Group Title: solution to the Schwinger-Dyson equations of quantum electrodynamics
Title: A solution to the Schwinger-Dyson equations of quantum electrodynamics
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Title: A solution to the Schwinger-Dyson equations of quantum electrodynamics
Alternate Title: Schwinger-Dyson equations of quantum electrodynamics
Physical Description: v, 197 leaves : ill. ; 28 cm.
Language: English
Creator: Cartier, Joan F., 1950-
Copyright Date: 1983
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Subject: Quantum electrodynamics   ( lcsh )
Integral equations   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
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Statement of Responsibility: by Joan F. Cartier.
Thesis: Thesis (Ph. D.)--University of Florida, 1983.
Bibliography: Bibliography: leaves 195-196.
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
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notis - ACB8571

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A SOLUTION TO THE SCHWINGER-DYSON EQUATIONS
OF QUANTUM ELECTRODYNAMICS









BY

JOAN F. CARTIER










A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1983













ACKNOWLEDGMENTS


I would like to express my sincere thanks to all of those who

have helped me. I would especially like to thank Charles Reid for

his constant support and interest, Arthur Broyles for his thoughtful

criticism without which no progress could have been made, H. S. Green

for his invaluable suggestions and guidance, Ruben Mendez Placito

for being a strong ally and good friend, and Robert Coldwell for

his resourceful presence which saw me through many computational

crises with wisdom, insight and humor. This work was substantially

assisted by the Northeast Regional Data Center which donated a MUSIC

account and valuable technical assistance,and the Division of Spon-

sored Research which provided a Research Assistant Fellowship Award.














TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . . . . . . ... . . . . ii

ABSTRACT . . . . . . . . ... . . . . . v

CHAPTER
I INTRODUCTION . . . . . . . . ... .. .. 1

II THE GENERAL PROCEDURE FOR THE SOLUTION TO THE SCHWINGER-
DYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS . . . 9

2-1 The Schwinger-Dyson Equations . . . . . 9
2-2 Initial Approximations . . . . . . .. 14
2-3 Approximating the Vertex Equation with
Green's Generalized Ward Identity . . . ... 19
2-4 Converting the Integral Equations into
Differential Equations . ... . . ..... 23

III THE ELECTRON PROPAGATOR EQUATION . . . . .. 27

IV THE VERTEX EQUATION . . . . . . . .... .33

4-1 Introduction . . . . . . . . . 33
4-2 The Main Computer Program . . . . . . 37
4-3 The Left-Hand Sides of the Eight Equations .... . 39
4-4 The Right-Hand Sides of the Eight Equations . . 44

V CHECKING THE ALGEBRA . . . . . . . . . 57

VI THE MASS SHELL SOLUTION . . . . . . . .. 75

6-1 An Approximation Solution . . . . . . 75
6-2 The F and I Functions . . . . . . . 80
6-3 The Go, G1 and G2 Functions . . . . .... 82
6-4 The Ho, H1 and H2 Functions . . . . .... 84
6-5 Summary of the Mass Shell Solution . . . ... 86

VII VERIFICATION OF THE MASS SHELL SOLUTION . . . ... 91

7-1 The Mass Shell Program . . . . . . .. 91
7-2 Contributions to Error in the Main Program .... . 93
7-3 The Mass Shell Program . .. .. . . .. . 95
7-4 Summary of Results of Mass Shell Program . . .. 104
iii










TABLE OF CONTENTS (Continued)

CHAPTER Page

VIII EXTENDING THE MASS SHELL SOLUTION . . ..... . 108

8-1 A Scaling Symmetry .. . .......... 108
8-2 The Large p2 Region . . . . . . 112
8-3 The Large k2 Region .. . . . . . . 119

IX THE CONCLUSION ...... ........... . . . . 133


APPENDIXES
A DIRAC GAMMA MATRICES: DEFINITIONS AND PRODUCT RULES . 139

B DERIVATION OF GREEN'S MASS SHELL SOLUTION ...... 143

C FORTRAN PROGRAMS .. . ......... ..... 157

D CALCULATION OF FOUR DIMENSIONAL INTEGRALS .. . ... 190

REFERENCES .... .... .. . . .. .... ..... 195

BIOGRAPHICAL SKETCH . .. ..... .. . . 197













Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


A SOLUTION TO THE SCHWINGER-DYSON EQUATIONS
OF QUANTUM ELECTRODYNAMICS

By

Joan F. Cartier

Chairman: Charles E. Reid
Major Department: Chemistry
A non-perturbative solution to the unrenormalized Schwinger-Dyson

equations of Quantum Electrodynamics was obtained by using combined

analytical and numerical techniques. The photon propagator is approx-

imated by its form near the mass shell. The vertex equation is cut off

at the second order in the coupling constant and the remainder is

approximated by H. S. Green's generalization of the Ward Identity for

higher order diagrams. Under these approximations a functional form

for the electron propagator, S(p), and the vertex function, Fr(p,p+k)

was obtained for all magnitudes of moment. Both functions were found

to be finite. No infinities were subtracted to obtain the solutions.













CHAPTER I


INTRODUCTION



A physical theory must furnish an accurate description of observable

quantities. Beyond this supreme requirement a physical theory is judged

by its simplicity and ease of comprehension. Another important measure

of a theory is how widely the approach could be applied. There is a

sense of a theory being more truthful if it is applicable to more than

a single subject. If the same approach can be applied to several diverse

problems then the problems lose their diversity. To be a really success-

ful theory it needs to be one that unifies a broad spectrum of ideas.

Quantum Electrodynamics (Q.E.D.) has been a successful theory to

the extent that it enjoys a number of these characteristics. One of its

attributes is that the theory had its beginnings in the bringing together

of several separate fields of study. This unification formed Relativ-

istic Quantum theory and then as a natural extension the study of dynam-

ics of the interaction of particles with electromagnetic radiation

evolved.

Quantum Theory (Q.T.) was developed in the 1930's from the ideas of

wave mechanics. It accomplished the explanation of atomic structure,

molecular structure, the structure of solids and the symmetry and energy

bands of crystals. However, Q.T. was insulated from the effects of

special relativity in all these endeavors because the ordinary effects

of chemistry arise from interactions with only the outermost part of
1









the atom. The calculation of reaction rates, surface potentials or

scattering cross sections requires a detailed description of only the

outermost electrons which require relatively low energies to excite.

Relativistic corrections represent a very small fraction of these cal-

culations.

Successful as Q.T. was at describing in terms of fundamental

principles the workings of chemistry, it represented a break in the

general pattern of explaining physical phenomena because it had not

incorporated the principles of special relativity. Quantum Theory

needed to be formulated in a Lorentz covariant form. The relativistic

descendant of Schrodinger's equation is the Klein Gordon equation.

This equation allows for the relativistic variation of mass with the

velocity of the particle. However such effects as the fine structure

of the hydrogen atom were only partly accounted for by the relativistic

Q.T. of Schrodinger. It took Dirac's work on the fully relativistic

theory of particles with spin 1/2 to complete the analysis of the fine

structure and to explain the Zeeman effect.

A complete description of the relativistic electron would have to

include the interaction of the electron with its own electromagnetic

field. Feynman1 and Schwinger2 formed calculational methods in Q.E.D.

in two separate mathematical languages. Dyson3 demonstrated these

languages were equivalent. As early as 1930, Waller, Weisskopf and

Oppenheimer4 had calculated the self energy of the electron and found

it to be disappointingly, quadratically divergent. Later Weisskopf

established that the divergencies were only logarithmic. These inex-

plicable divergencies that occurred in the calculation of measurable









quantities (though at the time such measurements were not practically

feasible), held the theory in a quandary for quite a while.

Real impetus was given to Q.E.D. when Lamb and Retherford5 succeeded

in measuring the splitting between the 2S- and the 2P energy levels

of the hydrogen atom. Acting on a suggestion of Lamb's, Bethe6 circum-

vented the divergence problem by simply cutting off the range of inte-

gration over the divergent integrals. Surprisingly, Bethe came up with

a very close calculation of the "Lamb shift," as it has come to be known.

Other attempts were made at trying to eliminate the divergencies in

a more rigorous manner. Schwinger and Tomonaga29 developed the first

Lorentz covariant scheme designed to make the elimination of the diver-

gencies more acceptable. But, by whatever the justification, calcula-

tions in Q.E.D. have enjoyed remarkable practical success. Because of

the small coupling constant for the electromagnetic interaction, per-

turbation techniques have resulted in impressive calculations of exper-

imental values of the Lamb shift, separation of the ground state doublet

of positronium, the hyperfine structure of the hydrogen atom, the line

shape of emitted radiation in atomic transitions and other relativistic

but measurable phenomena.

The road which connects Quantum Theory with special relativity

leads to the theory of Quantum Electrodynamics. The road continues on

today to connect Q.E.D. with further frontiers. Quantum Electrodynamics

has become a model for other field theories. An appreciation of the

special role of Q.E.D. is one way to provide for a better understand-

ing of the mathematical structure of field theories in general. The

recent success of the Non-Abelian gauge theories in unifying the










electromagnetic interaction with the weak interaction further motivates

efforts to understand the ambiguities of Q.E.D. The current theories of

electro-weak and strong interactions have been shown to have the same

underlying structure based on assumptions about global and local invar-

iance. The unification has its dark side since as a part of the bargain

comes the problem of the divergencies

It is apparent that Q.E.D. sits as a bridge between many well-

travelled roads of thought. Thisis why it is particularly frustrating

that the theory should be flawed by unnatural infinities which rear up

in the evaluation of physically observable quantities. If it were a

complete and satisfactory theory they should never have occurred.

There are four different kinds of divergencies. The following

classification of them has been paraphrased from the text by Janch and

Rohrlich,8

(a) divergencies associated with the description of the vacuum

(b) infrared divergencies

(c) divergencies associated with closed loops

(d) serious divergencies.

The type (a) occurs only in the form of a phase factor multiplying a

particle amplitude. Since this does alter the probability density it

can have no observable effect. It is possible to ignore type (a)

divergencies. Type (b) is an artifact of the mathematical procedure.

An analysis of the problem has shown it can be eliminated by an improved

mathematical treatment. Type (c) is associated with the photon self

energy. This type of divergence has been handled by invoking the invar-

iance of the theory under gauge transformations. Type (d), the










"Serious" divergencies, comes up in the calculation of the vacuum polar-

ization, electron self energy and the vertex function. These serious

divergencies are the object of this whole discussion.

The redressing of these infinite quantities is called Renormaliza-

tion. In this process the amplitudes are expanded in a power series of

the coupling constant. Many terms in the series may contain divergent

integrals. It is possible to remove these infinities in a relativisticly

covariant way by redefining the parameters of mass and charge. The

finite terms which remain in the series are taken to be the renormalized

expressions for the amplitudes. Renormalization is neither simple nor

straightforward. It brings in a new and complicated set of rules which

are not properly anticipated by the initial understanding of the problem.

