribution with
time:
pt+1 = T tl,xtpt. (43)
If S is finite (which it is in any computer simulation), then T is a matrix whose rows sum
to 1. An equilibrium distribution 7 is defined by 7 = TT. A very important result of
the mathematical theory of Markov chains is that an periodic, irreducible and positive
A probability space is a triplet (Q, A, P) where Q represents the set of possible
outcomes, A is the set of events represented as a collection of subsets of Q and P : A
[0, 1] is the probability function.
t Equilibrium distributions are sometimes called invariant or stationary.
recurrent transition matrix T has a unique equilibrium distribution 7 and will converge
to it irrespective of its starting distribution, more precisely:
T N N+00
(T)ijPi s 7. (44)
Moreover, such a transition matrix satisfies the Ergodic Theorem which states that for any
bounded function f : S  R we have:
N1
SN1 N+o0 
+Z f(xk)  f (45)
k0
with probability one,) where = ExaEs 7(x)f(x). This result is a statement that one
may find intuitively sensible: that the proportion of time which the Markov chain spends
in a state x approaches 7(x), the value of the equilibrium distribution in that state. It
is a common strategy to approximate 7 with a finite sum kJ 0 f(xk) using the Ergodic
Theorem. This strategy is called importance sampling because it can be thought of as
summing over the important elements of 5: those that sample 7.
By construction, the Markov chain can simulate many physical processes, simply
because the defining "lack of memory" property, is seen so widely in nature. Markov
chains in general have many applications in the field of physics outside of stochastic
numerical simulations. But Monte Carlo simulation are the topic of interest here, and
we are now in a position to clarify what is meant by Monte Carlo simulations in the
first place. Let us consider the case where the state space S is very large and where each
element of it is a complicated object. Let us further assume that we want to calculate the
Irreducibility is sometimes called ergodicity, and essentially means that the transition
matrix must be able to go everywhere in state space. We will see this better later. The
conditions are not of great concern, positive recurrency is automatic in a finite state space.
i Rigorously this means that the events {Xk Xk} for k = 1, 2,... N such that (45) is
true, have probability one, or conversely that (45) is false on a subset of Q with measure
zero.
value of .s (x) f(x), the < value of an operator f defined on the statespace,
under some distribution t. 'I : full sum i: not be feasible to : .orrn when S is large
(as we shall see, this is most certainly true in the u work!). T. idea is, then,
to construct a. Markov chain which has as its equilibrium distribution and perform
ortance : pling to obtain an approximation of the expectation value. So in Monte
Carlo simulations we are given a distribution Fr and it is the Markov chain we want to
find. Ti : stands in contrast to .. other .. plications of the Markov theory, where
the transition matrix is known, and the equilibrium distribution is the object of interest.
Of course we will not be able to find the Markov chain in itself; instead we find what we
shall call a Markov sequence: a sequence of actual states, rather than random variables in
the statespace. T : is : table because if the sequence really represents the Markov
chain, then the states will distribute as dictated '. the random variables T.
central trick in a Carlo simulation is how to obtain the Markov sequence from
the distribution rr. Although sometimes very difficult in practice, the idea is almost
embarr;. '.. simple. Since Fr satisfies the Ergodic .....em, rr(x,) 74 0 for all x .
P 1 the most straight forward ansatz would be to select states x G S at random
and ,ending each to the Markov sequence with probability yr( r). I.. result would
certainly be a. Markov sequence with equilibrium distribution t. T : method, however,
would be hopeless in i i cases because of the computational complexity of calculating
7r(x)), for an arbitr x, as many times as would be :' 1 to obtain an acceptably
long Markov .... Fortunately, many important cases in pl exhibit what we
shall call Monte Carlo ..'. or me' '" for short. T: is the property that the
ratio T(x')/(,T(x) is drastically simpler to evaluate ... :: '.onally if x is close to x' in
a sense that is simulation specific. It suffices to that they are close precisely when
x x' and x (x')/r(x) are .:. . ::i !:. nally simple operations. We :: lively
write x' = x I+ and talk about Ax being small, even though there  not be an
additionoperation, metric or measure defined on state i. For an minclocal m we
now construct a sequence (not necessarily a Markov sequence yet) with any state x EG as
the first element. We then change x a little bit: x' = x + Ax, and automatically accept x'
as the next element in the sequence if 7(x') > 7(x). If however 7(x') < 7(x) we accept it
with probability p = 7(x')/w(x). In practice this is done by having the computer generate
a (pseudo)random number r uniformly distributed between 0 and 1 and accepting the
change if r < p. If the change is rejected then by default the next element is x, the same
as the current one. This way new elements are generated, or rather selected, one by one
from statespace. Notice that Ax is in general different for each time the state is changed
and in some sophisticated models it is generated stochastically using information about
7 to maximize acceptance rates. The procedure generates a sequence (xk)O
construction is distributed according to 7. But it is not, in general, a Markov sequence
because successive elements are quite possibly correlated, especially if Ax is very small
(to be made clear in the context of a specific system). It becomes a matter of statistical
testing to construct a new sequence (yk) out of (xk) by throwing out "intermediate"
states to ensure that y, depends only on y,i ("lack of memory") The mathematical
justification for this is the following: Assuming that Xk corresponds to yj and Xk+m+l to
yj+l (meaning we threw out m intermediate states) we can in regard the entire collection
of states xk, Xk+, k+m as the Markov sequence element. This means that one term in
the sum on the left hand side of (45) would really be the average of the operator over all
the intermediate states. After constructing the Markov sequence (yk), which certainly is
distributed according to 7, we are ready to apply (45) to calculate expectation values of
operators.
The above is a general overview of Monte Carlo simulations and the basic concepts of
the underlying mathematical theory of Markov chains. Markov chains are a simple case
of a more general phenomenon called Markov processes where time evolves continuously
In practice one checks that yT, and yn2 are uncorrelated for all n.
Table 41: Monte Carlo concepts in mathematics and physics.
Concept Markov chain symbol Physics symbol
State Space S Q
State x q
Probability Distribution e S/Z
rather than in discrete steps. The results are almost the same, although the mathematical
framework becomes a little bit more involved.
To continue the discussion, it is appropriate at this point to narrow the scope and
consider the physical context in which we will work.
4.1.2 Expectation Values of Operators
Let us consider a physical system with dynamical variables denoted collectively by
q and let q live in some space Q which will have any mathematical structure needed to
perform the operations that follow. If S is the euclidian action then the expectation value
of an operator is written:
(F) dqF(q)eS(q), (46)
where Z = fQ dqeS(q) and F is the operator in question. Clearly, from the form of (46)
the integral is supported by the regions in Q where es is large. Standard optimization
yield the classical equations of motion for q. In quantum field theory a traditional next
step would be normal perturbation techniques, to expand the nongaussian part of es
in powers of the coupling constant. But in anticipation for the application of stochastic
methods we interpret (46) as the statistical average of F weighted with the probability
S
distribution  which it of course is. With Q finite we are in an exact application of the
Monte Carlo methods discussed earlier, with a translation of notation summarized in table
41.
We are interested in the special case of a quantum field theory represented as
a Lightcone World Sheet. In this case the space Q is the set of all (allowed) field
configurations on the two dimensional lattice representing the discretized world sheet.
'. will  'y the :. etropolis Algorithm to construct the i : ')v sequence and it works
as follows: Given a field configuration q, we
1. Visit a site in q, and alter the field values there and possibly in the immediate
vicinity to obtain a new configuration q'.
2. Calculate x = exp{(S(q ) S(qi)} and have a computer calculate a random number
y between 0 and 1.
3. If y < x accept the change and go back to the first. with q[ as qg, otherwise return
to the first I ix without modification.
As before, we cannot take the Markov sequence to be (. ): q[ is too correlated with
qi. In the terminology of the last section we can : that updating only a single site
constitutes too small a difference Ax between successive Markov ... elements.
Swe continue and repeat the three steps above. ".'.h.en they have been repeated
the same number of times as there are lattice sites we consider a sweep of the lattice to
be complete. Sites can be visited either at random or sequentially. When enough sweeps
have been done on a lattice configuration for it, to be ::ci( 1 uncorrelated with the
original one we accept it as and proceed again from this configuration to obtain m
another one: .. By sufficiently we mean that .' and qi+2 are completely uncorrelated
and we find the number of s  required for this with extensive testing. In '.'e this
choice of sample rate is not rigorous but as mentioned in the last section, we still average
over the intermediate states. ' efore, this ": out" of states will not affect the
mean value of the operator but only the uncertainty anal of it.
4.1.3 A Simple Example: Bosonic Chain
In order to test the application of the Monte Carlo method in this setting, and the
computer code in .. ticular, we start with an extremely :. .'. example. Because of the
the :::>uter code is organized, this test applies to a large extent the same code as
will later be used in more complex ins. S :.. : fic functions reside in a neatly
separated set of files and can be replaced without consequences to the rest of the computer
program.
Consider the following action:
M1
S = a (qk+l qk)2 with qo q= M = 0. (47)
k0
As before, use the following:
q= (qi, q2, ... qM1), Dq= dx dx2... dxM1, Z = qe S(q). (48)
With the very simple observable operator q > Ok (q) = qk q1 the expectation value
becomes
(Okl) = qeS(q)Okl qkql}) = min(k, 1) (M max(k, 1)) (49)
We use this result to find the expectation value F(k, 1) = ((qk q,)2
F(k, 1) ((qk 2) (q 2(qk qi) + (q
SMIk II(M Ik 1) (410)
2aM
so we can consider the function of the difference only f(m) = F(k, k + m). Without any
loss of generality we take k = 0 and the behavior of f as m ranges from 1 to M 1. The
results for a MCsimulation are shown in Figure 41 together with the above exact answer.
The relative agreement allows us to consider this test passed by the simulation software.
To see the computer implementation of the bosonic chain presented here see Appendix A.
4.1.4 Another Simple Example: ID Ising Spins
The spin system si, which plays a vital role in the Lightcone World Sheet formalism,
has often been likened to an Ising spin system. This is of course true because the spins, as
in Ising's model for ferromagnetism take on the two values s' =T and s' = implemented
on a computer with s\ = +1 and s = 1. A ID Ising spin system is therefore exactly as
2
I4 1 III
0 2 4 6 8 10 12
Chain separation Ik 11
Figure 41.
14 16 18 20
Test results for the simple example of a bosonic chain. An MC simulation
with 2 104 sweeps and parameters M 21, a 1. The figure shows the
results for the expectation value F(k,1) ((qk q)2) where the spacing
Ik l1 is shown on xaxis. The two graphs share the xaxis and in the upper
plot the stems are MC results and the solid line is the exact answer for F(k, 1)
from Eqn. 410. The lower plot shows the residuals, meaning the difference
between the MC and exact results. By fitting the MC results to a functional
form as for F above we obtain M = 20.49 and a = 0.9926. Notice that
the difference plot has some structure. We believe that this is an indication
of correlation between MC errors along the chain, which is an effect we see
clearly on the lattice and shall discuss in more detail later.
our Lightcone World Sheet setup except with a different interaction. The Ising spin system
only has a local interaction. Although the Lightcone World Sheet interaction is also local,
we intend to treat the 1/p factors nonlocally, i.e., we do not employ the local b, c ghosts
but instead put in the 1/p+ by hand making the interaction at least mildly nonlocal. The
Table 42.
Test results for ID Ising system. Here we have exact results to compare to the
MC numbers. The Efit is obtained by fitting the correlation to an exponential
and reading off the exponent as shown on graph 42. We can see that the
overall error decreases with increasing number of sweeps, but once the error is
pretty small this decrease is not very consistent. This is an inherent property of
a stochastic method such as this one.
Numerical Results
Coupling g Exact E MC Efit
104 sweeps 4 104 sweeps 1.6 105 sweeps
0.1 0.20067 0.25724 0.22188 0.21900
0.2 0.40547 0.38471 0.41673 0.41903
0.3 0.61904 0.64785 0.61391 0.61466
wholly local Ising interaction is given by
= ( A\sis]r+ msi)
(411)
with A a matrix with only nearest neighbor interaction. For simplicity of the test we took
m 0 and Aij I= ./ i1.
All the resemblance almost drives us to test the software and methods on just a
ID Ising spin system, which we did. The MC implementation involved using simply the
software for the Lightcone World Sheet with a simplified interaction. The exact results are
very simple to obtain for the case described here. For a given time the state of the system
is represented by the vector of spins s= (si, S2,... SM). Since the interaction is so simple
then the transition matrix is given by
(412)
i4J 9 ( ,j+l + 6J+l,j)
so that s(t + dt) = Ts(t). Taking t A= t/N and s(t) TN(0) then finite time
propagation corresponds to N > +oo. The eigenvalues of T are = 1 g and for
large enough N then E = Int and therefore the splitting is AE ln((1 + g)/(1 g'
The results for the exactvsMC comparison are shown in Table 42 and the fitting is
exemplified by the plot in Figure 42. Again, the relative agreement with known exact
answers gives the computer code a "pass" for this test as well. Having passed both of
03 
0
02 A
position of spin si
10 20 30 40 50 60 70 80 90
Figure 42. Test results for ID Ising system. The graph shows the correlation C between
spins si and S20 so that the yaxis shows: C(i) = (sis20) and the xaxis labels
the spin index i. There are three sets of data plotted together, one dataset
for each value of the Isingcoupling g. The graph also depicts the exponential
fits done so that the exponential falloff of the correlation can be read off. The
results are systematically organized in a table below.
these simple tests, or examples of application, we can allow ourselves to spend some time
to prepare for the real application of the Monte Carlo method presented in this thesis.
4.1.5 Statistical Errors and Data Analysis
Being a stochastic numerical method, the Monte Carlo approach gives only
statistically significant results. We saw this clearly in the last section where even in
the very simple cases of a bosonic chain and a ID Ising spin system, where deterministic
methods would probably have served better (in fact, exact answers were available!). One
might think that since the Monte Carlo method indeed performed relatively poorly in the
simplest cases, it is likely to fail utterly when a more complicated system is considered.
This however is by no means the case. The statistical inaccuracies of the Monte Carlo
method are inherent in the algorithm and remain in the more complicated applications,
however, there is nothing which indicates that this effect should increase in any way just
because the system under consideration is complicated. To understand this, we will discuss
briefly some of the statistical observations which are standard in the application of Monte
Carlo methods [19, 20]. (The book by Lyons [21] contains many useful discussions on
statistics in general scenarios.) Although structurally this section belongs here with the
general discussion of Monte Carlo simulations and ..,ilications of Markov chains, it is not
a required reading to continue the chapter. In fact, the discussion of spin correlations in
the next section are instructive before reading this section. At the end of the day however,
the considerations here apply for any observable and not just ':: correlations.
When .i1 ':. Monte Carlo simulations to study a physical 'm we sample an
observable in various states of the system. We will work extensively with the correlations
between ':: on various sites of the lattice. ': data which we will have available is
the full spin configuration of the lattice for each sweep. Let us denote by n the sweep
number and assume we have performed a total of K We then have the data: s (T)
for i L [1, MJ, j [ LI, N] and n e [1, K], where each value s'(n) is an up or down
spin, represented by ss (n) = 1 or s'(n) 0 r(e ''ely. Here [ki, k2j means the set of
all integers between ki and k2. We will use the expectation notation ( '), when we are
taking averages over sweeps n, i.e., (s,),,, .,() r, where the surn on n runs over
various subsets of [1, K]. When there is no risk of confusion, we omit the i)t n
and write simply ( '). We shall denote by s(n) the entire configuration of spins at sweep
n, s(n) is in some sense a large matrix of spins. We are interested in the correlations
Corr (s, s') = ,) (s{)( ) and mostly their dependence on the action n j j'. So
tackling the issue topdown we can break the data analysis into two parts:
1. Obtain from the raw data (s(n)) another set of data points of the form (, )
such that x ~ j j' and y ~ Corr (s, s') up to additive constants. Because of
the statistical nature of MC simulations the data will have an inherent distribution,
so that along with the data itself we need the variance and covaariance of the data
>ints. In other words, we need the crrormfalrix: F = cov ( ..)
2. T. new data ( ,. ) is fitted to an onentialt ;.e function fa(x) where A
(A, 2... ) are fitting parameters. 'i fit is then performed by simply minimizing
the function F(A) where
F(A) r (f ,A(. ))( f( )). (413)
k.1
Recall that since E is symmetric, then F will be nonnegative
In practice we run MC simulations for a given lattice first to obtain estimates for
E but then run a much longer simulation to obtain the data points (xk, Uk) using the
previous estimate for the error matrix. We start the discussion with point 1) above.
In obtaining an estimate for the ij j' dependence of Corr (s, s') we calculate only
(si s ) since we are not interested in the additive constant which the second term in
the correlation is. This corresponds to keeping (i', j') fixed and using only the data for
which s, = +1. In practice this is achieved by simply not updating the spin at (i',j')
and calculating only (sj) on this lattice. The data set mentioned above then becomes
yj = iE(s') and xj = j'.
As was explained in section 4.1.1 we may need to throw out some of the configurations
in the sequence (s(n)) because <,., v are too correlated to generate a Markov chain. This
is done by calculating the "latticemean" autocorrelation:
Aut (m) M sZ(n + m)s((n)).
i,j
Typically Aut(m) is a falling exponential as a function of m. The reciprocal of the
exponent is called the autocorrelationlength, denoted by LAUT and in our case we have
LAUT = 10 20. This means that by using only every LAUTth sweep, i.e., by generating
a reduced data set: s(k) for k = n LAUT + ko with n = 0, 1, 2,... we are certain the
spin configurations are not correlated sequentially. For calculating the data points yk it
is correct to use all of the data II but when estimating the variance and covariance of
the statistical errors we must consider the reduced data set. These are calculated using
II This is simply because the average over (s(n), s(n+ 1),..., s(n+LAUT)) can be used in
the reduced data set instead of the value s(n) only, and this is equivalent to just averaging
over all the spin configurations.
standard data analysis formulas [21].
A YA, 1i, B = B 1i
cov(A,B) (AA,)(B B,) / 1
SAB AB
given the data sets (A1, A2,..., A, ...) and (BI, B2, ....
Turning next to point 2) above. The interpretation for the expression for F is
the following: We consider the values ', = (yk fA(Xk)), for k = 1,... K, to be
normally distributed random variables with given variance and covariance. Recall that the
underlying n (sweep label) dependence of Xk and Yk has now been discarded and replaced
by an "allowable" distribution in the "errors" ,,' If ,; for k = 1,..., K, were normally
distributed random variables with mean zero and variance one and among themselves
uncorrelated, we could relate the vs and ws by: Cv = w with the matrix C given by
the Cholesky factorization of the error matrix E and v = (v,... vK) But since v is so
simple we can calculate the norm as follows
Iv = V TV (C w) T(C w)
wTE w F (414)
so that F is just the squared distance between the data and the fitting function fA.
Minimizing F(A) as a function of A is just the well known leastsquares fitting, with
the extension that we use statistical information about the fitted data. We minimize F by
use of the LevenbergMarquardt algorithm, which is an optimized and robust version of the
NewtonRhapson class of methods. These optimization methods "search" for the minima
by starting at an initial guess for the parameters A and travelling in Aspace, with certain
rules determining the step AA, evaluating F at each step until convergence is obtained.
The rather advanced LevenbergMarquardt method uses a combination of the following
two step determinations:
* Steepest descent method: Take simply
AA t VAF(A)
and use one dimensional optimization to determine the step length t (called a line
search).
* Use a quadratic approximation for F:
F(A + AA) F(A) + AA VF(A) + AATU (A)AA
2
with R the Hessian matrix for F, and use the step AA which minimized this
approximation AA = H1 VF.
We implement all of the statistical calculations and data analysis procedures in
MATLAB@.
4.2 Application to 2D Trf3
A scalar matrix field theory in 1+1 spacetime dimensions described by the action
(21) can of course be written in the Lightcone World Sheet form, just as was done in
3+1 dimensions. However, with only two dimensions there are no transverse bosonic q
variables living in the bulk of the world sheet. It is also a well known fact that this theory
is ultraviolet finite meaning that we can without considering counterterms on the world
sheet, proceed directly with simulating the theory using the methods developed in this
chapter. This very simple choice is motivated by this fact and also by the fact that the
theory has been used widely as a toy model and the results can therefore be verified and
compared. The Monte Carlo method can in this context be verified in its own right.
With this preamble, we turn now to the simplest possible case where the only
dynamical variable left in the system are the spins which designate the presence of solid
lines. The b, c ghosts can be eliminated by simply putting in by hand the factors of 1/M
which ,.r v were designed to produce. Although this does introduce a nonlocal effect,
and therefore a seemingly dramatic slowdown of the Monte Carlo simulation there are
shortcuts that can be used as will be explained.
In this simple case a world sheet configuration consists only of NM spins labelled by
st, where i = 1, 2,. M and j 1, 2,. N label the space and time lattice coordinates
respectively. An up spin could be represented by st = 1 and a down spin by st = 0.
We employ periodic spacial and temporal boundary conditions so that the field, in this
case only st, lives on a torus. In principle the Metropolis algorithm now just proceeds
analogously to the Ising spin system, where each site in the lattice is visited and a spin is
flipped with a corresponding probability to complete a full sweep of the lattice which is
then repeated. Lattice configurations Ck, where k labels the sweeps are generated and then
used to calculate observables of the physical system.
4.2.1 Generating the Lattice Configurations
The basic ingredient for the Metropolis algorithm of generating lattice configurations
is to determine the change in the action under the possible local spin flips that may occur.
When a spin is flipped as to make a new solid line the mass term of the propagator goes
from e6p2/ (Mi+M2) to being e62/,mM, Ca2/mM2 where the M, and I.[. denote the lattice
steps to the nearest up spin to the left and right respectively. Further when a spin flip
results in a solid line being lengthened upwards (downwards) the factor for the fusion
(fission) vertex will be moved upwards (downwards) possibly resulting in a change in
which M1, 3 [_, to use. Then there is the appearance or disappearance of fission and fusion
vertices as solid lines split or join. These basic ingredients are summarized in Table 43.
However the table (43) does not tell the whole story because there are a number of
subtleties that need to be addressed. These can be summarized as follows:
1. Particle Interpretation.
2. No Four Vertex.
3. Ergodicity.
Table 4 3.
Basic spinflip probabilities. T .: right column has the basic probabilities
for the (i,j) ; '; to flip given the local lattice configurations shown in the
left column. A filled dot at a lattice location indicated a. spin value of up
there, no dot. indicates a ':. value of down. Notice that the reverse "',
(from down to up) has thle reciprocal probability. M(JMW) is thie spatial
lattice steps to the next :: spill to tih left (right) at t ime slice j. Here we
us Al 12 1 + 12
', : ,
d+1
M
j1
M M
SJ P= M "(M1 r)ef
ji
4. Nonlocal Effects.
We now turn to these issues in order.
Particle Interpretation. On a periodic lattice there cannot be a time slice which does
not have any up spin. This has no interpretation in the Feynman rules from which our
world sheet description comes.
No Four Vertex. Two solid lines cannot end on the same time slice if the lines are
in clear sight from each other, i.e., there is no solid line separating the two ends in the
space direction. This is illustrated in figure (43). The reason for this is that this would
in correspond to a four point interaction which is not present in the field theory. In full
Lightcone World Sheet formulation these configurations are automatically avoided because
a double ghost deletion on the same time slice results in a value of zero for the action.
Here however the ghosts have here been treated by hand and we must therefore ban such
configurations by hand also.
** *
*
disallowed disallowed
*. *
*5 9 *
allowed allowed
Figure 43. Examples of allowed and disallowed spin configurations. Configurations which
must be avoided by hand because of the manual application of the b, c ghosts.
F,.q..l. :.;, By banning the configurations mentioned above we introduce a certain
nonergodicity into the Metropolis scheme. If all such configurations are banned it is
impossible for a solid line to grow past the end of another solid line residing at a nearby
space slice, unless we allow two simultaneous updates of spins. This way a solid line is
allowed such a growth without going through the banned configuration. This is illustrated
in figure (44). The probabilities for double updates are obtained in a similar way as those
for single updates.
*
*
Figure 44. Basic double spinflip. If the two spins indicated by nonfilled circles are
allowed to be flipped at the same time the system is able to evolve from one
configuration to the other without going through the banned configuration.
Nonlocal Effects. Changes in the action involving M1 and [. are inherently nonlocal
but are easy to deal with in a simulation and do not overly slow down the execution of the
program. When double updates have been allowed there are a number of scenarios which
will lead to potentially worse nonlocal changes in the action even though the spin flips
themselves are local. An example is illustrated in figure (45). If we denote by Ml(i,j)
([_ (i,j)) the distances to the left (right) of spin (i,j) then in the case shown in the figure
the fusion vertex factor at (i1,j) change by a factor:
M i(i,j) + _(i, j)( M(i,j + 1)
(415)
Mi(iI,j) + MI(i2,j) + f_ (12,j 1)Mi( + _(2,j + 1)
It is interesting that the resulting change in action due to the fusion vertex at (i1,j)
depends only upon the factors of M derived from one of the sites which undergo the flip,
namely (i2,j + 1). Going through all the different scenarios this turns out to be generally
true. The nonlocal effects of this type therefore do not require any large scale lattice
inspections by the simulation program and therefore do not slow down execution.
Table 4 4: World Sheet spin pictures.
II I
Ii
1,l ,
:1 I I I~ iI ,I
i I .,, I I ,
1,1 1, I I :
11.1 I
I I
.1 i
I I
.1
Very strong coupling, g/p2
Strong coupling, g/p2
Weak coupling, g/1p2
1.2649
0.4472
0.3464
Very weak coupling, g/p2
I
0.2000
0
0
*
*
1 2
;ure 45. Double spinflip, with distant vertex modification. When the two
indicated '. nonfilled circles are flipped down there is break in the solid
line at i2 with corresponding local fission and fusion vertex factors. However,
there is also a change in the factors b for the fusion vertex at i\. ':.
distance to the right of spin (i, j) to the next up spin has changed as
well. in other words, "' (it,j) has changed.
4.2.2 Using the Lattice Configurations
We are interested in the physical observables of the, : In :. :'ticular those which
have a direct int. ) on in terms of the field theory described by the Lightcone World
Sheet. Such an observable would be for example the energy levels of the system. To
understand how these will emerge from the world sheet let us consider the system in terms
of time evolution in the discretized Lightcone time x i nair. l our lattice time j. At a
given time j the min can be characterized as a 2M state ..um .m with a time
evolution operator T which takes it to the next time j + 1. i negative logarithm of this
:ator is the Hamiltonian of the system, namely the Lightcone P < .. *ator and the
energy levels are given '. the eigenvalues. '..'.. write T H, and consider the correlator
between two states (5) and I} :at ed j time steps on a i dic lattice with total
time steps N
(G T T) T (416)
Take m) for m N to be the eigenbasis for T with .) = ) and write 0) = In)
and ) I in) then
(E a(b 17)
G(jI I n abti (1A 17)
Notice that by dividing through equation (417) with to assuming the t,s are in
decreasing size order we see that the j dependence of G is entirely of the form
G(j) A B + Cne a^"N B* + ( Czj) (418)
where AT is the energy gap between the ground state and the nth excited state and
A, B and the C,'s are constants depending on the overlap of the states 0), Q) with each
other and the eigenstates of T. By judiciously choosing states Q), Q) and going near
the continuum limit (equivalently choosing large enough N and j) we see that the lowest
energy gap of the theory can be read off as the exponential coefficient in the j dependence
of G(j).
A very important fact is that the full 2M state quantum system that describes the
lattice at each time has a large redundancy in terms of the field theory we are simulating.
Let us for example consider the simple case of a propagator in the two languages. On the
periodic lattice this propagator can be represented by a solid line at any of the M different
spatial points. But in the field theory there is no such labelling of which of the M different
propagators to chose from. The propagator state is really the linear combination of all
the states on the spaceslice with one upturned spin somewhere. In the more complicated
cases where there are a number of up spins, again the field theory does not distinguish
between where the spin combination lies but only between the different combinations
of down spins in a row and the order of these. In other words the field theory state
is the cyclicly symmetric linear combination of the spaceslice states. Notice that the
transition operator T preserves the cyclicly symmetric sector, so if it were possible to
project out the noncyclicly symmetric contributions by a choice of 1) and y) we would
obtain a nonpolluted transition amplitude. The problem is that on the lattice only M
of the 2M states at each timeslice are available, namely the "pure" states with each spin
either up or down. Out of these there are only two purely cyclicly symmetric states,
the allspinsdown and the allspinsup, and the allspinsdown is forbidden as described
above. The allspinsup state would be a bad choice since it has a very small overlap with
the cyclicly symmetric ground state, in other words, it is very far away from being the
configuration preferred by the probability distribution es
So proi. ,tii in* out the unwanted states is not a viable solution. Instead we try
to consider cyclicly symmetric observables, for example the sum of the spins at a
timeslice. Although such an amplitude would still get contributions from the intermediate
noncyclicly symmetric states, these can be suppressed. Consider for example cyclicly
symmetric operators 01 and 02 at times ji and j2, in the presence of a state 1p). The
quantity in question is
Q(i2) = iTJ 21) (419)
Q(jlj2) TrTjl+j2
As before we write Q) = a, In) but now we assume that even and odd n label the
cyclicly symmetric and ..i,iin,. I ic sectors respectively. The tas are now ordered in
decreasing size within each sector separately. We write O = (mnOi In) for all n, and
i 1,2 and we consider the case where El < E2 (see figure (47)). We have
QJ1,J2) Tanakr nOnkt0t^
TT1J2 (AtJ+2 + B't t2 + Btt]2 + C +tj+j2 + DI 2 + DT )
A + B1ea2 + B2eaAIjI + CeaAI(j1+j2) + DieaA2J2 + D2aAZ1j + ... (420)
(we use capital Latin letters for constants with respect to ji and j2). But we have
O',2m+i = 0 because of cyclic symmetry and therefore B1 = B2 = 0. We can see
that the unphysical gap A1 only contributes to Q as an exponent with ji + j2. In the limit
where the sheet is large N = ji + j2 compared to interspacing of operators j = \j* j1
we see that artifacts of A1 are suppressed compared to A2. Of course, this relies on A1
not being too small. We will see how a degeneracy between E0 and the unphysical El can
compromise this argument and render the method unusable in that case. Fortunately this
case corresponds to small coupling where perturbation theory is a better method anyway.
The above arguments show how we can obtain the energy gaps of the field theory
from the correlations of spins on the Lightcone World Sheet. Even if contributions from
noncyclically symmetric states may act as noise in our data there is hope that when the
coupling is large, the energy gaps can be read off from the exp behavior of the correlation
as a function of time (j e [1, N]). In Figure (46) an example of the correlation obtained
from the system simulation. The figure shows a number of interesting and representative
facts about the methods employed in this work. It shows the fit (solid line) plotted
together with the data points (with error bars). The fit is nonlinear and obtained using
specialized methods which we developed for this purpose, in order to capture the specifics
of equation (420). The points on the graph are not really the correlation between spins,
but rather, the value of R(j) = (sl ,), with (i', j') (1,500) a fixed upspin. The
correlation is equal to R (s) (sf,), which, since (i',j') is fixed, is just equal to R less
the average of spins on the lattice. This average has no lattice dependence (no (i,j)
dependance), which is why we work with R rather than the correlation itself. From the
graph we can see that even though correlations are longer ranged in the weak coupling
(that is, long lines predominant) the overall average of spins is much lower, there is much
less happening in the low coupling regime, which is what we could see qualitatively from
the world sheet figures (45).
Before data fits such as the one above could be tried, we had to relax the system as
explained earlier. When we relax the system we are essentially finding a state in which
to start the simulation off in, i.e., a state which is representative for the states near the
action minima. Of course, we cannot ever rigorously prove that we are in a truly relaxed
state of the system, but in practice we achieve relaxation by observing system variables,
preferably true physical observables such as the correlations or the total magnetization
(the average value of the spins) of the system. We observe these variables starting off from
0 100 200 300 400 500 600 700 800 900 1000
Figure 46. Fitting MonteCarlo data to exp. The data from the MonteCarlo simulations
is fitted to an exponential in order to read off the massgap. Each point on the
graph gives the correlation between a fixed upspin at (i, j) = (1,500) and a
spin at various location on the spaceslice i = 1. The simulated system here
had M=40 and N 1000 and we used about 106 sweeps. The The top plot is
from a low coupling MonteCarlo run (g/p2 = 0.9), whereas the lower plot
is from strong coupling (g/p 2 = 0.4). Notice how much weaker the overall
correlation is for low coupling, i.e., the signal strength in the exp falloff is
weak.
various starting states such as the "almost empty" state (all spins down except for a long
line of up spins through the entire lattice), or some random spin state. We saw how these
observables, although starting at some values converged to the same "equilibrium" value
irrespective of the starting point. When this value had been reached within statistical
accuracy, we claimed the system to be relaxed. A systematic and detailed analysis was
done by observing magnetization and correlations and we found that about 105 or 106
sweeps was usually more than enough, even if starting from the "almost empty" state, a
state which we believe had little overlap with the "ground" state, or "equilibrium" state.
The equilibrium state of course depends on the coupling so we took up the standard
of running at least 105 sweeps before data was sampled, even when we started from
the ground state of the simulation in the previous run. Since, in practice, we often ran
simulations with similar coupling right after the other, so this should be a very safe end
accurate standard.
4.2.3 Comparison of small M results: Exact Numerical vs. Monte Carlo
Let us first look at the simplest of cases, namely the M = 2 case. This can be solved
exactly with arbitrary N. At every timeslice we have a 3 state system corresponding to
SIT), I T) and I T). Recall that the [{) state is not allowed on a periodic lattice. In this
ordered basis the transfer matrix is
/ ep /4To 0 e5p2/s8T
as can be read off from the rules 43).
to make 72 hermitian. It is illustrative
V21o
e 2/4To 9 g 5p/8To (42
__ / To
) 5p2/8TO p2/IO
,/5% e /
The offdiagonal elements have been symmetrized
to use the basis:
1
I) (I IT) I ))
v2
1
2+) (I I T) + I Ti))
2) iT )
where the transfer matrix becomes
e2 /4To
T 0
0
0
ep /4To
0 e5p2/s8T
TO
0
_kg5p /8To
To
e'/rTo
1)
(4 22)
(423)
(424)
(425)
And we see how the noncyc( '. 1 symmetric state ..... les. Let us denote by A1, A2 and
As the eigenvalues of T in decreasing size order. ",'. have
A e e ( /4I 2/ 2 e52/47" ) (4
A ) (4
To/
A2 .e^/7 (4
27)
As 2 eU/ + (e , 2 + C. (41
I..: energy levels are given In Ak and we see that A2 is the one :ding to the
nor licly symmetric sector. From the simulation we will extract energy and we
have the unphysical one G, = In i'2 and the 1. I. 1 one C = In Unfortunately
the smaller gap is the uninteresting one and moreover, the unphysical gap goes to zero in
the small coupling limit because A, and A2 are degenerate. We shall see how the general
argument given above about the suppression of such v.. ,. 1 '. J relics works in this simple
case.
jure 47.
Energy levels of a typical TT. A schematic diagram showing the en.
levels of a nical .... Ti levels are labelled for 1n 0, 1, 2,... ordered
in ascending size within each sector, even and odd n labelling the cly
symmetric and asymmetric sectors respectively. Ti I.. ( ... level must
always be the lowest lying. We consider here the potentially troublesome
situation where EL < E2. 1.: ordering of the remaining "r . levels is
ort ant in the context of the current discussion.
Now let us consider an observable that can be calculated easily with the simulation.
Namely the average value of ':. at various sites given a fixed :: at (ij) = (1,1).
Let us denote by E,,, the sum over states n') = IT) and In') I= I) and let Z
)
Z,,/(n'7NTn') then the average value of a spin at a site (i,j) is given by
(P) + Z (n 2N I ') ( 'T2 I) (429)
n mn
This can easily be evaluated by simply diagonalizing (425). We extract the j dependence
a) ar + a2SN + 3 + a4(rJ + rNj) + a5(sj + sNj) + a6(JrNj + rs NJ)
a(Pr) a aN +(4 +30)
vTr" + sN + v /
air N + a2N 3 a4rj Nj) a5 + Nj) a6 (SjrNj + N ,
(P2i) = aa2s 4 31)
VrT + sT + ^ \ J
where r = eG, and s = e are the exponentials of the j dependence and the aks
are known although complicated functions of and . Clearly there is a exponential
drop off with j with several exponents corresponding to each energy gap. It is impossible
practically to extract G, from statistical data for (P/) because GT is 1, .. But if we look
at the cyclically symmetric observable R = i Pi we have
(R) 2 a _rN + a2S +a3 + 2 + (,j + rNJ) (432)
(R)rN + s + ,a3 +2/,rN + aar S + /a3
In this simple case not only is the dependence of G, suppressed in (Ri), but actually
drops out completely.
(3g2+22 2 x2+g2
3y2 2 +2 x2 2 x Ix2 + 2 y2g2)
a1
a2
fI
4 (2 y2 2 + +x2 2 + 2y22)
1
4
(3 y2 2+2x2+2x x2+2y2g2)
(433)
(434)
(435)
9y4g4
a4 = y (436)
4 (2 y2 2 + X2 2 + 2 y2g2) 2y2 2 + X2 + x X2 + 2 y2g2)
as = y2 (437)
2
a6 2g (438)
4 2 y2 + 2 2 + 2 y22)
e , /TO1 2/4TO e 52/STO
where x C 2 T and y = v2
We can now compare the results of the Monte Carlo simulation with the exact results
given by (430) as functions of j for given values of g,To and p2. In figure (48) and (49)
we see the results of a sample calculation using the Monte Carlo simulation for M = 2 and
N = 1000 in conjunction with the exact results obtained above. Although the results are
disp] y.I without proper error analysis the reader should be convinced that the simulation
is in qualitative agreement with the exact results. The idea is that in more complicated
cases where it is intractable to do the exact calculation one should be able to fit the
statistical curve for (Ri) with an exponential and read off the energy gap. Doing this for
this simple case gives:
A very similar procedure can be done for M = 3 in which case there is an eight state
system at each timeslice, namely  TTT), I ITT), I TIT), I TTI), I tIT), I TI), I ITI), III).
The 73 matrix can also be diagonalized (although this time it was done numerically) and
exact results can be obtained and compared to the Monte Carlo simulation as shown in
Figure (410). Clearly the size of the matrix TM increases exponentially with M which
4 (2 y2 2+ X2 2 + 2 y2g2 2

Full Text 
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STUDIESOFLIGHTCONEWORLDSHEETDYNAMICS INPERTURBATIONTHEORY ANDWITHMONTECARLOSIMULATIONS By SKULIGUDMUNDSSON ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2006
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Copyright2006 by SkuliGudmundsson
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IdedicatethisworktomychildrenDanel,Lsaandsar.
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ACKNOWLEDGMENTS IwouldliketothankCharlesB.Thorn,mysupervisor,forhis manysuggestionsand patientsupportandguidanceduringthisresearch.Ialsoth ankhimforhisfriendshipand understandingwhentimeswerehardbutespeciallyforhisab ilitytokeepmeinterested andmotivated. TheIFTFellowship,whichwasawardedtomefortheperiod200 12003,wascrucial tothesuccessfulcompletionofthisproject. IshouldalsomentionthatmygraduatestudiesintheUSAwere supportedinpart bytheIcelandicStudentLoanInstituteaswellastheFulbri ghtInstitutewhichsupported measaFellowandnallytheAmericanScandinavianAssocia tionsupportedmewithits "ThorThorsGrant."Ithanktheseinstitutionsfortheirna ncialsupport. MostlyIamgratefulformybelovedchildrenDanel,Lsaand sar,whomImissmore thanwordscanexpress.Throughallthediculttimesanddar kestmomentsmychildren werealwaysmyanchor,myhopeandmylife.Tomyrealfamily,t hosewhomIcould alwaysfallbackon:Mamma,pabbi,Inga,Kalli,Tobbiandthe irfamiliesIamforever grateful.Theystoodbymethroughdicultiesattimeswheni twasthemostdicult. Finally,Iwishtothankthefollowing:AlbertoandBobby(fo ratruefriendshipthat willneverbeforgotten);Hjalti(forhiscalmandsensitive advice,butmoreimportantly, forhisloyaltyasafriend),Gulli(forhisuncompromisingh elpandsupport)andGsti(for alwaysbeingmylifelongbestfriend). 4
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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................7 LISTOFFIGURES ....................................8 ABSTRACT ........................................10 CHAPTER 1INTRODUCTION ..................................11 1.1LightconeVariablesandOtherConventions .................13 1.2'tHooft'sLarge N c Limit ............................15 1.3WorkonPlanarDiagrams ...........................18 2LIGHTCONEWORLDSHEETFORMALISM ..................24 2.1IntroductiontotheLightconeWorldSheetFormalism ............24 2.2SupersymmetricGaugeTheories ........................31 2.2.1SUSYYangMillsQuantumFieldTheory ...............31 2.2.2SUSYYangMillsasaLightconeWorldSheet ............33 3PERTURBATIONTHEORYONTHEWORLDSHEET .............40 3.1GluonSelfEnergy ...............................41 3.2OneLoopGluonCubicVertex:InternalGluons ...............44 3.2.1AFeynmanDiagramCalculation ...................45 3.2.2Simplication:TheWorldSheetPicture ...............53 3.3AddingSUSYParticleContent:FermionsandScalars ............53 3.4DiscussionofResults ..............................55 3.5DetailsoftheLoopCalculation ........................58 3.5.1FeynmandiagramCalculation:Evaluationof ^^_B 1 ..........58 3.5.2FeynmandiagramCalculation:Evaluationof ^^_B 2 ..........59 3.5.3FeynmandiagramCalculation:Evaluationof ^^_II ..........61 3.5.4FeynmandiagramCalculation:DivergentPartsofInt egralsandSums 63 3.5.5DetailsofSUSYParticleCalculation .................68 4THEMONTECARLOAPPROACH ........................72 4.1IntroductiontoMonteCarloTechniques ...................72 5
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4.1.1Mathematics:MarkovChains .....................73 4.1.2ExpectationValuesofOperators ....................77 4.1.3ASimpleExample:BosonicChain ...................78 4.1.4AnotherSimpleExample:1 D IsingSpins ...............79 4.1.5StatisticalErrorsandDataAnalysis ..................82 4.2Applicationto2DTr 3 .............................86 4.2.1GeneratingtheLatticeCongurations .................87 4.2.2UsingtheLatticeCongurations ....................92 4.2.3Comparisonofsmall M results:ExactNumericalvs.MonteCarlo .97 4.3ARealTestof2DTr 3 .............................101 APPENDIX ACOMPUTERSIMULATION:DESIGN .......................111 A.1ObjectOrientedApproach,SoftwareDesign .................111 A.1.1BasicIdeas ................................111 A.1.2ObjectOrientedProgramming:GeneralConceptsandNo menclature 113 A.1.3OrganizationoftheSimulationCode .................117 A.2DescriptionoftheComputerFunctions ....................121 BCOMPUTERSIMULATION:EXAMPLEHEADERFILE ............127 REFERENCES .......................................130 BIOGRAPHICALSKETCH ................................132 6
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LISTOFTABLES Table page 21LightconeFeynmanrules ...............................27 41MonteCarloconceptsinmathematicsandphysics .................77 42Testresultsfor1 D Isingsystem. ..........................81 43Basicspinipprobabilities ..............................88 44WorldSheetspinpictures. ..............................91 45MonteCarlovs.Exactresultsfor M =2 ......................101 A1Softwarelestructure. ................................117 7
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LISTOFFIGURES Figure page 11Exampleof'tHooft'sdoublelinenotation. .....................17 12VariouswaystodrawFeynmandiagrams. .....................20 21WorldSheetpictureoffreescalareldtheory. ...................29 22GrassmanneldsnakingaroundtheWorldSheet. .................36 23Quarticfromcubicsforasimplecase. ........................39 31Oneloopgluonselfenergy. ..............................42 32Cubicvertexkinematics. ...............................45 33Oneloopdiagramsforxed l intheLightconeWorldSheet. ...........55 41Testresultsforthesimpleexampleofabosonicchain. ..............80 42Testresultsfor1 D Isingsystem. ..........................82 43Examplesofallowedanddisallowedspincongurations. ..............89 44Basicdoublespinip. ................................90 45Doublespinip,withdistantvertexmodication. .................92 46FittingMonteCarlodatatoexp. ..........................96 47EnergylevelsofatypicalQFT. ...........................98 48MonteCarlovs.Exactresultsfor M =2 .....................102 49MonteCarlovs.Exactresultsfor M =2 ,cycliclysymmetricobservable. ....103 410MonteCarlovs.Exactresultsfor M =3 .....................104 411 M =12D&KandMCcomparison. ..........................107 412Dataanalysisanddatattingfor M =12 simulations. ..............108 413 E asafunctionof 1 =a ...............................109 414 E asafunctionof x ................................109 8
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415 E asafunctionof M ...............................110 A1Organizationofthecomputercode. .........................119 A2OperationsoftheMonteCarlolayer. ........................123 9
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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy STUDIESOFLIGHTCONEWORLDSHEETDYNAMICS INPERTURBATIONTHEORY ANDWITHMONTECARLOSIMULATIONS By SkuliGudmundsson December2006 Chair: CharlesThorn MajorDepartment:Physics Inthisthesisweapplycomputerbasedsimulationtechniquestotacklelarge N c quantumeldtheoryonthelightcone.ThebasisfortheinvestigationwassetbyBardakci andThornin2002whentheyshowedhowplanardiagramsofsuchtheoriescouldbe mappedintothedynamicsofaworldsheet.Wecallsuchareformulatedeldtheorya lightconeworldsheet TheLightconeWorldSheetformalismoersanewviewonlightconeeldtheories whichallowsforapplicationsofstringtheorytechniques,therebygreatlyaugmentingthe machineryavailableforinvestigatingsuchtheories.AsaninvestigationoftheLightcone WorldSheetitselfthecurrentworkpresentsstochasticmodellingtechniquesasameansof clarifyingtheformalismanddevelopingintuitionforfurtherwork.Seenasaninvestigation oftheunderlyingquantumeldtheory,theapproachtakenhereistocapitalizeonthe onlytwodimensionsoftheworldsheetasopposedtothefourdimensionsoftheeld theory. 10
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CHAPTER1 INTRODUCTION Inthisdissertationwewillpresentaconnectionbetweene ldtheoriesandstring theorywhichhasenjoyedanincreasinginterestbyresearch ersintheeldoftheoretical highenergyphysics.ThisistrueespeciallysinceMaldacen aproposedthatIIBsupersting theoryonanAdS 5 S 5 backgroundisequivalenttosupersymmetricYangMillsel dtheory withextended N =4 supersymmetry(aconjecturecalledtheAdS/CFTcorrespond ence [ 1 ]).Thisistheviewthatalthoughstringtheoryhassofarbee nunsuccessfulinitstaskof predictingexperimentalresultsasanallinclusivetheory ofeverythinginitsownright,its techniquesarepotentiallyapplicableindescribingmathe maticallylimitsorviewsofolder eldtheorieswhichthemselveshaveastrongexperimentalf oundation. Thestringtechniquesineldtheoryhaveamyriadofpossibl eapplications.Wherever thestandardperturbativeadvancesfail,someothertechni queisneeded,theconnement probleminQCDbeinganobviousexample.Mechanismswhichha vebeenproposed toconnethequarksofQCDincludecoloruxtubesandgluonc hains.Acomplete descriptionofthemechanismwouldofcourseeventuallyinv olveastringylimitof theunderlyingeldtheory.Itmaybethatastringdescripti onofparticularphysical mechanisms,andnotofthetheoryasawhole,iswhatwillprev ail.Theideadatesback to1970[ 2 ]and[ 3 ]andwasfurtherestablishedbyt'Hooftin1974[ 4 ].t'Hooft'sapproach wastoconstructasystematicexpansionin 1 =N c forageneraleldtheory,wherethe meaningof N c couldbemadedenitefor"nonGauge"theoriesbymeansofag lobal SU ( N c ) symmetrygroupactingonamatrixofelds.The 1 =N c expansionturnedoutto singleout planar Feynmandiagramsinexpressionsforobservablesofthetheo ry, planar inthesensethatinhisdoublelinenotationtheycouldbedr awnonaplanei.e.,withno linesintersecting.Whentheplanardiagramsint'Hooft'sor iginalworkwereselected,the 11
PAGE 12
approachsuggestedaworldsheetdescriptionwhichwasnonlocalonthesheet.Asimilar, butlocal,worldsheetwasmuchlaterexplicitlyconstructe dbyBardakciandThorn[ 5 ] andtheirformalismisessentiallythebackdropandfoundat iontotheworkpresentedhere. Wewillinthisdissertationpresentworkthathasbeendonei ntheframeworkof thisexplicitlocalworldsheetdescriptionofwellknown eldtheories.InChapter1we discussthissettingingeneral,theplanardiagramapproac handthelightcone.InChapter 2wepresentinsomedetailtheLightconeWorldSheetestabli shedbyBardakciand Thornandlaterdevelopedbythemandothers.Theapproachis tomaintainclarityand leavedetailsandtechnicalityintheoriginalwork.Nextwe turntothetwothemesof thedissertation.InChapter3wediscussstandardeldtheo reticperturbationtheoryin theLightconeWorldSheetsetting;sincetheworldsheetmap pingisdonediagramper diagramthisshouldyieldentirelyknownresultsbutnowint hecontextofadierent regulator.Thischapterisalmostentirelyarecapfromarti cleswhichIcoauthored [ 6 ]and[ 7 ].Itservesbothtofamiliarizethereaderwiththesettinga ndtotestthe formalismintheperturbativelimit.Chapter4dealswithth eformalisminadierent andcomplementaryway,namelybyuseofnumericalmethodswh icharemosteective inthestrongcouplingregime.Itwasoutsidethescopeofthe currentthesistocomplete thenumericalinvestigationforothertheoriesthanthesim plestonesthattheLightcone WorldSheetformalismhadbeendevelopedfor,namelytwodim ensional 3 scalareld theory.WeemployMonteCarlomethodssimilartothoseoflat ticegaugetheorybuton theworldsheetsystemdescribingtheeldtheory.Although thegeneralmethodologyof MonteCarlosimulationshasbeenwellestablished,notonly inlatticegaugetheoryand statisticalphysicsbutinawiderangeofappliedmathemati calsettings,theapplication hereiscompletelynovelandrequiredthedevelopmentofspe cializedcomputerprograms. Itwaspartlyforthisreasonthattimedidnotpermitanumeri calinvestigationofmore interestingeldtheories,butmainlythereasonwasthatat thetimewhenthisworkwas done,therenormalizationprocedureforthemorecomplicat edtheorieshadnotyetbeen 12
PAGE 13
developed.Sucharenormalizationisofcoursenecessaryfo radirectnumericalapproach sinceinnitiescannotbehandledbyoatingpointnumbersi nacomputer.Theprogram couldstillbeemployedtothetwodimensional 3 scalareldtheory,sinceitisniteto beginwith.Thedissertationconcludeswithappendicestha tdescribetheimplementation ofthecomputersimulation.Theydonotcontaintheexplicit sourcecode,sincethiswould makeforcountlesspages,butratherexplainsanddescribes thecomputercodeandits developmentandorganization.Theactualsourcecodecanbe obtainedfrommeviaemail shouldthereaderbeinterested. 1.1LightconeVariablesandOtherConventions Onestartsoutwithsocalled Lightconevariables whichfor D dimensionalMinkowski vectors X aredenedas X + = 1 p 2 X 0 + X D 1 X = 1 p 2 X 0 X D 1 X k = X k (11) The transverse components X k areoftendenotedbyvectorboldfacetype,sothatthe Minkowskivectorispresentedbycoordinates. ( X + ;X ; X ) .TheLorenzinvariantscalar productofvectorshastheform X Y = X Y = X Y X + Y X Y + inthe Lightconevariables. When D =4 ,acasewendourselvesdealingwithfromtimetotime,theni tis sometimesconvenienttouse polarization denedby X ^ = 1 p 2 X 1 + iX 2 X = 1 p 2 X 1 iX 2 : (12) Thenthecoordinatesare ( X + ;X ;X ^ ;X ) butwedonotuseaspecialindexbecause thischoiceofcoordinatescanbetreatedinthesamefashion asCartesianones,sothat X k 13
PAGE 14
canmean k =1 ; 2 butalso k = ^ ; .HeretheLorentzinvariantproductcanbewritten X Y = X ^ Y + X Y ^ X + Y X Y + Toclarifythemeaningofthese Lightconecoordinates welookalittlecloserata simpleexample.ConsiderascalareldtheorywithLagrangi andensity: L = 1 2 @ @ 1 2 m 2 V ( ) : (13) Wewishtostudythistheoryinthecanonicalformalism,with x + representingtime,i.e., theevolutionparameter.Wewrite = @ + sothat @ @ =( r ) 2 2 @ andthen = L = @ (14) where istheconjugatemomentumto .Weshallsometimeswanttoworkwiththe LightconeHamiltoniandensity,writtenas P = L = 1 2 ( r ) 2 + m 2 2 + V ( ) (15) ThisproceduregoesthroughanalogouslyforGaugetheory,a lthoughthealgebraisa bitmorecumbersome,asweshallseeinsection 2.2.1 .Letusconsiderapure U ( N c ) Gauge TheorywithLagrangian: L = 1 4 Tr ( G G ) (16) where G = @ A @ A + ig [ A ; A ] (17) A = X a A a a 2 (18) andthe sarethegeneratorsof U ( N c ) (thequantumoperatornatureof A residesinthe coecients A a ).Nowtheindex a labelsthematricesinthechosenrepresentationofthe 14
PAGE 15
Liealgebra U ( N c ) whichisdeterminedbythestructureconstants f abc bytherelation: a ; b = if abc c (19) Sincethematrices a formabasisforthe U ( N c ) algebratheindicesmustrunfrom 1 to N 2 c thedimensionofthealgebra.Arepresentationof U ( N c ) isasetofmatriceswhich satisfyrelation( 19 ),andthesocalled fundamental representationisthesetof N c N c unitarymatrices(hencethelabel U ( N c ) ).Anothersuchset(whichsatises( 19 ))are thestructureconstantsthemselves,namely f abc thoughtofasamatrix,withtheindex a labellingwhichgeneratorandindices b and c labellingtherowsandcolumnsofthematrix. Thestructureconstantsareinthiswaysaidtoformtheadjoi ntrepresentationofthe algebra U ( N c ) .Noticethatinthisrepresentationthematricesare N 2 c N 2 c butthereare stilljust N 2 c degreesoffreedom,i.e. U ( N c ) isfoundhereasasmallsubalgebraamongall matricesofthissize. 1.2'tHooft'sLarge N c Limit In1974apaper[ 4 ]waspublishedby'tHooftexplaininghowasystematicexpan sion in 1 =N c foreldtheorieswith U ( N c ) symmetriescouldbeachieved.Thesheetsonwhich Feynmandiagramsaredrawnwereclassiedintoplanar,onehole,twohole,etc.surfaces anditwasshownthatthistopologicalclassicationcorres pondstotheorderof 1 =N c intheexpansion.Inhisarticle'tHooftpointedoutthatthi swasalsothetopologyof theclassesofstringdiagramsinthequantizeddualstringm odelswithquarksatits ends.HefurtherpursuedthestringanalogybygoingtoLight coneframeandproposed aworldsheetnonlocalHamiltoniantheorywhichwouldsum allFeynmandiagrams perturbativelywith N c + 1 and g 2 N c xed.Thecouplingof g 2 N c hassincebeen calledthe 'tHooftcoupling andhisworldsheetmodelwasthebeginningofextensive considerationsalongthesamelinesbytheoreticalphysici sts.Sincethisworkby'tHooft mostcertainlywasthefoundationonwhichtheLightconeWor ldSheetpicturelaterbuilt, 15
PAGE 16
wepresentherethebasisof'tHooft'sresultsandnotionswh ichbecamestandardinthis eld. 'tHooft'sargumentsworkequallywellwithpure U ( N c ) gaugetheoriesastheydo with U ( N c ) gaugetheoriescoupledtoquarksorevenscalarmatrixeldt heorieswith global U ( N c ) symmetry.'tHooftpresentedhisargumentsforatheorycoup ledtoquarks withLagrangiangivenby L = 1 4 Tr G G X r r r D + m ( r ) r with(110) D r = @ r + g A r (111) wheretheindex r =1 ; 2 ; 3 labelledthequarkfamily: 1 = p ; 2 = n ; 3 = ; (112) eachonebeingavectorwhichthefundamentalrepresentatio nof U ( N c ) canacton;i.e. theboldfacetype indicatesacolumnvector(andthe alinevector)whichthematrix indicesof A canacton.ThecovariantderivativeinEqn.( 111 )forexamplecontainsthe matrixmultiplicationof A with r .The G areasinEqn.( 17 ). TheFeynmanrulesareobtainedintheusualmannerandtheya renicelyassembled withguresin'tHooft'soriginalarticle.Toillustratehi ssocalled doubleline notation weconsiderthepureGaugepropagatorandcubicvertex,assh ownintoptwopicturesin gure 11 .Thelowertwopicturesinthatsamegureshowthesameobjec tsinthedouble linenotation.Thisnotationisessentiallyaconvenientwa yoforganizingandkeepingtrack ofthecolorgroup U ( N c ) .Since A isinthefundamentalrepresentationof U ( N c ) ,then thereare N 2 c degreesoffreedomassociatedwithit,representedbythema trixelementsof the A with A = A y ,thedaggerbothtransposingthematrixandtakingtheconju gate ofallelements.Thevectorandmatrixindices i;j ofthefundamentalrepresentation, goingfrom1to N c ,aredenotedbyanarrowedsingleline,incomingarrowforro wvectors andoutgoingforcolumnvectors.Itisthenclearthatfermio nsbeing N c vectorswillbe 16
PAGE 17
representedbysinglelinesonlybutgaugebosonsbeing N c N c matrices,bydouble lines,incomingandoutgoing.TheKroneckerdelta'sidenti fyingthematrixindicesinthe traditionalFeynmanrulesaredirectlyandmanifestlyimpl ementedbyconnectingthe singlelinesatthevertices,asshowningure 11 below. g ik jl = ( k 2 i ) ( i;j ) ( k;l ) ig lj mk ni g ( k q ) + g ( p k ) + g ( q p ) k p q ( n;j ) ( l;k )( m;i ) i i j j GaugePropagatorasabovewithout s. i i j j k k Cubicvertexasabovewithout s. Figure11.Exampleof'tHooft'sdoublelinenotation.Eac hlineholdasinglematrix index i or j resultingintheGaugepropagatortoberepresentedbytwo suchlines.ThismanifestlyimplementstheKroneckerdelta 'spresentinthe propagatorsasshown.Anincomingarrowonasinglelinedeno testhatthe indexitholdsreferstoarowandanoutgoingarrowtoacolumn index.This meansthatfermionpropagators(notshown)wouldbereprese ntedbysingle lines. ThedoublelinenotationcanalsobeimplementedforFadeevPopovghostsinthe Feynmangauge.Inthisnotation,consideralargeFeynmandi agramwiththeKronecker delta'smanifestlyimplementedgraphically.Ageneralsuc hdiagramwouldoftenrequire 17
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thedoublelinestotwist,i.e.,onegoingovertheother.Ifw eimaginethatlinecrossingis forbidden,thenthetwistcanstillbeachievedbyinserting a"wormhole"inthesheeton whichthediagramisdrawn.Itisthereforeclearthatwithth eFeynmanrulesaccording tothedoublelinenotation,andwithlinecrossingforbid den,thesumoveralldiagrams wouldhavetoincludethesumoveralltopologiesofthesheet onwhichthediagrams aredrawn.Noticealsothatifasinglelineclosesinacontra ctibleloop,theresulting amplitude,howevercomplicated,willhaveafactorof P i ii = N c .Iftheloop,however, isnotcontractible,meaningthatthesurfaceonwhichitisd rawnisnotplanar,thenthe indexsumdoesnotdecouplefromtheexpressionstoyieldthe factorof N c .Arigorous proofthatamplitudesdrawnon H holesurfaceswillbesuppressedfromtheplanarones by 1 =N H c ,isgivenin'tHooft'spaper[ 4 ]usingEuler'sformula: F P + V =2 2 H relating thenumberoffaces( F ),lines( P ),vertices( V )andholes( H )ofaplanarshape.Thelimit N c + 1 withthe'tHooftcouplingheldconstantthereforesingleso utplanardiagram. Furthermore,thetopologyofthesheetsonwhichthediagram saredrawnconstitutesa meansofsystematicallyimprovingthezerothorder N c + 1 limit. Theconceptof"thesheetonwhichFeynmandiagramsaredrawn "isnotperhaps fullyusefuluntiltheLightconeworldsheetconceptintrod ucedinthenextchapter identiesthissheetwiththeworldsheetitself.Thentheto pologyisnotthatof"the surfaceonwhichtodrawtheFeynmandiagrams"butratherthe topologyofaworldsheet whichitselfdirectlyholdsthedynamicalvariables.TheLi ghtconeparametrizationenables ustoidentifythe p + componentofpropagatorswithaspacecoordinateonthisshe etand x + withthetimecoordinate.Thismeansthatrstorder 1 =N c correctionstothework presentedherethishaveaprescriptionintermsofthetopol ogyoftheLightconeWorld Sheet. 1.3WorkonPlanarDiagrams Theapproachdueto'tHooft,consideringquantumeldtheor ieswith U ( N c ) symmetryperturbativelyin 1 =N c ,isparticularlyintriguingbecauseitorganizestheorder 18
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inperturbationtheoryaccordingtothetopologyofasheetw hichimmediatelycallsfor ananalogywithastringtheoryworldsheet.Lightconevaria blesoeraparametrization ofthesheetonwhichonewoulddrawFeynmandiagrams,using x + and p + andfurther promoting x + toanevolutionparameter.EachFeynmandiagramcaninsucha waybe seenasaworldsheetasshowninFigure 12 TheabovewayofidentifyingeachFeynmandiagramwithaworl dsheetwasfurther implementedbyBardakciandThornandisthebasisfortheLig htconeWorldSheet formalism,whichistheframeworkonwhichtheworkinthisth esisisbased.Aswas discussedabove,thestringeldtheoryconnectionhasbee nanactivesubjectofresearch eversincetheMaldacenaAdS/CFTcorrespondenceemerged.T hedevelopmentofthe LightconeWorldSheetideaisanrareexampleofthetheoreti calattemptat constructing astringworldsheetwhichsumsplanardiagramsofeldtheor y.Moreemphasishas beenputintotheoppositeview,i.e.,torecovereldtheore ticphysicsfromthestring formulation.AswaspointedoutinThornandTran[ 8 ]thenexttaskistodevelopa tractableframeworkforextractingthephysicsfromthisne wworldsheetformalism.The rststepsoforganizingtherenormalizationinthispictur eweretakenalongsidedeveloping thesupersymmetricLightconeWorldSheet[ 7 ]andarefurtherrepeatedhereaspartofthe graduateworkoftheauthor.Thenecessaryconclusionwasla terpublishedinanarticle whichsummarizedthe renormalized 3 worldsheet[ 9 ]. Previousworkonsummingplanardiagramsshouldnotbeforgo tten.Ideasand methodsofconsideringspecialcasesofplanardiagrams,th osewhichwoulddescribe simplemodelsforforcesbetweenquarks,havebeenpursued. Amongthose,shnet diagramsareparticularlyinteresting[ 10 ]becauseofhowthey,oratleastshnetlike diagrams,seemtobeemergingfromtheMonteCarlostudiespr esentedlaterinthis dissertation. Theconclusionisthatshnetscomeaboutasparticularsubsumsinthesumover allplanardiagramsifviewedintheLightconeWorldSheetpi cture;inotherwords,they 19