Renormalization is an after the fact reaction to something unforeseen and

undesirable. This leaves two possible interpretations of the problem:

(1) There is something wrong with the basic theory of Q.E.D.

(2) Some mathematical procedure has been inappropriately applied.

A new method of calculation must be tried.

It is hard to argue that there is something wrong with the founda-

tions of the theory. Q.E.D. has enjoyed a huge practical success in

calculating various physical measurements. For example, from renormal-

ized perturbation calculations of the anomolous magnetic moment of the

electron the following results for the inverse of the fine structure

constant were derived:9


- = 137.03549(21).








Presently the best experimental value is10

Sxp = 137.03604(11).
a exptl

It would seem that there could be little wrong with a theory that makes

such accurate predictions.

The unqualified success of Q.E.D. in calculating (by whatever means)

precise experimental results, and the success Q.E.D. has enjoyed in

linking quantum theory to special relativity, and its position central

to general unifying theory of forces are forceful evidence that the

fundamentals of Q.E.D. are sound.

The implication therefore must be that some mathematical procedure

has been inappropriately applied. The response to this implication has

been a long search by many persons for a self-consistent and finite

approach to Q.E.D. In 1954, Gell-Mann and Lowl2 sought to demonstrate

that the renormalizing constants which relate the bare mass and charge

are infinite. They found they could not rule out the possibility of infin-

ite coupling constants but they isolated a necessary condition for the

vacuum polarization to be finite. Johnson, Baker and Willeyl3 took up

the interesting problem in a long series of papers. Under a certain

set of approximations they solved for the renormalizing constants and

concluded that in order for the self energy of the electron to be finite,

the bare mass of the electron must be zero and a special gauge must be

used. These results spawned interest in a number of people14 for find-

ing asymptotic but non-perturbative evaluations of the self energy of

the electron, the vertex function and the vacuum polarization.

Chapter III will describe the paper in which the author was involved in










some of the effort of repeating, in a new way, the determination of the

electron self energy. The work reaffirmed the results of Johnson,

Baker and Willey by using an inventive non-perturbative approach of

H. S. Green's with less restrictive approximations than had been used

before. This work extended the results of Johnson, Baker and Willey by

finding a complete solution instead of an asymptotic one.

These three basic functions, the electron propagator, the vertex

and the photon propagator are identified by their role in the three-

linked nonlinear integral equations known as the Schwinger-Dyson equa-

tions. It is of great interest if a method of solution could be found

which would yield no divergent function forms for the three basic func-

tions. Encouraged by the success of Green's method in extracting a

finite solution for the electron propagator, we decided to see if the

same non-perturbative procedure of H. S. Green would work to provide

a vertex function.

The description of the project is arranged in the following way.

Chapter II is a general description of the overall method of taking the

nonlinear linked integral equations and transforming them into a set of

differential equations. An outline is given of the proposed method for

solving the equations for the electron propagator and the vertex.

Chapter III provides a description of the electron propagator solution.

Chapter IV provides a preparation of the differential equations for the

vertex function. The tensor equation is broken down into eight linked

third-order differential equations for the eight scalar functions which

comprise the transverse part of the vertex. In Chapter V an algebraic

technique is described which makes the product of the complicated gamma





8



matrix function easier to obtain. In Chapter VI, Green's method is used

to obtain an approximate solution to the eight linked differential

equations. In Chapter VII this solution is tested and its range of

applicability is defined. In Chapter VIII some alterations are made to

the solution which extends its viability. The last chapter summarizes

the solutions which were obtained to the Schwinger-Dyson equations.

It is felt that the work described here will contribute to the

idea that Q.E.D. is a complete and satisfactory theory; a theory which

is a faithful rendition of experimental results, a theory which lends

clarity by unifying several fields of study, and a theory which is

unambiguously expressed.













CHAPTER II


THE GENERAL PROCEDURE FOR THE SOLUTION OF THE SCHWINGER-
DYSON EQUATIONS OF QUANTUM ELECTRODYNAMICS



2-1 Schwinger-Dyson Equations

Quantum Electrodynamics is a description of the interaction of

light with matter. A classical charged particle generates an electro-

magnetic field around it with which it can self interact. Calculations

of this interaction have traditionally led to infinities. To demonstrate

how these infinities arise in Q.E.D., it is convenient to express the

theory in terms of the propagators of the particles.

The amplitude of an electron at some point in space-time is related

to its amplitude at a different point in space-time by its Feynman

propagator or Green's function. The Green's function is determined by

the equation of motion that governs the wave function. For the free

electron the differential equation is

(i0 m): = 0 (2-1)


where W is a shorthand notation for yu and yW is a Dirac gamma

matrix.t The photon wave equation is

[] A' = 0 (2-2)


where ] is the D'ALambertian, --- The propagator for the
ax axin

tSee Appendix A for representations of the y1.









electron satisfies a corresponding equation,

(i-mo)S (x' ,x) = 64(x'-x). (2-3)

The solution to this equation in momentum space determines the Fourier
transform of the free electron propagator to be

S(p) -m (2-4)

Similarly the photon propagator satisfies

Do (x-x') =i64(x-x'), (2-5)


so that the Fourier transform of the photon progagator is

D(q2) =- (2-6)
q

Thus the propagators for the free particles are explicitly known.
When it is allowed that a source term may be present, the inter-

action between the electron and photon will lead to nonhomogeneous dif-

ferential equations. The exact electron and photon propagators are then
determined by these nonhomogeneous differential equations but the solu-

tions are not explicitly known. The equations which determine the Fourier

transform of the photon and electron propagators are an open set of inter-
locked integral equations. This hierarchy of integral equations was
formulated by Dyson3 and Schwinger.16 Using the notational practices of

Bjorken and Drell,17 these integral equations appear as:


tThe 4 vectors are denoted by a bar over the symbol and matrices
are distinguished by a bar under the symbol.









(a) S(p) = So(p) + S (p)_(p)S(p) (2-7)
or equivalently

S- (p) = S (p) _(p) (2-8)
where

ie2
(p) = p(pq)S(q)D (p-q)yd4q.- (2-9)
S (2T)4

(b) D (k2) = D (k2) +Do(k2)~R B(k2)D (k2) (2-10)

where
ie 2
t (k2 0e Tr [yS(q) (qq+)_q+)] d4q. (2-11)
(27Tr)

(c) r'(pq)- rl+A.:p,q) (2-12)
where

A(p,q) =ie2 fD )(k2) (p p-p)S(p-k)

r (p-k,q-)S(- (q--k,,q)-d4k

+ ... f... f (p p- p-k) ... ...

rn (-q-n l ,q-n)S(q- )r. (q-vq)

d4k ... d4k (2 )-4n + ... (2-13)

The zero subscript follows all bare quantities, that is, those functions
or constants which are associated with the free particles. The I'(p,q)
is called the vertex function. It represents the sum of all nodeless
diagrams which are connected to two external electron lines and one









external photon line. If a closed expressed for the vertex function

could be formed, then a complete knowledge of the interaction propaga-

tors would depend only on a solution to the three linked nonlinear

equations; the electron propagator equation, Eq. (a), the photon propa-
gator equation, Eq. (b), and the vertex equation, Eq. (c). However,

Eq. (c) gives the vertex equation in terms of an infinite series.
Therefore the Schwinger-Dyson equations, though simple in form, possess

no simple solution.
One method for attempting a solution is to assume the interaction

propagators differ from the bare propagators by only a small variation.
Perturbation theory might then lead to at least an asymptotically con-

verging series. The first iteration would replace the propagators.on

the right-hand side of Eqs. (a), (b) and (c) by the bare propagators.
When this is done the equation for the electron propagator function

Z(p) becomes
_eo 1 v gv1 d4k
S(p) 4 (-)-m 2
(2fl 0 k

e2 4(0+mo) ) 4
4 [(p 2-m d k. (2-14)


By a power counting of k it can be seen that for k-~ the integrand
behaves like k3dk/k4 which would yield a logarithmic divergence. A more
careful consideration of this integral would take into account the

hyperbolic metric. The four-dimensional integral can be performed by
transforming into hyperspherical coordinates but the logarithmic divergence

persists. A corresponding divergence appears in the vertex function

See Appendix D on the subject of calculation of four-dimensional
integrals.










and in the photon propagator function. These are of the type (d)

category of divergences of Janch and Rohrlich8--the so-called "serious

divergences."

These divergences can be handled by any of a number of methods

grouped under the title of Renormalization methods. The first proof of

renormalizability was provided by Dyson, Salam and Ward.7 Basically the

idea is that although absolute calculations cannot be made with the for-

malism as it stands, still relative calculations can be made. The param-

eters of the theory such as mass and charge are redefined to absorb the

terms which contribute the infinite quantities. The renormalized mass

and charge are taken to have the experimentally observed values. Renor-

malization, as such, was a huge but puzzling step forward. It provided

the tools to make impressive calculations of relativistic corrections

in the spectrum of the atom but the meaning and the value of the bare

quantities remained a mystery.

The very fact that renormalization works is an indication that there

ought to be a mathematically consistent way to solve the hierarchy

without encountering undefinable quantities. The practical success of

renormalized theory argues against a fundamental flaw in the theory.

For these reasons the following scheme was developed to solve the equa-

tions without recourse to renormalization methods.
The procedure which has made possible an unrenormalized solution

of the Swinger-Dyson equations has three elements to it. The first is

the assumption of a reasonable starting point in terms of an approx-

imate form for the photon propagator. The second is the generalization










of the Ward Identity which provides a neat formula for systematically

truncating the hierarchy of the vertex equation at increasing levels of

accuracy. The last element consists of transforming the electron and

vertex integral equations into linked differential equations. The dif-

ferential equations are more tractable than the integral equations to

numerical and analytic approaches to the solution. A description of

these three elements is the object of the following three sections.


2-2 Initial Approximations

The question which must be raised first is whether it is best to

solve first for the electron propagator, S(p), or first for the photon

propagator, D (k2). This can be decided by a consideration of which

of the two is easiest to approximate. Since Lorentz30 first offered the

idea, as early as 1909, it has been a popular view to consider the mass

of the electron as mechanical in origin. This was based on the observa-

tion that the electron, when accelerated by interaction with the electro-

magnetic field, behaves as though it were gaining mass. It is appeal-

ing to imagine that the bare mass of the electron may be zero and the

self interaction with its own electromagnetic field is what "dresses"

the electron in its apparent mass. If the bare mass tis zero or very

small the self energy of the electron will be of theorderof magnitude

of its rest energy. In units of inverse time, the rest energy of the

electron is 1.2x1014 megacycles. On the other hand, renormalized per-

turbation type calculations of the Lamb shift show the vacuum polariza-

tion contributes only about 27 megacycles. The disparity between these

tObviously the assumption the bare mass of the electron is zero is
predicated on the additional requirement that there be no other forms of
interaction which contribute to the mass of the electron.









two numbers argues that the self energy of the electron may be more
important than the vacuum polarization of the photon. Therefore, it
is reasonable to start with an approximate form of the photon propagator
and solve first equation (a) for the electron propagator.
The electron propagator is a function of the scalar I, (y pP), and
as such its most general form is expressible in terms of two functions,
one the coefficient of unity and the other the coefficient of 0.
Because of the simplicity of Eq. (2-8) relative to Eq. (2-7), it is of
interest to focus on the form of S(p)-,

S-1(p) = A(p2) B(p2)O. (2-15)

The vertex function is a matrix function of the Dirac gamma matrices
and the four vectors, p and k, the electron and photon momentum respec-
tively. Its most general form is
A G
( (p+k,p) = F +y G + pX l +- 2
V % 0 pk 2







+ kJ + kX 2 K + kXK L + iBp ak kM (2-16)

2 2
where F, G Gl, G2, H H H, and I are functions of p2, k and u.
The u is the angle variable defined as

u = (2-17)
P2 Ik I2











They are coefficients of that part of r which is transverse to k.