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1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 2 3 3 5 5 4 4 7 7 6 6 8 8 9 9 11 11 10 10 12 12 Figure12.VariouswaystodrawFeynmandiagrams.Thisgur eshowsvariousways acertainperturbativeexpressionforanamplitudecanbegr aphically represented,atechniqueinventedandnamedafterFeynman. Totherightthe traditionalFeynmandiagramisshownwithpropagators,i.e .eldcorrelations, depictedbycurlyandstraightlines.Inthegureontheleft thesamegraph isdrawnusing'tHooft'sdoublelinenotation.Inthemiddl eisthescheme, alsoby'tHooft,whichdrawstheFeynmandiagramasaworldsh eetwitheach numberedsquareareaofthegraphcorrespondstothepropaga tors. arecontainedinthatdescription.Incontrasttotheworkpr esentedinthisthesis,the LightconeWorldSheethasbeeninvestigatedwithanentirel ydierentapproximation scheme,namelythesocalled meaneld approach.Thisapproachconstitutesthenextstep towardsdevelopingthenonperturbativephysicsfromthen ewworldsheetformalism. WewillbrieydiscussthismethodandtheresultsfortheLig htconeWorldSheet obtainedfromit.AlthoughsomeknowledgeoftheLightconeW orldSheetisnecessary tounderstandthisdiscussion,itstillbelongsherebecaus eitisreallynotpartofthemain themesofthethesis. ThemeaneldapproximationeorttoanalyzingtheLightco neWorldSheetmodel foreldtheories(forsimplicity,thescalar 3 eldtheory)startedsoonaftertheoriginal 20
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formalismwasputforthin[ 11 ].Intheirpaper,theauthorsreviewtheLightconeWorld Sheetformalismasameansofrealizingthesumoverplanargr aphsbycouplingtheworld sheeteldstoatwodimensionalIsingspinsystem.Theythen regardtheresultingtwo dimensionalsystemasanoninteractingstringmovinginaba ckgrounddescribedbythe Isingspinsystem.Themeaneldapproximationistherefor eappliedtothespinsystem sothataqualitativeunderstandingofthephysicsofthesum sofplanargraphscouldbe achieved.Thismeaneldworkwasdoneinstages,whereren ementsandimprovements wereconstantlybeingpublished,anditisofcourseanongoi ngeortsincethemore complicatedtheories,suchasQCDandSupersymmetricGauge theory,haveyettobe tackled.Therearemorerecentviewsofthisproject[ 12 13 ].Wewillconcentrateonthe earlydevelopmentsforsakeofsimplicity;therenementsa ndespeciallythestringtheory implicationsarebeyondthescopeofthisthesis. AswillbeclearaftertheintroductiontotheLightconeWorl dSheetformalism, theIsingspinsystemdescribestheFeynmandiagramtopolog yofeachgraph.For nowitissucienttonotethattheIsingsystemrepresentsth elinesmakingupthe smallerrectanglesinthecentergraphofFigure 12 .Noticethataswegotohigher orderperturbativeFeynmandiagrams,thesolidlinesbecom emorenumerousandinthe asymptoticregimewecanimaginealimitwherethelinesacqu ireanitedensityonthe worldsheet.Theauthorsofthemeaneldpapers[ 11 14 ]refertothismechanismasthe condensationofboundaries,anddependingonthedynamicsi nvestigationsonwhether stringformationoccursthroughthiscondensationwascons ideredinsomelimitingcases. Furtherworkonstringformationandtheimplicationisalso foundinlaterarticles[ 12 13 ]. Togivethereaderatasteofthemeaneldidea,wewillelabo ratebrieyonthe originalattempts[ 11 ].Tokeeptrackofthesolidlineswhichinturnkeeptrackoft he splittingofmomentumbetweenpropagators,i.e.,thetopol ogyofFeynmandiagrams,a 21
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scalareld ontheworldsheetisintroduced whichtakesthevalue1onlinesand0 awayfromthem.TheauthorsthenbuildaLagrangiandensityt hroughtheLightcone WorldSheetformalismonacontinuousworldsheetwithasim plecuto(seechapter 2 for athoroughdiscussionoftheformalism).Acontinuouseld replacestheIsinglikespin systemintroducingacutosothattheworldsheetboundary conditionsonlyexactlyhold whenthecutoisremovedwhilemaintainingitamountstoimp osinganinfraredcuto. Theconstraints =0 and =1 areimplementedbyaLagrangemultiplier ( ; ) ,so thattheyendupwithtwoelds ( ; ) and ( ; ) ontheworldsheet.Thesetwoelds aretreatedasabackgroundonwhichthequantumeldslive,t heauthorsthencompute thegroundstateofthequantumsysteminthepresenceoftheb ackgroundandthensolve theclassicalequationsofmotionforthebackgroundeldst herebyminimizingthetotal energy.Inlaterrenements[ 14 ]themeaneldsweretakentobescalarbilinearsofthe targetspaceworldsheetelds,whichmakesthemeanelda pproximationmoreclearly applicablebutobscurestheinterpretationofthemeanel dasbackground,representing thesolidlines.Afurtherextensiontothemeaneldapproa chwaslaterpublished[ 8 ]. Here,insteadofauniformeld ( ; )= ontheworldsheetandrepresentingthe "smearedout"solidlines(condensation),twoelds ( ; ) and 0 ( ; ) areintroducedat alternating y sitesonadiscretizedworldsheetlattice.Theseelds,al thougheachistaken tobehomogenousontheworldsheet,allowforinhomogeneit yinthedistributionofsolid lines.Noteinparticularthat = 0 correspondstothepreviousworkbut =1 and 0 =0 correspondstoanorderingofsolidlinesreminiscentofan antiFerromagnetic Ising spinarrangement.Suchanarrangementofsolidlinesyields asocalledshnetdiagram Nottobeconfusedwiththequantummatrixeld intheoriginalscalar 3 Lagrangiandensity. y Notexactlyalternatingsitesisrequiredtoreproducethe shnetdiagrams,butrather spincongurationsofthetype ; ; ; # ; ; ; ; # ;::: 22
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ifwrittenasatraditionalFeynmandiagramaccordingtogu re 12 .Suchdiagrams havebeenselectivelysummedbefore,butthenoveltyoered inthispictureisthat congurationsawayfrom =1 ; 0 =0 take all otherplanardiagramsintoaccount,in anaverageway.Inotherwords,thetreatmentofsolidlinesa sinthemeaneldapproach allowsforadierentwayoforganizingFeynmandiagramsand approximatingdierently, namelyacrossallFeynmanlooporders. TheMonteCarloapproach,presentedinchapter 4 ,oersyetanotherapproach.It treatsthesolidlinesmentionedaboveinastochasticway.T heterms selectivesummation or importancesampling ,oftenseeninMonteCarloapplicationsofstatisticalphys ics, describequitewellhowthisapproachworks.Acloserlook,a ndamoremeaningfulone,is reservedforchapter 4 23
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CHAPTER2 LIGHTCONEWORLDSHEETFORMALISM Inthischapterwegiveanoverviewoftheconstructionwecal lthe LightconeWorld Sheet .TheformalismwasoriginallyinventedbyBardakciandThor n[ 5 ]andwas presentedasaconcretemechanismtoseetheworldsheetbeha vioroflarge N c matrix quantumeldtheory.Inthatpapertheauthorsestablishthe rstLightconeWorldSheet fromascalarmatrixquantumeldtheorywithaninteraction termof g Tr 3 = p N c and indicatehowthetheapproachmightbeextendedtomoregener aleldtheories.This extensioniscarriedoutinanumberoflaterpapersbyThorna ndcollaborators.The formalismisthefundamentalbackdroptotheworkpresented inthisthesis,sincethe worldsheetpictureallowsfornewinterpretationsandmeth odologyforinvestigatingthe eldtheory,amongotherstheMonteCarlotechniquesofchap ter4.TheLightconeWorld Sheetdescribesawidecollectionofeldtheoriesintermso fapathintegraloverelds thatliveinatwodimensionalspacereferredtoasthe worldsheet .Themechanicsofthe procedureisdescribedinthischapter. 2.1IntroductiontotheLightconeWorldSheetFormalism ConsiderthesimplestcaseofaLightconeWorldSheet,namel ytheonepresentedby BardakciandThornintheirpaperfrom2002[ 5 ],describingascalarmatrixeldtheory withcubicinteraction.Inordertoshedlightonthemaincon ceptsofthemechanismit isinordertosketchinsomedetailthestepspresentedthere in,sinceasaeldtheoryit isthesimplestpossiblecase,andbecausethematrixscalar eldtheorywillbeapplied laterwhenwelookatMonteCarlostudies.Considertherefor ethedynamicsoftheplanar diagramsofalarge N c matrixquantumeldtheorywithactiongivenby 24
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S = Tr 1 2 @ @ + 1 2 m 2 2 g 3 p N c 3 : (21) Here isa N c byN c matrixofscalareldsandthederivativeistakenelementw iseinthe matrix.Thepropagatorisgivenby r ( p )= i r ( p )= i r p 2 i where ;;r; 2f 1 ; 2 ;:::;N g .Usingthedoublelinenotationduetot'Hooftasexplained earliertheGreekindicescorrespondtothe"color"oftheli nes.Ast'Hooftpresribed,we dealwiththecolorsdiagrammaticallyandsincewewillbeta kingthelarge N c limit,thus ignoringallbutplanardiagrams,wesuppressthecolorfact ors r altogetherinwhat follows.IntroducingnextLightconecoordinatesdenedfo ra D dimensionalMinkowski vector x as x = x 0 x D 1 = p 2 : (22) Thereisnotransformationoftheremainingcomponents,and wedistinguishthem insteadbyLatinindices,orasvectorboldfacetype.Theco ordinatesare x + ;x ;x k or ( x + ;x ; x ) andtheLorentzinvariantscalarproductbecomes x y = x y x + y x y + .Bynowchoosing x + tobethequantumevolutionoperator,or"time",itsHamilto nian conjugate p = p 2 = 2 p + becomesthemasslessonshell"energy"ofaparticle.Wecho ose thevariables ( x + ;p + ; p ) torepresenttheFeynmanrules andarriveatthefollowing expressionforthepropagator: Thisiscalled"mixedrepresentation",i.e.,Fouriertrans formingbackthe p variable butretainingthemomentumrepresentationintheothercomp onents. 25
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D ( x + ;p + ; p )= Z dp 2 i e ix + p ( p ) = ( x + ) 2 p + e ix + p 2 = 2 p + wherewehaveassumed p + > 0 .Next,discretize"imaginarytime"and"momentum" ( p + = lm;ix + = ka with l =1 ; 2 ;:::;M and k =1 ; 2 ;:::;N )anddene T 0 = m=a .The expressionbecomes D ( x + ;p + ; p ) ( k ) 2 lm e k p 2 = 2 lT 0 Sofarthestepsmayadmittedlyseemadhocandreminiscento facookbookrecipe,but noticethatthispropagatorcanbeassociatedwitharectang ulargridwithwidth M and length N .Furthermore,theimaginarytimetranscriptionhasbeensh owntobeanalogous totheanalyticextensionoftheSchwingerrepresentationt oarealexponential[ 15 ], whichcorrespondstothenormalWickrotation.Wewishtoass ociatethemathematical expressionforthepropagatorwithapathintegraloflocalv ariablesonthisgrid.Todoso, noticethatapropagatoralwaysconnectstwoverticessoasl ongassomenalexpression foranamplitudeorotherFeynmandiagramcalculationconta insalltermsandfactors, wearefreetoredenetherulesforthediagrammaticconstru ctionoftheexpressions.In thisline,wecanassignthefactorof 1 =l presentinthepropagator,tooneoftheverticesit connects.SinceLightconeparametrizationonlyallowsfor propagationforwardintime,it ismeaningfultoassignthefactortotheearliervertexconn ectedbythepropagator.This createsanasymmetrybetween ssion and fusion verticeswhichnowrequireindependent treatment.Instead,propagationbecomesasimpleexponent ialwhichwasthegoal.The resultingreprescribedFeynmanrulesaresummarizedinTa ble 21 26
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Table21:LightconeFeynmanrules.Thearrowsdenotetheo woftime ix + 2 1 g 8 3 = 2 T 0 1 M 1 + M 2 2 1 g 8 3 = 2 T 0 1 M 1 M 2 e k p 2 = 2 lT 0 Thelocalworldsheetvariablesalludedtoabove,arenowwit hinreach.Writethe totalmomentumasadierence p = q M q 0 anddene: S = S g + S q S q = 1 2 T 0 X j M 1 X i =0 ( q ji +1 q ji ) 2 (23) S g = 1 T 0 X j b j1 c j1 + b jM 1 c jM 1 + M 2 X i =1 ( b ji +1 b ji )( c ji +1 c ji ) # (24) Then exp N 1 T 0 ( q M q 0 ) 2 2 M = Z N Y j =1 M 1 Y i =1 dc ji db ji 2 d q ji e S g S q (25) where b ki ;c ki areapairofGrassmanneldsforeachpoint ( i;k ) onthegrid(orlattice). Weimplement q kM = q M and q k0 = q 0 forall k byputtingin functions.Proofofthe aboveidentityanddiscussionofusefulintermediateresul tsarefoundintheoriginalpaper [ 5 ],butneitherisnecessarytoappreciatethefactthatwehav ehereaworldsheetlocal representationofthefreeeldtheory.Toputitlessdramat ically,wehavespreadoutthe verysimpleexponentialmomentumpropagationoverthewidt hofaworldsheet.Wecan evenwritetheabovementioned functionsasdiscretizedpathintegralsofanexponential emphasizingtheinterpretationof( 25 )asapathintegraloveran N M discretizedworld sheetgrid.Eventhoughtheconstructionisabitcumbersome consideringthatwestill onlyhavethefreeeldtheory,theimportantpointtonotice isthathereisacompletely 27
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rigorousmechanismbywhichthedynamicsofparticleisdesc ribedinthelanguageofa string.Byusingthesamelanguage,comparisonbetweenthea ctualdynamicsoftheoriesis possible.Weshallsee,inthecaseofGaugetheory,thatdram aticsimplicationoccursand thattheworldsheetlocalityistrulyastrongcondition. Theformalcontinuumlimitofexpression( 25 )forthepropagatoris T free fi = Z D q DbDce S 0 (26) where D q DbDc N Y j =1 M 1 Y i =1 dc ji db ji 2 d q ji S 0 = Z 0 d Z p + 0 d b 0 c 0 1 2 ( q 0 ) 2 S Theexpression( 26 )ispreciselytheinnitetensionlimitofthebosonicLight conestring. LetuspauseandsummarizethisLightconeWorldSheetpictur eofapropagatorin freescalareldtheory.Figure( 21 )showshoweasilythisdescriptionlendsitselftoa graphicalrepresentation:Wehaveaworldsheetwithlength T = Na andwidth p + = Mm organizedasalattice.Oneachsite ( i;j ) 2b 1 ;M cb 1 ;N c livesamomentumvariable q ji andapairofghostelds b ji ;c ji .Thediscretizationservesasaregulatorofthetheory. Wecontinuenowtoexplainthescalareldtheorybutinlessd etailandreferthe interestedreadertotheoriginalpaper.IntheLightconeWo rldSheetinterpretation,cubic verticesoftheeldtheoryareplaceswheretheworldsheets plitsintotwoworldsheets. Thisisimplementedbydrawingsolidlinesonthegridwhereb oundaryconditionsfor b;c and q aresuppliedasfortheoriginalworldsheetboundaries.Ver texfactorsthatmust bepresentaccordingtotheFeynmanrulesshouldsomehowbei nsertedatbeginningsand endsofsolidlines.Itturnsoutthatthiscanbedonebylocal lyalteringtheaction.For 28
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M i 2 2 1 1 j q ji b ji c ji N Figure21.WorldSheetpictureoffreescalareldtheory.T hegureshowsthe diagrammaticsetupforfreescalareldtheoryintheLightc oneWorldSheet formalism.ThepicturedescribesapropagatorinLightcone variableswith p + = mM and p accordingtotheboundaryvaluesoftheelds q .The propagatorevolvesin"time" = ix + = aj andtheghostelds b and c makesurethattheworldsheetinteractsproperlywithother worldsheetsonce interactionsareadded. example,toobtainthefusionvertexabovewesubtractone b;c linkatatimeslice k just aftertheendofthesolidlinegivingrisetoafactorof 1 =M 1 M e p 2 = 2 MT 0 = Z M 1 Y i =1 dc ki db ki 2 d q ki exp ( 1 T 0 b k1 c k1 + b kM 1 c kM 1 + M 2 X j =1 ;j 6 = l ( b kj +1 b kj )( c kj +1 c kj ) # 1 2 T 0 M 1 X i =0 ( q ki +1 q ki ) 2 ) Thisdetailedmanipulationofalteringtheactionandinser ting functionsto implementboundaryconditionscanbedonedynamicallybyin troducinganIsingspin systemontheworldsheetgrid.Consideryetanothersetofwo rldsheetdynamical variables s ji whichareequalto+1ifthereisasolidline(aboundary)atsi te ( i;j ) and1 ifthereisnone.Thelocalactionmanipulationterms,creat ingboundariesandinserting 29
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1 =M andcouplingsforvertices,arethenmultipliedwithapprop riatecombinationsof s ji 's sothattheyarepresentwheretheyshouldbe.Forexample,at endpointsofsolidlineswe have s ji = s j +1 i soafactorof (1+ s j +1 i )(1+ s ji )(1 s j 1 i ) = 8 is1atthebeginningofasolid linebut0otherwise,whereasthefactor (1 s j +1 i )(1+ s ji )(1+ s j 1 i ) = 8 is1attheendofa solidlinebut0otherwise.Wethenputanoverallsumoverall s ji congurationsinfrontof thewholepathintegralandtheresultingexpressionthenre presentsthesumofallplanar Feynmandiagrams: T fi = X s ji = 1 Z N Y j =1 M 1 Y i =1 dc ji db ji 2 d y ji d q ji (2 ) 2 exp ( S + X i;j b ji c ji V j 0 i P j i + V j 0 i P j i ) exp ( X i;j i y ji ( q ji q j 1 i ) P j i P j 1 i +(1 P j i P j 1 i )ln 4 2 V +2 X i;j P j i P j 1 i P j +1 i b ji c ji #) exp ( a m X i;j P j i b ji c ji b ji c ji +1 b ji +1 c ji + b ji +1 c ji +1 P j i +1 + b ji c ji b ji c ji 1 b ji 1 c ji (1 P j i 1 ) ) ; where P j i =(1+ s ji ) = 2 and P ( P )isacombinationof s ij 'swhichis1atbeginnings(ends) ofsolidlinesthatareatleast2timestepslong,and0otherw ise. TheIsinglikespinsystem s ji isclearlyintroducedsimplytomanipulatethepresence orabsenceoftermswhichhavetodowithboundariesofthewor ldsheet.Thespinsystem iscompletelynewandhasnocounterpartintheoriginaleld theory.Inawayitis preciselywhatmakesthestrongcouplingregimereachableb ythisworldsheetapproach ascomparedwithFeynmandiagramperturbationtheoryorlat ticequantumeldtheory. Thevariousschemestotackletheworldsheet,byMonteCarlo simulationaswillbedone here,orbyintroducingameaneld[ 11 14 ]ortheantiferromagneticlikeconguration consideredbyThornandTran[ 8 ],allinvolveaparticularchoiceforthetreatmentofthe spinsystem,andofcourse,thetreatmentofrenormalizatio n.Whathappenstothespin 30
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systeminthecontinuumlimitisalongwithrenormalization themostinterestingquestion tobeaskingtheLightconeWorldSheet. 2.2SupersymmetricGaugeTheories CastingpureYangMillstheoryintoLightconeWorldSheetf ormhasbeendoneby Thorn[ 16 ]shortlyaftertheappearanceoftherstarticle.Sinceaun iedandslightly renedrepresentationforgeneralGaugetheorieslaterapp eared[ 7 ]weskipoverthe otherwisecrucialstepinthedevelopmentoftheLightconeW orldSheetconstructions,and turnrighttoSupersymmetricGaugeTheories.2.2.1SUSYYangMillsQuantumFieldTheory Intheirpaper,Gudmundssonet.al.[ 7 ]buildtheextended N =2 and N =4 supersymmetricGaugetheorybymeansof dimensionalreduction .Thismethodstartso withan N =1 supersymmetricGaugetheoryinhigherdimensionsandthen reducethe theoryto D =4 bymakingtheeldsindependentontheextra D 4 dimensions.This automaticallycreatesthecorrectnumberofeldsfortheex tendedsupersymmetry.The Gaugebosonsassociatedwiththeextradimensionsbecomeju stthescalarswhentheir Gaugesymmetryintheextradimensionsbecomesaglobalsymm etryandsimilarlythe higherdimensionalrepresentationoftheCliordalgebrag eneratesjusttherightnumber offermions.ThismethodisparticularlyusefulontheLight coneWorldSheetbecause makingtheeldsindependentupontheextradimensionscane asilybeimplementedby settingtheextra q componentsequaltozeroontheboundariesofthesheet.They arestill allowedtouctuateinthebulk,whichallowsthemtopartici pateinthecrucialgeneration ofquarticsfromcubicsaswillbedescribedinthenextsecti on.Inordertocarryoutthe mappingoftheories ( N ;D )=(1 ; 6) 7! ( N ;D )=(2 ; 4) and ( N ;D )=(1 ; 10) 7! ( N ;D )=(4 ; 4) 31
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weneedtoformulatethe N =1 supersymmetricGaugetheoryin D dimensions.The Lagrangiandensitywestartowithisgivenby L = 1 4 Tr F F + i Tr y 0 @ ig [ A ; ] (27) F = @ A @ A ig [ A ;A ] ; (28) where arethe D dimensionalDiracgammamatrices.Lightconegaugedictate s A =0 and A + iseliminatedusingGauss'law.The"time"evolutionoperat or P isobtainedin ordertoreadotheLightconeFeynmanrules: P = Z d x dx T + C + Q (29) wheretheindividualtermswhichgivethevariousverticeso ftheFeynmanrulesaregiven bytheexpressionsbelow.UsingtheLightconeDiracequatio nandGauss'lawinthe A =0 Gaugeonearrivesat T = 1 2 Tr @ i A j @ i A j i Tr b y @ + r 2 2 @ b C = ig Tr @ A k A j @ k @ A j A k @ i @ A i A i @ i @ A k # + g 2 Tr @ n @ c y ( nk + i nk ) cb [ A k ; b ] g 2 Tr [ c y ;A n ]( nk + i nk ) cb @ k @ b + g Tr b y 1 @ r A; b Q = g 2 2 Tr 1 @ [ A k ;@ A k ] 2 + A i A j [ A i ;A j ] # ++ g 2 2 Tr f a ; a y g 1 @ 2 f b ; b y g (210) ig 2 Tr 1 @ 2 [ @ A k ;A k ] f b ; b y g + ig 2 2 Tr [ c y ;A n ]( nk + i nk ) cb 1 @ [ A k ; b ] where X @ i @ Y = X @ i @ Y @ i @ X Y: (211) Althoughwedointendtoreferthereadertotheoriginalwork [ 7 16 ]forthedetailsofthe dimensionalreductionandformulationoftheindividualve rticesofthetheory,wepresent thisexpressionfortheLightconeGaugeHamiltonianinorde rforthereadertobeable toreadotheFeynmanrulesandvertexfactors.Inthenextse ctionweindicate(albeit 32
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bythesamemeansofqualitativehandwavingasupuntilnow) howthecompletesetof verticesaregeneratedbylocalinsertionsontheworldshee t. 2.2.2SUSYYangMillsasaLightconeWorldSheet Theconstructionoftheworldsheetlocalactionisanalogou stothescalarmatrixeld case.Thepathintegralreproducesrstthemixedrepresent ationpropagatorandthen addsinteractionverticesbymeansofthespinsystem s ji asbefore.Recallthatthe q s nowhave D 2 componentswithallbuttwoidenticallyzeroonallboundari es.Clearly, iftherichparticlestructureofothereldtheoriesistot intothepicture,thenalleld theoreticpropagatorsmust(anddo)havethesamesimpleexp onentialformasthescalar propagator,times,atmost,Kroneckerdeltasthatdescribe theowofspinandother internalquantumnumbers.Thereforetheexpression( 26 )isuniversalfortheLightcone WorldSheetformofaeldtheory.Thegeneralconstructiono ftheSUSYLightconeWorld Sheetfollowsverymuchthesameprocedureasthescalarcase withafewfundamental dierences.IntheLightconeWorldSheetpicturethefundam entalpropagationisthatof momentum,representedbythewidthofthestrip.Suchthings astheeldtheories'rich collectionofparticlesmustbepropagatedthroughtheshee tbymeansof"avoring"the strip(latticesites)withdynamicalvariables.Furthermo re,theverticesmustcomeoutof localalterationsontheworldsheet,ifthepictureistoret ainitseleganceanditsrelation tostringtheory.Insteadofrepeatingthesystematicconst ructionoftheworldsheettheory fromtheoriginalwork[ 7 16 ],andaswasdoneforthescalartheoryabove,wecomment onthemainissuesbriey.Thediscussionhereisintheformo fanexistenceargument, weshowhowaLightconeWorldSheetdescriptionofsupersym metricYangMillstheory couldbeconstructed.Theinterestedreaderisreferredtot heoriginalworkforamore thoroughandfullpicture. AfterseeingtheconstructionoftheLightconeWorldSheetf orthescalarmatrix theory,therstquestionsoneneedstoaddresswhenturning toSUSYYangMillsarethe following: 33
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1.Propagationofthegaugebosonpolarizationonthestrip.2.Propagationofthefermionspinorinformationonthestri p. 3.Creatingthecorrectverticesfromtheinformationpropa gatingmechanism. 4.Localrepresentationofallthecubicvertices.5.Quarticvertices,howdoesonejoinfourstripsinonepoin t?Mustweabandonworld sheetlocality? Itturnsoutthatissues1),2)and3)aresolvedsimultaneous lyinaratherelegantway byintroductionofworldsheetGrassmannvariablesandintr eatingissue4)theproblem withquarticvertices,issue5),issolvedautomatically. Considerforthemoment,theworldsheetasinthelastsectio n,with i 2b 1 ;M c and j 2b 1 ;N c labellingthesitesasbeforeandwiththesystemofthe q scalarsand foreachsuchcomponenta b;c ghostpair.Witheachsite,wefurthermoreassociatefour Grassmanneldpairs S a p ; S a p .The p sarelocationlabellingindicesexplainedinamoment, and a isan O ( D 2) spinorindexoftheCliordalgebraofthefermions.Thespin or indexallowstheGrassmanneldtocarryallthefermionicin formationandbycreation ofbilinearssuchas J n =2 ( D 2) = 4 S a p r n ab S b p thevectorandscalarquantumnumbersare mitigated.ToshowhowletusrstexplainhowthefourGrassm anneldsareplacedand linkedbetweensites.ReferringtoFigure 22 wedrawthefourGrassmannpairsaround eachvertexasshown.TheGrassmannactionisoftheform: A = 2 K 1 X p =1 S a p S a p +1 + 2 K 1 X p =1 S a p S a p +1 (212) andwith (213) D S = Y a dS a 2 K dS a 2 K 1 dS a 1 d S a 2 K d S a 2 K 1 d S a 1 : (214) inotherwords,itisasumoflinktermsbetweenthepairs.Thi sway,wecandynamically remove"space"linksoverboundariesandadd"time"linksto theaction,simplyby removingthecorrectterms.Thisisdonebyimposingconditi onsonthe S a sand S a s. 34
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Withthecorrectlinkingstructure,weusethefactthat Z D Se A S a 1 S b 2 K = Z D Se A S a 1 S b 2 K = ab (215) Z D Se A S a 1 S b 2 K = Z D Se A S a 1 S b 2 K =0 : (216) Thisensuresthatthecorrectfermionicinformationtravel sfromvertextovertex. Similarly y Z D Se A S a 1 J j 2 K = Z D Se A S a 1 J j 2 K = Z D Se A J j 1 S a 2 K = Z D Se A J j 1 S a 2 K =0 (217) Z D Se A J i 1 J j 2 K = ij : (218) takescareofthevectorandscalarparticleinformation.At thevertices,alongwithother insertions,Diracgammamatricesconnectthefermionicind ices a;b withthevectorindices k .Thedetailsarecarefullypresentedintheabovementioned article[ 7 ].Itisrather interestingtonoticethat,aswiththetransversemomentum elds q ontheworldsheet, thetimederivativeoftheGrassmann'scomesuponlyalongbo undaries.Themeaneld approach,wheretheboundariesreachanitedensity,shoul dthereforeexhibit S and S dependence. Letusnextturntoitems4)and5)fromthelistofissuesweexp ectfromthe LightconeWorldSheetdescriptionofGaugetheory.Inthesc alartheorytheplain Feynmanverticescontainednomomentumdependentfactorsb utthetreatmentofthe propagatorscreatedfactorsof 1 =M inthevertices.IntheGaugetheoryhowever,the verticesaresomewhatmorecomplex.Recallingnowthelongex pression( 210 )forthe LightconeHamiltonian,theFeynmanrulescontainanumbero f A 3 and y A cubicsas wellas A 4 y A 2 and ( y ) 2 quarticsandthederivationoftheircouplingisfoundinthe y Thecaseof N =4 extendedsupersymmetryrequiresaslightlydierenttreat ment sincethenthespinorsaresimultaneouslyMajoranaandWeyl making S a and S a thesame. Thisdierenttreatmentstillproducesequivalentequatio ns 35
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1 1 2 2 3 3 M M i i 2 2 1 1 j N M i Figure22.GrassmanneldsnakingaroundtheWorldSheet.T hegureshowshow, byimposingsimplerulesforconnectingthefourGrassmannp airsateach sitetogether,onecan"sew"theGrassmannchainintothewor ldsheetstrip andtherebypropagatingtheinformationcarriedbytheGras smann'sfrom onevertextothenext.Atthevertex(orrather,justbelowth evertex)the initialchain(solidcoloredchainnumber 1 )terminatesandtwonewchains arestarted(greycoloredandwhitecoloredchains 2 and 3 ).Itispossiblevia localinsertionsatthevertextobreaktherequiredlinksan datboundaries theGrassmann'sconnectintimeratherthanspace.TheKrone ckerdelta identitiesinthetextshowthattheGrassmannpathintegral sguarantees thatthesamespinororvectorindicesappearsonbothendsof thechains,in thegurefrom 1 ; 2 ; 3 to 1 ; 2 ; 3 respectively.Ontheright,thecorresponding Feynmandiagramisshown,witheachleglabelledbyits p + momentum. originalpaper[ 7 ]andforpureGaugetheoryinpapersbyThornandothers[ 10 16 ].In shorttheyareingeneral,rationalfunctionsofthe p + enteringintothevertex.Justasthe simplerationalfunction 1 =p + wasgeneratedbyuseofthe b;c ghostpairstheserational functionsmustbecreatedbylocalinsertionsontheworldsh eet.Considerthepure A 3 36
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cubicvertex,showninthepaperonSupersymmetricLightcon eWorldSheetconstruction [ 7 ]tobegivenby V n 1 n 2 n 3 = g 4 3 = 2 n 1 n 2 K n 3 p +3 + n 1 n 3 K n 2 p +2 + n 2 n 3 K n 1 p +1 where(219) K = p +2 p 1 p +1 p 2 = p +3 p 2 p +2 p 3 = p +1 p 3 p +3 p 1 Notethatherethelabels1,2and3denotethethreeparticles comingintothevertex,and n k particle's k polarization.Themomenta q k arethoseofworldsheetstripsmeetingat thevertexandeachisthereforeadierence p k = q A q B where A;B areboundariesofa strip.Thefollowingidentitiesholdirrespectiveof l ,i.e.,irrespectiveofwhereonthestrip betweentheboundariesthe q l insertionismade I = Z D q e S q q M q 0 M I = Z D q q l q l 1 e S q = Z D q q l e S q (220) where S q isthe q actionunchangedfromthescalarcaseEq.( 23 )andthemeasure D q isjustasinthatcase: D q = d 2 q 1 :::d 2 q M 1 .ComparingwithEq.( 219 )weseethat theseidentitiesaresucientforconstructingthecouplin gsfortheall A svertices.With anadditionofmoreghostvariableslikethe b;c stheyarethebasisforconstructingallthe rationalfunctionsrequiredforsupersymmetricgaugethe orycubicverticesinLightcone gauge. Nowtheonlythinglefttoexplainishowthequarticvertices arehandled.A'priori, thiswouldseemthebiggestobstacletondingaworldsheetl ocaldescriptionofthe theory.Thereasonisthatitseemsnotpossibletorecreates uchvertices,asfourstripsof arbitrarywidthandheightcannotingeneralbejoinedinapo int.Itseemsthatlocality wouldhavetobeabandonedandthestripsjoinedalongawhole line.Onehope,isthat onecould construct localcubiclikeinsertionswhichwouldreproducethequar ticvertices whenoccurringonthesametimeslice,buteventhiscana'pr iorinotbeguaranteed.It 37
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isthereforetrulyremarkablethatthequarticverticesdro poutoftheveryexpressions forthecubicverticesalreadypresentinthetheory,whenth esearetakentooccuronthe sametimeslice.ThisisnotonlytruefortheGaugequartics ,butalsoforthefermionic quarticsofequation( 210 ).Aswasbrieytouchedonbefore,thisoccursbecauseofthe uctuationsofthe q sjustwhentwo q insertionsaremadeonthesametimeslice. RecallthatthelefthandsideofEq. 220 wasindependentuponwhereonthetimeslice theinsertionwasmade(independentupon l ).Twoinsertionat k and l give Z D q ( q k q k 1 )( q l q l 1 ) e S q = q M q 0 M 2 I + m a ij 1 M I (221) wherenotationfromEq. 220 hasbeenborrowed.Thesecondtermisthe"quantumuctuati on" termandisanadditiontothesimpleconcatenationoftwocub ics.Noticethat M above isthetotalwidthofthestripswhichhavethetwoinsertions .Considerasituationasin Figure 23 ,with M k labellingthewidthsofthevariousstrips.Thedoubleinser tionsat i 1 and i 2 produceaquantumuctuationof 1 = ( M 1 + M 4 ) .Takingintoaccountthe M 2 M 4 prefactorofthetwocubicverticesandthe 1 = ( M 1 + M 4 ) oftheintermediatepropagator givesthecombination M 2 M 4 = ( M 1 + M 4 ) 2 .Addingthecontributionforwherethearrow ofLightconetimeontheintermediateparticlegoestheoppo siteway,wehavethetotal expression: M 2 M 4 + M 1 M 3 ( M 1 + M 4 ) 2 = 1 2 ( M 1 M 4 )( M 2 M 3 ) ( M 1 + M 4 ) 2 +1 (222) whichispreciselythemomentumdependenceinexpression(3 1)inThorn[ 16 ]foroneof thequarticinteractions. z Thisverymuchsimpliedexampleservestoshowhowthe quarticseemstojustmiraculouslyfalloutfromthealgebra .Theaboveargumentgoes z Inthepaper[ 16 ]theauthorderivestwoLightconequarticverticesforthep ure YangMillstheory,theaboveonewhichcontainsthe"Coulom b"exchangeand contributionfromthecommutatorsquared,andanotherones lightlymorecomplex.The verticesdenedtheredependonthepolarizationoftheinco mingparticles. 38
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throughintermsofallprefactorsandforallparticlesand congurations.Asbeforein thissection,thereaderisreferredtotheoriginaltextfor details. 1 i 1 i 2 M 1 M 2 M 3 M 4 M j p 1 p 2 p 3 p 4 Figure23.Quarticfromcubicsforasimplecase.Thegures howstwoworldsheet strips(boundariesat 1 i 1 and M )breakingintoanothersetoftwostrips (boundariesat 1 i 2 and M )attimeslice j .Theincomingworldsheetstrips have M 1 = i 1 and M 2 = M i 1 unitsof p + momentumrespectivelywhereas theoutgoingstripshave M 3 = M i 2 and M 4 = i 2 units.Totheright isshownthecorrespondingFeynmandiagramasaconcatenati onoftwo cubicswiththeLightconetimedierencebetweenthemequal tozero.The legsoftheFeynmandiagramarelabelledbytheirrespective fourmomenta. Forsimplicitytheexampleshownshowshowtheconcatenatio noftwopure bosoniccubicsbecomeaquartic,butthesameappliestoallo therquarticsof thetheory. 39
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CHAPTER3 PERTURBATIONTHEORYONTHEWORLDSHEET AswehaveseentheLightconeWorldSheetformulationissetu pintheframeworkof theLightcone.The = ix + and p + latticeasconstructedbyBering,RozowskyandThorn [ 10 ]isofcoursethestartingpointfortheworldsheetandthepe rturbativeissuesfacedin thatwork,ofcourseremainpresentintheworldsheetpictur e. Whenusingthediscretizedworldsheettocalculateprocesse stoagivenorderin perturbationtheorytheinsertionshavebeendesignedtoex actlyreproducethecubic verticesoftheLightconeFeynmanrulesinthecontinuumlim it.Theprecisemeaningof thislimitisthateverysolidlineinthediagramismanylatt icestepslongandalsoismany latticestepsawayfromeveryothersolidline.Clearlyadia graminwhichoneofthese criteriaisnotmetissensitivetothedetailsofourdiscret izationchoice.Intreediagrams onecanalwaysavoidthesedangeroussituationsbyrestrict ingtheexternallegssothat theycarry p +i sothatthedierences j p +i p +j j areseveralunitsof m foranypair i;j ,and sothatthetimeofevolution, ,betweeninitialandnalstatesarealsoseveralunitsof a .However,adiagramcontainingoneormoreloopswillinvolv esumsoverintermediate statesthatviolatetheseinequalities,andbecauseofeld theoreticdivergencesthe dangerousregionsofthesesumscanproducesignicanteec tsinthecontinuumlimit. Inparticularweshouldexpecttheseeectstoincludeaviol ationofLorentzinvariance, inadditiontotheusualharmlesseectsthatareabsorbedin torenormalizedcouplings. Indeed,whenasolidlineisoforderafewlatticestepsinlen gth,itproducesagapin thegluonenergyspectrumthatisforbiddenbyLorentzinvar iance.Thiseectcanbe cancelledbyacountertermthatrepresentsalocalmodica tionoftheworldsheetaction. Thehopeisthatallcountertermsneededforaconsistentre normalizationprogramcan beimplementedbylocalmodicationsoftheworldsheetdyna mics.Aslightweakeningof 40
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thelaststatement,thatallcountertermswillbeconsiste ntwithworldsheetlocality,isin factaconjectureputforwardbyThorninmanyofhispaperson thesubject.IfThorn's conjecturewerenottrue,itwouldmaketheformalismconsid erablylessinteresting. Weconsiderinthischapterrenormalizationtooneloopinth econtextofthe LightconeWorldSheet,forageneralclassofSupersymmetri cgaugetheoriesformulated earlier.Wewillseethatwiththeratherunusualregulators ,withdiscrete and p + are (equivalently,cutosin p and ix ),westillobtainthecorrectwellknownresultfor gaugecouplingrenormalization.Inthecalculationsthatf ollowweshallrstrevisitolder workdoneonthesubjectwhereclosecontactwiththeFeynman diagrampicturehasbeen kept,because,eventhoughtheworldsheetdiagramsarecomp letelyequivalent,organizing diagramaccordingtolooporderisonlynaturalandapparent intheFeynmandiagrams. Followingthisratherthoroughtreatmentweturntoaworlds heetorganizationwhich allowsforsomeinsightstheFeynmanpicturedoesnot.Wesha llalsolaterseethatthe worldsheetdiagramsorganizethemselvesinawaymuchbette rsuitedforMonteCarlo methodswhicharethesecondsubjectofthisthesis. Wepresentherewithpermission,theworkwhichwaspublishe dbeforeinapaper titled"Oneloopcalculationsingaugetheoriesregulatedo nanx+p+lattice"[ 6 ].This workconstitutesinpartthegraduateworkwhichwasdonetow ardsthecompletionof thisthesisandisthereforepresentedherewithoutadditio nsormajormodication.Italso relatesverystronglytothesecondthemeofthethesis,name lytheMonteCarloapproach oftheLightconeWorldSheetformalism. 3.1GluonSelfEnergy Inadditiontosettingupthebasicformulationofthe x + ;p + lattice,Beringet.al. [ 10 ]alsocalculatedtheoneloopgluonselfenergydiagramas acheckofthefaithfulnessof thelatticeasaregulatorofdivergences.Werecaptheirres ultsbutwiththeworldsheet conventionsusedhere.Inadditionweconsiderageneralpar ticlecontentwith N f number offermionsand N s numberofscalars.Thegluonselfenergytooneloopcanbeext racted 41
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fromthelowestordercorrectiontoagluonpropagatorrepre sentedbyasinglesolidline segmentonaworldsheetstripasinFig. 31 lM k 2 k 1 Figure31.Oneloopgluonselfenergy.Becauseoftimetrans lationinvarianceonlythe dierence k = k 2 k 1 isimportant. Withtheconventionsusedhere,theresultanalogoustoEqs.( 52)and(53)ofthat article,forxed k 1 ;k 2 ;l with k = k 2 k 1 > 1 and 0
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propagatorcorrection: D 1 ( p ;M;K )= K 3 X k 1 =1 K 1 X k 2 = k 1 +2 u k 1 u K k 2 g 2 8 2 u k 2 k 1 ( k 2 k 1 ) 2 M 1 X l =1 A ( l;M ) (32) = g 2 8 2 K 3 X k 1 =1 K 1 X k 2 = k 1 +2 u K 1 ( k 2 k 1 ) 2 M 1 X l =1 A ( l;M ) (33) = u K g 2 8 2 2 6 1 K ln K + O ( K 0 ) M 1 X l =1 A ( l;M ) : (34) Thetermlinearin K comesfromtermswheretheloopisshort( k 2 k 1 K )andthe sumisoverthepossiblelocationsofit.Itisclearthatwhen n shortloopsaresummed overtheirlocationswegetfactorsproportionalto C n K n =n where C isthecoecientof K intheabovelinearterm.Theshortloopbehaviorthereforee xponentiatesandcausesa shiftoftheenergy, a p 2 = 2 Mm ,intheexponentofthefreepropagator.Thisshiftcauses agapinthegluonenergyspectrumthatisforbiddeninpertur bationtheorybyLorentz invariance.Wemustthereforeattempttocancelthislinear termin K orderbyorderin perturbationtheorywithasuitablechoiceofcounterterm .Onesimplechoiceisatwo timestepshortloopofexactlythestructurethatwentintot hebareselfenergy.Thenat onelooporderitwillbeproportionaltothe k =2 termandwillhavetheform: e ka p 2 = 2 mM g 2 4 2 k 4 M 1 X l =1 A ( l;M ) ; (35) whereweadjust tocancelthetermproportionalto k inthepropagatorcorrection. Choosing =4(1 2 = 6) doesthejobandweareleftwithalogarithmicdivergencewhi ch willcontributetothewavefunctioncontributiontocoupli ngrenormalization.Wehave: ln k =ln(1 =a )+ln( T ) ,with T = ka ,thetotalevolutiontime.Wecanthereforeabsorbthe divergenceinthewavefunctionrenormalizationfactor: Z ( M )=1 g 2 8 2 ln(1 =a ) M 1 X l =1 2 l + 2 M l + F ( l=M ) M ; (36) 43
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where F ( x )=2 f g ( x )+ N f f f ( x )+ N s f s ( x ) (37) f i ( x )= 8>>>>>><>>>>>>: x (1 x ) 2for i = g (gluons) x (1 x )for i = s (scalars) 1 = 2 x (1 x )for i = f (fermions) : (38) Thersttwotermsinthe l sumproducea ln(1 =m ) divergenceandwenoticethefamiliar entanglementofultraviolet( a 0 )andinfrared( m 0 )divergences[ 17 ].Ithasbeen explainedhowthesedivergencesdisentangle[ 6 ]andwewilldiscussthisfurtherinthe nextsection.In( 36 ) N f countsthetotalnumberoffermionicstates,so,forexample ,a singleDiracfermionin4spacetimedimensionshas N f =4 .WeseethatSupersymmetry, N f = N b =2+ N s ,killsthe l dependentterminthesummand.If N f =8 aswell,thewave functioncontributiontocouplingrenormalization(apart fromtheentangleddivergences) vanishes.Thisistheparticlecontentof N =4 SUSYYangMillstheory. 3.2OneLoopGluonCubicVertex:InternalGluons Nowweturntothecontributionofthepropervertextocoupli ngrenormalization. Theproperoneloopcorrectiontothecubicvertexisreprese ntedbyaFeynmantriangle graphappearingintheworldsheetasshowninFig. 32 .Withtheexternalparticlesof Fig. 32 restrictedtobegluons(vectorbosons)theonelooprenorma lizationofthegauge couplingrequirescalculatingthetrianglegraphforthedi erentparticlesofthetheory runningaroundtheloop.Inthefollowingitwillbeusefulto employthecomplexbasis x ^ = x 1 + ix 2 and x = x 1 ix 2 forthersttwocomponentsofanytransversevector x andasthenameofthesectionsuggestsweconsiderrstonlyg luonsrunningaroundthe loop. 44
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p ;l p 1 ;M 1 p 2 ;M 2 p 3 ;M 3 k =0 k = k 1 k = k 2 lM 3 M 1 Figure32.Cubicvertexkinematics.Basickinematicsetup fortheoneloopcorrectionto thecubicvertex.Themomenta p 1 and p 2 aretakentopointintothevertex whereas p 3 pointsout,sothatmomentumconservationreads p 1 + p 2 = p 3 .By timetranslationinvariancewetakeoneoftheverticestobe at =0 .Wetake theexternalgluonlinestohavepolarizations n 1 ;n 2 ;n 3 3.2.1AFeynmanDiagramCalculation Forsimplicityinpresentationweconsideratrstthecase n 1 = n 2 = ^ ;n 3 = Omittingtermswhichareconvergentinthecontinuumlimit, i.e.,retainingonlythose termsthatcontributetothechargerenormalization,wehav e ( g1 ) ^^_ = g 3 K ^ 16 2 T 3 0 M M 1 M 2 X l;k 1 ;k 2 e H j l j ( M 2 + l )( M 1 l ) T 1 T 2 T 3 K 2 A M 2 ( T 1 + T 2 + T 3 ) 4 + T 1 B 1 + T 2 B 2 + T 3 B 3 ( T 1 + T 2 + T 3 ) 3 (39) where g1 denotesthe"triangle"contributiontochargerenormaliza tion,atoneloop,from gluonsinternally.Inthissectionwewillomitthisdescrip tivenotationandreferto g1 as simply .Wehaveusedthefollowingdenitions: T 1 = 1 2 T 0 k 1 l ;T 2 = 1 2 T 0 k 2 k 1 M 2 + l ;T 3 = 1 2 T 0 k 2 M 1 l : (310) T = T 1 + T 2 + T 3 (311) H = 1 T T 1 T 3 p 21 + T 1 T 2 p 22 + T 2 T 3 p 23 (312) K ij = M i p j M j p i (313) 45
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and A = M 2 M 2 1 l 2 ( M 1 l ) 2 + M 2 M 2 2 l 2 ( M 2 + l ) 2 + ( M 2 + l ) 2 ( M 1 l ) 2 + ( M 1 l ) 2 ( M 2 + l ) 2 (314) B 1 = M 2 M 3 1 l 2 ( M 1 l ) 2 + M 1 M 3 2 l 2 ( M 2 + l ) 2 (315) B 2 = MM 3 2 l 2 ( M 2 + l ) 2 M 2 ( M 2 + l ) 2 M ( M 1 l ) 2 M 2 ( M 1 l ) 2 M ( M 2 + l ) 2 (316) B 3 = MM 3 1 l 2 ( M 1 l ) 2 M 1 ( M 2 + l ) 2 M ( M 1 l ) 2 M 1 ( M 1 l ) 2 M ( M 2 + l ) 2 (317) Notetheconstraint lT 1 +( M 2 + l ) T 2 ( M 1 l ) T 3 =0 ,whichimpliesthatforxed l ,onlytwo ofthe T 'sareindependent.Also,momentumconservationimpliesth at K ij iscyclically symmetricandwethereforeuse K K 12 = K 23 = K 31 .Weintroducethefollowing notationthatwillhelpstreamlinesomeoftheformulae: P i p 2i =M i ;P p 2 =M = P 3 Forexample, K 2 = M 1 M 2 M 3 ( P 1 + P 2 + P 3 )= M 1 M 2 M ( P 1 + P 2 P ) : (318) Wearenowdealingwithpotentiallyultravioletdivergentd iagrams.Torevealthe ultravioletstructureweconsiderthecontinuumlimitinth eorder a 0 followedby m 0 .Recallthat a 6 =0 servesasourultravioletcuto.Inthe a 0 limitwecan attempttoreplacethesumsover k 1 ;k 0 2 ( k 0 1 ;k 2 )for k 1 > 0 ( k 1 < 0 )byintegralsover T 1 and T 2 ( T 3 ).Sincewewishtokeep l xedinthisrststep,forthecase k 1 > 0 weexpress T 3 intermsof T 1 and T 2 : T 3 =( lT 1 +( M 2 + l ) T 2 ) = ( M 1 l ) .Forthecase k 1 < 0 ,itismore convenienttoexpress T 2 intermsof T 1 and T 3 : T 2 =( l 0 T 1 +( M 1 + l 0 ) T 3 ) = ( M 2 l 0 ) .Wend T =( MT 2 + M 1 T 1 ) = ( M 1 l )=( MT 3 + M 2 T 1 ) = ( M 2 l 0 ) .Forthe A term,thisprocedure 46
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encountersnoobstacle,andweobtain(displayingexplicit lythecontributionfor k 1 ;l> 0 ) ^^_A g 3 K ^ 4 2 T 0 M M 1 M 2 ( X l Z dT 1 dT 2 ( M 1 l ) 2 T 1 T 2 ( lT 1 +( M 2 + l ) T 2 ) K 2 A M 2 ( M 1 T 1 + MT 2 ) 4 e H ( T 1 ;T 2 ) +(1 $ 2) = g 3 K ^ 4 2 T 0 M M 1 M 2 ( X l Z dT ( M 1 l ) 2 T ( lT +( M 2 + l )) K 2 A M 2 H ( T; 1)( M 1 T + M ) 4 +(1 $ 2) ) : (319) Itwillbeusefultonotethat H canbewritteninthealternativeforms H =( M 2 + l ) T 2 P + lT 1 P 1 + K 2 M 1 M ( M 1 l ) T 1 T 2 MT 2 + M 1 T 1 (320) =( M 1 + l 0 ) T 3 P + l 0 T 1 P 2 + K 2 M 2 M ( M 2 l 0 ) T 1 T 3 MT 3 + M 2 T 1 ; (321) wheretherstisusefulwhen k 1 ;l> 0 andthesecondfor k 1 ;l< 0 Howeverthe B termsproducelogarithmicallydivergentintegralswithth isprocedure, sotheymustbehandleddierently.Todealwiththeselogari thmicallydivergentterms, werstnotetheidentities: T 1 ( T 1 + T 2 + T 3 ) 3 = @ @T 2 ( M 1 l ) 3 2 M T 1 ( MT 2 + M 1 T 1 ) 2 (322) = @ @T 3 ( M 2 l 0 ) 3 2 M T 1 ( MT 3 + M 2 T 1 ) 2 (323) T 2 ( T 1 + T 2 + T 3 ) 3 = @ @T 2 ( M 1 l ) 3 2 M 2 M 1 T 1 +2 MT 2 ( MT 2 + M 1 T 1 ) 2 (324) T 3 ( T 1 + T 2 + T 3 ) 3 = @ @T 3 ( M 2 l 0 ) 3 2 M 2 M 2 T 1 +2 MT 3 ( MT 3 + M 2 T 1 ) 2 ; (325) wherethepartialderivativesaretakenwith T 1 xed. Becauseofthedivergenceswecan'timmediatelywritetheco ntinuumlimitofthe B termsasanintegral.Howeverwecanmakethesubstitution e H ( e H e H 0 )+ e H 0 where H 0 ischosentobeanappropriatesimpliedversionof H ,whichcoincideswith H at T 2 =0 .For k 1 ;l> 0 ,itisconvenienttochoose H 0 =( lT 1 +( M 2 + l ) T 2 ) P 1 ,whereasfor k 1 ;l< 0 H 0 0 =( l 0 T 1 +( M 1 + l 0 ) T 3 ) P 2 ismoreconvenient.Thenthefactor ( e H e H 0 ) regulatestheintegrandatsmall T i sothatthesumsmaythensafelybereplacedby 47
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integrals.Weshalldenotethecontributionsfromtheseter msby ^^_B 1 .Thenusingthe aboveidentities,anintegrationbyparts(forwhichthesur facetermvanishes)makesthe integrandsimilartothatin ^^_A andsimplicationscanbeachieved.FordetailsseeSec 3.5 ^^_A + ^^_B 1 g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 Z 1 0 dT TK 2 ( M 1 l ) 2 A 0 M 2 H ( M + M 1 T ) 3 + lT ( H 0 H )( M 1 l ) M 1 A 0 M 2 H ( lT + M 2 + l )( M + M 1 T ) lT ( H 0 H )( M 1 l ) M 1 A M 2 H ( M + M 1 T ) 2 +(1 $ 2) ; (326) where A 0 = M 2 M 2 2 lM 1 ( M 2 + l ) 2 M 2 ( M 2 + l ) 2 M 1 M ( M 1 l ) M 2 ( M 1 l ) 3 M 1 M ( M 2 + l ) 2 : (327) Sincetheintegrandof( 326 )isarationalfunctionof T thelastintegralcanalsobedone. TheevaluationissketchedinSec 3.5 Thereremainsthecontributionoftheterm e H 0 whichwouldgiveadivergent integral.However,becausethe T 1 ;T 2 ( T 1 ;T 3 )dependenceintheexponentialisdisentangled byourchoiceof H 0 ,thesumscanbedirectlyanalyzedinthe a 0 limit,givingan explicitexpressionforthedivergentpartintermsofthela tticecuto.Wedenotethis contribution,containingtheultravioletdivergenceofth etriangles,by ^^_B 2 .Referringto Sec 3.5 fordetailsweobtain ^^_B 2 = g 3 K ^ 8 2 T 0 M M 1 M 2 (" M 1 1 X l =1 M 1 l MM 1 N 1 l M 1 + N 2 ( M 2 + l ) M ln 2 p +1 ap 21 + f M 1 1 X l =1 M 1 l MM 1 N 1 l M 1 N 2 ( M 2 + l ) M f 0 # +(1 $ 2) ) = g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 B 0 M ln 2 p +1 ap 21 + f + AM 2 ( M 1 l ) 2 M 3 M 1 f 0 +(1 $ 2) ; (328) 48
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wherewehavedened B 0 = ( M 1 l ) 3 M 2 ( M 2 + l ) + MM 1 l ( M 1 l ) + ( M 2 + l ) 3 M 2 ( M 1 l ) : (329) and N 1 B 1 ( M 1 l ) l + B 3 ;N 2 B 2 ( M 1 l ) M 2 + l + B 3 (330) f ( x )= ln x 1 x x Z 1 0 dte xt 1 xt e xt (1 e xt ) 2 ln(1 e t ) (331) M 1 l ; M M 2 + l (332) In ^^_B 2 wecanfurthersimplifythetermproportionalto ln(2 p +1 =ap 21 ) ,whichcontains theultravioletdivergenceofthetrianglediagrams.Weobt ain ^^_div = g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 1 M ( M 1 l ) 3 M 2 ( M 2 + l ) + MM 1 l ( M 1 l ) + ( M 2 + l ) 3 M 2 ( M 1 l ) ln 2 p +1 ap 21 +(1 $ 2) = g 3 K ^ 4 2 T 0 M M 1 M 2 ln 2 p +1 ap 21 ( M 1 + M 2 ) ( M 2 +1)+3 ( M 1 )+3 r + M 1 1 M 3 11 3 M 2 1 7 M 1 M 2 4 M 2 2 + M 1 3 +(1 $ 2) : (333) where isthedigammafunction: ( x )= d dx ln(( x )) : (334) 49
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Writingoutthetermsfrominterchanging 1 $ 2 inthisexpression,andsimplifyingwe obtain ^^_uv = g 3 K ^ 16 2 T 0 M M 1 M 2 ln 2 p +1 ap 21 +ln 2 p +2 ap 22 4( ( M )+ ( M 1 )+ ( M 2 )+3 r ) +2 11 3 8 M + M M 1 M 2 + 2 M 1 M 2 M 3 + 1 3 M 2 ln p +1 p 22 p + p 21 8( ( M 1 ) ( M 2 ))+ 2( M 1 M 2 ) 3 M 3 ( 11 M 2 +2 M 1 M 2 1) M 1 M 2 M 1 M 2 g 3 K ^ 16 2 T 0 M M 1 M 2 ( 2 3 X i =1 ln 2 j p +i j ap 2i 4(ln j M i j + r )+ 22 9 + ln p 2 p +1 p + p 21 +ln p 2 p +2 p + p 22 4(ln M + r )+ 22 9 4 ln M 1 M 2 ln p 21 p +2 p +1 p 22 ln p +1 p 22 p +2 p 21 8ln p +1 p +2 + 2( p +1 p +2 ) 3 p + 11+ 2 p +1 p +2 p +2 : (335) wherethenalexpression,validatlarge M;M 1 ;M 2 ,hasbeenarrangedsothatthe uv divergenceappearssymmetricallyamongthethreelegsofth evertex. Puttingeverythingtogether,theamplitudeforthevertexf unctiontooneloopis giveninthecontinuumlimitby ^^_ = g 3 K ^ 4 2 T 0 M M 1 M 2 ( 1 M 1 M 1 1 X l =1 Z 1 0 dTI 1 + S 1 +(1 $ 2) ) + g 3 K ^ 16 2 T 0 M M 1 M 2 ( 2 3 X i =1 ln 2 j p +i j ap 2i 4(ln j M i j + r )+ 22 9 + ln p 2 p +1 p + p 21 +ln p 2 p +2 p + p 22 4(ln M + r )+ 22 9 ln p +1 p 22 p +2 p 21 4ln p +1 p +2 + 2( p +1 p +2 ) 3 p + 11+ 2 p +1 p +2 p +2 (336) 50
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where I 1 = I T;p 1 ;p 2 ; l M 1 M 1 ( M 1 l ) 2 TK 2 A 0 M 2 H ( T; 1)( M 1 T + M ) 3 + lT ( H 0 H )( M 1 l ) M 2 1 A 0 M 2 H ( lT + M 2 + l )( M + M 1 T ) lT ( H 0 H )( M 1 l ) M 2 1 A M 2 H ( M + M 1 T ) 2 (337) S 1 = S M 1 ;M 2 ; l M 1 M 1 B 0 M f + AM 2 ( M 1 l ) 2 M 3 f 0 (338) I 2 = I T;p 2 ;p 1 ; l M 2 ;S 2 = S M 2 ;M 1 ; l M 2 (339) andwherewerecall,forconvenience,ourdenitions(appro priatetothecase k 1 ;l> 0 ) A = M 2 M 2 1 l 2 ( M 1 l ) 2 + M 2 M 2 2 l 2 ( M 2 + l ) 2 + ( M 2 + l ) 2 ( M 1 l ) 2 + ( M 1 l ) 2 ( M 2 + l ) 2 A 0 = M 2 M 2 2 lM 1 ( M 2 + l ) 2 M 2 ( M 2 + l ) 2 M 1 M ( M 1 l ) M 2 ( M 1 l ) 3 M 1 M ( M 2 + l ) 2 B 0 = ( M 2 + l ) 3 M 2 ( M 1 l ) + ( M 1 l ) 3 M 2 ( M 2 + l ) + MM 1 l ( M 1 l ) H = H ( T; 1)=( M 2 + l ) P 3 + lTP 1 + ( M 1 l ) T M + M 1 T K 2 M 1 M H 0 = H 0 ( T; 1)=( M 2 + l ) P 1 + lTP 1 = M 1 ( M 2 + l ) lM : (340) Tocompletethecontinuumlimitweassume M;M 1 ;M 2 largeandattempttoreplace thesumsover l byintegralsoveracontinuousvariable = l=M 1 ,with 0 << 1 .This procedureisobstructedbysingularbehavioroftheintegra ndfor near0or1.Whenthis occurs,weintroduceacuto << 1 ,andonlydothereplacementfor << 1 dealingwiththesumsdirectlyinthesingularregions.Thed etailedanalysisispresented intheappendicesofthepaperonwhichthischapterismostly based[ 6 ].Referringto 51
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Eq.B.56ofthatpaper,weseethatwecanwrite ( g1 ) ^^_ = g 3 K ^ 16 2 T 0 M M 1 M 2 ( 2 3 X i =1 ln 2 j p +i j ap 2i 4(ln j M i j + r )+ 22 9 +ln p 2 p +1 p + p 21 4ln M M 1 + 22 9 +ln p 2 p +2 p + p 22 4ln M M 2 + 22 9 + g 3 K ^ 4 2 T 0 2 2 3 + M M 1 M 2 (ln M 1 + r ) M 1 p 2 + Mp 21 M 1 p 2 Mp 21 + M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 1 p 2 Mp 21 3+ 2 6 +(ln M 2 + r ) M 2 p 2 + Mp 22 M 2 p 2 Mp 22 M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 2 p 2 Mp 22 3+ 2 6 (341) Comparingthezerothordervertex, 2 gK ^ M=M 1 M 2 T 0 ,toEq. 341 ,weseethatthe ultravioletdivergenceofthetriangleiscontainedinthem ultiplicativefactor 1+ g 2 16 2 ln 2 a 4(ln M +ln M 1 +ln M 2 +3 r ) 22 3 : (342) Notetheentanglementofultraviolet( ln(1 =a ) )andinfrared( ln M i )divergences,typicalof Lightconegauge.The ln M 'smultiplying ln(1 =a ) mustcanceltogivethecorrectcharge renormalization.Toseehowthishappens,dothe l suminthegluonwavefunction renormalizationfactorfrombefore Z ( Q )=1 g 2 N c 16 2 8(ln M + r ) 22 3 ln 2 Q + aQ 2 4 3 : (343) Thustheappropriatewavefunctionrenormalizationfactor forthetriangle, p Z ( p 1 ) Z ( p 2 ) Z ( p ) containstheultravioletdivergentfactor 1 g 2 N c 16 2 [4(ln MM 1 M 2 +3 r ) 11]ln 2 a ; (344) sothedivergencefortherenormalizedtriangleiscontaine dinthemultiplicativefactor 1+ 11 3 g 2 N c 16 2 ln 2 a (345) implyingthecorrectrelationofrenormalizedtobarecharg e g R = g 1+ 11 24 s N c ln 2 a ; (346) 52
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where s = g 2 = 2 3.2.2Simplication:TheWorldSheetPicture Itishowevermoreinterestingtodothecalculationabovein aslightlydierent manner.Itissuggestedbytheworldsheetpicture,thatonec ombineselfenergyand vertexdiagramsateachvalueofthediscrete p + oftheloop.Ifwebackuptheabove calculationandconsiderthecontributiontochargerenorm alizationbeforethe p + sumis donewehave ( gluons1 ) ^^_ = g 3 4 2 a m ln(1 =a ) M 3 M 1 M 2 K ^ M 1 1 X l =1 ( 2 l + 1 M 3 l + 1 M 1 l + 2 f g ( l=M 3 ) M 3 ) (1 $ 2) ; (347) Therstthreelogarithmicallydivergenttermsinthe l summandsagainrepresentthe entanglementofinfraredandultravioletdivergences.The setermswillcancelagainstterms fromtheselfenergy,sothattheentangleddivergencesnev erarise.Weshallseethisbetter inthenextsection,whenthefullparticlecontentistakeni ntoaccount. 3.3AddingSUSYParticleContent:FermionsandScalars Theproperoneloopcorrectiontothecubicvertexisreprese ntedbyaFeynman trianglegraphappearingintheworldsheetasshowninFig. 32 WiththeexternalparticlesofFig. 32 restrictedtobegluons(vectorbosons)theone looprenormalizationofthegaugecouplingrequirescalcul atingthetrianglegraphforthe dierentparticlesofthetheoryrunningaroundtheloop.In thefollowingsubsectionsit willbeusefultoemploythecomplexbasis x ^ = x 1 + ix 2 and x = x 1 ix 2 fortherst twocomponentsofanytransversevector x Fermions 53
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ReferringtoSection 3.5 fordetailsofthecalculationtheresultforthediagramdep ictedin Fig. 32 withfermionsontheinternallinesisgivenby ( fermions1 ) ^^_ = N f ag 3 K ^ 4 2 m ln(1 =a ) M 1 M 2 M 1 1 X l =1 f f ( l=M 3 ) (1 $ 2) ; (348) forpolarizations n 1 = n 2 = ^ ;n 3 = GluonsThiscalculationhasbeendonefor n 1 = n 2 = ^ ;n 3 = inthepaper[ 6 ]anditisvery similartothefermioncalculation.Thecontributiontocha rgerenormalizationisgivenby: ( gluons1 ) ^^_ = g 3 4 2 a m ln(1 =a ) M 3 M 1 M 2 K ^ M 1 1 X l =1 ( 2 l + 1 M 3 l + 1 M 1 l + 2 f g ( l=M 3 ) M 3 ) (349) (1 $ 2) ; (350) Therstthreelogarithmicallydivergenttermsinthe l summandsagainrepresentthe entanglementofinfraredandultravioletdivergencesandw ewillseeinsection 3.4 how theycancelagainstsimilartermsfromtheselfenergycontr ibution. ScalarsNowconsiderscalarsoninternallinesandthesameexternal polarizationsasbefore. Recallthattheindices n i inEq.( 219 )runfrom1to D 2 .Letususeindices a;b for directions3to D 2 .Thendimensionalreductionisimplementedbytaking p ai =0 for all i and a .Usingtheseconventionswewillbeinterestedinthespecia lcaseofEq.( 219 ) with n 1 = a;n 2 = b and n 3 = ab 0 = 1 8 3 = 2 ag m ab 2 K M 1 + M 2 ; (351) 54
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andsimilarlyfor ab ^ 0 .Theevaluationofthediagramisanalogoustotheprevious calculationsandtheresultcorrespondingto( 348 )is ( scalars1 ) ^^_ = N s ag 3 4 2 m ln(1 =a ) K ^ M 1 M 2 M 1 1 X l =1 f s ( l=M 3 ) (1 $ 2) : (352) 3.4DiscussionofResults Thephysicalcouplingcanbemeasuredby Q i p Z i ,therenormalizedvertexfunction, where isthepropervertexand Z i isthewavefunctionrenormalizationforleg i .Toone loopwewritethisintermsofourquantitiesas: Y 1 + 1 2 0 X i ( Z i 1) ; (353) where 0 isthetreelevelvertexand 1 isouroneloopresultforthevertex: ^^_1 = fermions1 + gluons1 + scalars1 ^^_ (354) = g 3 4 2 a m ln(1 =a ) p +3 p +1 p +2 K ^ M 1 1 X l =1 2 l + 1 M 1 l + 1 M 3 l + F ( l=M 3 ) M 3 (1 $ 2) : (355) BecauseofhowloopsaretreatedintheLightconeWorldSheet formalismweare motivatedtocombinetheoneloopvertexresultandthewavef unctionrenormalization foraxedpositionofthesolidlinerepresentingtheloop.I notherwordswerenormalize locally ontheworldsheet.Toclarifythis,notethethreedierentw aystoinsertaoneloop correctiontothecubicvertexatxed l ontheworldsheet,asinFig. 33 .Noticethat M 3 M 1 l k =0 k = k 1 k = k 2 M 3 M 1 l k =0 k = k 1 k = k 2 M 3 M 1 l k =0 k = k 1 k = k 2 Figure33:Oneloopdiagramsforxed l intheLightconeWorldSheet. 55