Similarly J, K, L and M are functions of p2, k2 and u. They are coef-
ficients of the longitudinal part of F.


The general form of the photon propagator is known from its relativ-
istic covariant properties. Since D (k2) is a second rank tensor which
depends only on the four-vector R, the photon momentum, D (k2) can be

taken to have two components. One component is proportional to k k

and the other is proportional to g Thus


D (k2) = dl(k2) g /k2 + d2(k2) k k /k4. (2-18)




In addition to knowing the general forms of the three functions,

the electron propagator, the vertex and the photon propagator, the
limiting form of the functions is known on the mass shell. In the
2 2 2
limit as the moment approach the mass shell where p -2 m k2 0,
the functions approach the following limits,


S-(p) Z (g-m) (2-19)


X(p+kp) -1 X (2-20


D (k2) Z (2-21)
P'Ou 3 k2









The constants of proportionality to be determined by the theory are
Z2 and Z3. First order perturbation theory finds them to be zero or

infinite. They are called renormalization constants because the

standard procedure is to use them to redefine the series expansions

so that the divergences are absorbed.
The initial step of solving for the electron propagator requires

a reasonable first estimate of the dressed function D and iF. The

approximation that was used for the photon propagator was

S(k2)= Z3[- (1-b) -1 ]. (2-22)
v k2 k4
13,14
This choice was motivated by the results of earlier investigations.1314
Studies of the asymptotic forms of the propagators which were not incon-

sistent with finite renormalization constants found the photon propagator
to be in the Landau gauge for at least the lowest order in perturbation

theory. The Landau gauge is a special case (b=0) of the general form

given in Eq. (2-22).
The approximation that was used for the vertex function was obtained

from the Ward Identity which related the vertex to the inverse electron

propagator,

k rF(p+f,p) = S- (+) S-1p). (2-23)

In the limit as the photon momentum, k, grows small this can be expressed
in a differential form,

as'I (p)
_(p,p) --- (2-24)









As illustrated by Eq. (2-14), when the bare form of the functions

is used to generate a first approximation to the dressed function, it
is found that "serious" divergences occur. It is desirable then to avoid
the substitution of bare quantities for dressed quantities. Therefore
if the general form of the electron propagator were used in the differ-

ential Ward Identity,it would be possible to generate a general form for
the vertex at vanishing k,

S(p,p) =- [A(p2) + jB(p2)]. (2-25)


To solve the electron propagator equation a knowledge of the vertex
function r (p+k,p) is needed. Equation (2-25) represents a very good

approximation to the vertex function where Ik21 is small. In consider-
ing the electron propagator Eqs. (2-7) and (2-9), it can be seen that
this approximation is at its best when, as the argument of D vanishes,
the integrand is at a maximum. This coincidence of the region of best
approximation with the region of most importance argues that

F (p+k,p) -- [A(p2) + 0B(p2)] (2-26)

is a reasonable first approximation for the vertex.
By using the approximations in equations (2-22) and (2-26) for the

photon propagator and vertex, a solution to the electron propagator
equation can be found. A description of the method of solution and its
conclusions are found in the next chapter. Once a solution to the elec-
tron equation had yielded a functional form for the electron propagator
functions A(p2) and B(p2), a solution to the vertex equation was sought.
The next section describes the method by which the vertex equation









was approximated so that it no longer depended on the higher levels of

the hierarchy.

2-3 Approximating the Vertex Equation with
Green's Generalized Ward Identity
A construction of the infinite hierarchy of equations can be selected
so that a highly repetitive pattern emerges which can be used to system-

atically separate the equations at any chosen level of complexity.
Instead of the infinite series which appears in Eq. (c) the vertex equa-

tion can be written as

= +A(p,q)

where
2





.2 r
A(peq)= -- -e BD (k2)S(_p-k)"(p-_,p-4_-_)


S(p-q-k)Fa(p-q-kp- )d k

ie2 f 2
+ Jo DaB (k (p- ja(p --q)d4k. (2-27)
(211)

A new function has appeared, E"a, a function of three external
moment, which corresponds to four independent points in space-time;
hence Ela is a four-point function. Just as the two-point functions

S(p) and D (k2) were related to the three-point function, r A ,p+R),
and as the three-point vertex, r was related to the four-point func-
tion Ea(p-R,qq-k), so the four-point function, El", can be related to
a five-point function and so forth. The greater the number of moment

involved the lower the contribution from such a cross section should be.









Each new n-point function is created by pulling a dressed photon and
dressed electron propagator out of an n-l point diagram, creating in

this way a new vertex or point.
In Table 2-1 the equations for the two-point electron propagator,
the three-point vertex, and the four-point E1 are given to show the

pattern that is emerging.


The repeated structure in the relations that link an n point diagram
to an n+l point diagram suggests there should be a generalization of the

well-known Ward Identity which states

q A1(p-q,p) = -z(P-q) + E(p). (2-28)

A generalization of this for the four- and five-point diagrams would be

q EUV(p+,p,k) = AV(p+q,p-R,k) A"(p,p-k,R) (2-29)

q F'uv(p+q,p,k,) = E'a(p+q,R+i-p-q,kl)- E'(p,k+7-p,k,L). (2-30)

This generalization of the Ward Identity was first proven by H. S. Green18
in 1953. These identities exactly define the longitudinal components
of the n+l point diagram in terms of the difference of two n-point
diagrams. These identities can be used to truncate the hierarchy of
equations by approximating any given diagram by its longitudinal compo-
nent. In this manner the vertex function could be solved for if EV was
approximated through an application of Eq. (2-29).
In conclusion, it has been proposed the electron propagator can be
found by approximating the three-point vertex by Ward's Identity.











TABLE 2-1. RELATIONSHIP BETWEEN N-POINT DIAGRAMS
Two point diagram
S-1
S-1(p) = S (p) Z(p)

where So = v m
-ie
= 4 0 -)S(-q)D C2)y vd4.
(2)v q q

Three point diagram

"(p,p-q) = r + AP(p,p-q)


where o = y


A- = (p-E,,p-q)S(p- )o D(k2 yvd4k
(2) TJ

Four point diagram

(p-,,p- = + E'l(p-K,q,-p-q)
0

where D]a = yp(-p-q-A()S(p-q-R)(p-q-,-)
ie
E = 4 0~ O- p,p (p-) (Z2) yVd4
(27F) v

Five point diagram

(. ,E,-,-_) -01=00 B + FP- -^,,q,-q)

whereOopa -
0 2 P4()
ie2 f2
Fp 0- (p-k-_-s,l,k,q,p-q)S(p-s)D v(s )y.d4s
(2)I s





















10-















1I--
+





II







r\

1&


'4


I-


a
i;L
II

'a-
0.^
1-


11o-










a-
I I







lO .-I




\0--
aV









Although the Ward Identity provides an exact relationship for the

longitudinal component of the vertex, it only yields the limit of the

transverse part for small photon momentum. To obtain the transverse part

for large values of the photon momentum a solution to the vertex equation

must be found. This can be done by using the first-order generalized

Ward Identity to determine an approximation for the four-point diagram.

This process has two very important aspects to it. One is that at all

levels in the solution to the Schwinger-Dyson equations the Ward Identity

is exactly preserved. The second is never are bare functions substituted

for dressed functions--a procedure which has always been associated with

divergences.


2-4 Converting the Integral Equations
into Differential Equations

Another essential step in the whole procedure of reducing these

equations to a tractable form without giving in to the drastic approxima-

tions, which have characterized earlier attempts at a solution, was the

conversion of the integral equations into differential equations. This

method was developed by H. S. Green19 in connection with the Bethe-

Saltpeter equation. It was first used for a study of the Schwinger-

Dyson equations for the electron by Bose and Biswas.20

Whenever the photon propagator appears under the integral it can

be used to eliminate the integration. The photon propagator was taken

to have the form,

D (k2) =- 3[ + (-b k ]. (2-31)
k k









Because
2 g k k
S-- n (k2) 2 9 4 (2-32)
3k Tk 2- 4
J v k k
the photon propagator can be put into the alternate form,

D (k2) = z23 (-b) k n (k2) (b+1) (2-33)


Notice also that the D'Alemberian operator,

- (2-34)
k oku Sk '


will operate on -L to yield a delta function,
k

k = 4T2i5(k2). (2-35)

This delta function can be used to trivially perform the integration
over integrals of the form,

S(k2) ie2 fDV((_-)2)ylF\F (q2) d4q. (2-36)

Substituting Eq. (2-33) into Eq. (2-36) results in

( k ) = i e 2 k-b ) pl q k -


in((k-q)2) (b+1) y F(q2)d4q (2-37)
nK -q )J

Apply the D'Alembertian

n-I kl(k2)= ie2 (l-b) (k)2
0 4bD k) 4i k6 (k-q )2

(b+1)g 4Ti6 (k-q )I1" FXV(q2)d4q









= 2e2(b+l)i2y F (k2)

+ ie2f- (1-b) -- Fv(q2)d4q. (2-38)
v (k-q)

There is one remaining part under the integral. This can be removed by
either of two ways. The first is operate on the equation with

Y -- = f. This yields the following third-order differential equation
aka

lkIDk(k2) = 2e2 2(b+l ky FXv(k2)

+ 4e272(1-b) F(k2). (2-39)


The second method is to identify the remaining integral as some func-
tion GX(k2) such that

G (k2) = i (1-b) -- F (q2)q. (2-40)
v (k-q)

Then the problem of solving Eq. (2-38) becomes the problem of solving
the pair of equations,

SkI (k2) = 2e2(b+1) 2 y F (k2) + e2(k2)

VGX(k2) = 42 (1-b) -- FX(k2). (2-41)

By either of these routes a solution to an integral equation of the
form (2-36) is equivalent to a solution to the differential Eq. (2-39)
or the pair of differential Eqs. (2-41) when the appropriate boundary
conditions are satisfied.