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therstandlastgurescorrespondtoselfenergydiagramsf orthelegswithmomenta ( p 3 ;M 3 ) and ( p 1 ;M 1 ) respectively.However,themiddlegurecorrespondstoatr iangle diagramwithtimeordering k 1 > 0 .Socombiningourpreviousresultscancalculatethe Y correspondingtothispolarizationforaxed l
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Wewriteouttheresultsfortherenormalizedvertex Y whereasubscript refersto thetwodierenttimeorderings, k 1 > 0 ; ( lM 1 ) respectively. Y ^^^ = Y ___ =0 ; (361) Y ^^_ + = g 3 8 2 a ln(1 =a ) p +3 K ^ p +1 p +2 M 1 1 X l =1 F ( l=M 3 ) M 3 F ( l=M 1 ) M 1 ; (362) Y ^_^ + = g 3 8 2 a ln(1 =a ) p +2 K ^ p +1 p +3 M 1 1 X l =1 F ( l=M 3 ) M 3 F ( l=M 1 ) M 1 ; (363) Y _^^ + = g 3 8 2 a ln(1 =a ) p +1 K ^ p +2 p +3 M 1 1 X l =1 F ( l=M 1 ) M 1 F ( l=M 3 ) M 3 ; (364) Y ^^_ = g 3 8 2 a ln(1 =a ) p +3 K ^ p +1 p +2 M 3 1 X l = M 1 +1 F ( M 3 l=M 3 ) M 3 F ( M 3 l=M 2 ) M 2 ; (365) Y ^_^ = g 3 8 2 a ln(1 =a ) p +2 K ^ p +1 p +3 M 3 1 X l = M 1 +1 F ( M 3 l=M 2 ) M 2 F ( M 3 l=M 3 ) M 3 ; (366) Y _^^ = g 3 8 2 a ln(1 =a ) p +1 K ^ p +2 p +3 M 3 1 X l = M 1 +1 F ( M 3 l=M 3 ) M 3 F ( M 3 l=M 2 ) M 2 : (367) Theexpressionsforthe Y 'swith ^$_ arethesamewith K ^ $ K .Westressthatthe summandsintheaboveexpressionsfor Y areexactlycontributionofthethreediagramsin Fig. 33 withtheloopxedat l Denethecouplingconstantrenormalization ( N f ;N f ) by: Y n 1 n 2 n 3 = Y n 1 n 2 n 3 + + Y n 1 n 2 n 3 = n 1 n 2 n 3 0 ( N f ;N s ) : (368) Wethenhaveinthelimit M i + 1 : ( N f ;N s )= g 2 8 2 ln(1 =a ) + 11 3 N f 3 N s 6 ; (369) whichisthewellknownresult.Inparticularwehaveasympto ticfreedomwhen > 0 and vanishesfortheparticlecontentof N =4 SupersymmetricYangMillstheory, N f =8 and N s =6 57
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ForsomecasessuchastheSupersymmetric( N f =2+ N s )orpureYangMills ( N f = N s =0 )thesummandsintheexpressionsforthe Y 'sdonotchangesign.When N f < 8 ,sothatthesecasesareasymptoticallyfree,thesummandso ntherightsides of( 364 )and( 366 )haveasignwhichworksagainstasymptoticfreedom.Sincet he fullsumexhibitsasymptoticfreedomforeachpolarization ,thismeansthatthatthe complementarytimeorderings,( 363 )and( 367 ),mustcontributemorethantheirshare toasymptoticfreedom.Thisfactmaybeusefulforapproxima tionsinvolvingselective summation. 3.5DetailsoftheLoopCalculation Wepresentheresomeofthecalculationaldetailsthatwereo mittedfromtheabove sectionsforclarity.3.5.1FeynmandiagramCalculation:Evaluationof ^^_B 1 Inthecalculationof ^^_B 1 westartbyintegratingbyparts.Thistransfersthe derivativetothefactor ( e H e H 0 ) .Fordenitenesstakethecase l> 0 .Thenwe compute @ @T 2 ( e H e H 0 )= e H H T 1 lP 1 T 2 + e H 0 H 0 T 1 lP 1 T 2 + e H K 2 M 1 ( M 1 l ) T 1 T 2 ( MT 2 + M 1 T 1 ) 2 : (370) Thersttwotermsonther.h.s.partlycancelafterintegrat ionover T 1 ;T 2 .Thisisbecause theintegralsareseparatelynite,soonecanchangevariab les T 1 = T 2 T ineachterm separately.Forthersttermwend Z 1 0 dT 1 dT 2 I ( T 1 ;T 2 ) H ( T 1 ;T 2 ) T 1 lP 1 T 2 e H ( T 1 ;T 2 ) = Z dTdT 2 I ( T; 1)[ H ( T; 1) TlP 1 ] e T 2 H ( T; 1) = Z dT I ( T; 1) 1 TlP 1 H ( T; 1) ; (371) 58
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andthesecondtermyieldsthesameexpressionwith H ( T; 1) H 0 ( T; 1) ,sothetwoterms combineto Z 1 0 dT I ( T; 1) Tl H 0 ( T; 1) H ( T; 1) H ( T; 1)( lT + M 2 + l ) : (372) Simplifyingthecontributiontothe T integrandfromthesetermsleadstothecontinuum limit ^^_B 1 g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 Z 1 0 dT lT ( H 0 H ) H ( lT + M 2 + l ) + ( M 1 l ) TK 2 HM 1 ( M + M 1 T ) 2 ( M 1 l ) 2 ( M + M 1 T ) 2 T 2 M B 1 + M 1 T +2 M 2 M 2 B 2 + M 2 T ( M 1 l )+2 MlT 2 M 2 ( M 1 l ) B 3 + M 2 + l M ( M 1 l ) B 3 +(1 $ 2) = g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 Z 1 0 dT lT ( H 0 H ) H ( lT + M 2 + l ) + ( M 1 l ) TK 2 HM 1 ( M + M 1 T ) 2 ( M 1 l ) 2 ( M + M 1 T ) 2 M 1 M 2 M + M 1 T M 1 l A 0 lT + M 2 + l M 1 l A +(1 $ 2) ; (373) wherewehavedened A 0 = M 2 M 2 2 lM 1 ( M 2 + l ) 2 M 2 ( M 2 + l ) 2 M 1 M ( M 1 l ) M 2 ( M 1 l ) 3 M 1 M ( M 2 + l ) 2 : (374) Noticethatthisresultcombinesneatlywith ^^_A togive ^^_A + ^^_B 1 g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 Z 1 0 dT TK 2 ( M 1 l ) 2 A 0 M 2 H ( M + M 1 T ) 3 + lT ( H 0 H )( M 1 l ) M 1 A 0 M 2 H ( lT + M 2 + l )( M + M 1 T ) lT ( H 0 H )( M 1 l ) M 1 A M 2 H ( M + M 1 T ) 2 +(1 $ 2) : (375) 3.5.2FeynmandiagramCalculation:Evaluationof ^^_B 2 Weanalyzethecontinuumlimitofthe ^^_B 2 contributiontothe B terms,whichwill beretainedasdiscretesumsoverthe k 's.Againfordenitenesswedisplaythecase l> 0 59
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indetail: ^^_B 2 = g 3 K ^ 16 2 T 3 0 M M 1 M 2 8<: M 1 1 X l =1 1 j l j ( M 2 + l )( M 1 l ) X k 1 ;k 0 2 T 1 B 1 + T 2 B 2 + T 3 B 3 ( T 1 + T 2 + T 3 ) 3 e H 0 +(1 $ 2) 9=; = g 3 K ^ 4 2 T 0 M M 1 M 2 8<: M 1 1 X l =1 M 1 l j l j ( M 2 + l ) X k 1 ;k 0 2 k 1 N 1 + k 0 2 N 2 ( k 1 + k 0 2 ) 3 u k 1 + k 0 2 1 +(1 $ 2) 9=; : (376) where M 1 l ; M M 2 + l ;N 1 B 1 ( M 1 l ) l + B 3 ;N 2 B 2 ( M 1 l ) M 2 + l + B 3 u 1 e p 21 = 2 M 1 T 0 = e ap 21 = 2 p +1 ;u 2 e p 22 = 2 M 2 T 0 = e ap 22 = 2 p +2 ; (377) where u 2 istobeusedinthecase k 1 ;l< 0 insteadof u 1 .Clearlythecontinuumlimit entails u 1 ;u 2 1 ,causingthe k sumstodivergelogarithmically.Tomakethisexplicit,we rstnotetheintegralrepresentation X k 1 ;k 0 2 u k 1 + k 0 2 1 ( k 1 + k 0 2 ) 2 = Z 1 0 tdt u 21 ( e t u 1 )( e t u 1 ) (378) Z 1 tdt 1 ( e t 1)( e t 1) + Z 0 tdt (1 u 1 + t )(1 u 1 + t ) : (379) wheretheapproximateformisvalidfor 1 u 1 ; .Doingtheintegralinthe secondtermleadsto X k 1 ;k 0 2 u k 1 + k 0 2 1 ( k 1 + k 0 2 ) 2 1 ln 2 p +1 ap 21 + ln ln + 1 ln + Z 1 tdt 1 ( e t 1)( e t 1) : (380) Thesumswerequirecanbeobtainedfromthisidentitybydie rentiationwithrespectto or .Topresenttheresultsitisconvenienttodeneafunction f ( x ) by f ( x ) 1 2 1+ x 1 x ln x +lim 0 ln + Z 1 tdt ( e t p x 1)( e t= p x 1) (381) = ln x 1 x x Z 1 0 dte xt 1 xt e xt (1 e xt ) 2 ln(1 e t ) (382) 60
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wherethesecondformisobtainedbyintegrationbyparts.It isevidentfromtherstform that f ( x )= f (1 =x ) .Alsoonecaneasilycalculate f (1)= 2 = 6 .Fromthesecondform oneeasilyseesthat f ( x ) ln x for x 0 ,whencefromthesymmetry, f ( x ) ln x for x !1 .Exploitingthefunction f anditssymmetries,wededuce X k 1 ;k 0 2 u k 1 + k 0 2 1 ( k 1 + k 0 2 ) 2 1 ln 2 p +1 ap 21 + f (383) X k 1 ;k 0 2 k 1 u k 1 + k 0 2 1 ( k 1 + k 0 2 ) 2 1 2 2 ln 2 p +1 ap 21 + f 1 2 2 f 0 (384) X k 1 ;k 0 2 k 0 2 u k 1 + k 0 2 1 ( k 1 + k 0 2 ) 2 1 2 2 ln 2 p +1 ap 21 + f + 1 3 f 0 1 2 2 ln 2 p +1 ap 21 + f 1 2 f 0 : (385) InsertingtheseresultsintoEq. 376 produces ^^_B 2 = g 3 K ^ 8 2 T 0 M M 1 M 2 (" M 1 1 X l =1 M 1 l MM 1 N 1 l M 1 + N 2 ( M 2 + l ) M ln 2 p +1 ap 21 + f M 1 1 X l =1 M 1 l MM 1 N 1 l M 1 N 2 ( M 2 + l ) M f 0 # +(1 $ 2) ) = g 3 K ^ 4 2 T 0 M M 1 M 2 ( M 1 1 X l =1 B 0 M ln 2 p +1 ap 21 + f + AM 2 ( M 1 l ) 2 M 3 M 1 f 0 +(1 $ 2) ; (386) wherewehavedened B 0 = ( M 1 l ) 3 M 2 ( M 2 + l ) + MM 1 l ( M 1 l ) + ( M 2 + l ) 3 M 2 ( M 1 l ) : (387) 3.5.3FeynmandiagramCalculation:Evaluationof ^^_II Forlarge M i ,thesumover l canbeapproximatedbyanintegralover = l=M 1 from 1 to 1 ,plussumsfor 1 l M 1 and M 1 (1 ) l M 1 1 whichcontain thedivergences.Thesedivergencesareonlypresentinthe rstsumonther.h.s.ofEq. ?? for l M 1 andinthelastsumfor M 1 l M 1 .Themiddlesumcontainsnodivergence 61
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andcanbereplacedbyanintegralfrom 0 to 1 withno cuto.Toextractthesedivergent contributions,wecanusethelargeargumentexpansionof 0 0 ( z +1) 1 z 1 2 z 2 + O ( 1 z 3 ) ; (388) toisolatethem.Itisthusevidentthattheircoecientswil lbeproportionaltothe moments P f k =k n ; P h k =k n for k =0 ; 1 ,whicharepreciselythemomentsconstrainedby therequirementthatthegluonremainmasslessatoneloop. Fortheendpointnear l =0 ,weput z = kM 1 ( M 2 + l ) =lM andwrite k l 0 ( z +1) M M 1 ( M 2 + l ) l 2 k M 2 M 2 1 ( M 2 + l ) 2 + :::; (389) sothesummandforsmall l becomes X k ( f k M (2 M 1 l )(2 M 2 + l ) l 2 M 1 ( M 2 + l ) + h k M M 1 ( M 2 + l ) f k 2 k M 2 (2 M 1 l )(2 M 2 + l ) lM 2 1 ( M 2 + l ) 2 h k 2 k lM 2 M 2 1 ( M 2 + l ) 2 ) M (2 M 1 l )(2 M 2 + l ) l 2 M 1 ( M 2 + l ) + M M 1 ( M 2 + l ) 2 12 M 2 (2 M 1 l )(2 M 2 + l ) lM 2 1 ( M 2 + l ) 2 + 2 ( l 1) 36 l lM 2 M 2 1 ( M 2 + l ) 2 4 M l 2 2 M 2 lM 1 M 2 2 3 M 2 lM 1 M 2 : (390) Summing l upto M 1 gives M 1 X l =1 4 M l 2 2 M 2 lM 1 M 2 2 3 M 2 lM 1 M 2 4 M 2 6 4 M M 1 M 2 M 1 M 2 (ln M 1 + r ) 2+ 2 3 ; (391) InsertingtheseresultsintoEq. ?? andwritingoutexplicitlythe 1 $ 2 termsforthe divergentpartgives ^^_C 1 g 3 8 2 T 0 K ^ M ( 1 X k =1 Z 1 d k 0 1+ k ( M 2 + M 1 ) M f k (2 )(2 M 2 + M 1 ) 2 M 1 + h k +(1 $ 2) 4 M 2 M 1 M 2 + 8 M 2 6 M 2 M 1 M 2 (ln 2 M 1 M 2 +2 r ) 2+ 2 3 : (392) Onlythethirdsumcontributesnear l = M 1 .Weagainusethelargeargument expansionof 0 .Butthistimeoneonlygetsalogarithmicdivergence,becau sethe dierenceof 0 'sisoforder ( M l ) 2 asistheexplicitrationalterm.Putting z 1 = 62
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kM 1 ( M 2 + l ) =M 2 ( M 1 l ) and z 2 = klM=M 2 ( M 1 l ) ,wehave 0 ( z 1 +1) 0 ( z 2 +1) 1 z 1 1 z 2 1+ 1 2 z 1 + 1 2 z 2 M 2 2 ( M 1 l ) 2 klMM 1 ( M 2 + l ) (1+ O ( M 1 l )) M 2 2 ( M 1 l ) 2 kM 2 1 M 2 : (393) Thusthe l M 1 endpointdivergenceisjust g 3 8 2 K ^ M 4 M 2 M 1 M 2 1 X k =1 f k k f k M 1 1 X l = M 1 (1 ) 1 M 1 l = g 3 8 2 4 MK ^ M 1 M 2 2 6 1 (ln M 1 + r ) : (394) Puttingeverythingtogetherweobtainforthecontinuumlim itofthetrianglewithinternal longitudinalgluons ^^_II g 3 8 2 T 0 K ^ M ( 1 X k =1 Z 1 d k 0 1+ k M 2 + M 1 ) M f k (2 )(2 M 2 + M 1 ) 2 M 1 + h k + 1 X k =1 Z 1 0 d k (1 ) 2 M M 2 + M 1 0 1+ k M M 2 + M 1 f k ( M 2 M 1 )( M + M 1 (1 )) ( M 2 + M 1 ) 2 + h k + 1 X k =1 Z 1 0 d k ( M 2 + M 1 ) 2 M (1 ) M 3 2 M 2 2 (1 ) 2 k 2 ( M 2 + M 1 ) 2 + 0 1+ k M 2 + M 1 M 2 (1 ) 0 1+ k M M 2 (1 ) f k (1+ )( M + M 2 + M 1 ) M 1 (1 ) 2 + h k 2 M 2 M 1 M 2 +(1 $ 2) ) g 3 4 2 T 0 K ^ M + 2 M 2 3 M 2 M 1 M 2 (ln 2 M 1 M 2 +2 r ) 3 2 6 : (395) 3.5.4FeynmandiagramCalculation:DivergentPartsofIntegralsandSums The T integralinEq. 326 canbeevaluatedbyexpandingtheintegrand I M 1 TK 2 ( M 1 l ) 2 A 0 M 2 H ( M + M 1 T ) 3 + lT ( H 0 H )( M 1 l ) M 2 1 A 0 M 2 H ( lT + M 2 + l )( M + M 1 T ) lT ( H 0 H )( M 1 l ) M 2 1 A M 2 H ( M + M 1 T ) 2 : (396) inpartialfractions.Firstnotethatsince ( M 1 T + M ) H isaquadraticpolynomial,itmay befactoredas ( M 1 T + M ) H = lp 21 ( T T + )( T T ) (397) 63
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where T ( K 2 + M 1 M 2 p 2 ) =lMp 21 and T + MM 2 p 2 = ( K 2 + M 1 M 2 p 2 ) when l M 1 Thenthepartialfractionexpansionreads I = R 1 T T + + R 2 T T + R 3 lT + M 2 + l + R 4 ( M 1 T + M ) 2 + R 5 M 1 T + M ; (398) withthe R i independentof T .Ofcoursethe R i aresuchthat I fallsoatleastas 1 =T 2 forlarge T ,i.e., R 1 + R 2 + R 3 =l + R 5 =M 1 =0 .Thisidentityishelpfulfordetermining R 5 Thuswehave Z 1 0 dTI = R 1 ln( T + ) R 2 ln( T )+ R 3 l ln l M 2 + l + R 4 MM 1 + R 5 M 1 ln M 1 M : (399) The R i aregivenexplicitlyby R 1 = M 1 ( M 1 l ) T + lM 2 p 21 ( T + T ) ( M 1 l ) K 2 A 0 ( M 1 T + + M ) 2 + lp 21 A 0 lp 21 ( M 2 + l + lT + ) A M 1 T + + M (3100) R 2 = M 1 ( M 1 l ) T lM 2 p 21 ( T + T ) ( M 1 l ) K 2 A 0 ( M 1 T + M ) 2 + lp 21 A 0 lp 21 ( M 2 + l + lT ) A M 1 T + M (3101) R 3 = l ( M 2 + l ) M 2 1 A 0 M 2 M 2 (3102) R 4 = M 2 1 ( M 1 l ) A 0 M lM 1 ( M 1 l ) A M (3103) R 5 = lM 1 ( M 1 l ) A M 2 1 p 21 MM 2 K 2 + lM 2 1 A 0 MM 2 M 2 1 ( M 1 l ) A 0 M 2 1 M M 1 T + + M M M 1 T + M : (3104) Whenthe M 'sarelarge,thesumover l canbereplacedbyanintegralover = l=M 1 as longas iskeptawayfromtheendpoints =0 ; 1 .Wecanisolatethetermsthatgiverise tosingularendpointcontributionsandsimplifythemconsi derably.Weshallthenseparate thedivergentcontributionsanddisplaythemindetail. Firstnotethattheworstendpointdivergenceis 1 ln near =0 or 1 = (1 ) near =1 .Thuswecandropalltermsdownbyafactorof l=M i forsmall l orby ( M 1 l ) =M i for l near M 1 .Thusfor l M i ,wenotethat lT ( M + M 1 T + ) K 2 =p 21 and 64
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T + = ( M + M 1 T + ) M 2 p 2 =K 2 andobtain R 1 p 2 M 2 1 M 2 lK 2 K 2 + M 1 M 2 p 2 2 M 2 Mp 21 ( K 2 + M 1 M 2 p 2 ) = p 2 M 2 1 M 2 lK 2 M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 (3105) R 2 M 1 l + 2 M 1 M 2 p 21 l ( M 1 p 22 + M 2 p 21 ) = M 1 l M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 (3106) R 3 M 1 (3107) R 4 MM 2 1 l (3108) R 5 M 2 1 l 1+ MM 1 p 22 MM 2 p 21 K 2 : (3109) Combiningthe l 0 endpointcontributionsgives Z 1 0 dTI R 1 ln MM 2 p 2 K 2 + M 1 M 2 p 2 R 2 ln K 2 + M 1 M 2 p 2 lMp 21 + R 3 l ln l M 2 + R 4 MM 1 + R 5 M 1 ln M 1 M M 1 l M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 M 1 M 2 p 2 K 2 ln MM 2 p 2 K 2 + M 1 M 2 p 2 +ln K 2 + M 1 M 2 p 2 lMp 21 ln l M 2 + 1+ MM 1 p 22 MM 2 p 21 K 2 ln M 1 M 1 M 1 l M M 1 p 22 M 2 p 21 K 2 ln M 1 M 2 p 2 K 2 + M 1 M 2 p 2 + M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln ( K 2 + M 1 M 2 p 2 ) 2 lM 2 M 2 p 2 p 21 ln lM M 1 M 2 1 ; for l M i : (3110) Fortheotherendpoint, M 1 l M i ,therootsofthepolynomial ( M + M 1 T ) H approach T 1 = M=M 1 and T 2 = p 2 =p 21 .Whichoftheserootsisapproachedby T dependsonthe parametervalues,butsincetheformulaearesymmetricunde rtheirinterchange,wecan choosetousetherstinplaceof T + andthesecondinplaceof T .Sincethedenominator M + M 1 T 1 =0 inthislimit,weneedtocarefullyevaluate M 1 l M + M 1 T 1 M 1 ( Mp 21 M 1 p 2 ) K 2 : 65
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Thenweobtainforsmall M 1 l R 1 M 1 M 2 ( Mp 21 + M 1 p 2 ) ( M 1 l ) K 2 2 MM 1 p 21 ( M 1 l )( Mp 21 M 1 p 2 ) (3111) R 2 2 p 2 M 2 1 ( M l )( M 1 p 2 Mp 21 ) (3112) R 3 l M 1 M 1 l (3113) R 4 2 MM 2 1 M 1 l (3114) R 5 M 2 1 ( M 1 l ) M 2 1 M 2 ( Mp 21 + M 1 p 2 ) ( M 1 l ) K 2 : (3115) Combiningthe l M 1 endpointcontributionsgives Z 1 0 dTI R 1 + R 3 + R 5 M 1 ln M 1 M R 2 ln p 2 p 21 + R 4 MM 1 ; for M 1 l M i 2 M 1 ( M 1 l ) M 1 p 2 M 1 p 2 Mp 21 ln M 1 p 2 Mp 21 1 : (3116) Inwritingthe l sumasanintegraltheseendpointdivergencescanbeseparat edby picking 1 andsumming l intheranges 1 l M 1 and M 1 (1 ) l M 1 1 .For thesepartsofthesumtheaboveapproximationscanbemadean dthesumevaluated: M 1 X l =1 1 l = M 1 1 X l = M 1 (1 ) 1 M 1 l = (1+ M 1 )+ r ln M 1 + r M 1 X l =1 ln l l = r ( (1+ M )+ r ) Z 1 0 dt ln t e t e M 1 t 1 e t 1 2 ln 2 ( M 1 )+ (2) r 2 2 + 1 2 Z 1 0 dt t ln 2 t e t 1 1 2 ln 2 ( M 1 )+ C (3117) 66
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Therestofthesumisreplacedbyanintegralover 1 1 M 1 M 1 1 X l =1 Z 1 0 IdT Z 1 d Z 1 0 IdT +(ln M 1 + r ) 2 M 1 p 2 M 1 p 2 Mp 21 + M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 1 p 2 Mp 21 3 + M M 1 p 22 M 2 p 21 K 2 2 M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 1 M 2 p 2 K 2 + M 1 M 2 p 2 ln M M 1 M 2 (ln M 1 + r )+ 1 2 ln 2 ( M 1 )+ C 2 M 1 p 22 M 1 p 22 + M 2 p 21 (3118) Finally,wemustextractthedivergentcontributionsthata risefromreplacingthesums M 1 1 X l =1 S l = M 1 1 X l =1 M 1 B 0 M f + AM 2 ( M 1 l ) 2 M 3 f 0 (3119) inEq. 336 byanintegral.First,for l M 1 = 1 andonlythersttermgivesa singularendpointcontribution, M 1 1 X l = M 1 (1 ) S l 2 M 1 f (1)[ (1+ M 1 )+ r ] 2 3 M 1 [ln M 1 + r ] : (3120) Ontheotherhand,for l 0 ,wehave f M 1 M 2 lM ln M 1 M 2 lM Z 1 0 dte t ln t 1 t e t (1 e t ) 2 =ln lM M 1 M 2 (3121) M 1 M 2 lM f 0 M 1 M 2 lM 1+ lM M 1 M 2 2 12 +ln M 1 M 2 lM 1 : (3122) Theintegralintherstlineiszerobecausetheintegrandis aderivativeofafunction vanishingattheendpoints.Insertingtheseapproximation s,weobtain M 1 X l =1 S l M 2 1 M 2 M M 1 X l =1 2 l 2 + M 1 ( M 1 X l =1 1 l ln lM M 1 M 2 [ (1+ M 1 )+ r ] 2 6 ) (3123) 67
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PuttingEqs. 3118 3120 3123 together,somesimplicationoccursandweobtain 1 M 1 M 1 1 X l =1 S ( l=M 1 )+ Z 1 0 IdT Z 1 d Z 1 0 IdT + Z 1 dS ( ) + M 1 M 2 2 3 M 2 M 2 M +(ln M 1 + r ) 2 M 1 p 2 M 1 p 2 Mp 21 + M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 1 p 2 Mp 21 + M M 1 p 22 M 2 p 21 K 2 2 M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 1 M 2 p 2 K 2 + M 1 M 2 p 2 3+ 2 6 ln M M 1 M 2 (ln M 1 + r )+ 1 2 ln 2 ( M 1 )+ C M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 : (3124) Whenweaddthecontributionwith 1 $ 2 ,theantisymmetryofsomeofthecoecients leadstofurthersimplicationaswellasareductioninthed egreeofdivergenceofsomeof theterms 1 M 1 M 1 1 X l =1 S ( l=M 1 )+ Z 1 0 IdT +(1 $ 2) Z 1 d Z 1 0 IdT + Z 1 dS ( )+(1 $ 2) 2 ln p +1 p 22 p +2 p 21 p +1 p 22 + p +2 p 21 ln p +1 p +2 +ln 2 p +1 p 2 p +1 p 2 p + p 21 ln p +1 p 2 p + p 21 + p +1 p 22 p +2 p 21 p +1 p 22 + p +2 p 21 ln p +1 p 22 p +2 p 21 + 2 p +2 p 2 p +2 p 2 p + p 22 ln p +2 p 2 p + p 22 6+ 2 3 +ln p +1 p +2 M M 1 p 22 M 2 p 21 K 2 2 p +1 p 22 p +2 p 21 p +1 p 22 + p +2 p 21 ln M 1 M 2 p 2 K 2 + M 1 M 2 p 2 + 1 2 ln p +1 p +2 M 2 p +1 p 22 p +2 p 21 p +1 p 22 + p +2 p 21 + 2 2 M 1 M 2 3 M + (ln M 1 + r ) 2 M 1 p 2 M 1 p 2 Mp 21 + M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 1 p 2 Mp 21 3+ 2 6 +(ln M 2 + r ) 2 M 2 p 2 M 2 p 2 Mp 22 M 1 p 22 M 2 p 21 M 1 p 22 + M 2 p 21 ln M 2 p 2 Mp 22 3+ 2 6 : (3125) As 0 thersttwolinesonther.h.s.approachanite independentanswer.The thirdlineisexplicitlynite.Alldivergencesareshownin thelasttwolines.As M i !1 thereisaleadinglineardivergenceaswellasasinglelogar ithmicsubleadingdivergence. 3.5.5DetailsofSUSYParticleCalculation WeconsiderthediagramdepictedinFig. 32 withfermionsoninternallinesand gluonswithpolarizations n 1 ;n 2 and n 3 onexternallines.Insection 2.2 themethodof constructingtheworldsheetverticesisoutlinedandrefer ringtothepaperitself[ 7 ]forthe 68
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detailedexpressionsweget M 1 1 X l =1 X k 1 ;k 2 Z d q (2 ) 3 exp a 2 m k 1 ( p 1 p ) 2 M 1 l + k 2 p 2 l + ( k 2 k 1 )( p 3 p ) 2 M 3 l Tr ( ag 2 m ( r n 1 r r ) p l p 1 p M 1 l r + ag m p 1 p M 1 l p 1 M 1 n 1 ag 2 m ( r n 3 r s ) p 3 p M 3 l p l s + ag m p l p 3 M 3 n 3 ag 2 m ( r n 2 r t ) p 1 p M 1 l p 3 p M 3 l t + ag m p 3 p M 3 l p 2 M 2 n 2 #) (3126) Noticethatthisistheexpressionassociatedwithfermiona rrowsrunningcounterclockwise aroundtheloop.Theotherdiagramcontributesthesameamou ntasthisone.Also, thisexpressionisfor k 1 > 0 ,theothertimeordering k 1 < 0 isobtainedbymakingthe substitution p 1 $ p 2 asinthegluoncalculationofthepaper[ 6 ].Wenowproceedmuchas inthatcalculationbycompletingthesquareintheexponent ofEq.( 3126 )andshifting momentum M 1 1 X l =1 X k 1 ;k 2 Z d q (2 ) 3 exp t 1 + t 2 + t 3 2 m=a q 2 e H Tr ( g 2 ( r n 1 r r ) r1 l ( M 1 l ) + g n 1 1 M 1 ( M 1 l ) g 2 ( r n 3 r s ) s3 l ( M 3 l ) + g n 3 3 M 3 l g 2 ( r n 2 r t ) t2 ( M 1 l )( M 3 l ) + g n 2 2 ( M 3 l ) M 2 ) (3127) with n1 = t 3 K n t 1 + t 2 + t 3 M 1 q n ; n2 = t 2 K n t 1 + t 2 + t 3 M 2 q n ; n3 = t 1 K n t 1 + t 2 + t 3 + M 3 q n (3128) t 1 = k 1 M 1 l ;t 2 = k 2 l ;t 3 = k 2 k 1 M 3 l (3129) H = a 2 m t 1 t 3 p 22 + t 1 t 2 p 21 + t 2 t 3 p 23 t 1 + t 2 + t 3 (3130) 69
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Inthe q integralonlythetermsproportionalto q 2 timestheGaussianwillexhibit a 0 divergencessoweretainonlythose.Thegeneralloopintegr alisgivenby Z d q i1 k3 j2 exp t 1 + t 2 + t 3 2 m=a (2 m=a ) 2 2( t 1 + t 2 + t 3 ) 3 M 1 M 2 t 1 K k ij M 1 M 3 t 2 K j ik M 2 M 3 t 3 K i jk (3131) (Thearrowmeansthat a 0 nitetermshavebeendropped.)Somesimplication canbedonerightaway,forexamplethetermproportionalto Tr ( r n 1 r r r n 3 r s r n 2 r t ) after contractingwiththemomentumintegralisproportionalto X r Tr( r n 1 r r r n 3 r r r n 2 ( r K ))=(4 D 0 )Tr( r n 1 r n 3 r n 2 ( r K )) (3132) where D 0 isthespacetimedimensionalityoftheloopmomentumintegr al,thatisthe reduceddimension D 0 =4 sothistermvanishes.Furthersimplicationscanbeseenwh en aparticularexternalpolarizationischosen. Thedetailed a 0 behaviorbecomesapparentwhenthesumover k 1 and k 2 isdone. Withalittleworkitcanbeshownthat X k 1 ;k 2 t 1 ( t 1 + t 2 + t 3 ) 2 e H ln(1 =a ) l 3 ( M 3 l )( M 1 l ) 2 M 2 1 M 3 (3133) X k 1 ;k 2 t 2 ( t 1 + t 2 + t 3 ) 2 e H ln(1 =a ) l 2 ( M 3 l )( M 1 l ) 2 M 1 M 3 M 1 l M 1 + M 3 l M 3 (3134) X k 1 ;k 2 t 3 ( t 1 + t 2 + t 3 ) 2 e H ln(1 =a ) l 3 ( M 3 l )( M 1 l ) 2 M 1 M 2 3 (3135) Carryingthisthroughforthepolarization n 1 = n 2 = n 3 = n yields N f ag 3 K n 32 2 m ln(1 =a ) M 1 M 2 M 3 M 1 1 X l =1 M 3 1 2 l M 3 1 l M 3 + M 1 1 2 l M 1 1 l M 1 (3136) Inthecontinuumlimitwehave P M i 1 l =1 f ( l=M j ) M j R M i =M j 0 dxf ( x ) foranycontinuous function f .Therefore,afteraddingthe k 1 < 0 contributionandmultiplyingby2forthe 70
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otherorientationofthefermionloopweobtain N f ag 3 K n 16 2 m M 2 1 + M 2 2 + M 2 3 M 1 M 2 M 3 ln(1 =a ) 2 3 : (3137) Incontrast,thecalculationandresultforthe n 1 = n 2 = ^ ;n 3 = polarizationis quitealotsimpler.Theexpressionanalogousto( 3136 )is N f ag 3 K ^ 16 2 m ln(1 =a ) M 1 M 2 M 1 1 X l =1 1 2 l M 3 1 l M 3 : (3138) Theresultshownin( 348 )isobtainedfromthisonebyaddingthe k 1 < 0 contribution andmultiplyingbytwowhichaccountsfortheotherorientat ionofthefermionarrowsin theloop. 71
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CHAPTER4 THEMONTECARLOAPPROACH Thischapterpresentsthesecondthemeofthethesis,namely thenumerically implementedMonteCarloapproachtostudyingtheTr ( 3 ) eldtheoryintwodimensions. Wewillseewhy,inthiscase,suchastochasticapproachispr eferredoveradeterministic numericalinvestigation.ThemathematicalframeworkforM arkovchainsiswell establishedandisdiscussedindetailinbasictextbookson thesubject[ 18 ]. 4.1IntroductiontoMonteCarloTechniques Sincetheadventofeasy,ubiquitousaccesstopowerfulcomp uters,numericalmethods havecometobewidelyusedinphysics.Quantumeldtheoryis noexceptiontothis. Numericalmethodscan,inprinciple,becategorizedas deterministic or stochastic .Both typesyieldapproximateanswersbutinthecaseofdetermini sticmethodstheerrorcan betracedbacktotheniteprecisionrepresentationofrea lnumbersusedincomputers. Asthenameimplies,deterministicmethodsoperateinapred enedmanner.Stochastic methods,ontheotherhand,relyheavilyonstatisticsander rorsoriginatenotonlyfrom oatingpointrepresentationofnumbers,butalsofromthes tatisticalinterpretationofthe results,sincerandomnumbersareusedasinputsofthesimul ation.MonteCarlomethods areexamplesofstochasticnumericalmethodsandtheyareso widelyusedthattheterm "MonteCarlo"issometimestakentomeananystochasticnume ricalmethod. 72
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4.1.1Mathematics:MarkovChains TheunderlyingmathematicalconstructionforMonteCarlot echniquesisthatofthe Markovchain.Givenaprobabilityspace (n ; A ;P ) aMarkovchainisasequenceof randomvariables ( X k ) k 0 distributedinsomewayovera statespace S withtheproperty that: P ( X k = x k j X k 1 = x k 1 ;X k 2 = x k 2 ;:::;X 0 = x 0 )= P ( X k = x k j X k 1 = x k 1 )= T x k ;x k 1 : (41) forsome transfermapping T .Noticethattherandomvariable X k dependson X k 1 only, andnottheearlierrandomvariablesinthesequence.Thisis sometimesreferredtoasthe Markovchain'slackofmemory.A distribution : S [0 ; 1] foranelement X k ofthe Markovchainisdenedtobetheprobabilitythat X k takesthevalue x or: k ( x )= P ( X k = x ) : (42) Clearly P x 2 S ( x )=1 foranydistribution(hencethename).Itisusefultothinko fthe Markovindex k in ( X k ) astimeandinvestigatetheevolutionofthedistributionwi th time: t +1 = T x t +1 ;x t t : (43) If S isnite(whichitisinanycomputersimulation),then T isamatrixwhoserowssum to 1 .An equilibriumdistribution y isdenedby = T .Averyimportantresultof themathematicaltheoryofMarkovchainsisthatanaperiodi c,irreducibleandpositive Aprobabilityspaceisatriplet (n ; A ;P ) where n representsthesetofpossible outcomes, A isthesetofeventsrepresentedasacollectionofsubsetsof n and P : A! [0 ; 1] istheprobabilityfunction. y Equilibriumdistributionsaresometimescalled invariant or stationary 73
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recurrent z transitionmatrix T hasauniqueequilibriumdistribution andwillconverge toitirrespectiveofitsstartingdistribution,morepreci sely: ( T ) Nij i N + 1 j : (44) Moreover,suchatransitionmatrixsatisesthe ErgodicTheorem whichstatesthatforany boundedfunction f : S R wehave: 1 N N 1 X k =0 f ( x k ) N + 1 f (45) withprobabilityone, x where f = P x 2 S ( x ) f ( x ) .Thisresultisastatementthatone mayndintuitivelysensible:thattheproportionoftimewh ichtheMarkovchainspends inastate x approaches ( x ) ,thevalueoftheequilibriumdistributioninthatstate.It isacommonstrategytoapproximate f withanitesum P N 1 k =0 f ( x k ) usingtheErgodic Theorem.Thisstrategyiscalled importancesampling becauseitcanbethoughtofas summingovertheimportantelementsof S :thosethatsample Byconstruction,theMarkovchaincansimulatemanyphysica lprocesses,simply becausethedening"lackofmemory"property,isseensowid elyinnature.Markov chainsingeneralhavemanyapplicationsintheeldofphysi csoutsideofstochastic numericalsimulations.ButMonteCarlosimulationarethet opicofinteresthere,and wearenowinapositiontoclarifywhatismeantbyMonteCarlo simulationsinthe rstplace.Letusconsiderthecasewherethestatespace S isverylargeandwhereeach elementofitisacomplicatedobject.Letusfurtherassumet hatwewanttocalculatethe z Irreducibilityissometimescalledergodicity,andessent iallymeansthatthetransition matrixmustbeabletogoeverywhereinstatespace.Wewillse ethisbetterlater.The conditionsarenotofgreatconcern,positiverecurrencyis automaticinanitestatespace. x Rigorouslythismeansthattheevents f X k = x k g for k =1 ; 2 ;:::;N suchthat( 45 )is true,haveprobabilityone,orconverselythat( 45 )isfalseonasubsetof n withmeasure zero. 74
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valueof P x 2 S ( x ) f ( x ) ,theexpectationvalueofanoperator f denedonthestatespace, undersomedistribution .Thefullsummaynotbefeasibletoperformwhen S islarge (asweshallsee,thisismostcertainlytrueinthepresentwo rk!).Theideais,then, toconstructaMarkovchainwhichhas asitsequilibriumdistributionandperform importancesamplingtoobtainanapproximationoftheexpec tationvalue.SoinMonte Carlosimulationswearegivenadistribution anditistheMarkovchainwewantto nd.Thisstandsincontrasttomanyotherapplicationsofth eMarkovtheory,where thetransitionmatrixisknown,andtheequilibriumdistrib utionistheobjectofinterest. OfcoursewewillnotbeabletondtheMarkovchaininitself; insteadwendwhatwe shallcallaMarkov sequence :asequenceofactualstates,ratherthanrandomvariablesi n thestatespace.Thisisacceptablebecauseifthesequence reallyrepresentstheMarkov chain,thenthestates x k willdistributeasdictatedbytherandomvariables X k .The centraltrickinaMonteCarlosimulationishowtoobtainthe Markovsequencefrom thedistribution .Althoughsometimesverydicultinpractice,theideaisal most embarrassinglysimple.Since satisestheErgodicTheorem, ( x ) 6 =0 forall x 2 S Perhapsthemoststraightforwardansatzwouldbetoselects tates x 2 S atrandom andappendingeachtotheMarkovsequencewithprobability ( x ) .Theresultwould certainlybeaMarkovsequencewithequilibriumdistributi on .Thismethod,however, wouldbehopelessinmanycasesbecauseofthecomputational complexityofcalculating ( x ) ,foranarbitrary x ,asmanytimesaswouldberequiredtoobtainanacceptably longMarkovsequence.Fortunately,manyimportantcasesin physicsexhibitwhatwe shallcall MonteCarlolocality or mclocality forshort.Thisisthepropertythatthe ratio ( x 0 ) = ( x ) isdrasticallysimplertoevaluatecomputationallyif x iscloseto x 0 in asensethatissimulationspecic.Itsucestosaythatthey areclosepreciselywhen x 7! x 0 and x 7! ( x 0 ) = ( x ) arecomputationallysimpleoperations.Wesuggestively write x 0 = x + x andtalkabout x beingsmall,eventhoughtheremaynotbean additionoperation,metricormeasuredenedonstatespac e.Foranmclocalsystemwe 75
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nowconstructasequence(notnecessarilyaMarkovsequence yet)with any state x 2 S as therstelement.Wethenchange x alittlebit: x 0 = x + x ,andautomaticallyaccept x 0 asthenextelementinthesequenceif ( x 0 ) ( x ) .Ifhowever ( x 0 ) ( x ) weacceptit withprobability p = ( x 0 ) = ( x ) .Inpracticethisisdonebyhavingthecomputergenerate a(pseudo)randomnumber r uniformlydistributedbetween 0 and 1 andacceptingthe changeif r
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Table41:MonteCarloconceptsinmathematicsandphysics. Concept Markovchainsymbol Physicssymbol StateSpace S Q State x q ProbabilityDistribution e S =Z ratherthanindiscretesteps.Theresultsarealmostthesam e,althoughthemathematical frameworkbecomesalittlebitmoreinvolved. Tocontinuethediscussion,itisappropriateatthispointt onarrowthescopeand considerthephysicalcontextinwhichwewillwork.4.1.2ExpectationValuesofOperators Letusconsideraphysicalsystemwithdynamicalvariablesd enotedcollectivelyby q andlet q liveinsomespace Q whichwillhaveanymathematicalstructureneededto performtheoperationsthatfollow.If S istheeuclidianactionthentheexpectationvalue ofanoperatoriswritten: h F i = 1 Z Z Q dqF ( q ) e S ( q ) ; (46) where Z = R Q dqe S ( q ) and F istheoperatorinquestion.Clearly,fromtheformof( 46 ) theintegralissupportedbytheregionsin Q where e S islarge.Standardoptimization yieldtheclassicalequationsofmotionfor q .Inquantumeldtheoryatraditionalnext stepwouldbenormalperturbationtechniques,toexpandthe nongaussianpartof e S inpowersofthecouplingconstant.Butinanticipationfort heapplicationofstochastic methodsweinterpret( 46 )asthestatisticalaverageof F weightedwiththeprobability distribution e S Z whichitofcourseis.With Q niteweareinanexactapplicationofthe MonteCarlomethodsdiscussedearlier,withatranslationo fnotationsummarizedintable 41 Weareinterestedinthespecialcaseofaquantumeldtheory representedas aLightconeWorldSheet.Inthiscasethespace Q isthesetofall(allowed)eld 77
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congurationsonthetwodimensionallatticerepresenting thediscretizedworldsheet. Wewillemploythe MetropolisAlgorithm toconstructtheMarkovsequenceanditworks asfollows:Givenaeldconguration q i we 1.Visitasitein q i andaltertheeldvaluesthereandpossiblyintheimmediate vicinitytoobtainanewconguration q 0 i 2.Calculate x =exp f ( S ( q 0 i ) S ( q i )) g andhaveacomputercalculatearandomnumber y between0and1. 3.If y
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separatedsetoflesandcanbereplacedwithoutconsequenc estotherestofthecomputer program. Considerthefollowingaction: S = M 1 X k =0 ( q k +1 q k ) 2 with q 0 = q M =0 : (47) Asbefore,usethefollowing: q =( q 1 ;q 2 ;:::;q M 1 ) ; D q = dx 1 dx 2 :::dx M 1 ;Z = Z D qe S ( q ) : (48) Withtheverysimpleobservableoperator q 7!O kl ( q )= q k q l theexpectationvalue becomes hO kl i = Z D qe S ( q ) O kl = h q k q l i = min ( k;l )( M max ( k;l )) (49) Weusethisresulttondtheexpectationvalue F ( k;l )= h ( q k q l ) 2 i F ( k;l )= h ( q k q l ) 2 i = h q 2 k i 2 h q k q l i + h q 2 l i = 1 2 M j k l j ( M j k l j ) (410) sowecanconsiderthefunctionofthedierenceonly f ( m )= F ( k;k + m ) .Withoutany lossofgeneralitywetake k =0 andthebehaviorof f as m rangesfrom1to M 1 .The resultsforaMCsimulationareshowninFigure 41 togetherwiththeaboveexactanswer. Therelativeagreementallowsustoconsiderthistestpasse dbythesimulationsoftware. Toseethecomputerimplementationofthebosonicchainpres entedhereseeAppendix A 4.1.4AnotherSimpleExample:1 D IsingSpins Thespinsystem s ji ,whichplaysavitalroleintheLightconeWorldSheetformal ism, hasoftenbeenlikenedtoanIsingspinsystem.Thisisofcour setruebecausethespins,as inIsing'smodelforferromagnetismtakeonthetwovalues s ji = and s ji = # implemented onacomputerwith s ji =+1 and s ji = 1 .A1 D Isingspinsystemisthereforeexactlyas 79
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2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 0.4 0.2 0 0.2 Chainseparation j k l jF ( k;l ) ResidualsFigure41.Testresultsforthesimpleexampleofabosonicc hain.AnMCsimulation with 2 10 4 sweepsandparameters M =21 ; =1 .Thegureshowsthe resultsfortheexpectationvalue F ( k;l )= h ( q k q l ) 2 i wherethespacing j k l j isshownon x axis.Thetwographssharethe x axisandintheupper plotthestemsareMCresultsandthesolidlineistheexactan swerfor F ( k;l ) fromEqn. 410 .Thelowerplotshowstheresiduals,meaningthedierence betweentheMCandexactresults.ByttingtheMCresultstoa functional formasfor F aboveweobtain M =20 : 49 and =0 : 9926 .Noticethat thedierenceplothassomestructure.Webelievethatthisi sanindication ofcorrelationbetweenMCerrorsalongthechain,whichisan eectwesee clearlyonthelatticeandshalldiscussinmoredetaillater ourLightconeWorldSheetsetupexceptwithadierentinter action.TheIsingspinsystem onlyhasalocalinteraction.AlthoughtheLightconeWorldS heetinteractionisalsolocal, weintendtotreatthe 1 =p + factorsnonlocally,i.e.,wedonotemploythelocal b;c ghosts butinsteadputinthe 1 =p + byhandmakingtheinteractionatleastmildlynonlocal.Th e 80
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Table4.Testresultsfor1 D Isingsystem.Herewehaveexactresultstocomparetothe MCnumbers.The E tisobtainedbyttingthecorrelationtoanexponential andreadingotheexponentasshownongraph4.Wecanseethatthe overallerrordecreaseswithincreasingnumberofsweeps,b utoncetheerroris prettysmallthisdecreaseisnotveryconsistent.Thisisan inherentpropertyof astochasticmethodsuchasthisone. NumericalResults Coupling g Exact E MC E t 10 4 sweeps 4 10 4 sweeps 1 :6 10 5 sweeps 0.1 0.20067 0.25724 0.22188 0.21900 0.2 0.40547 0.38471 0.41673 0.41903 0.3 0.61904 0.64785 0.61391 0.61466 whollylocalIsinginteractionisgivenby H = X i;j )Tj/T1_4 11.955 Tf5.481 9.684 Td()Tj/T1_1 11.955 Tf9.297 0 Td( ij s 1 i s 1 j + ms i (4) with amatrixwithonlynearestneighborinteraction.Forsimpli cityofthetestwetook m =0 and i;j = g i;j )Tj/T1_3 7.97 Tf6.588 0 Td(1 Alltheresemblancealmostdrivesustotestthesoftwareand methodsonjusta 1 D Isingspinsystem,whichwedid.TheMCimplementationinvol vedusingsimplythe softwarefortheLightconeWorldSheetwithasimpliedinte raction.Theexactresultsare verysimpletoobtainforthecasedescribedhere.Foragiven timethestateofthesystem isrepresentedbythevectorofspins ~s =( s 1 ;s 2 ;:::;s M ) .Sincetheinteractionissosimple thenthetransitionmatrixisgivenby T ij = ij )Tj/T1_1 11.955 Tf13.149 8.082 Td(g 2 ( i;j +1 + i +1 ;j ) (4) sothat ~s ( t + dt )= T~s ( t ) .Taking t = t=N and ~s ( t )= T N ~s (0) thennitetime propagationcorrespondsto N + 1 .Theeigenvaluesof T are t =1 g andfor largeenough N then E =ln t andthereforethesplittingis E =ln((1+ g ) = (1 )Tj/T1_1 11.955 Tf12.276 0 Td(g )) TheresultsfortheexactvsMCcomparisonareshowninTabl e4andthettingis exempliedbytheplotinFigure4.Again,therelativeagreementwithknownexact answersgivesthecomputercodea"pass"forthistestaswell .Havingpassedbothof 81
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Figure42.Testresultsfor1 D Isingsystem.Thegraphshowsthecorrelation C between spins s i and s 20 sothattheyaxisshows: C ( i )= h s i s 20 i andthexaxislabels thespinindex i .Therearethreesetsofdataplottedtogether,onedataset foreachvalueoftheIsingcoupling g .Thegraphalsodepictstheexponential tsdonesothattheexponentialfalloofthecorrelationca nbereado.The resultsaresystematicallyorganizedinatablebelow. thesesimpletests,orexamplesofapplication,wecanallow ourselvestospendsometime topreparefortherealapplicationoftheMonteCarlomethod presentedinthisthesis. 4.1.5StatisticalErrorsandDataAnalysis Beingastochasticnumericalmethod,theMonteCarloapproa chgivesonly statisticallysignicantresults.Wesawthisclearlyinth elastsectionwhereevenin theverysimplecasesofabosonicchainanda1 D Isingspinsystem,wheredeterministic methodswouldprobablyhaveservedbetter(infact,exactan swerswereavailable!).One mightthinkthatsincetheMonteCarlomethodindeedperform edrelativelypoorlyinthe simplestcases,itislikelytofailutterlywhenamorecompl icatedsystemisconsidered. Thishoweverisbynomeansthecase.Thestatisticalinaccur aciesoftheMonteCarlo methodareinherentinthealgorithmandremaininthemoreco mplicatedapplications, however,thereisnothingwhichindicatesthatthiseectsh ouldincreaseinanywayjust becausethesystemunderconsiderationiscomplicated.Tou nderstandthis,wewilldiscuss brieysomeofthestatisticalobservationswhicharestand ardintheapplicationofMonte Carlomethods[ 19 20 ].(ThebookbyLyons[ 21 ]containsmanyusefuldiscussionson statisticsingeneralscenarios.)Althoughstructurallyt hissectionbelongsherewiththe 82
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generaldiscussionofMonteCarlosimulationsandapplicat ionsofMarkovchains,itisnot arequiredreadingtocontinuethechapter.Infact,thedisc ussionofspincorrelationsin thenextsectionareinstructivebeforereadingthissectio n.Attheendofthedayhowever, theconsiderationshereapplyforanyobservableandnotjus tspincorrelations. WhenapplyingMonteCarlosimulationstostudyaphysicalsys temwesamplean observableinvariousstatesofthesystem.Wewillworkexte nsivelywiththecorrelations betweenspinsonvarioussitesofthelattice.Thedatawhich wewillhaveavailableis thefullspincongurationofthelatticeforeachsweep.Let usdenoteby n thesweep numberandassumewehaveperformedatotalof K sweeps.Wethenhavethedata: s ji ( n ) for i 2b 1 ;M c j 2b 1 ;N c and n 2b 1 ;K c ,whereeachvalue s ji ( n ) isanupordown spin,representedby s ji ( n )=1 or s ji ( n )=0 respectively.Here b k 1 ;k 2 c meansthesetof allintegersbetween k 1 and k 2 .Wewillusetheexpectationnotation h s ji i n whenweare takingaveragesoversweeps n ,i.e., h s ji i n = P n s ji ( n ) = P n 1 wherethesumon n runsover varioussubsetsof b 1 ;K c .Whenthereisnoriskofconfusion,weomitthesubscript n andwritesimply h s ji i .Weshalldenoteby s ( n ) theentirecongurationofspinsatsweep n s ( n ) isinsomesensealargematrixofspins.Weareinterestedint hecorrelations Corr ( s;s 0 )= h s ji s j 0 i 0 ih s ji ih s j 0 i 0 i andmostlytheirdependenceontheseparation j j j 0 j .So tacklingtheissuetopdownwecanbreakthedataanalysisin totwoparts: 1.Obtainfromtherawdata s ( n ) n anothersetofdatapointsoftheform ( x k ;y k ) suchthat x j j j 0 j and y Corr ( s;s 0 ) uptoadditiveconstants.Becauseof thestatisticalnatureofMCsimulationsthedatawillhavea ninherentdistribution, sothatalongwiththedataitselfweneedthevarianceandcov arianceofthedata points.Inotherwords,weneedthe errormatrix : E kl = cov ( y k ;y l ) 2.Thenewdata ( x k ;y k ) isttedtoanexponentialtypefunction f A ( x ) where A = ( A 1 ;:::;A n ) arettingparameters.Thetisthenperformedbysimplymin imizing thefunction F ( A ) where F ( A )= X k;l E 1 kl y k f A ( x k ) y l f A ( x l ) : (413) Recallthatsince E issymmetric,then F willbenonnegative. 83
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InpracticewerunMCsimulationsforagivenlatticersttoo btainestimatesfor E butthenrunamuchlongersimulationtoobtainthedatapoint s ( x k ;y k ) usingthe previousestimatefortheerrormatrix.Westartthediscuss ionwithpoint1)above. Inobtaininganestimateforthe j j j 0 j dependenceofCorr ( s;s 0 ) wecalculateonly h s ji s j 0 i 0 i sincewearenotinterestedintheadditiveconstantwhichth esecondtermin thecorrelationis.Thiscorrespondstokeeping ( i 0 ;j 0 ) xedandusingonlythedatafor which s j 0 i 0 =+1 .Inpracticethisisachievedbysimplynotupdatingthespin at ( i 0 ;j 0 ) andcalculatingonly h s ji i onthislattice.Thedatasetmentionedabovethenbecomes y j = 1 M P i h s ji i and x j = j j j 0 j Aswasexplainedinsection 4.1.1 wemayneedtothrowoutsomeofthecongurations inthesequence s ( n ) n becausetheyaretoocorrelatedtogenerateaMarkovchain.T his isdonebycalculatingthe"latticemean"autocorrelation : Aut ( m )= 1 MN X i;j D s ji ( n + m ) s ji ( n ) E n : TypicallyAut ( m ) isafallingexponentialasafunctionof m .Thereciprocalofthe exponentiscalledthe autocorrelationlength ,denotedby L AUT andinourcasewehave L AUT =10 20 .Thismeansthatbyusingonlyevery L AUT thsweep,i.e.,bygenerating a reduced dataset: s ( k ) for k = n L AUT + k 0 with n =0 ; 1 ; 2 ;::: ,wearecertainthe spincongurationsarenotcorrelatedsequentially.Forca lculatingthedatapoints y k it iscorrecttouseallofthedata k butwhenestimatingthevarianceandcovarianceof thestatisticalerrorswemustconsiderthereduceddataset .Thesearecalculatedusing k Thisissimplybecausetheaverageover s ( n ) ;s ( n +1) ;:::;s ( n + L AUT ) canbeusedin thereduceddatasetinsteadofthevalue s ( n ) only,andthisisequivalenttojustaveraging overallthespincongurations. 84
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standarddataanalysisformulas[ 21 ]. A = X n A n X n 1 ; B = X n B n X n 1 cov ( A;B )= X n A A n B B n X n 1 = AB A B giventhedatasets A 1 ;A 2 ;:::;A n ;::: and B 1 ;B 2 ;:::;B n ;::: Turningnexttopoint2)above.Theinterpretationfortheex pressionfor F is thefollowing:Weconsiderthevalues w k =( y k f A ( x k )) ,for k =1 ;:::;K ,tobe normallydistributedrandomvariableswithgivenvariance andcovariance.Recallthatthe underlying n (sweeplabel)dependenceof x k and y k hasnowbeendiscardedandreplaced byan"allowable"distributioninthe"errors" w k .If v k ,for k =1 ;:::;K ,werenormally distributedrandomvariableswithmeanzeroandvarianceon eandamongthemselves uncorrelated,wecouldrelatethe v sand w sby: C v = w withthematrix C givenby theCholeskyfactorizationoftheerrormatrix E and v = v 1 ;:::;v K T .Butsince v isso simplewecancalculatethenormasfollows k v k = v T v = C 1 w T C 1 w = w T E w = F (414) sothat F isjustthesquareddistancebetweenthedataandthettingf unction f A Minimizing F ( A ) asafunctionof A isjustthewellknown leastsquares tting,with theextensionthatweusestatisticalinformationaboutthe tteddata.Weminimize F by useofthe LevenbergMarquardt algorithm,whichisanoptimizedandrobustversionofthe NewtonRhapson classofmethods.Theseoptimizationmethods"search"fort heminima bystartingataninitialguessfortheparameters A andtravellingin A space,withcertain rulesdeterminingthestep A ,evaluating F ateachstepuntilconvergenceisobtained. 85
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TheratheradvancedLevenbergMarquardtmethodusesacomb inationofthefollowing twostepdeterminations: Steepestdescentmethod:Takesimply A = t r A F ( A ) anduseonedimensionaloptimizationtodeterminethestepl ength t (calleda line search ). Useaquadraticapproximationfor F : F ( A + A )= F ( A )+ A r F ( A )+ 1 2 A T H ( A ) A with H theHessianmatrixfor F ,andusethestep A whichminimizedthis approximation A = H 1 r F Weimplementallofthestatisticalcalculationsanddataan alysisproceduresin MATLAB r 4.2Applicationto2DTr 3 Ascalarmatrixeldtheoryin1+1spacetimedimensionsdes cribedbytheaction ( 21 )canofcoursebewrittenintheLightconeWorldSheetform,j ustaswasdonein 3+1dimensions.However,withonlytwodimensionsthereare notransversebosonic q variableslivinginthebulkoftheworldsheet.Itisalsoawe llknownfactthatthistheory isultravioletnitemeaningthatwecanwithoutconsiderin gcountertermsontheworld sheet,proceeddirectlywithsimulatingthetheoryusingth emethodsdevelopedinthis chapter.Thisverysimplechoiceismotivatedbythisfactan dalsobythefactthatthe theoryhasbeenusedwidelyasatoymodelandtheresultscant hereforebeveriedand compared.TheMonteCarlomethodcaninthiscontextbeveri edinitsownright. Withthispreamble,weturnnowtothesimplestpossiblecasew heretheonly dynamicalvariableleftinthesystemarethespinswhichdes ignatethepresenceofsolid lines.The b;c ghostscanbeeliminatedbysimplyputtinginbyhandthefact orsof 1 =M whichtheyweredesignedtoproduce.Althoughthisdoesintr oduceanonlocaleect, 86
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andthereforeaseeminglydramaticslowdownoftheMonteCar losimulationthereare shortcutsthatcanbeusedaswillbeexplained. Inthissimplecaseaworldsheetcongurationconsistsonly of NM spinslabelledby s ji ,where i =1 ; 2 ;:::;M and j =1 ; 2 ;:::;N labelthespaceandtimelatticecoordinates respectively.Anupspincouldberepresentedby s ji =1 andadownspinby s ji =0 Weemployperiodicspacialandtemporalboundarycondition ssothattheeld,inthis caseonly s ji ,livesonatorus.InprincipletheMetropolisalgorithmnow justproceeds analogouslytotheIsingspinsystem,whereeachsiteinthel atticeisvisitedandaspinis ippedwithacorrespondingprobabilitytocompleteafulls weepofthelatticewhichis thenrepeated.Latticecongurations c k ,whereklabelsthesweepsaregeneratedandthen usedtocalculateobservablesofthephysicalsystem.4.2.1GeneratingtheLatticeCongurations ThebasicingredientfortheMetropolisalgorithmofgenera tinglatticecongurations istodeterminethechangeintheactionunderthepossiblelo calspinipsthatmayoccur. Whenaspinisippedastomakeanewsolidlinethemasstermoft hepropagatorgoes from e a 2 =m ( M 1 + M 2 ) tobeing e a 2 =mM 1 e a 2 =mM 2 wherethe M 1 and M 2 denotethelattice stepstothenearestupspintotheleftandrightrespectivel y.Furtherwhenaspinip resultsinasolidlinebeinglengthenedupwards(downwards )thefactorforthefusion (ssion)vertexwillbemovedupwards(downwards)possibly resultinginachangein which M 1 ;M 2 stouse.Thenthereistheappearanceordisappearanceofs sionandfusion verticesassolidlinessplitorjoin.Thesebasicingredien tsaresummarizedinTable 43 Howeverthetable( 43 )doesnottellthewholestorybecausethereareanumberof subtletiesthatneedtobeaddressed.Thesecanbesummarize dasfollows: 1.ParticleInterpretation.2.NoFourVertex.3.Ergodicity. 87
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Table43.Basicspinipprobabilities.Therightcolumnh asthebasicprobabilities forthe ( i;j ) spintoipgiventhelocallatticecongurationsshowninth e leftcolumn.Alleddotatalatticelocationindicatedaspi nvalueofup there,nodotindicatesaspinvalueofdown.Noticethatther everseip (fromdowntoup)hasthereciprocalprobability. M j 1 ( M j 2 ) isthespatial latticestepstothenextupspintotheleft(right)attimesl ice j .Herewe use r = 2 2 T 0 1 M 1 + 1 M 2 1 M 1 + M 2 j+1 j j1 i P = g 2 e r M j +1 1 M j +1 2 ( M j 1 1 + M j 1 2 ) j+1 j j1 i P = M j 1 + M j 2 M j 1 1 + M j 1 2 e r j+1 j j1 i P = M j 1 M j 2 M j +1 1 M j +1 2 e r j+1 j j1 i P = 1 g 2 M j 1 M j 2 ( M j 1 + M j 2 ) e r 88
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4.NonlocalEects. Wenowturntotheseissuesinorder.ParticleInterpretation. Onaperiodiclatticetherecannotbeatimeslicewhichdoes nothaveanyupspin.ThishasnointerpretationintheFeynma nrulesfromwhichour worldsheetdescriptioncomes. NoFourVertex. Twosolidlinescannotendonthesametimesliceifthelinesa re inclearsightfromeachother,i.e.,thereisnosolidlinese paratingthetwoendsinthe spacedirection.Thisisillustratedingure( 43 ).Thereasonforthisisthatthiswould incorrespondtoafourpointinteractionwhichisnotpresen tintheeldtheory.Infull LightconeWorldSheetformulationthesecongurationsare automaticallyavoidedbecause adoubleghostdeletiononthesametimesliceresultsinaval ueofzerofortheaction. Herehowevertheghostshaveherebeentreatedbyhandandwem ustthereforebansuch congurationsbyhandalso. disallowed disallowed allowed allowed Figure43.Examplesofallowedanddisallowedspincongur ations.Congurationswhich mustbeavoidedbyhandbecauseofthemanualapplicationoft he b;c ghosts. Ergodicity. Bybanningthecongurationsmentionedaboveweintroducea certain nonergodicityintotheMetropolisscheme.Ifallsuchcon gurationsarebanneditis impossibleforasolidlinetogrowpasttheendofanothersol idlineresidingatanearby 89
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spaceslice,unlessweallowtwosimultaneousupdatesofspi ns.Thiswayasolidlineis allowedsuchagrowthwithoutgoingthroughthebannedcong uration.Thisisillustrated ingure( 44 ).Theprobabilitiesfordoubleupdatesareobtainedinasim ilarwayasthose forsingleupdates. Figure44.Basicdoublespinip.Ifthetwospinsindicate dbynonlledcirclesare allowedtobeippedatthesametimethesystemisabletoevol vefromone congurationtotheotherwithoutgoingthroughthebannedc onguration. NonlocalEects. Changesintheactioninvolving M 1 and M 2 areinherentlynonlocal butareeasytodealwithinasimulationanddonotoverlyslow downtheexecutionofthe program.Whendoubleupdateshavebeenallowedthereareanum berofscenarioswhich willleadtopotentiallyworsenonlocalchangesintheactio neventhoughthespinips themselvesarelocal.Anexampleisillustratedingure( 45 ).Ifwedenoteby M 1 ( i;j ) ( M 2 ( i;j ) )thedistancestotheleft(right)ofspin ( i;j ) theninthecaseshowninthegure thefusionvertexfactorat ( i 1 ;j ) changebyafactor: M 1 ( i 1 ;j )+ M 2 ( i 1 ;j ) M 1 ( i 1 ;j )+ M 1 ( i 2 ;j )+ M 2 ( i 2 ;j ) = M 1 ( i 2 ;j +1) M 1 ( i 2 ;j +1)+ M 2 ( i 2 ;j +1) (415) Itisinterestingthattheresultingchangeinactionduetot hefusionvertexat ( i 1 ;j ) dependsonlyuponthefactorsof M derivedfromoneofthesiteswhichundergotheip, namely ( i 2 ;j +1) .Goingthroughallthedierentscenariosthisturnsouttob egenerally true.Thenonlocaleectsofthistypethereforedonotrequi reanylargescalelattice inspectionsbythesimulationprogramandthereforedonots lowdownexecution. 90
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Table44:WorldSheetspinpictures. Verystrongcoupling, g= 2 =1 : 2649 Strongcoupling, g= 2 =0 : 4472 Weakcoupling, g= 2 =0 : 3464 Veryweakcoupling, g= 2 =0 : 2000 91
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j i 1 i 2 Figure45.Doublespinip,withdistantvertexmodicati on.Whenthetwospins indicatedbynonlledcirclesareippeddownthereisbrea kinthesolid lineat i 2 withcorrespondinglocalssionandfusionvertexfactors. However, thereisalsoachangeinthefactorsappearingforthefusion vertexat i 1 .The distancetotherightofspin ( i 1 ;j ) tothenextupspinhasjustchangedas well,inotherwords, M 2 ( i 1 ;j ) haschanged. 4.2.2UsingtheLatticeCongurations Weareinterestedinthephysicalobservablesofthesystem. Inparticularthosewhich haveadirectinterpretationintermsoftheeldtheorydesc ribedbytheLightconeWorld Sheet.Suchanobservablewouldbeforexampletheenergylev elsofthesystem.To understandhowthesewillemergefromtheworldsheetletusc onsiderthesysteminterms oftimeevolutioninthediscretizedLightconetime x + namelyourlatticetime j .Ata giventime j thesystemcanbecharacterizedasa 2 M statequantumsystemwithatime evolutionoperator T whichtakesittothenexttime j +1 .Thenegativelogarithmofthis operatoristheHamiltonianofthesystem,namelytheLightc one P operatorandthe energylevelsaregivenbytheeigenvalues.Wewrite T = e H ,andconsiderthecorrelator betweentwostates j i and j i separatedby j timestepsonaperiodiclatticewithtotal timesteps N G ( j )= h jT N j j ih jT j j i Tr ( T N ) (416) Take j m i for m 2 N tobetheeigenbasisfor T with Tj n i = t n j n i andwrite j i = P n a n j n i and j i = P n b n j n i then G ( j )= P n a n b n t N j n ( P n a n b n t jn ) P n t Nn (417) 92
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Noticethatbydividingthroughequation( 417 )with t N0 assumingthe t n sarein decreasingsizeorderweseethatthe j dependenceof G isentirelyoftheform G ( j )= A 0@ B + 2 M X n =1 C n e a n ( N j ) 1A 0@ B + 2 M X n =1 C n e a n j 1A (418) where n istheenergygapbetweenthegroundstateandthe n thexcitedstateand A;B andthe C n 'sareconstantsdependingontheoverlapofthestates j i ; j i witheach otherandtheeigenstatesof T .Byjudiciouslychoosingstates j i ; j i andgoingnear thecontinuumlimit(equivalentlychoosinglargeenough N and j )weseethatthelowest energygapofthetheorycanbereadoastheexponentialcoe cientinthe j dependence of G ( j ) Averyimportantfactisthatthefull 2 M statequantumsystemthatdescribesthe latticeateachtimehasalargeredundancyintermsoftheel dtheorywearesimulating. Letusforexampleconsiderthesimplecaseofapropagatorin thetwolanguages.Onthe periodiclatticethispropagatorcanberepresentedbyasol idlineatanyofthe M dierent spatialpoints.Butintheeldtheorythereisnosuchlabell ingof which ofthe M dierent propagatorstochosefrom.Thepropagatorstateisreallyth elinearcombinationofall thestatesonthespaceslicewithoneupturnedspinsomewhe re.Inthemorecomplicated caseswherethereareanumberofupspins,againtheeldtheo rydoesnotdistinguish betweenwherethespincombinationliesbutonlybetweenthe dierentcombinations ofdownspinsinarowandtheorderofthese.Inotherwordsthe eldtheorystate isthecycliclysymmetriclinearcombinationofthespaces licestates.Noticethatthe transitionoperator T preservesthecycliclysymmetricsector,soifitwerepossi bleto projectoutthenoncycliclysymmetriccontributionsbyac hoiceof j i and j i wewould obtainanonpollutedtransitionamplitude.Theproblemis thatonthelatticeonly M ofthe 2 M statesateachtimesliceareavailable,namelythe"pure"s tateswitheachspin eitherupordown.Outofthesethereareonlytwopurelycycli clysymmetricstates, theallspinsdownandtheallspinsup,andtheallspins downisforbiddenasdescribed 93
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above.Theallspinsupstatewouldbeabadchoicesinceith asaverysmalloverlapwith thecycliclysymmetricgroundstate,inotherwords,itisve ryfarawayfrombeingthe congurationpreferredbytheprobabilitydistribution e S Soprojectingouttheunwantedstatesisnotaviablesolutio n.Insteadwetry toconsidercycliclysymmetricobservables,forexampleth esumofthespinsata timeslice.Althoughsuchanamplitudewouldstillgetcont ributionsfromtheintermediate noncycliclysymmetricstates,thesecanbesuppressed.Co nsiderforexamplecyclicly symmetricoperators O 1 and O 2 attimes j 1 and j 2 ,inthepresenceofastate j i .The quantityinquestionis Q ( j 1 ;j 2 )= h jT j 1 O 1 T j 2 O 2 j i Tr T j 1 + j 2 (419) Asbeforewewrite j i = P n a n j n i butnowweassumethatevenandodd n labelthe cycliclysymmetricandasymmetricsectorsrespectively.T he t n sarenoworderedin decreasingsizewithineachsectorseparately.Wewrite O i mn = h m jO i j n i forall n;m and i =1 ; 2 andweconsiderthecasewhere E 1