To briefly summarize this chapter, a scheme has been laid out by

which the first two unrenormalized Schwinger-Dyson equations could be

solved. The first step of the scheme involved solving for the electron

propagator. To solve the electron equation required using Eq. (2-22)

and (2-26) to initially approximate the vertex function and the photon

function. The second stage of the scheme involved solving the vertex

equation to determine the form of the transverse part of the vertex for

other than very small photon moment. To make this possible the higher

order terms were also approximated using a generalization of the Ward

Identity. Finally the integral equations relating these functions were

to be converted into differential equations to make an analytic solution

easier.

In the next chapter, Chapter III, a survey of the work that was

done on the electron equation is given. In Chapter IV, an introduction

to the details of solving the vertex equation is given.












CHAPTER III


THE ELECTRON PROPAGATOR EQUATION


Recall the form of the electron propagator previously given in
Eq. (2-7) where S the Fourier transform of the bare propagatator, is
given by

S = 1- (3-1)


The m0 is the bare mass of the electron, that is, the mass the electron
would have if the electromagnetic interaction could be turned off.

The z(p) equation was given as
ie2 r
S(p) o f ( ,)S(q()DV (p-tq)yVd4q. (3-2)
(2Tr)J

On the basis of rationalizations detailed in section 2-2 two

approximations are invoked to sever the connection of the electron equa-
tion from the vertex equation and the photon propagator equation. These

were,

D ) + (1-b) p 1-
(p-q)2 (-q)
and

rX p+q,p) a S- (p). (3-4)

When these substitutions are made the electron propagator Eq. (3-1)
becomes









ie2 aS-l1p)
S (p)= S(p) + (q)
0 (2-)4Jf 3p

S- (1-b) ( 4 d] 4qyV. (3-5)
(p-q) (p-q)

Because the electron propagator is a scalar function of only the electron
momentum, its most general form is given by

S-(p) = A(p2) + B(p2). (3-6)

So that

S (p)-= 2pA'(p2) + y (p2) + 2pl B'(p2) (3-7)
P,

where

A' (p2) = and B' (p2) = d (3-8)
dp2 dp2

Substituting Eqs. (3-2), (3-7) and (3-8) into the electron propagator
Eq. (3-6) yields,

A(p2) + lB(p2) m + ie 2 J 2p4 A'(p2)

+ yvB(p2) + 2plB' (p2))(A(q2) + gB(q2))

-- (1-b) P d4q (3-9)
(p-q) (p-q)

The equation was converted into a differential equation by
application of the D'Alembertian operator, as described in section 2-4.











For the purpose of perfonning all of the needed matrix multiplications,

a table of products of gamma matrices was prepared. This table appears

in Appendix A. Equation (3-10) then separates into twolinearly inde-

pendent differential equations, the coefficient equations for the unit

matrix and V.

The unit matrix equation is

A"2 + 2A e2 4-b) AB (2+b)p (B'A-A'B)
At p 2 2- 4 22 -
4 22 4 (A -p2B2) 4 A p B
(3-10)

where

S 6 B'A-A'BA (3-11)


The V equation is
B"p2 + 3B' e2 (2+b) (A'A-B'B 2) (1-b)B2 (3-12)
4n2 4 (A2p2B2) (A2_-p2)j

where
A' = dA and B' =dB (3-13)
dp dp

SA" d2A and B" d2B (3-14)
d(p2)2 d(p2)2

Numerical solutions to these equations were formed using a fourth

order Runge Kutta method which generated the values for the functions

A, B and T. The functions A and B were found to be very slowly chang-

ing functions. In fact, for most practical purposes, B is essentially
2
equal to 1. The function A very slowly declines as p










A description of the solution and a simple analytic determination

of the asymptotic behavior of these functions was presented in the paper21

on the electron propagator equation. In that paper a different approach

was taken. The integral equations were not converted into differential

equations by the action of the D'Alembertian operator. Instead the

variables of integration were converted to hyperspherical coordinates;

a Wick rotation was performed; then the (p-q)-2 factor was expanded in

terms of Gegenbaur polynomials, Cn(p). These polynomials have an orthog-

onality condition which was used to simplify the integration. This

useful procedure is illustrated in detail in Appendix D.

The solutions to the differential equations had to be restricted

to be particular solutions of the integral equations by the boundary

conditions. The boundary conditions for the four momentum, p2, approach-

ing the mass shell is known. There the electron propagator is propor-

tional to the bare propagator. The asymptotic boundary conditions,where

the magnitude of p2 is indefinitely large, are not explicitly known.

This has been an object of study of a large number of papers.14 Inter-

esting things can be determined about the asymptotic form of the solution

when the differential equations are substituted into the integral equa-

tions. This was carried out in the third section of that same paper.

There it was found that the functions A and B, of the electron propa-

gator, must approach constants for large p2. In order that they could

approach constants the gauge parameter b had to be set to zero. With

b equal to zero, the photon propagator was set in the Landau gauge.

At the same time it was demonstrated that a finite solution required

the bare mass to be zero.









An expression for the function A was fitted to the tabulated

numerical solution over a finite range of moment. The accuracy of

the fit was around 0.1 percent. To an even greater accuracy the func-
tion B was observed to equal the constant one. Thus the functions A

and B appeared to be well represented by

A(p2) = jl-p21(-P)p2 2 (3-15)

B(p2) = 1.0 (3-16)
where e = 1.74517 x 10-3

The tabulated values of A then predicted an asymptotic form of

A = Ip21-E (3-17)

and an asymptotic B of
B = 1. (3-18)


When these asymptotic expressions were substituted back into the inte-

gral equation the power law for A was explicitly determined. It was
found that

e = (3a/4i) + (3a/4T)2 + 3(3a/4r)3 + ... (3-19)

where a is the fine structure constant. Baker and Johnson13 obtained
almost the same expression for the power law of A. They concluded

E = (3a/4n) + (3a/4)2 + ... (3-20)

Last of all it was possible to see by comparing the limit of the electron
propagator as the mass shell is approached, to the propagator expressed
in terms of the renormalized propagator that the renormalization constant,
Z2, was equal to unity.









All of these result were in agreement with the results given by
14
Johnson, Baker and Willey.4 They had also concluded that the bare

mass was zero and they had determined a very similar value for the

power law of the asymptotic expression for A.

This paper represented an extension to the understanding of the

electron propagator because it went beyond trying to determine an

asymptotic form of the propagator which was consistent with finite

renormalization constants. This paper actually determined an approx-

imate expression for the electron propagator which was good for all

moment. The electron propagator was determined to be

S- (p) = g + l-p21 c(-p2)/p2 (3-31)

where e = 1.74517 x 10-3.

In this manner it was shown that the electron propagator could be

determined using approximations of a far less drastic nature than had

been tried before. No infinite quantities were encountered. Encour-

aged by the success of this first stage of the project, an attempt to

solve the vertex equation was ventured.













CHAPTER IV


THE VERTEX EQUATION



4-1 Introduction

The next step in the process of seeking a solution to the Schwinger-

Dyson equations is solving the vertex equation, restated here,


S(p,q) = y + A (p,q).

where
2
Aie
AX(pq) 0o4 fyD1 (k2)S(p-R)/(p-kq-')S(q-k)r (q-kq)d4k
(2n)4 a

2
+ le /DB(k)S(p-K)E'A(p-kq-k,q)d4k. (4-1)
(2iT)


As with the electron propagator equation, the solution to this equation

is preceded by three elements of preparation.

The first, the photon propagator is taken to have the same form

as was utilized in the electron propagator equation. However, it is now

fixed in the Landau gauge so that
2-gpv k k\7
DU(k2) Z3 (4-2)


The second element is to recognize that the four-point diagram,

E which is defined in terms of the five-point diagram, which in turn

is defined in terms of all higher order diagrams, an infinite progression,









must be expressed in some closed and approximate form. Just as the

vertex function could be related to the electron function through

Ward's Identity, so the four-point diagram can be related to the vertex

through a Generalized Ward's Identity. There are two possible longi-

tudinal components of the four-point diagram, one is longitudinal with

respect to k and the other which is longitudinal with respect to q.
The relationships between the longitudinal components and the vertex

function as given by the generalized Ward Identity are

k EX(p-k,q,p-q) = A'(p,p-q)- _A(p-k,p-q) (4-3)


q E (p-k,q,p-q) = A (p-k,p) A(p-,p-q). (4-4)

It is possible to substitute r for AX and Fr for Av in these relations

because the difference between F and A is a constant. Using this fact

and putting Eqs. (4-4) and (4-5) in the differential form it can be

seen that

EX(p ,qp-q) r (- pp-q) (4-5)
ap V


EX(p-k,0,p) = (p-V,p). (4-6)
3p


Thus in this procedure the complete E V will be approximated by these

two parts,

E (p-k,q,p-q) -- F (p,p-q) + V(_p-k,p). (4-7)
,v


For the purposes of simplification it is observed that the expression

r (p-k,p-q)S(p-q-k)r(P-q-k,p-q) r(p,p-q) (4-8)
-Pv









can be reduced to

a [(p- k,p-q-)(p-q-R)] S-(p-q-). (4-9)
aP,

To obtain this simplification use is made of the fact that

[(p-q) S p-q) S(p-q) -q), (4-10)
) 3ap

and the vertex function was again approximated by

v a -1 -
r (p-q-k p-q-) S (p-q). (4-11)
pv -
ap

By substituting the expressions in Eqs. (4-8) and (4-9) into the vertex
Eq. (4-1), one obtains a simplified vertex equation,
2
-p,q) = + ie YD(k2)S(p-k) rV (p-k,p)d4k
(an7T) f

2
+ 4 yD (k2) S(p-c) 3- [r(p-k,p-q-k)S(p-q-k)]
(2n) 4p4 -

S- (p-q-k)d4k. (4-12)

The last step in preparing the vertex equation was to operate with
the D'Alembertian,

02 = (4-13)
'v ap

The D'Alembertian has the desirable feature of operating on the Fourier
transform of photon propagator, here in the Landau gauge, to produce
a Dirac delta function,









2 D (p-r) = 22i g, 6(p-r)Z3

-Z a 1 (4-14)
3 3(p~-r) D(p-rV) (p-r)2

Therefore,
2 2
l2 r(prl+ ie 24 f [-2 2i g 6(p-r)]S(r)
(27T

[ r(r,r+q)S(r+q)- ]_S(F+q)d4r

ie2 _4 S()
(27T)4 a(p _-r) (pv-r ) (p-r) -


[_ (r,r+q)S(r+q)-] S(r+q)d r


S22 S(p) [X(p,p+)S(p+q)-1] S(p+q
82 p S( )

-ie2 f[- 3 1 ]S)
(2n S(p -r") a(pv-r",) (p-r)2

a [rA,(+q)S(Fr+)-1] S(Fr+)d4r. (4-15)
v

In the above the definition Z e = e2 was used.