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compromisethisargumentandrenderthemethodunusableint hatcase.Fortunatelythis casecorrespondstosmallcouplingwhereperturbationtheo ryisabettermethodanyway. Theaboveargumentsshowhowwecanobtaintheenergygapsoft heeldtheory fromthecorrelationsofspinsontheLightconeWorldSheet. Evenifcontributionsfrom noncyclicallysymmetricstatesmayactasnoiseinourdata thereishopethatwhenthe couplingislarge,theenergygapscanbereadofromthe exp behaviorofthecorrelation asafunctionoftime( j 2b 1 ;N c ).InFigure( 46 )anexampleofthecorrelationobtained fromthesystemsimulation.Thegureshowsanumberofinter estingandrepresentative factsaboutthemethodsemployedinthiswork.Itshowsthet (solidline)plotted togetherwiththedatapoints(witherrorbars).Thetisnon linearandobtainedusing specializedmethodswhichwedevelopedforthispurpose,in ordertocapturethespecics ofequation( 420 ).Thepointsonthegrapharenotreallythecorrelationbetw eenspins, butrather,thevalueof R ( j )= h s ji 0 s j 0 i 0 i ,with ( i 0 ;j 0 )=(1 ; 500) axedupspin.The correlationisequalto R h s ji 0 ih s j 0 i 0 i ,which,since ( i 0 ;j 0 ) isxed,isjustequalto R less theaverageofspinsonthelattice.Thisaveragehasnolatti cedependence(no ( i;j ) dependance),whichiswhyweworkwith R ratherthanthecorrelationitself.Fromthe graphwecanseethateventhoughcorrelationsarelongerran gedintheweakcoupling (thatis,longlinespredominant)theoverallaverageofspi nsismuchlower,thereismuch lesshappeninginthelowcouplingregime,whichiswhatweco uldseequalitativelyfrom theworldsheetgures( 45 ). Beforedatatssuchastheoneabovecouldbetried,wehadtor elaxthesystemas explainedearlier.Whenwerelaxthesystemweareessentiall yndingastateinwhich tostartthesimulationoin,i.e.,astatewhichisrepresen tativeforthestatesnearthe actionminima.Ofcourse,wecannoteverrigorouslyproveth atweareinatrulyrelaxed stateofthesystem,butinpracticeweachieverelaxationby observingsystemvariables, preferablytruephysicalobservablessuchasthecorrelati onsorthetotalmagnetization (theaveragevalueofthespins)ofthesystem.Weobservethe sevariablesstartingofrom 95
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Figure46.FittingMonteCarlodatatoexp.Thedatafromth eMonteCarlosimulations isttedtoanexponentialinordertoreadothemassgap.Ea chpointonthe graphgivesthecorrelationbetweenaxedupspinat ( i;j )=(1 ; 500) anda spinatvariouslocationonthespaceslice i =1 .Thesimulatedsystemhere had M =40and N =1000andweusedabout 10 6 sweeps.TheThetopplotis fromalowcouplingMonteCarlorun( g= 2 =0 : 9 ),whereasthelowerplot isfromstrongcoupling( g= 2 =0 : 4 ).Noticehowmuchweakertheoverall correlationisforlowcoupling,i.e.,thesignalstrengthi nthe exp fallois weak. variousstartingstatessuchasthe"almostempty"state(al lspinsdownexceptforalong lineofupspinsthroughtheentirelattice),orsomerandoms pinstate.Wesawhowthese observables,althoughstartingatsomevaluesconvergedto thesame"equilibrium"value irrespectiveofthestartingpoint.Whenthisvaluehadbeenr eachedwithinstatistical accuracy,weclaimedthesystemtoberelaxed.Asystematica nddetailedanalysiswas donebyobservingmagnetizationandcorrelationsandwefou ndthatabout 10 5 or 10 6 sweepswasusuallymorethanenough,evenifstartingfromth e"almostempty"state,a statewhichwebelievehadlittleoverlapwiththe"ground"s tate,or"equilibrium"state. Theequilibriumstateofcoursedependsonthecouplingsowe tookupthestandard ofrunningatleast 10 5 sweepsbeforedatawassampled,evenwhenwestartedfrom 96
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thegroundstateofthesimulationinthepreviousrun.Since ,inpractice,weoftenran simulationswithsimilarcouplingrightaftertheother,so thisshouldbeaverysafeend accuratestandard.4.2.3Comparisonofsmall M results:ExactNumericalvs.MonteCarlo Letusrstlookatthesimplestofcases,namelythe M =2 case.Thiscanbesolved exactlywitharbitrary N .Ateverytimeslicewehavea3statesystemcorrespondingt o j#"i j"#i and j""i .Recallthatthe j##i stateisnotallowedonaperiodiclattice.Inthis orderedbasisthetransfermatrixis T = 0BBBB@ e 2 = 4 T 0 0 g p 2 T 0 e 5 2 = 8 T 0 0 e 2 = 4 T 0 g p 2 T 0 e 5 2 = 8 T 0 g p 2 T 0 e 5 2 = 8 T 0 g p 2 T 0 e 5 2 = 8 T 0 e 2 =T 0 1CCCCA (421) ascanbereadofromtherules 43 ).Theodiagonalelementshavebeensymmetrized tomake T 2 hermitian.Itisillustrativetousethebasis: ji = 1 p 2 ( j#"ij"#i ) (422) j + i = 1 p 2 ( j#"i + j"#i ) (423) j 2 i = j""i (424) wherethetransfermatrixbecomes T = 0BBBB@ e 2 = 4 T 0 00 0 e 2 = 4 T 0 g T 0 e 5 2 = 8 T 0 0 g T 0 e 5 2 = 8 T 0 e 2 =T 0 1CCCCA (425) 97
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Andweseehowthenoncycliclysymmetricstatedecouples.L etusdenoteby 1 2 and 3 theeigenvaluesof T indecreasingsizeorder.Wehave 1 = 1 2 e 2 = 4 T 0 + e 2 =T 0 + s ( e 2 = 4 T 0 e 2 =T 0 ) 2 + 4 g 2 T 2 0 e 5 2 = 4 T 0 (426) 2 = e 2 = 4 T 0 (427) 3 = 1 2 e 2 = 4 T 0 + e 2 =T 0 s ( e 2 = 4 T 0 e 2 =T 0 ) 2 + 4 g 2 T 2 0 e 5 2 = 4 T 0 (428) Theenergylevelsaregivenby ln k andweseethat 2 istheonecorrespondingtothe noncycliclysymmetricsector.Fromthesimulationwewill extractenergygapsandwe havetheunphysicalone G n = ln 2 1 andthephysicalone G s = ln 3 1 .Unfortunately thesmallergapistheuninterestingoneandmoreover,theun physicalgapgoestozeroin thesmallcouplinglimitbecause 1 and 2 aredegenerate.Weshallseehowthegeneral argumentgivenaboveaboutthesuppressionofsuchunphysic alrelicsworksinthissimple case. E 0 E 1 E 2 E 3 Figure47.EnergylevelsofatypicalQFT.Aschematicdiagr amshowingtheenergy levelsofatypicalsystem.Thelevelsarelabelled E n for n =0 ; 1 ; 2 ;::: ordered inascendingsizewithineachsector,evenandodd n labellingthecyclicly symmetricandasymmetricsectorsrespectively.The E 0 energylevelmust alwaysbethelowestlying.Weconsiderherethepotentially troublesome situationwhere E 1
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P n 0 h n 0 jT N j n 0 i thentheaveragevalueofaspinatasite ( i;j ) isgivenby h P j i i = 1 Z X n 0 X m 0 h n 0 jT N j 2 j m 0 ih m 0 jT j 2 j n 0 i (429) Thiscaneasilybeevaluatedbysimplydiagonalizing( 425 ).Weextractthe j dependence h P j 1 i = a 1 r N + a 2 s N + a 3 + a 4 ( r j + r N j )+ a 5 ( s j + s N j )+ a 6 ( s j r N j + r j s N j ) p a 1 r N + p a 2 s N + p a 3 (430) h P j 2 i = a 1 r N + a 2 s N + a 3 + a 4 ( r j + r N j ) a 5 ( s j + s N j ) a 6 ( s j r N j + r j s N j ) p a 1 r N + p a 2 s N + p a 3 (431) where r = e G s and s = e G n aretheexponentialsofthe j dependenceandthe a k s areknownalthoughcomplicatedfunctionsof g T 0 and 2 T 0 .Clearlythereisaexponential dropowith j withseveralexponentscorrespondingtoeachenergygap.It isimpossible practicallytoextract G s fromstatisticaldatafor h P j i i because G n isbigger.Butifwelook atthecyclicallysymmetricobservable R j = P i P j i wehave h R j i =2 a 1 r N + a 2 s N + a 3 p a 1 r N + p a 2 s N + p a 3 +2 a 4 p a 1 r N + p a 2 s N + p a 3 ( r j + r N j ) : (432) Inthissimplecasenotonlyisthedependenceof G n suppressedin h R j i ,butactually dropsoutcompletely. 99
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a 1 = 3 y 2 g 2 +2 x 2 2 x p x 2 +2 y 2 g 2 2 4 2 y 2 g 2 + x 2 x p x 2 +2 y 2 g 2 2 (433) a 2 = 1 4 (434) a 3 = 3 y 2 g 2 +2 x 2 +2 x p x 2 +2 y 2 g 2 2 4 2 y 2 g 2 + x 2 + x p x 2 +2 y 2 g 2 2 (435) a 4 = 9 y 4 g 4 4 2 y 2 g 2 + x 2 x p x 2 +2 y 2 g 2 2 y 2 g 2 + x 2 + x p x 2 +2 y 2 g 2 (436) a 5 = y 2 g 2 2 (437) a 6 = y 2 g 2 4 2 y 2 g 2 + x 2 x p x 2 +2 y 2 g 2 (438) where x = e 2 =T 0 e 2 = 4 T 0 2 and y = e 5 2 = 8 T 0 p 2 T 0 WecannowcomparetheresultsoftheMonteCarlosimulationw iththeexactresults givenby( 430 )asfunctionsof j forgivenvaluesof g T 0 and 2 .Ingure( 48 )and( 49 ) weseetheresultsofasamplecalculationusingtheMonteCar losimulationfor M =2 and N =1000 inconjunctionwiththeexactresultsobtainedabove.Altho ughtheresultsare displayedwithoutpropererroranalysisthereadershouldb econvincedthatthesimulation isinqualitativeagreementwiththeexactresults.Theidea isthatinmorecomplicated caseswhereitisintractabletodotheexactcalculationone shouldbeabletotthe statisticalcurvefor h R j i withanexponentialandreadotheenergygap.Doingthisfor thissimplecasegives: Averysimilarprocedurecanbedonefor M =3 inwhichcasethereisaneightstate systemateachtimeslice,namely j"""i j#""i j"#"i j""#i j##"i j"##i j#"#i j###i The T 3 matrixcanalsobediagonalized(althoughthistimeitwasdo nenumerically)and exactresultscanbeobtainedandcomparedtotheMonteCarlo simulationasshownin Figure( 410 ).Clearlythesizeofthematrix T M increasesexponentiallywith M which 100
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Table45.MonteCarlovs.Exactresultsfor M =2 .Tableshowingtted G s froma MonteCarlorunwith M =2 vs.exactvaluesthatcanbecalculatedinthis specialcase. tted G s exact G s remark 0.2404 0.21333 Sweeps: 7 10 5 x =0 : 6 and T 0 =4 0.1602 0.13291 Sweeps: 7 10 5 x =0 : 6 and T 0 =6 0.3672 0.39814 Sweeps: 7 10 5 x =0 : 8 and T 0 =3 0.1602 0.16821 Sweeps: 1 : 5 10 6 x =0 : 6 and T 0 =6 makesthismethodforsolvingtheproblemintractableas M startstoreallygrow.In contrast,theMonteCarlosimulationsonlygetlinearlymo recomputationallyintensiveas M grows.Weshalldiscussthisfactmoreinthenextsection. 4.3ARealTestof2DTr 3 TheprevioustestshaveshownthattheMonteCarlosimulatio ncapturesatleast someofthefeaturesofthemodelitissetouttosimulate.How ever,thecomparisonso farhasonlybeenbetweentwoviewsoftheLightconeWorldShe etpicture.Asalasttest wesearchtheliteratureforknownresultsfor2DTr 3 matrixquantumeldtheoryto compare.Suchatestnotonlyconrmsthatthecomputersimul ationworks,butthatthe LightconeWorldSheetpicturedescribestheeldtheorytha tisshould. In1992DalleyandKlebanov[ 22 ]studiedatwodimensional,Lightconequantized, large N c matrixeldtheory.Theyshowedthatinthetheorypossessed closedstring excitationswhichbecamefreeinthe N c + 1 limit.AsisdoneintheLightconeWorld Sheetpicture,theydiscretizethelongitudinalmomentuma ndobtainalinearSchrdinger typeofequationforthestringspectrumwhichtheytacklenu mericallybydiagonalizing thematrixrepresentingtheHamiltonian.Thediscretizati onmakesthematrixformulation possibleandthematrixisofnitesizeinthepresenceofthe cutowhichisimposedby consideringastringstatewithagivenlongitudinalmoment um.Theirworkhasanentirely dierentagendathantheworkpresentedhere;theyareinter estedinstringtheoretical implicationsofthestringspectrum.AsisdoneintheLightc oneWorldSheetpicturethe cutoisremovedbytakingthemomentumresolutiontoinnit y,holdingthemomentum 101
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0 100 200 300 400 500 600 700 800 900 1000 0 0.2 0.4 0.6 0.8 1 1.2 0 100 200 300 400 500 600 700 800 900 1000 0.02 0.01 0 0.01 0.02 j h Pji j Figure48.MonteCarlovs.Exactresultsfor M =2 .Theuppergureshowstwocurves, theupperonecorrespondingto h P j 1 i andtheloweroneto h P j 2 i .Thecircles areresultsfromtheMonteCarlosimulationbutthesolidlin esarefrom ( 430 ).Thelowergurealsohastwocurvescorrespondingto h P j 1 i and h P j 2 i butshowsthedierencebetweentheexactresultsandthesim ulation.Clearly theerrorsarequitecorrelatedin j .Thedataisobtainedwith g =0 : 3 =1 and T 0 =6 .Thesimulationhadatotalof1.5millionsweeps,discardin gthe initial150ksweepsforrelaxation.Thexedspinwasat ( i;j )=(1 ; 500) ofthestringstatexed.Ifthespectrumbecomesdensetheys uggestthatasthecuto isremovedthespectrumwouldbecomecontinuousimplyinga third dimensionofthe stringtheory.Theseresultsassucharenotofgreatinteres ttousintermsofthecurrent work.Nonetheless,intestingtheLightconeWorldSheetfor malismagainsttheirwork,we address,inpassing,anumberofissueswhichareputforthin thearticle. 102
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0 100 200 300 400 500 600 700 800 900 1000 1.1 1.12 1.14 1.16 1.18 1.2 1.22 j h RjiFigure49.MonteCarlovs.Exactresultsfor M =2 ,cycliclysymmetricobservable.This gureshows h R j i forthesamecaseasgure( 48 ).Astherethesolidline correspondstotheexactsolutionandthecirclestothesimu lation.Clearlythe gapismuchsmallerthanthatwhichdominatesboth h P j 1 i and h P j 2 i Tounderstandthecomparisonitishelpfultobrieyrecapth eresultsofthatpaper. Forbrevityinthissection,werefertoresultsandconventi onsofthatpaperassimply D&K.Startingwithanaction: S DK = Z d 2 x Tr ( 1 2 ( @ ) 2 + 1 2 DK 2 DK 3 p N c 3 ) (439) 103
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0 100 200 300 400 500 600 700 800 900 1000 0.2 0.4 0.6 0.8 1 0 100 200 300 400 500 600 700 800 900 1000 0.02 0.01 0 0.01 0.02 0.03 j h Pji j Figure410.MonteCarlovs.Exactresultsfor M =3 .Theuppergureshowsthree curves,theupperonecorrespondingto h P j 1 i andtheloweronesto h P j 2 i and h P j 3 i whichbysymmetryshouldbeexactlythesameastheyareinthe exact solution.ThecirclesareresultsfromtheMonteCarlosimul ationbutthe solidlinesarefromthenumericaldiagonalizationof T 3 .Thelowergure alsohasthreecurvescorrespondingto h P j 1 i and h P j 2 i and h P j 3 i butshows thedierence j ,betweentheexactresultsandthesimulation.Clearlythe errorsarequitecorrelatedin j asforthe M =2 case.Thedataisobtained with g =0 : 4 =1 and T 0 =4 .Thesimulationhadatotalof700,000sweeps usingthelast90%fordatagathering.Thexedspinwasat ( i;j )=(1 ; 500) with an N c byN c hermitianmatrixeldintwodimensionsasinEq.( 21 ),theauthors workouttheLightconetimeevolutionoperator: P ( x + )= Z dx Tr ( 1 2 DK 2 DK 3 p N c 3 ) (440) 104
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Theeld isthengivenaLightconequantizationoneachsliceof x + and x ismade compactleadingtothediscretizationoftheconjugate k + = nP + =K .Bywritingoutthe (nowdiscrete)oscillatormodeexpansionfor theauthorsarriveatthefollowingnormal orderedexpressionfortheLightconeHamiltonian: P = DK K 2 P + ( V xT ) (441) where V and T aregivenasseriesinthe(discrete)creationandannihilat ionmatrix operators A ij ( n ) and A yij ( n ) ,and x = DK = 2 DK p .Outofallstatestheauthors concentrateontheglobal SU ( N ) singlets: 1 N B= 2 c p s Tr A y ( n 1 ) A y ( n B ) j 0 i : (442) Thestatesaredenedbyorderedpartitionsof K into B positiveintegers,modulo cyclicpermutations.Here s denotesthemultiplicityofthestatewithrespecttocyclic permutationsofthesequenceof n sintothemselves,and P Bk =1 n k = K isthetotalnumber ofmomentumunitsofthestates.Thesestatescorrespondtoo riented,closedstrings withtotalmomentum P + =2 K=L where L isradiusof x space.Foraxedvalue ofthecuto K thereareonlyanitenumberofstatesofthestring,andthea uthors wereabletowriteouttheactionoftheoperator P fromEq.( 441 )onthesestatesas a(nitesize)matrix,whichwassubsequentlydiagonalize d.Theauthorsarriveatthe energyspectrumandshowthatthereisanindicationthatthe separationbetweenlevels getsdenseras K isincreased.Theproblemwiththeirresultsistheverylimi tedsizeof K whichistractableusingtheirmethods.Thesizeofthematri xbeingdiagonalizedgrowsas exp K ,soevenifonewouldchooseanextremelyeectivediagonali zationtechnique,then theproblemiswhatisreferredtoincomputerscienceasnum ericallyintractable.Weshall seethatthemethoddevelopedheretoobtaintheenergygapis infactacomputationally tractableproblemattheexpenseofbeingstochastic.Event houghtheresultsarenotas 105
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accuratethentheMonteCarlomethodscomputationtimeonl yincreasespolynomiallyas afunctionofthesizeoftheproblem K ComparingtheresultsofD&KtotheMonteCarlosimulationi sespeciallyeasy,since theirsetupissomuchliketheLightconeWorldSheetformula tionofthesametheory. Theactionisthesame,sothemassandcouplingcanbedirectl yrelatedandfurthermore, themomentum P + isevendiscretizedinthesameway,sothattheD&K K isprecisely our M .However,D&Kworkincontinuous"time"sothatwemusttake N verylarge comparedto M tobeinthesameregionofparameterspaceforthecomparison (recall thattheworldsheetintheMonteCarlosimulationhas N byM sites).Anotherissueof greatimportancetothecomparisonisD&K'sconcentrationo nthestringystates,i.e., thosewhicharesingletsundertheglobal SU ( N c ) ofthematrixindicesandunderthe cyclicpermutationsmentionedabove.Forthecomparisonto bemeaningfulwemust alsobeworkingwiththeverysamestates,whichbyconstruct ionweare.Firstly,the singlestringstateisautomaticbecauseofourchoiceofext ernallegs,i.e.,thesimulation startswith,andconserves,asinglesheetcominginandasin glesheetgoingout.And secondly,becausewetakeperiodicboundaryconditionsfor thespinsinthe p + direction, i.e.,wrappingthesheetupintoaroll.Thisensuresthatwea reworkingwiththesame verystatesasintheD&Kwork.Noticehowever,thatwedidnot dothisintheexact treatmentofthe M =2 and M =3 casesabove.Inthosecasesweworkedwithallstates, cycliclysymmetricornot,whichaccountsforthespecialtr eatmentthere. Intheirwork,D&Kplottheenergylevelsfoundfromdiagonal izingEq.( 441 )as wellasthesplittingofthersttwoenergylevelsasafuncti onoftheparameter x inthat equation.Theydothisfor K =8 ; 10 and 12 .TheirplotisreproducedinFigure( 411 ) togetherwithourMonteCarloresultsplottedonthesamegr aph.Theresultsarein relativelygoodagreement,especiallywhentakingaccount ofthestatisticalaccuracyofthe stochasticsimulation. 106
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TheMonteCarloresultsareobtainedinasimilarwayasbefo re.Thespincorrelations areusedasaproxytogettotheenergygapofthesystembecaus eitcanbecompared toD&K.Ashasbeendiscussedearlierhowever,itunfortunat elydoesnotgiveextremely accurateanswers,andreliesheavilyupondatattingandha saratherweaksignalto noiseratio.ThetypicalttingwasexempliedinFigure 46 andtheprocedurewas employedheretoobtaintheenergygapforanumberofruns. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 4 6 8 10 12 14 16 18 MonteCarlo D&K Figure411. M =12D&KandMCcomparison.Comparisonbetweentheenergyspl itsof theD&KpaperandthatofLightconeWorldSheetMonteCarlos imulation. Bothcasesshownhave M =12(equivalently K =12inthenotationofD&K), theMonteCarlosimulationisdonewith N =1000whereastheD&Kresults aredonewithcontinuous"time".Inthiscontexttheresults for M =10and M =8arestatisticallyindistinguishableintheMonteCarlo simulation,which iswhywedonotbotherwiththecomparisonforthosevaluesof M Asanalnoteonthecomparisonofresultspresentedhere,we shouldpointout thateventhoughtheMonteCarlomethodallowsforestimatio nsoftheenergygapfor muchbroaderrangeof M sthenthereisnowayofobtainingtheactualenergylevelso f thetheory.Moreover,onlythegapbetweenthegroundstatea ndtherstexcitedstate hasbeenobtainedwiththeMonteCarlomethod,andalthoughh ighergapscertainly leaveasignatureintheMonteCarlodata,theirquantitativ esizeareverydicultto obtain,atleastusingthespincorrelationsasisdonehere. TheMonteCarlomethods asimplementedhere,arenotappropriateforlowcoupling,b ecauseinthisregimethe 107
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Figure412.Dataanalysisanddatattingfor M =12 simulations.Weshowherethe ttingprocedurefortheD&Kcomparisonwith M =12 andweakcoupling. Atlowcouplingthesignaltonoiseratioisunfavorable.The tisstillrather goodandisachievedbyttingtotwoexponentials.Bydoings oweareare takingintoaccountpossiblecontributionsfromthenextga pwhichwould comeoutastheexponentofthesecondexponentialwhichiswh athappened. Thismakesthereadofortherstgapbetter( E inthegureisscaledto agapof2.9693whichisconsistentwiththelowcouplinglimi tof3).The dierenceplotshowsinterestingoscillationindicatings omeremnantbehavior inthecorrelationswhichwehavenotstudiedsofar. acceptancerate,orequivalentlythesamplingrate,islowa ndapproacheszero.This howevershouldnotbeareasontoabandonMonteCarloapproac hessincethisregimeis preciselytheonewhereperturbativeresultsareaccurate. 108
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0 2 4 6 8 10 12 14 16 1 1.5 2 2.5 3 3.5 4 E1 =a Figure413. E asafunctionof 1 =a .MonteCarloresultsfor M =40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 Ex Figure414. E asafunctionof x .MonteCarloresultsfor M =40 109
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Figure415. E asafunctionof M .MonteCarloresultsforvariousvaluesof M .The readershouldbearinmindthattheperturbativelowcouplin gresultis E =3 .ThestarsareMonteCarloresultswhereasthecrossesareo btained fromtheD&Kpaper. 110
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APPENDIXA COMPUTERSIMULATION:DESIGN InthischapterwediscussthesoftwareandcodeusedintheMo nteCarlosimulation. Wellknowndeterministicnumericalmethodsarenotdiscuss edinanydetailassuch studiesaresystematicallydescribedinregulartextbooks onthesubject[ 23 ].Similar systematictreatmentoftheuseofstochasticmethodsinphy sicscanbefoundin textbooksaswell[ 19 20 ]butsinceourimplementationisnovelwedodiscussitin quitesomedetail. A.1ObjectOrientedApproach,SoftwareDesign A.1.1BasicIdeas OurcomputerrepresentationofaLightconeWorldSheetmode lwillinvolvethe lattice,labelledby i =1 ; 2 ;:::;M and j =1 ; 2 ;:::;N andtheeldslivingonit.One eldwillbepresentinanyLightconeWorldSheetmodel,name lythe"spins"whose statedeterminesthetopologyoftheunderlyingFeynmandia gram.Thisisthebasic buildingblockofthecomputermodel.Thesimulationmethod usedhere,basedonthe Metropolisalgorithmasexplainedinsection 4.1 istotraversethelatticeinsomeway visitingsitessequentially.Ateachsiteoneoerslocali psofspinsandchangesinthe eldcongurationwithaprobabilityobtainedfromthephys icalaction.Thecalculationof thisprobabilityisrathercomputationallyintensiveasaf unctionofthevaluesofeldsand spinsinthevicinityofthesiteinquestion.Onefulltraver seofthelattice,whenallsites havebeenvisitedonce,iscalledasweep .Thestateofthesystembetweensweeps,that Sometimesasweepmeansthatasmanyvisitshavebeenmadeast herearelattice sites.Ifsitesarevisitedinarandomorderthiscouldmeant hatasweepisdonewhile somelatticesiteshaveneverbeenvisitedandothershavebe envisitedmorethanonce, 111
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isthevaluesofalleldsandspinsonthelattice,isusedtoc alculateaphysicalproperty ofthesysteminthatstate(thecorrelationbetweenspinval uesatdierentlatticesitesis atypicalpropertytoinvestigate).Werefertothisastakin gasampleofthesystem,or simply,samplingthesystem.Aftersamplingweproceedtoco mpleteanothersweep,the objectivebeingtocollectalargenumberofsampleswhichwe usetoobtainstatistically relevantinformationaboutthesystem. Eventhoughchangesinthephysicalactiondeterminetheevo lutionofthesystemas weprogressivelyperformmoreandmoresweeps,itispossibl ethatwestartoutfroma veryunphysicalstate.Thismeansthatitmaytakemanysweep s,i.e.,alongMonteCarlo time,forthesystemtobeinaphysicallyfavoredstate.Wedo notwantthearbitrary (andquitepossibly"highlystrung")initialstateofthes ystemtoaectthestatistical integrityofthesampledistributionweusetocalculatephy sicalpropertiesofthesystem. Topreventthis,itiscommonpracticeto"relax"thesystemb eforesamplesaretaken. Thisisdonebyperformingsweepsuntilthestateofthesyste misinaregionofstate spacewhichisfavoredbythephysicalaction. Havingdescribedinbroadtermsthebasicideasitisinorder toexplaininmore detail,thebeforementionedMetropolisalgorithm:Itisin principleaverysimplemethod (althoughdetailsquicklybecomesomewhatinvolved)andwe cansummarizeitasfollows: 1.Initialization.Constructthesystemmodelobject S whichmanagesandstoresthe completestateofthesystem,i.e.,valuesofalleldsandsp ins. 2.Relaxation.Performsweepson S untilrelaxationcriteriaismet.Ineachsweep: 1.Atagivensitesuggestachangeineldconguration,andc alculatetheacceptance probability. 2.Acceptordeclinethechangebycomparingarandomnumbert otheprobability. 3.Gotoanothersiteandrepeatthesestepsuntilsweepisdon e. butinthecurrentworkwevisitsitesinadeterministicorde r,allowingeachsitetobe visitedexactlyonceinasweep 112
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3.Datasampling.Continuetoperformsweepson S andcalculatethephysicalproperty ofinterestforthestateatregularintervals.Thesampling dataisaccumulatedandstored. ThisverysimplestructureoftheMonteCarlosimulationall owsforarathergeneral implementationofthesimulationprogram,morespecicall y,dierentLightconeWorld Sheetmodelscanbehandledwiththesamebasiccomputercode .Infact,byusingobject orientedprogrammingtechniques,itispossibletoallowmo delspecicissues,suchas howtheactioniscalculated,whicheldsliveontheworldsh eetetc.toresideinthe specicationsofthemodelobject S .Thisway,aninstanceofasystemmodelobject S of aparticularclass(forexample2DTr 3 )isconstructed,andpassedintothesimulation software,whichitselfisnotconcernedwithwhatkindofmod el S is.Inmodernsoftware designobjectorientedprogramminghasbecomeacompletely dominatingstandard.Itis inordertobrieydescribehowthismethodologyisusedinou rcontext. A.1.2ObjectOrientedProgramming:GeneralConceptsandNomenclature Todescribetherelevanceofobjectorientedcomputerprogr ammingtothepresent computersimulationwetakeanexamplefrommathematics.Co nsidertherealnumbers, R asasubsetofthecomplexnumbers, C .Letustakearealnumber x 2 R andacomplex number z =( a + ib ) 2 C andconsiderthetwooperations: abs r : x 7!k x k = 8><>: x; if x 0 x; if x< 0 (A1) abs c : z 7!k z k = p a 2 + b 2 (A2) Itsimportanttoemphasizethatitisofinteresttothecompu terprogrammertomakea distinctionbetweenthesenumbers(oringeneral,betweend atatypes)sincestoringsay, anintegerismuchmorememoryecientthanstoringitasaspe ciccaseofacomplex 113
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number,whichcertainlyisapossibility.Tostoreacomplex number,acomputersimply storestworealnumbers y Letusnowconsidertheoperationsabovefromacomputerspoi ntofview.Onewould liketobeabletousetheabsolutevaluefunctionwithoutha vingtoknow apriori howthe argumentofthefunctionisstoredinthecomputersmemory.T hecomputershoulditself determinebywhatsortofinputissuppliedwhichmethodtous e.Thegeneralizedfunction abscallsforanexecutionofabs r orabs c dependingontheinput.Moreover,onewould perhapsliketobeabletosupplyastructurefor hermitiannumbers andallowtheuser ofthatstructuretousethefunctionabswithahermitianinp ut.Inthesameway,inour presentwork,westructuretheMonteCarlosimulationinsuc hawaythatitisperformed inthesamemannerirrespectiveofthedetailednatureofthe underlyingmodel.Aslongas anewmodelbuildersupplies some sortofimplementationofthenecessaryfunctions(in theaboveexample,animplementationoftheabsfunction)t henaMonteCarlosimulation ofitispossibleusingthesoftwaredevelopedhere.Inorder tomakethecomputercode suppliedusabletoothers,inparticular,newmodelbuilder s,weexplainindetailthe mechanicsofhowthisworksinC++inparticularandinobject orientedprogramming languagesingeneral. Althoughtheaboveexampleisverysimpleandhaslongagobee nsolvedby computerprogrammersitwillserveillustrativelyhere.In theobjectorientedapproach, ageneral class calledsimply Number willbedened.Fortheclass Number special functions(called methods orsometimes memberfunctions )suchasaddition,subtraction, multiplication,divisionand x 7!k x k arenamed.Theclass Number iswhatiscalled aparent.Sometimesanotherclassisconsidereda childclass orsaidto inherit fromthe baseclass Number .Childrenmusthaveimplementationsofallthememberfunct ionsof y Namelyitsrealandimaginaryparts ( a;b ) ,orequivalently,itslengthandargument angle ( r; )=( p a 2 + b 2 ; arctan( y=x )) 114
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theirparents.Examplesofchildrenof Number areofcourseclassessuchas Integers Reals or Complex .Sometimesonereferstoachildofsay Number asalsobeingof class Number .Thechildrenof Number willknowhowtobeaddedandsubtracted etc.Itisthereforesmartsometimestodeneanabstractpar entclass,suchas Number above,whichdoesnothaveameaningfulimplementationofsa yaddition.Instead, themerepresenceofthememberfunctionintheparent,force sotherprogrammersto createanewimplementationforitiftheywanttocreateachi ldof Number .Ifwe lookalittlecloseratforexamplethechildclass Complex of Number weseethatit musthavetwo membervariables eachadoubleprecisionoatingpoint(oftenjustcalled adouble).Tooutsidefunctionsthesemembervariablesaren otvisible z .The Complex class'privateimplementationofforexamplethefunction x 7!k x k isgivenby z 7! p z.re*z.re + z.im*z.im (inC++thesyntaxtoaccessmembervariables x or y ofanobject A incodehandlingtheimplementationofitsmemberfunctions ,is A.x or A.y respectively.).Wemakeapointofthefactthatthesemember variablesarenot accessiblefromcodeotherthanthememberfunctions.There asonisthatitwouldallow externalprogrammerstotakespecialnoticeofhow,forexam plethe Complex class,is implemented.Thiswoulddrawfromtheindependenceofthecl ass,thecrucialpointbeing thatafteranobjecthasbeenconstructedwehaveseparatedi mplementationfromusage, inotherwords,weneednotknowexplicitly how multiplicationisdoneandinsteadjust usemultiplication,trustingtheclass'privatememberfun ctions x .Afteraclasshasbeen denedan instance ofitcanbeconstructed,inthecase x 2 C thenthevariable x is consideredaninstanceof(oranobjectof)class Complex z Sometimesitiswisetoallowtheextractionofmembervariab leswithfunctionssuch as im() and re() whichimplementtheextractionoftherealandimaginarypar tofa complexnumber. x infact,weshouldactuallynotbe allowed toknowthis,becauseitwouldtempt programmerstousetheknowledgewhenusingtheclass 115
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Tonishthenumbersystemanalogy;addinganewtypeoftheor ytotheLightcone WorldSheetMonteCarlosimulationcodehere,issimilarto deninganewalgebraic numbersystem.Firstonemustdecideuponthedataofthenewt heory,i.e.,theelds. Thesecorrespondtoconstructingtheactualnumbersofthes ystem.Nextonemustmake surethatalltheoperationsoftheMCsimulationwork,which isanalogoustodeninghow thealgebraicoperationsactonthenewnumbersystem.Theac tualMCsimulationisthen somethinglikecalculatingthevalueofananalyticalfunct ion f ( z )= P k a k z k ,something whichcaninprinciplebedoneforanythinginplaceof z aslongasmultiplication( z k ), multiplicationbycomplex'( a k z k )andaddition( a k z k + a k +1 z k +1 )aredened. Asanalnotewebrieyexplainhowtheimplementationofthe classesand inheritanceexplainedaboveisorganizedinsourcecodele s.Weusetheprogramming languageC++exclusivelyfortheMonteCarlosimulationsof tware(althoughfordata analysistheMATLAB r environmentwasused.).InbasicC++programmingthereare twobasicletypesforsourcecode,the header ledistinguishedbytheextension .h and the implementation ledistinguishedby .cpp .Thedierenceismainlysemantic,inthat headerlesareusedtodeclarefunctions,classesorvariab leswhereastheimplementation lesareusedfortheactualprogramimplementation.Itisne cessarytodeclareafunction forittoberecognizedbythecompiler { .Ifthefunctionisdeclaredinaleitcanbe usedandthecompilercreatesareferencewhichordersacall tothatfunctionname.The actualimplementationcanthereforeresideelsewhere,i.e .,inaseparatele.Eachclass oftheprogramwillhavean .h leassociatedwithit,andforthemostsimpleclassesor fortheabstractbaseclasses,thisissucient.Forthemore involvedclassesaseparate .cpp leexistswhichincludestheimplementationoftheverymet hodsdeclaredinthe .h { Acompilerisan engine (reallyjustaprogram)whichconvertstheprograms descriptiveC++sourcecodeintoexecutablebuthumanlyinc omprehensiblemachinecode. 116
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TableA1:Softwarelestructure.Thelenameconventions usedintheproject. Class headerle implementation LattConf mytools.h noneBosonic mytools.h noneSpins mytools.h noneSimpleChain simplechain.h simplechain.cpp TwoD1 twod1.h twod1.cpp le.Table A1 showshowthisisorganizedfortheprojectathandandweseet hatthe simplestclasseshaveasinglecommonheaderle mytools.h associatedwiththem. A.1.3OrganizationoftheSimulationCode Abovewehaveexplainedthegeneralideaofhowobjectorient edsoftwaredesign mightbeusefulforconstructingacomputersimulationofaL ightconeWorldSheet modelofaeldtheory.Itistheeasewithwhichthecomputerc odecanbe"recycled", maintainedandaugmentedwhichdrivesustothisapproach.B elowthespecicsofthe simulationdesignareoutlinedandforcompleteness,adesc riptionoftheactualcomputer codeispresentedinthenextsection.Themotivationistofa cilitatetheuseofthe currentlydevelopedsoftwareinfutureattemptsatsimulat ingLightconeWorldSheets, hopefullyapplyingthesetechniquestosheetswithrichers tructuresthanhavebeentested here. Itisimportanttounderstandthatthephilosophyunderlyin gthistypeofcomputer programisthateachbitofcodeisthoughtofandcreatedwith theintentionsinmind ofbeingusedforaspecicpurpose.Wearebuildingupatoolb oxwhichisthenusedby ashortandsimple"main"programwhichshouldbeunderstand ableandtransparent. Therstbuildingblocksaretheeldswhichliveonthelatti cewhichappearasclasses, tobeusedbyan"upperlayer"ofcomputercode,namelythemo dels.Themodelsmake availabletotheirrespectiveusers,functionswhichimple mentthespecicphysicalaction. Theactioncomesinasthechangeinprobabilityofgoingfrom onestatetothenext.The "toplayer"iscomputercodewhosepurposeistocalculates omephysicalobservablesand 117
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propertiesofthesystem,examplesarecorrelationortotal magnetization.Theblockscan becategorizedbytheirpurposeandlocationinthecodeasfo llows: 0.Userinterface Sincethesoftwareisessentiallystillonadevelopmentals tagethis levelstillhardlyexists.TheprogramisexecutedfromtheU NIXcommandline wherekeyparameterscanbepassedintotheprogram. 1.Thetoplayer Calculateaphysicalpropertybygoingthroughthebasicste psofthe MonteCarlosimulationasdescribedinsection A.1.1 2.Middlelayer TheimplementationofthevariousMonteCarlosimulationst eps.This iswhereonedetermineshowexactlytheacceptanceprobabil itiesarecalculated, whicheldsliveonthelatticeandsuch.Tosimulateadiere ntmodelcorresponds toreplacingthislayer. 3.Primitivelayer Createthevarioustoolsandeldclassesmakingupthelatti ce. Twonecessarydatastructureshavebeendevelopedandimple mentedforuseinthe programmingthatfollows.Thesedatastructuresarethebui ldingblocksofaLightcone WorldSheetandrepresenttheeldslivingonthebasiclatti ce.Thedatastructuresinherit fromanabstractbaseclasscalled LattConf (latticeconguration),andtheirrespective namesare Spins and Bosonic .Shouldothereldssuchassomeimplementationfor fermioniceldslaterbeaddedtothemodelthentheseshould alsoinheritfromthe LattConf baseclass,meaningessentiallythattheyareatypeoflatti ceconguration. Theonlyinformationwhichresidesineach LattConf objectisthesizeofthelattice, M and N .Anyfurtherspecicationoftheeldarecontainedinthesu bclassitself. Inthecaseof Bosonic ,aclasswhichdescribesabosoniceldonthelattice,aseto f M N doubleprecisionoatingpointnumbersarestoredandacces sibleasandarraywith indices i =1 ; 2 ;:::;M and j =1 ; 2 ;:::;N representingthelattice.Theclass Bosonic alsocontainsmethodstoaddandandequateobjectsofthecla ssandfurthermore,a methodtowriteout(orread)theentirecollectionofdouble stoale.Thisway,auserof Bosonic needsnottobotherhimselfabouthowthenumbersrepresenti ngtheeldsare 118
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U SER I NTERFACE L AYER 0 M ONTE C ARLO L AYER 1 M ODELS L AYER 2 B ASIC T OOLS L AYER 3 Calculation Correlation Relaxation Lattice BASEMODELSIMPLECHAINTWOD1 TWOD2 S PINS B OSONIC F ERMIONIC L ATT C ONF ToolBox UNIX command with input Output Files with results Shapes Key: Implemented class,instances of it can becreated Abstract parent class,unusable by itself Code piece withwell defined purposebut not implementedas a class FigureA1.Organizationofthecomputercode.Theguresho wsthecollectionofobjects makingupthesimulationsoftware.Eachpartofthecodecanb ecategorized intooneoffourlayersdependinguponitsrelationtotheres tofthecodeas wellasitspurposeinthesimulation.Dottedshapesareidea swhicharenot partofthefunctionalsoftwareyet. accessedorstored.The Bosonic hasonlybeenusedinthesimpleIsingmodeltestofthe simulationsoftware. Aclassofgreaterimportancetotheprojectis Spins whichisrequiredforany LightconeWorldSheetmodeltodescribethetopologyoftheu nderlyingFeynman diagram.The Spins classcontainsabooleanvariableforeachlatticesite,acc essed asinthecaseof Bosonic ,whereavalueof"true"("false")orequivalently1(0)in 119
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computerlanguage,meansup(down)spin.Therearealsosome veryusefulmethodsin theclasssuchas Mone and Mtwo ,whichyieldthedistancefromthecurrentspintoits nearestupspinintheleftandrightdirectionsrespectivel y.Thereisalsoamethodcalled magnetization whichasthenameimpliesgivesthetotalnumberofupspinson the lattice.Onecouldeasilyaddnewmethodstothe Spins classbutthisshouldonlybedone withoperationswhichbelongwiththisgeneralimplementat ionofspins.Thisbreaksthe programdownintounderstandableanduseableblocks.Model specicoperationsbelong thethemodelobjectswhichwewillnowexplain. Inthesamewayaseldsonalatticearesubtypesof LattConf ,modelsarebuilt inasimilarhierarchialstructure.Althoughthehierarchy hasyettostretchout,itserves asatemplateforfutureconstructionandmodelbuilding.E achmodelinheritsproperties, methodnamesandactualmethodsfromanabstractbaseclass named BaseModel Sofartherearetwosubclassesto BaseModel namely SimpleChain and TwoD1 Thelatterisanimplementationofthemodelwhichistheseco ndmainsubjectofthis thesis,whereastheformerisabitoftestcodeimplementing theMonteCarlosimulation ofasimpleonedimensionalIsingspinsystem(spinchain) asdescribedinsection 4.1.3 Theabstractclass BaseModel declarescertainuniversalmethodswhichallmodels withinthisframeworkmusthave.Thefunctions startsweep() prob() accept() and nextsite() areexamplesofsuchmethods.Thismeansthatanargumentoft ype BaseModel isexpectedasinputtotheMonteCarlosimulationprogram.T heprogram doesnotcarewhatkindofmodelitis,aslongasitisasubclas sof BaseModel .The simulationwillsimplyaskthemodeltoperformcertainacti onswhosenamesarecommon withall BaseModel s.Repeatingthethreecorestepsinperformingasweepfroms ection A.1.1 : 1.Calculatetheacceptanceprobability.Methodused: prob 2.Acceptordeclinethechangeusingarandomnumbergenerat or.Onemethodusedin thisstep: accept : 120
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3.Gotoanothersiteandtorepeatthesteps.Descriptivelyt hemethodusedisMethod nextsite Foramodeltobeanacceptablechildobjectof BaseModel itmustthushave implementationsofallmethodswhichtheparentclassdecla res.Somemethodsmay furthermorebeaddedtoachildclass,methodswhichwouldse tthemodelapartfromits siblings. A.2DescriptionoftheComputerFunctions Withtheaboveoverviewofthecomputercodeworkingswearein positionto schematicallyexplaineachfunctioningpartofthecode.We explaineachpartand itsfunctionalitywitha"topdown"approach,omittinghow everLayer0sinceitis notcompletetoasatisfactorylevelofaworkingsoftwarepr oduct.Tohaveafully userfriendlyinterfaceisnottobeexpectedofcomputerpro gramwhichisstillunder constructionasaresearchproject. WewillrstlookatLayer1.Itiscurrentlyimplementedwith asinglelewhichis compiledtogetherwiththelecreatingLayer0tomakeafunc tioningexecutableprogram. ThepurposeofthislayeristoperformtheactualMonteCarlo simulationbyuseofthe Metropolisalgorithm.Thismeansthathereiswherewegothr oughsweepsonamodel whichissuppliedassomeclassofa BaseModel .Torelaxamodel,i.e.,ndasuitably lowenergyinitialstate,andtosavethiscongurationtoa leforlateruse,wouldbea tasktobeimplementedinthislayer.Anothertaskwouldbeto actuallysamplestates forcalculating(ormeasuringonecouldsay)aphysicalprop ertysuchasthecorrelation betweenspinsontheworldsheet.Anothertaskwhichgaveuse fulinsightsduringthe developmentofthesoftwarewastocalculatetheinitialval ueoftheactionandtrackingit throughthechangesthatwereacceptedduringalargenumber ofsweeps.Thisvaluecould thenbecomparedtothevalueobtainedbycalculatingtheabs olutevalueoftheactionfor thesysteminthenalrelaxedstate. Thefollowingbasicstepsaretypicalofanimplementationo flayer1: 121
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Initialization Openlestowriteoutput,initializea BaseModel objectofaclass TwoD1 ,import(orcreate)arelaxedlatticeconguration. Sweeps Gothroughsweepingthelattice,i.e.,usingaforloop,gof romlatticesite tolatticesiteupdatingtheeldvaluesbyuseofthestandar d BaseModel methods.Makingachoiceastowhatrandomnumbergenerating methodtouse whenacceptingaspiniporsimplemovingalongtothenexts ite. SampleData Oneasucientnumberofsweepshavebeendonetoensurestati stical integrityasexplainedinpreviouschapters,oneexaminest helatticeconguration andobtainsasampleofthephysicalvariable,inthiscaseth espincorrelation,for thesysteminthestatecurrentlyoccupiedbythemodelobjec t. ReturnResults Afterenoughsampleshavebeentakenbyrepeatedsweeps,one returns theresultstolayer0.Thiscanbedoneeitherbywritingthes ampleddatatoale orbycalculatingpropertiesfromthesamplesandreturning thesepropertiesdirectly. Thislayerdoesnothaveafullyobjectorienteddesign.Ther easonissimplyprocedural, itwasoutsidethescopeofthecurrentworktofullycomplete thedesignofthislayeruntil theresultspresentedherehadveriedtheoverallvirtueof theapproach.Thishasnow beendoneanditwouldbeaworthytasktocompletethedesigno faclasshierarchyfor calculatingphysicalobservablesthroughtheMetropolis( orsimilar)algorithm.Anal designwouldimplementeachlayerwithanclassstructureco ntaininglogicalinheritance. Themodellayerconsistsofthesubclassof BaseModel whichimplementsthe modelunderinvestigation.Therstlayerexpectsa BaseModel input,andthemodel investigatedhereisthe TwoD1 subclassof BaseModel .Sincethisisbyfarthemost centralclassofthewholeprojectwepresenttheheaderlea ssociatedwithitinappendix B forclaritytothereader.Inthatlethereisalistwherethe classmethodsaredeclared (butnotimplemented).Someofthemethodsareperhapsselfe xplanatoryornotreally necessaryforrunningthesimulation(methodsmadeforanal ysisorotherspecialpurposes 122
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Relaxation Lattice w. fixed spin Correlation Calculate the Action value Autocorrelationvarious properties to calculate determine which data to sample MONTE CARLO LAYER 1 Initialization Sweeps Data Sampling Return Results FigureA2.OperationsoftheMonteCarlolayer.Theguresh owshowthecodeinlayer 1isorganized.Thereareanumberofdierentthingsthathav ecanbedone hereandtheyallamounttodeterminingwhichdatatosamplew hilelooping oversweeps. suchaserrorcheckingetc.).Theothermoreconceptuallyim portantmethodswillnowbe brieyexplained: Tounderstandthemethodsexplainedhereitisimportanttoh aveaclearpicture inmindofwhatisgoingon.Whenaninstanceoftheclass TwoD1 hasbeencreatedit containsonlyonesinglecopyofthespinsonthelattice.The instanceisthereforeina way the system,atagiveninstanceinMonteCarlotime.Onceithasbe enchangedthe informationaboutthepreviousstateof the systemislostandunlessithasbeensaved cannotberecovered.startsweep() Thismethoddoesalmostnothing.Itdoesnotreturnanyresul ts,shownby thekeyword void beforethefunctiondeclaration.Thefunctionsetsthe current site ofthelatticeto ( i;j )=(1 ; 1) andsetsthestatevariable sweepdone equalto FALSE Afterthisfunctionhasbeencalled,thestateofthesystemi sthusthefollowing:the spinsareinwhateverstatetheywerebefore,thecurrentsit eissetto (1 ; 1) andthe 123
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sweepofthesystemisnotnished,representedbythevariab le sweepdone beingset to FALSE .Thisstateofthesystemrepresentsthatanewsweephasbegu n. init(),init( int ) Thisfunctioniscalledattheverybeginning,beforeanyswe eps havebeendone.Ithastwodierentimplementations.One,ta kesnoinput argumentsandsetsallspinsofthesystemequaltozeroexcep tthespinsat i =1 whicharesettoone.Theotherimplementationtakesaninput argument,an integer,whichrepresents which initializationshouldbeemployed.Sofar,onlythe antiferromagneticorderinghasbeenimplemented,whichs etsalternatingspinsto one,theotherstozero.Becauseoftheperiodicboundarycon ditionsinbothtime andspacedirections,thisisonlypossibleif M and N areamultipleoffour. accept Thisfunctionsiscalledwhenaspiniphasbeenaccepted.I treferstothetwo statevariablesofthesystem update_hereup and update_heredown andifeitheris trueitipsthespinsatthecurrentsiteandatthesitewhich isupordownfromthe currentsiterespectively.Otherwiseitipsthespinatthe currentsiteonly.After ipping,thefunctionsetsthestatevariables update_hereup and update_heredown to FALSE .Thereasonfortheslightcomplicationofdoubleupdatingi sthatwedo notallowforsolidlinesofasinglelatticetimeinlength.T hismeansthatwemust allowforlinestoemergeordisappearwithouteverbeingofl engthone. singleupdate(),updatehereup(),updateheredown() Thesearefunctionswhich performnorealactionbutonlysetthestatevariables update_hereup and update_heredown thethecorrectvaluesdependingonwhichofthethreefuncti onsis called.Thisisanexcellentexampleofhowtoprotectstatev ariablesofthesystem. Theactualvariables update_hereup and update_heredown themselvescannot beaccessedfromoutsidetheclass.Onlyfromwithintheseme thodsexplained here.Thisguaranteesforexamplethatthesystemisneverin astatewhereboth update_hereup and update_heredown are TRUE bymistakeorotherwise. 124
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nextsite() Thisfunctionchangesthe current siteofthesystemtoanewsite.The functioncanhavetwodierentimplementationsdependingo nthetypeofMetropolis algorithminuse.Weemploythesimplerversion,wherewesim plytraversethe latticesequentiallyfromlefttorightandthengoingupaft ercompletingeachline. Afterthefulllatticehasbeentraversed,thestatevariabl e sweepdone issetto TRUE andthe update_hereup and update_heredown variablesaresetto FALSE prob() Thisismoreorlesstheheartofthe TwoD1 modelclass.Thefunctioncalculates theprobabilityofaspinipatthecurrentsiteofthesystem .Ifthestateofthe systemcallsforadoublespinipthefunctioncalculatesth eprobabilityofthat. Sincethisfunctioniscalledveryoftenandliesatthecoreo fthesimulation,itis veryimportantthatitisecientintermsofcomputationals peed.Itisthereforeset upasalistof if...else... statementswhichonceanappropriatecondition ismettheprobabilityiscalculatedandthefunctiontermin ates.Themostcommon systemstatesarecheckedforrstandthelesscommononesla ter,whichmeans thatthe prob() functionneedsonlycalculatethosevaluesrequiredforjus tthe probabilityneeded.Ifaspinipisdisallowed(ifitgenera tesatimeslicewithall spindownsforexample)thenthisisimmediatelyrecognized andtheprobability returnediszero,withoutcomputation.Also,thecomputati onsactuallydone,are optimizedsothattheyrequireasfewcomplexevaluations(s uchas exp )aspossible. absaction() Thisfunctioncalculatesthevalueoftheactionforthesyst eminitscurrent state.Thefunction prob() certainlycalculatesthe change intheactionforeach successivespinip,butthisfunctioninsteadcalculatest hevalueoftheaction straightfromthestateasitis.Thisfunctionispresentonl yforpurposesoferror checking,i.e.,bycomparingthesequentialchangeintheac tionobtainedthrough prob() totheabsolutevalueoftheactionatsomenalstate(afterp erhapsmillions ofsweeps)wehaveaverygoodcheckonhowaccuratethe prob() functionworks. Theresultsfromthisfunctionhelpedusndmanyerrorsunti lwenallystarted 125
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obtainingasatisfyingcomparison.Thisfunctionalsoserv edtoidentifywhena sucientrelaxationofthesystemhadbeenachieved. AsanalnotewetouchbrieyontheusageofdatafromtheMCsi mulations. TheprogramexplainedinthisAppendixmustalwaysyieldinf ormationaboutthe eldcongurationstobeanalyzedbysomeothermeans(inthe currentworkweused MATLAB r asexplainedinsection 4.1.5 ).Thefunctions printdata printclass and readclass allinvolvetheinteractionofthecurrentclassdatawithth eoutside,could beanexternalleforexample.Thesemethodsinthemselvese mploymethodswritten foreacheldclass.Thestreamoperators operator and operator whichcanbeseen declaredinAppendix ?? mustbedenedforeacheldclass.Theseoperatorsareused throughouttheC++languageto,forexample,writeinformat iontothescreenortoles. Theyareoverloadedfortheclassesrepresentingeldcong urationstomakeiteasyand simpletowriteeldinformationtoles.Whenneweldclasse sarecreatedtheymustalso haveimplementationfortheseclassesandshouldbedenedt oprintoutorreadthevalues oftheeldinalonglineseparatedbyspacesandparameteriz edby k = M ( j 1)+ i Thisisthebasisforthedataleformatused.Wespeakof"Spi nsdatale"meaningale whereeachlinecorrespondstooneentirecongurationofa Spins eld.Alldatalesof eldformatwillhave .dat endings. 126
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APPENDIXB COMPUTERSIMULATION:EXAMPLEHEADERFILE Belowistherawsourcecodeforthedeclarationle twod1.h ,presentedhereasan exampleoftheC++sourcecodeusedthroughouttheproject.S incethemodel TwoD1 is themainmodelsimulatedinthisworkthedeclarationleass ociatedwiththisclassisat theheartofthesimulationsoftware. Thecodepresentedbelowcontains remarks ,whichistextprecededbytwofront slashed: // orastextbetweenthetwosymbols /* and */ .Remarksarenotpartof thecodeandarejustignoredbythecompiler,theyserveonly toillustratethepurpose ofmeaningofnearbycode.Also,commandsprecededby # aresocalledprecompiler commandswhichdenetheplacementofthecurrentleintheh ierarchyofleswhich comprisethetotalsoftwarepackage.Theclassdeclaration followsthefollowingsyntax: classNAME_OF_CLASS:publicNAME_OF_PARENT_CLASS} {ACTUALDECLARATIONSTATEMENTS} #ifndef_TWOD1_H#define_TWOD1_H#include# include "mytools.h"#include"basemodel.h"//2Dscalarphi^3theoryinLightconeWorldsheetpicture//solvednumericallywithMonteCarloMetropolis//algorithm.////TwoD1usesnoexplicitqand1/p+byhand//DeclarationFile:classTwoD1:publicBaseModel{private: Spinsspin;//spinfieldofthemodelinti;//currentspacepositiononlatticeintj;//currenttimepositiononlattice//numberofparametersinthemodel: 127
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enum{NumberOfParameters=3};//arraycontainingtheparameters:realparameters[NumberOfParameters];//parameters[0]isT_0=m/a(discretizationratio)//parameters[1]isg^2(coupling)equaltofieldtheorycou pover //4*sqrt(Pi),goodfor2D,shouldbemodifiedfor//higherdimensions//parameters[2]ismu^2(themass)asinfieldtheoryLagran gianterm //1/2mu^2Trphi^2 public: boolupdate_hereup;//trueifTwoD1::accept()shouldupda te //hereandup boolupdate_heredown;//trueifTwoD1::accept()shouldup date //hereanddown interrorcheck;//Constructors:TwoD1(intM,intN,realpara[]):BaseModel(1.0),spin(M,N ),i(1),... ...,j(1),update_hereup(false),update_heredown(false ) { errorcheck=0;for(intcount=0;count
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virtualvoidnextsite();virtualvoidsetsite(int,int);virtualboolgets(int,int);virtualvoidrepara(real[]);virtualvoidsingleupdate();virtualvoidupdatehereup();virtualvoidupdateheredown();virtualintwhere_i();virtualintwhere_j();intMtot();intMone(int,int);intMtwo(int,int);virtuallongmagnetization();virtualrealabsaction(); };#endif 129
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[15]C.B.Thorn, Notesononeloopcalculationsinlightconegauge, [arXiv:hepth/0507213]. [16]C.B.Thorn, AworldsheetdescriptionofplanarYangMillstheory, Nucl.Phys.B 637 ,272(2002)[Erratumibid.B 648 ,457(2003)][arXiv:hepth/0203167]. [17]C.B.Thorn, Asymptoticfreedomintheinnitemomentumframe, Phys.Rev.D 20 (1979)1934. [18]J.R.Norris, MarkovChains ,CambridgeSeriesinStatisticalandProbabilistic Mathematics,CambridgeUniversityPress,Cambridge,1997 [19]DavidP.LandauandKurtBinder, AGuidetoMonteCarloSimulationsinStatisticalPhysics ,CambridgeUniversityPress,Cambridge,2000. [20]M.E.J.NewmanandG.T.Barkema, MonteCarloMethodsinStatisticalPhysics OxfordUniversityPress,Oxford,1998. [21]LouisLyons, Statisticsfornuclearandparticlephysicists, CambridgeUniversity Press,NewYork,1986. [22]S.DalleyandI.R.Klebanov, Lightconequantizationofthec=2matrixmodel, Phys.Lett.B 298 ,79(1993)arXiv:hepth/9207065. [23]MichaelT.Heath, ScienticComputing,AnIntroductorySurvey, McGrawHill InternationalEditions,1997. 131
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BIOGRAPHICALSKETCH IwasborninReykjavik,Icelandonthe28 th ofAugust1975.BeforeIreachedthe ageof1yearsoldmyfamilymovedtoStockholm,SwedenwhereI grewupandspentmy childhoodyearsuntiltheageof11.Atthistime,myfamilymo vedbacktoReykjavik, IcelandpartlybecausetheywantedmetogrowuptobeIceland icandpartlybecausethat hadalwaysbeentheplan.IenrolledinM.R.a ContinuationSchool ,whichinIceland iscalled'Framhaldsskli'attheageof15andalthoughIhad alwaysbeenfascinatedby astrophysicsandthenatureoftheworld,itwasonlyatthiss choollevelthatmyinterest inrealmathematicswaskindled.Itwasthereforenosurpris ethatwhenthetimecamein 1995IstartedstudyingMathematicsandlateralsoPhysicsa ttheUniversityofIceland anddidsountil1999whenIgraduatedwithtwoB.S.degrees,o neinMathematicsand oneinPhysics. AftergraduatingfromtheUniversityofIcelandIhadalread ybeguntheprocessof applyingtograduateschoolsintheUnitedStates.BecauseI graduatedatinthewinter howeverIworkedinanengineeringrminthespringandsumme randrelocatedto Gainesville,Floridainthefallof1999.Shortlyafterward smywifeandchildrenjoined meinGainesville.Ihadmetmywifein1995andshehadhersonD anielfromaprevious relationshipandhebecameaveryclosestepsontomeandmyda ughterLisawasborn inthesummerof1999.Thefallof1999markedthebeginningof mygraduatestudies attheUniversityofFlorida.Myinterestslayintheoretica lhighenergyphysicsandI wasfascinatedbytheworkofDr.CharlesB.Thornandsoonsta rtedworkingunder InIcelandthisisaschoollevelwhichcorrespondstoHighSc hoolandtherstyearof College. 132
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hisguidance.Inishedtheoralqualicationexamsintheye ar2002andwasformally acceptedasaPh.D.candidateattheUniversityofFloridawi thDr.CharlesThornasmy thesisadvisor. Inthefallof2002mydaughterandmyselfwereinaseriousaut omobileaccident whichhaltedmystudiessomewhatandlaterthatsameyearmys onIsarwasbornin Gainesville.Intheyear2004mywifeandIdecidedtogetdivo rcedandtherewasa subsequenthaltinmygraduatestudiesattheUniversityofF lorida.Imovedbackto Reykjavik,Icelandandstartedtoworkinnancialmathemati csforawhile.Inishedthis Ph.D.thesisinabsentiaandwasreadmittedtotheGraduat eSchoolattheUniversityof Floridatocompletemydegreeinthefallof2006. 133