At the cost of having to solve higher order differential equations
the last integral can be eliminated by the action of

S= ^ a p
Sp"
This yields the final form,

3 F(p,r+p)= -c[ Y V FV(p,-p+q) + FV(p,-p+q)] (4-16)










where

3= C, E = e2/47r2

and

F~ = S(p) [Fr(p,p+q) S(p+q)- ] S(p+q). (4-17)
PV


4-2 The Main Computer Program

It is only necessary to solve the vertex equation for the transverse

components of the vertex since by Ward's Identity the exact longitudinal

components are known in terms of the solution to the electron equation.

The main computer program is a realization of the equation


13 X =_ e 1 Xv
I tras [j V yv F + --L FXans
transPR 4 -v trans + -trans

where

Fa = S(p+k) [r (p+k,p) S(p)-1] S(p). (4-18)
-trans p -trans
v

Hereafter the subscript "trans" will be dropped and it will be under-

stood that any X superscript is taken to be transverse. Thus, for any

general function Q ,

Q = kA k (4-19)
k

In the vertex equation there appear two independent four moment,

k, the photon momentum and p, the electron momentum. Scalar functions

therefore will be functions of the variables p2 and


(4-20)


pk









where p = lp21 and k = lk2 2

2 2
and the functions will be parametrized by k The range of p is from
negative to positive infinity. The variable u has the same range owing
to the indefinite metric. The general expression for the vertex func-
tion in terms of such scalar functions is given by
2 H H
r(p+, F) =_ F(p2p) + poVX] H (p2,u) + [y,,l]-1 )p2 u)
p p k p

+ [y,] ~ (p2 u) + Y Go(p2u) + p (pu)


+ p G (p2u) + ^amvuyympk bk. (p2*u). (4-21)


In addition to the general expression for the vertex, the general
expression of the Fourier transform of the electron propagator, as given
in Eq. (2-16), is needed. So that


S(p) = A(p ) B(p 2 (4-22)
A2 (p2) p2B2(p2)
and

S(p+) = A((p+k)2) ( 2+ )B((p+k)2) (4-23)
A ((p+k)2) (+)2B2((p+k)2)

For simplicity the following notation will be observed throughout,

Al = A(p2) A2 = A((p+k)2)

B1 = B(p2) B2 = B((p+k)2)

D A2 p2B2 D = A2 (p+k)28 (4-24)
D1 =1 2 = A2 2









Thus S(p) AI- -- (4-25)
D1

A2- ( +X)B2
S(p+k) = 2- (4-26)
D2


Equation (4-18) presents the differential vertex equation in a

straightforward and simple form. Unfortunately when substitution of

the scalar functions of Eqs. (4-21), (4-22) and (4-23) is made and the

products of the gamma matrices are taken, thousands of terms need to be

resolved. Once expanded in this way it is clear the matrix equation is

equivalent to eight equations which are coefficients of the linearly

independent matrices: y p p K, E Bamy5 pak, icy ka, iua p

iou3p k pp and p The description of the process of identification of

these eight equations will consume the next several sections. In

section 4-3 the left-hand side of the eight equations will be explicitly

given, and in section 4-4 the right-hand side of the eight equations

will be given. These eight equations plus a set of boundary conditions

will be used to determine the eight transverse vertex functions F, GoGIG2'

HoH1H2 and I.

4-3 Left-Hand Sides of the Eight Equations

The formation of the left-hand sides of the eight equations which

comprise the matrix equation for the vertex is a relatively straight-

forward operation. The operation of ? on any general function, f, of

p2 and u will yield

Sf(p2,u) = 2f+ (427)
p









The prime denotes a partial derivative with respect to the scalar p2

and an asterisk denotes a partial derivative with respect to u.

When i3 was applied to rxthe coefficients of the eight linearly

independent matrices were obtained. These are given in Table 4-1.


TABLE 4-1 Left-Hand Side of the Eight Vertex Equations

Equation 1 Coefficient of p
** 2 2 ***
S2 3F + 10 u F u(l-u2) 14
9F + 23u 4 44
p p p p p P
**I
18 u- F + 2(1-u2) F2- + 20 F"- 4u F + 8 p 'F
p P
*
2 Ho u *** 2 2 2***
64 u H + (30- 86 u2) 4 + 26 (1l-u2)H* (1-u2)2 H0
p p p p
H **
+64 Y4 H + (44 u2-16) H- 4 -(l-u2)Ho 32 u H"
P P P

S(1-u2)H 16 p2uHo

**
H1 H1 2 6H1 ***
+ 18 + 46 u + (20 u -6) 2 -4(1-u2)H
p p P p
H *' 4 2 **'
28 - 36 H1 + (l-u )H1 + 40 H1 8 uH1
p P P


+ 16 pH1 p


(4-28)









Equation 2 Coefficient of p K


**
SF 7 ** + 8 7 + 4
P P P

8 4 H Ho 8 Ho 28
p p p


F* +-ui ) F***
F + F
P


2 Ho + (1-u2) Ho
p p


II 2 11
+ 64 H0 + 16 p Ho

+ H1 8 H*

p p p


+ 18 -4 H2 + [10
P


**
2 2 H2
u2-4(1-u2)] -
P


+ 48 H 8 uH + 16 p2
+ 48 H2 8 uH2 + 16 p H2


Equation 3 Coefficient of y
**
1 32 F +5 u F (1-u 2
p p p

*
8 H0 + 4(1-u2) + 8 uHo
p p
**
2 H1
-12 uH1 + 4(1-u2) + 20 H1
P

4 ,, 8u *
+ 16 p H1 2 1
P
**
2 H2 2 H
+ (18 u -6)- 14 u(1-u 2)
P P


***
22
- 2u(1-u ) 42
p
**'
+ 4(1-u2 H22
P


2 *1
28 H2
P


(4-29)


8 F' + 4 p2F"


2 **u 2 H
+ 4(1-u )H1 + 64 p Hl
; I I4pH


2 2 H2
+ 2(1-u2) H2 2-
p


6-28 2 + 28 p
+ (16-28u )H2' + 4u(1-u )H2 + 28 p uH2


+ p4 2 (-2 *"
+ 16 up H2 + 8p (l-u )H2 .


(4-30)









Equation 4 Coefficient of e Xa'5y yk pn


- 8 H Ho + 2 (l-u 2H ** 8 H +8 H
P P P P

8 10 ** +2 2*** 8 *' *"
-4 H1 104 H1 (-u2)H1 + H1 8 H1
P P P P


**
U 2. H2
- 18 u- H2 [10 u2-4(1-u2) -4 +
p P


***
2u(1-u2) H2
P


*1 **1
H2 *" 42 1 6u2 _H
+ 28 u - 48 H2 + 8 uH2 -16 p 4(-u2 ) 2
P P
Equation 5 Coefficient of ioa Bpk p8

*** **
k ** 2 G1 G1 *"
p p4 + 7 (1-u ) 4 4 G1
p 4 p 4 1p 4 p2 1


*2 G2 **
+ 9 + 23 u [3- 10 (u2) i --
9 p P
** '
G 2 u 2 G2
-14 2 18 G2 + 2(1-u2) -+20
P P P


***
u(1-u2) G2
P

S *" 2
G2-4 uG2 + 8 p G2


** ***
+ 9 + 23 u [3- 10 (u2) i (-u
P P P P

- 14 18 u -+2(1-u2 I -+20 I 4 ul
p P P


+ 8 pI I].


(4-31)


(4-32)









Equation 6 Coefficient of ioa.p
** *** *1
Go G G Go
9 u (c-7u2) u-u2) G 14 u -
p p P P
G**' 2
+ 2(1-u2) 2- + 24 GO 4 u GO + 8 p Go
P
**
G1 G1 G G1
+ 4 4 + 5 u (1-u ) 4 4G
p P p P
**
-9 u [23 (u2) 8] + 10(-u2)u I4
p P P
***
(1-u2)2 + 14 u -- (8-18u2) -
P P P

2 I 2 (4-33
2u(l-u2) 20 uI 4(1-u2)1 pul (4-33)
P
p


Equation 7 Coefficient of io ka

** ***
G G G *' 2 *"
[3 + 5 u (1-u2) G 4 p G
P P P
**

3 5 u + (1-u2) + G2
P P P
2 **
+4 p2G 3 5 u + (1-u2) 2
P P P

2 I + 10 u I 2(1-u2) I 28 p21 8 p41". (4-34)










Equation 8 Coefficient of p
** *** *1
G 2 G0 G 2
9 u [2 7 (u2 ) u(l-u2) - 14 u -
P p p p
**I
2 Go 2 '
p
+ 2(1-u2) ~2 + 24 G" 4 u Go + 8 p Go
P
** *I
G G G G' G1
12 15 u + 3(1-u2) -- + 12 - 10 u T
p P p p p
**'
G G2
+ 2(1-u2) ~ + 36 G" + 8 p2 G"' + 9 u
P P
** ***

[8 23 u] 2 2- 10 u(l-u2) 2 + (1-u2 2 2
*i 'l
G6 2 G2 2 G2
14 u +(8-18 u2) 6- + 2 u(1-u ) 2
P P- P

+ 20 u G" + 4(1-u2)G2 + 8 p2u G". (4-35)


4-4 The Right-Hand Sides of the Eight Equations

The right-hand side of the vertex equation,


y Fv + F ], (4-36)


is compiled through multiple layers of matrix multiplication. It is

best represented, not by an exhaustive itemization of each and every

term but by definition of the various layers.

The first layer is to define the matrix function F


FX = S(p+k) r (p+R,p) + S(p+k)r (p+k,p) S(p)S- (p)
ap, aV









A A' 2 BB2 B2 2


B2
v A2B2 B2 + 2p B2 B2A2
Y 2 + o1 pa + 2p 2 2 -D
2 2 2 2

= s(p+R)ZX

where

ZX = + X[-2pPPD+ 2p OMPD- yVABD+ yVBBD]

where


A2A2
PPD + 2
02

A2B2
MPD = -2-



ABD B2

BBD = 2
B2
BBD = D
2


2 B2B2
02


B2A2
02


(4-37)



(4-38)



(4-39)



(4-40)



(4-41)


(4-42)


The Z is a tensor with twenty-eight different linearly independent
combinations of the available matrices. These twenty-eight form a group,
fourteen elements of which are odd in gamma matrices, fourteen elements
are even.


odd = gZ1 + 4P pZ2 + p XkZ3 + gkvZ4


+ VKppVZ5 + Vp k Z6 + p 7y Z7 + Xp Z8









+ yk Zg + eva PY YkpbpZ + Ea a y 5 kaPp Zl

+ EhvB4 + y kvZ12

+ E Sx y 5y k kakvZ14 (4-43)


even = p p Ze + p +2 + v 5k p B3

+ io Ze4 + gv Ze5 + io kc0P p p Ze6

+ PAkvioaSk e7 + ioa V
caSe7 a e8
+ ioa XkVZe + iok pZlo + i i ap v Z
a e a ei p P ell
+ ica p k Z + ic pk + ilaBkp gvZ (4-44)
a e12 a 13 el4
where
G H1
Z = G0BBD + + 2 ABD ul BBD (4-45)
P P
MPD G1 2F G G- 2 2 G1 PPD
2 pk p 4 p 2 p 4 p 2

H4 H1
+ 4u MPD 4 MPD (4-46)
p p
kG1 k Ho k
Z3 3 p2 BBD 2 ABD + I BBD (4-47)
3 p p P

4 =- G2 + 2 H2 ABD + pk I BBD (4-48)


k k k k k
Z5 3- G2 u G2 + G2 BBD + 2 G2 2 G PPD

H
Ho k
+ 2 ABD 4 H MPD 4 H2 MPD - I BBD (4-49)
p p





47


G,
Z 2 (4-50)
6 2

F G1
7 = ABD + G BBD + 8- + G BBD + uG2 BBD (4-51)
p


P

H2


Go
H1-AD + 42P uH MPD + uI B8 (4-52)


Zg = k 2 H2 ABD pk I BBD (4-53)


10 = G BBD 2 ABD + I BBD (4-54)
S2p p


S= 4 H2 MPD I + 2 k I u 2 I PPD (4-55)
ll p2 3 p
p p p

H1
Z12 = G BBD 2 ABD + uI BBD (4-56)
p


13 = 2 H2 ABD +- I + pk I BBD (4-57)


14 (4-58)
14 7
i *
F F F F F
Z = + 2 2 --PPD + p BBD
e1 p3 p.k kp3 pk PT
kp kp
G1
+ 2 G MPD + 2 G1 MPD ABD + 2 uG2 MPD
p
H HI H
+ 2 u - BBD 2 -T BBD + 2 uG2 MPD + 2 u p BBD

HI
+ 2 -1 BBD (4-59)
pk









*
F k
Ze2 = F- G2 ABD 2 H0 BBD 2 H2 BBD
P

k
Z = I ABD + 2 H BBD
e3 p 2


Ze4 = G ABD + 2 8 + 2 t H, BBD + 2 8 u H2 BBD



Z5 k G ABD + 2 H1 BBD + 2 uH2 BD
e5 pk k 1 kH2
H' *
k H H Ho
Z6= G2 MPD 4 + 4 2 u
p p p

H
Ho k
4 PPD + 2 I MPD
2 p
P

Ho
Ze =- 2
e7 ~ p3k

i *
H1 H1 H1 H1
S =- 2 G MPD + 2 + 2 u + 4 PPD
e8 p3k k p 3k pk

2 u I MPD
*
H2
Z = 2
eg pk


Zel0 P G2 ABD + 2 -H + 2 H BBD + 2 H2 BBD
P

G H
Z1 BBD ABD + 2 _u H BBD 2 p BBD


el2 = 4 H2 2 -- H2 4 H2 PPD + 2 H2 BBD 2 pk I MPD
P
+ k I ABD
p


(4-60)



(4-61)



(4-62)



(4-63)






(4-64)



(4-65)





(4-66)


(4-67)



(4-68)



(4-69)


(4-70)








H1 k
S= 2 BBD + I ABD (4-71)

H
Ho k
e14 =2 2 2 H BBD k I ABD (4-72)
P

Since FXv = S(p2)ZX
A1 V B1 X
D1 Z D ( +K)ZV (4-73)
1 1

Even = P~P1 + pXkvW2 + XavE 5ka W3 + io W4

+ a 5 + p~P p Viaak 6W6 + P ciaBkapk W7

+ ioXa VW8 + ioaxk kVW9 + ioav pXka10

+ ip oa W + ioak VW12 + io Pak W13

+ ioaBkap g 14 (4-74)


odd = g9vV1 + p~p V2 + pkV3 + gKV4

+ KpXpVV5 + + pkV + pVV7 + ypVV8

+ yk V9 + va 5 y kagpp V10

+ E Yy aP p kpV + E y5 y P V12
+ Aav 5 , + F. p k k1V
+ E 5y kV13+ E 1y 5P kak V1V 4 (4-75)

where
W1 = AD Zel + BD[- (p2+pk u)Z2 (1 + )Z5 Z7 Z] (4-76)

W2 = AD Ze2 + BD[- (p2+pk u)Z3 (k2+pk u)Z6 k2 Z7 Z9 (4-77)

W3 = AD Ze3 + BD[Z13 + k2Z12] (4-78)








W4 = AD Ze4 + DB[(p2+pku)Z2 (1 +P u)Z13 (4-79)

Wg = AD Ze5 + BD[- (p2+pku)Z (1 + P u)Z4] (4-80)

W6 = AD Ze6 + BD[k2Z2 Z5 - Z 11] (4-81)

W7 = AD Ze7 + BD[Z Z6 Z10 14] (4-82)

W8 = AD Ze + BD[- Z8 + ( +R U)Z1 + Z12] (4-83)

W9 = AD Zeg + BD[Z9 + (p2+pku)Z14 + Z13] (4-84)

W10= AD Zel0 + BD[k2Z7 + (p2+pku)Z10 Z13] (4-85)

W11 = AD Zel + BD[- Z7 + (1 + u)Zlo Z12] (4-86)

W12 = AD Zel2 + BD[k2Z8 + (p2pku)Z1 + Z13] (4-87)

W13 = AD Z13 + BD[- Z9 + (k2+pku)Z14 + k2Z12] (4-88)

W14 = AD Z e4 + BD[k 2Z Z4] (4-89)

and where

V1 = AD Z + BD[- Ze5 + u e4+ Z4] (4-90)
e5 k e14+ e14

V = AD Z + BD[- Z + ( u+1)Z + Z + Z ] (4-91)
2 2 el k e6 e ell

V3 = AD Z3 + BD[ Z2 + pku Z + Ze + ekZ e+ kZ (4-92)

4 = AD Z4 + BD[- p2Ze4 k2Ze5 pku Ze l4 (4-93)

V = AD Z + BD[- p2Z Z Z k2Z pku Z ] (4-94)

V6 = AD Z6 + BD[- (p2+pku)Ze7 Ze9 Ze2 Zel] (4-95)









V7 = AD Z + BD[- Z + u Z (p2+pku)Ze + Zel (4-96)

V8 = AD Z + BD[Z (p2-pku)Ze8 + (1 + )Zel2] (4-97)

V9 = AD Z9 + BD[(k2+pku)Ze9 (p2+pku)Ze13 + k2Ze4] (4-98)

V10 = AD Z10 + BD[Ze3 + 0 + k2Z ell (4-99)

V11 = AD Z11 + BD[Z2 Ze3 + k2Ze8 (4-100)

V12 = AD Z12 + BD[(1 + u)Ze + Ze] (4-101)
12 k e3 e4

V13 = AD Z13 + BD[- k2Z + (2pku)Ze3 (4-102)

V14 = AD Z14 + BD[Ze9 + Ze3 Ze3 (4-103)

Another way to write the right-hand side of the vertex equation is
to notice
F X+
1 r yvXF, aFX
V 9p' V

= a{ [ g 3 av]F (4-104)

The next layer of definition is to choose

S 3 i Xv (4-105)
T = [ ( g a- ]F (4-105)

where
T even = p Tel + pXkvTe2 + cFX 5k pBTe3

+ i T e4 + gXvTe5 + p p Te6 + kpp k Te7


+ ia pTe + ioakc kVTe + iocvpXTe
*a0 e8 a eg em0









+ ioa pa p ell + i+a v el2 + ioa apk Tel3

+ ioUakcP bg Tel4


odd + 0~ppvTo2 + ~pkVTo3 + g To4

+ vp pVT + p Xk To6 + p YVTo7 + A p To8

+ ykVT 09 + vat 5Y 7ka Pp xToi 0

+ \^c ^ ^12
+ Xa y YmkaP p Toll + cNyBy T 012

+ xav y k Tol3+ ya 5y ka PTol4

where the

Tel = W1 2k 6 27 8 2 W

T 1 + p2 1W 3 1 + +
Te= 7 2 2 6 + pku W7 + 10 + 12 2+ 14

T 3 -W 1 N --1N
Te-3 3 12 W13 + W14

T W 1 W 1 2 12MW
e4 2 4 2W 5 2 W9 2 k 127 8 -7 k 13

Te5 =2 5 2 4 k12 2 p2W8 7 k 13- 9

Te = W6


Te7 = W7


e8 8

T = W
e9 9


(4-106)












(4-107)



(4-108)

(4-109)


(4-110)

(4-111)

(4-112)

(4-113)

(4-114)


(4-115)


(4-116)









Te =2 W10 + W2 + 26 + pku
ellO 11 2 W1 6


5 1 I 1
Te2 12 2 3 W14 2 13

T 3 1 1 1
Te3 2 13 - 3 W14 + 12
Te 3 1 w 1
Te14 = W14 2 W3 12 2 W13


W7 W14 + 2 W2

+2W3


To2 = V2

3 1 1
T3 =- 3 2 V5 10 2 11

T o =2V V- P2V1 V pku V14
3 1 1 2 1
04 2 4 2- 9 2 i- V13 -2 pku V14

T 1
05 7 7 S 7 V +lo 2 ii1

S6= V6




To8 =IV8-5 l- I k 11 127 14

To V9 V4 + p V1 V 13 + pku V14

3 1 1 1 v1+ V + v
T10 2 10 3 2 5 10

Toi Vl
011 1 +

Tol2 = 12 + V1 7 8 2 kV14


(4-117)

(4-118)

(4-119)

(4-120)

(4-121)


(4-122)

(4-123)

(4-124)

(4-125)

(4-126)

(4-127)

(4-128)

(4-129)

(4-130)

(4-131)

(4-132)

(4-133)









1 12 1 1
T13 2 V3 V p + Ppk V11 + p V9 (4-134)

To4 = V14 (4-135)


The right-hand side of the vertex equation,

E T (4-136)


is collapsed into a group of eight linearly independent matrix terms

by the summation over a. It is clear that in order to obtain the first-

order partial derivatives of T a type functions the first-order partial

derivativesof the W and Z had to be known. Initial work on this

main program tried to avoid this confrontation by seeking the deriva-

tives of Tha by a standard numerical process. This was found unsatis-

factory for two reasons. One was the numerical calculation added sig-

nificantly to the time parameter of the program. The other was the

precision which this time bought was inadequate. This method was

shortly abandoned in favor of compiling the algebraic forms of the

partial derivatives of Ta, W' and Z These are recorded in the

program included in Appendix C.

In conclusion the right-hand side of the vertex equation is

comprised of eight linearly independent matrix terms. The coefficients

of each of these terms are given in Table 4-2.









TABLE 4-2 Right-Hand Side of the Vertex Equations

Equation 1 Coefficient of pAO

E[2 T T2 + 6 To + 2 p 2P + 2 u T3
101 p z702 T2 0+ p 2 k o3


+ (1 T + 2 T T ]
pk T3 07 p2 o7

Equation 2 Coefficient of pY
u 2
E[To3 + 2 To4 p -u4 To4+ 5 T + 2 p To5

T 2k k *
+ 2 pku T + (1-u ) T + T ]
06 P 06 P 07

Equation 3 Coefficient of yA

01 07 0o 08 k 09

+ (1-u2) *
pk To]

Equation 4 Coefficient of F-asay k p

2 k* *
E[T + 4 T + 2 p T + T12 + 2 T13
010 011 011 p o12 ol3


Su T 2 pku T + k (1-u2) T ].
p2 013 014 P 014

Equation 5 Coefficient of io3Ska PpA

2' k -u2) *
E[6 T + 2 p2 T + 2 pku T + (1-u ) T

+2 T T + kT + 2 T T ].
e10 2 e10 P ell ep4 p e14
P P









Equation 6 Coefficient of il pa

u 2'
[2 T Te4 5 Te8 2 p Te8 Tell
p

-2 Pu T' (1-u2 1 T ].
k el13 pk el 3


Equation 7 Coefficient of ioL'k

k 2k*
E + 2 + T
p e4 e9 p eg T 10

2
+ 4 Te2 + 2 pT T + T 14].
el2 el2 el3 el4


Equation 8 Coefficient of p


E[5 T 2 p2 T 2 P T (1-u ) T 2 T
el 2 pk e2 e5


- u Te5].













CHAPTER V


CHECKING THE ALGEBRA



In performing all the myriad matrix operations necessary to

express the vertex equation we found that working out the algebra was

extensive, repetitive and subject to error whenever the practitioner's

strictest attention lapsed. After an unoriginal effort was made at

wading through the whole expression term by term,it was necessary to

confirm the results. Faced with the prospect of having to repeat the

monumental process, we drew the conclusion that this kind of work is

better done by machine. Efforts were made to solicit the use of a

computer language capable of symbolic algebra.

The software chosen was a programming language from the University

of Utah called Reduce. The program offered a great variety of general

algebraic calculating facilities. Amongst those of interest were

symbolic differentiation, automatic and user controlled simplification of

expressions, calculations with gamma matrices and tensor operations.

With all of these capabilities it offered a very promising approach to

the unwieldy task.

It was found that Reduce provided a direct and easily acquired check

on the left-hand side of the equation. The partial differentiation was

performed, including all nine possible mixed derivatives with respect to

p2 and u up to third order. A quick and accurate check of the original











left-hand side was achieved. Furthermore, this confirmation was achieved

with only a reasonable expenditure of time invested in becoming familiar

with the language.

However Reduce was less easy to make use of on the right-hand side

where, in addition to first and second order mixed partial derivatives,

there were three layers of matrix operations to undergo. The size of

the arrays quickly outgrew the allotted workspace in the machine. All

of the calculations had to be performed in steps and then the results

were summed afterwards. A further investment of time would have been

necessary to learn how to design the output to be displayed in a form

more amenable to easy checking. Despite this user related ineptitude,

a complete expansion of the right-hand side was obtained.

One of the reasons Reduce was found to be less useful than expected

on the right-hand side was the operations, though limited in variety,

involved a proliferation of terms. It was almost a waste to bring all

of the ingenious operational flexibility of Reduce to bear on what was

only a problem of tensor and matrix multiplication and large-scale sort-

ing of terms.

The right-hand side could be written down in a way that was better

designed for checking by giving up the luxury of exhibiting the right-

hand side in terms of the basic functions A, B, F, G G2, Ho, H1, H2

and I. The final form settled upon for the right-hand side,as it

appeared in the previous chapter, was expressed in a hierarchy of defin-

itions. The right-hand side was given in terms of the 28 components of

TXa tensor. The 28 components of T a were given in terms of the 28










components of the F v tensor. The F v tensor was defined in terms of

the 28 components of the Z v tensor. Finally the 28 components of the

Z v tensor were defined in terms of the basic functions, A, B, F, G G1,
G2, H H1, H2 and I. Each layer of redefinition represents the execu-
tion of another tensor operation.
Table 5-1 sunmarizes the five steps taken to define the right-
hand side. The first step was to define a tensor,

Z =- 2PPD FAp ABD r y + BBD Xyv + 2MPD FXp (5-1)

The abbreviations PPD, ABD, BBD and MPD, represent combinations of the
electron functions, A and B, which were defined in Eqs. (4-39), (4-40),
(4-41) and (4-42). Step 2 forms a new tensor,

WX = (0 + b)Z (5-2)

Step 3 forms yet another tensor out of the former two tensors,

A B
F =DZ D W (5-3)


where D1 = (A p2B2)

Step 4 forms the last tensor,

T [g Y + OYy]F. (5-4)

Finally, in Step 5, then index is contracted by a differentiation with
respect to pa. The right-side equals

c a TAa. (5-5)
apC0












TABLE 5-1 Formation of the Right-Hand Side of the Vertex Equation


Step 1.T Z -' = -2PPD FpV ABD FXy + BBD p 0yv
+ 2MPD rFip)

Step 2. W X = (0l + g)Z


A B1
Step 3. F =A Z B W


Step 4. Tr = [ga + Ya Y,]F


Step 5. Right-hand side = e-- TXa
apa


The abbreviations PPD, BBD, ABD, and MPD were defined in
Eqs. (4-39) through (4-42). The abbreviations Al and 01 were
defined in Eq. (4-24)










After the right-hand side is separated into layers so that the full

length and breadth of the right-hand side is disguised, a large number of
operations still have to be performed. Even a simple multiplication
like rA or (g+I)ZAV involves a large number of steps. Consider the
procedure necessary to perform the multiplication of a single element
of r say c 5 ky with p.


AaBQ 5 i I M# n 5 I
E y5k p6 p = ka p an pkpk



S kai 6i n r 55i+ TI 1I 3
= TW kpp[g +* Y ] -





=- [2g Ba+ 2gal + 2g 1a]k pp I k


=- [iokappXP + ipku o ap+ ip2oa ka I (5-6)

Table A-1 was used to obtain the product y5 n in the second line.
Equation A-12 was used to go from the third to fourth line.
It is easy to forget a sign or reverse the order of two indices
when a large number of such operations are performed. However it is
possible to express the multiplication rules for matrix operations in
a very simple way that allows the practitioner to do the same calculation
in his head without ever consulting Table A-1. This simplified method
can be used to supply an algorithm to enable a computer to do the same

kinds of manipulations in a common language like Fortran which does not
possess symbolic capabilities. The basis of this method was designed by
H. S. Green.










To explain the simple multiplication method it is necessary to alter
slightly the form of the definitions of vector functions like r and

tensor functions like Z v. The expressions used in Chapter IV were
evolved through a historical process that did not necessarily produce
the most symmetric arrangement. In this chapter it will be shown that
when some small changes are made, things become very much easier.
In Chapter IV,
= A F A A G1 G2
p= A p + 6 G + p 2 + p 2
P
Ho a Hla2 H
+ 2p ioa p k p 22 + 2ia P p + 2ia'xk p
a p 2k2 F k a pk

+ Eca Y 5 Y p k (5-7)

Now a slightly altered form will be used.

px = p + 2 + PA3 + PA4

+ p'ioa0kaB B5 + iojapc 6 + ioykLa 7

+ ca y kaB 8 (5-8)


wF 2Ho
where pk 5 k2

2H1
2 = 6 -pk

G1 2H2
3 p 7 k
P


(5-9)


I
p8kl .









The new form of rF is equivalent but notice that the order of the indices
in the sixth and seventh components has been changed. Throughout this
chapter it will be assumed that any general vector, say V is expanded
in terms of these same components.

V = pv1 + yv2 + pv3 + p v4

+ p ioR kapv5 + ioApav6 + ioakv7

+ aB5Sy kppv8" (5-10)

Similarly it will be required that any general tensor, TV, will be
expressed in terms of the following components only.

TX = pp t1 + p t2 + t3 + pioaBk t4

+ pv t5 + p iap t6 + pio akat7

+ P EXa Y 5 k t8 + pYt9 + p ioVCptlO

+ p k Ctll + pX VpyySk kP Bt12

+ i~Xtl3 + EA y5 yp t14 + :AcvtyS k atl5

+ vaBy 5ka Pt6 + p k tl7 + p kvotl8 + pAkvtl9

+ p k io BkPapt20 + y k t21 + kVioap t22

+ kVi akat23 + kvE ay5y kapg t24

+ gAVt25 + g t26 + gAVt27 + gAioaBka t28. (5-11)









Now that the groups of matrices have been carefully selected for
any general vector V or tensor T the following notation will prove
extremely useful. The components of a vector will be denoted by the
following brackets.

p = (p 0, 0, 0)

pg = (p X, 0, 0, 0)

p = (p, 0, 0)

pioa ka p = (pAI, 0, 0)

Y = (0, 0, y 0)

iAopc = (0, O, Y 0)

io Ak = (0, y, y O0)

ragSy ka B = (g, l, 0). (5-12)

The components of a tensor, T will be denoted by the following
brackets.
ppV = (p V, 0, 0, 0)

pp = (pp 0, 0)

pXp = (p p V, 0, 0)

p i aBkp B = (ppV, J, 0, 0)









pV = (p l 0, y, 0)

p ioh p = (pV~, A, y a 0)

pV Xaka = (p V y O)

pVCEXa6p 5y k p = (pVo i, -Y O)


p y = (p 0, 0,

pXiovp = (p p, 0,

piovca = (pX,

PvE 'YkaP =


io = (0, 0,

v y 5pa =

Xav5y 5ka =

EXAvaBykSkap


iV = (0, 0,
ur5 c =

Sav45yok =


E vay5k p =


SV)

0, YV)




i:va,)V


O, yA, V)
O, 4A, 0V)

~, yA, yV)
0, ', Y)
y V)



l y X, yV)


y vA, yv)










pXk = (p kV, 0, 0)

pXkV = (p V, kv, 0, 0)

p k\ = (p, kV', 0, 0)

pkviok pBka = (pV, k4, 0, 0)


ykv = (0, kv, 0)

kvia = ( X, k 0, y, )

kiX a yk a = (k0 y, 0), O )

kv k = (0, )g, 0)


g V = (0, 0, gA, 0)

gXV = (0, 0, gV, 0)


g ViaBao p = (g, i, gXV, 0). (5-13)

The rule for the mapping of each component into the bracket notation
is straightforward. The first position in each bracket holds all electron
moment terms; p p and V. The second position holds all photon moment
terms, kv and k. (The kX term does not appear here because only the trans-
verse part of the vertex equation has been considered. However it is
possible to extend the bracket notation to include longitudinal components
without any additional complications.) The third position holds yX and
gAv. The fourth and last position holds y.










Now to convert the component into its bracket merely tabulate the

occurrence of yP yk, y, y, p, kU and gWV and put them into
their appropriate "home" positions. For example, consider p iai k a. t
contains pA, y and k yQ so its bracket is (p /, 0, yV). Notice that
although there is a unique bracket notation for each component, it is not
necessarily possible to guess from the bracket notation what the original
choice was for the component.

( i, 0, 0) = ioapka

or

( 0, , 0) = iPk a g?

The originally selected meaning of (0, , 0, 0) must be preserved so it is
the latter relationship which is the correct one.
Now it will be shown that multiplying the components of the vector

V or tensor TAV by a term like ~, ,y p ,k etc. is simply done by
observing a few rules. To multiply by a unit matrix vector like p or
k merely add the vector to its correct position.

(~, O)pV = (pUV, /, 0) (5-14)

To multiply by a matrix like y V or i from the right, move the matrix
across the bracket from right to left. Each time the matrix crosses
another matrix take the scalar product of the two. Remove the scalar
products to the outside of the brackets.


(p g, p, 0, 0)g = p2(p 0, 0, 0).


(5-15)










One term will occur for each scalar product and one term will occur
when the multiplying gamma matrix reaches its "home" position,

(p kV, 0, 0)- = (pX, k 0, 0) + pku(p kI, 0, 0). (5-16)

Finally, reverse the sign of alternate scalar products,

(p, k, 0, y, ) = (p kA, 0, 0) k2p2 W 0, 0, y")

+ pku(p A, 0, y"). (5-17)

In order to multiply a tensor or vector from the left the rules
remain the same except that the multiplying term is moved across the
bracket from left to right.
Recall the example given earlier of the multiplication of

EC Y kPa (h by 0 from the right. This example can now be writtenas


(0, X A, Y 0) -- 0 = [(pV Ji, 0, 0) pku(g, O, y 0)

+ p2(0, X, yX, 0)] p1 (5-18)


Translating the bracket notation back to the original components it is
found that

C By5y kp p [pxi [ kap pku ioXpa

+ p2icAak k 1 (5-19)

This is the same result which was achieved in Eq. (5-6), yet it was
achieved without using Table A-1. With little practice the products
can be arrived at as quickly as they can be written down. This process










of bracket manipulation is not only many times easier for the human
mind, but it also lends itself to the writing of an algorithm for exe-
cuting similar products in Fortran.
It is also possible to predict easily the products of a contraction

of a vector Va or tensor T a with another vector like pa or gamma matrix
like y To contract with a momentum, merely perform the implied con-
traction. To contract with a gamma matrix it is necessary to first
move the gamma matrix across the bracket, taking alternate signs of
each possible scalar product. While doing so, perform the implied con-
tractions on each term in the product.
A summary of the rules for bracket operations is given in Table 5-2.
Returning to Table 5-1 it is easy to see that a knowledge of the
following operations is all that is needed to generate the right-hand
side, V~ VXp, VAyV, y, V, k T', Tv and y J where V stands
for any general vector function and TXV stands for any general tensor
function. Using the expressions for VA and T v given in Eqs. (5-10)
and (5-11) and using the method of bracket operations, these seven products
are easily written down.

(1) V = p (p2V2 + pku v3 + v5) + (p (v1 + pku v4 + v6)

+ p k(- p2v4 + v7) + pioBkp'(v8 v3)

+ y p2v6 pku v7) + ioa% (- v5 pku v8)


+ io'ak (p2v8) + EXB5Y kaYpB(v7)


(5-20)











TABLE 5-2 Bracket Operations

A. Multiplying by a momentum: Add the momentum to its

"home" position.

B. Contracting with a momentum: Merely contract the momentum

with its proper complement and move the scalar product

to the outside of the bracket.

C. Multiplying by a gamma matrix:

(i) Move the matrix across the bracket from right to
left in order to multiply from the right. Reverse
directions to multiply from the left.

(ii) Take alternate signs of each scalar product that
can be formed.

(iii) If there is no matrix in "home" position, include a
term with the multiplying matrix in the "home"
position.

D. Contracting with a gamma matrix:

(i) Perform steps (i), (ii).

(ii) Perform the implied contraction as the gamma matrix
is moved across the bracket.








(2) VpV = pAp(v ) + pVp (v2) + pApN(v3) + p pvica kap(v4)

+ ppx(v5) + p"io'apa(v6) + P'ioak (v8) + PV y y pB(v8)
(5-21)
(3) V y = pAp(v2) + p A (-v4) + Py(-v6) + p)ioaak(v8)
+ p yV(vl) + pxiop (v2) + pAi a k(v3)
+ pP VaySy k P (v4) + ioa(v5) + cavy5y oa(v6)

+ pXkV(v3) + pkAk(v4) + p kV(v7) + p Akv iak (v8)
+ yXkV(-v7) + kioXap (-v8) + g9(v5) + gX\)(v6)

+ gAV(V7) + gAiok ap(v8) (5-22)

(4) y VV = p (v2) + ppv (v4 () + p ) + pioj k a(v8)
+ p yV(vl) + p ioGpca(-v2) + p i(~k (-v3)
+ p A v Styyk p (v4) + io (-v5) + Yav5pa(v6)
+ xav y5yk a(v7) + Av SpBk (-v8) + p k(v3)

+ pAk (-v4) + yXk (v7) + kvioXpaa(-v8) + g (v5)
+ gA (-v6) + gA (-v7) + gX9ioABk p(v8N) (5-23)

(5) 4TXv = pp (pku t2+ k2t3) + pXpV (-k2t4)
+ pXp (tl+ pku t4) + pAP)ioaka p(-t2)








+ pVy (pku t6 + k2t7) + p iop p(-k2t8)

+ pvioak (t5 + pku t8) + pv cS y5ykas (-t6)

+ p yv(pku t10 + k2 t11) + p iovpa (-k t12)

+ p iovk (t9 + pku tl2) + pxE ySyk B(-tO0)
+ iov (pku t19 + k2 t5) + avYyS p(-k2t16)

+ F-avtSy ka(tl3 + pku tl6) + X 5p k BY (-t4)

+ p kv(tg+pku tl8 + k2t19) + pXkvO(-t0-k2t20)

+ p kv (tl7 + pku t20 tl) + p k i6 k (t12-ti8)

+ ykv(-tl3+pku t22 + k2t23) + k'iopa(t14-k2t24)

+ kviToXk (tl5+t21+pku t24)+ kVE AB Y5kaPB(-516-t22)

+ g (pku t26 + k2t27) + g (-k2t28)

+ gx(t25 + pku t28) + g AvioBk PB(-t26) (5-24)

(6) gTAv = p (p2t2 + t + t9 + pku t3)

+ p pV (tI pku t4 t6 t10)

+ pApVK(p2t4 t 11)

+ pXpVioa"kBkp(t3 + t + t12)

+ pV (p2t6 + pku t7 t3)









+ pvioXa (t5 pku tg + t14)

+ pioAak (p2t8 + t15)

+ p 5'yaB kp5(t7 t16)

+ pA (p2 t10 + t13 + pku tl)

+ piovpa (t9 pku t12 t14)

+ p iovak (p2t2 + t15)

+ pA EBay5y kap(tl + t16)

+ io (p2t14 + pku t15)

+ E vy5 pa(tl3 pku t16)

+ XavySy ka(p2 t16)

+ y5p k a(tl 5)

+ Pkv(p2tl8 + pku t19 + t21)

+ pxkv)tl7 pku t20 t22)

+ pxkV(p2t20 t23)

+ pAk ia kaV (t19 + t24)

+ yk"(p2t22 + pku t23)

+ kio p o(t21 pku t24)









+ kvioaak (p2t24)

+ k ea 5y k y pB(t23)

+ g~(p2t26 + pku t27)

+ gXV(t25 pku t28)

+ gxk(p2t28)

+ gXioaBka P(t27) (5-25)


(7) yT = pX(p2t2 + pku t3 + t5 + 4t9 + pku t18 + k2 t19 + t26)

+ p (tI pku t4 t5 3t0 p2t20)

+ pX(p2t4 t7 3tl + t17 + pku t20 + t28)

+ p ioBkap(t3 + t8 + 4t12 t18)

+ (p2t6 + pku t7 3t13 + pku t22 + k2t23 + t25)

+ ioP (t5 pku tg + 4t14 k2t24 + t26)

+ ioak (p2t8 + 4t15 + t21 pku t24 t27)

+ cxasSy k 6 (t7 3t16 + t2 +t28) (5-26)

These seven equations complete the work necessary to execute a
check on the correctness of the right-hand side as it was described
in Chapter IV.












CHAPTER VI


THE MASS SHELL SOLUTION


6-1 An Approximate Solution

In Chapter IV the transverse part of the vertex equation was
decomposed into a set of eight linked differential equations out of

the original integral expression for the matrix function. The eight

differential equations established relations for the eight unknown

scalar functions, F, G G G2, H, H2 and I. These scalar func-
tions exactly describe the transverse vertex for a given photon momentum-

squared, k2

X(p+k,p) -=- F(p2,u) + p[ _,K] (p2u)
p kp


+ EYgl] (p2,u) + [y yK] 2 (p2 u) + yG(p2

S x 1 1 2 + 2 2 + v 5 1 2( 6 -
+ p (p ,u) + pKT- (p ,u) + E k V (p ,u). (6-1)


The circumflex signifies the transverse component. The scalar functions
are shown as functions of the electron momentum-squared, p2,and the

relative angle function, u, where u = 2 The scalar functions
(ip ilk )I
are parametrized by the value of k2. The eight equations were checked

by a process described in Chapter V. The eight linked equations which
are third order in derivatives including all nine possible mixed
75









derivatives with respect to p2 and u, were complicated to write down,

difficult to check and promised to be significantly more difficult to

solve. Therefore it was important to start with a good guess of the
correct solution. From the experience of solving the electron equation,
where it was learned that the mass shell solution dominated a wide region
around the mass shell, it was hoped that the vertex equation would
similarly be a slowly changing function. If this were so, then a solu-
tion to the eight differential equations which worked near the mass shell
might be extended into the asymptotic region by simple variations of the

eight scalar functions.
In 1981 H. S. Green communicated an approximate solution to the

second order tensor equation,

V2FA = 1[ Fy + 3 V-Fv]. (6-2)


The third order tensor equation, Eq. (4-16), is obtained from this equa-
tion by the operation of 4. The tensor functions Green found to be
approximate solutions to Eq. (6-2) were used to identify the approximate
form of the eight scalar functions. The approximations employed were
commensurate with the known behavior of the vertex and the electron
propagator near the mass shell.
In Eq. (6-2) the tensor

F V= S(p) [r(pX 2)S(p2)]S-1 (p2) (6-3)

Av
was approximated by A v where


A 1 S( )F (Pl,2)S(p2).


(6-4)









This amounts to neglecting

S(p1)[- r pA(pl2) (Pp2)S(P2)(P2)(- S 2)- y)]. (6-5)


This term approaches zero as r approaches its mass shell limit of y.
Near the mass shell the appropriate form of the second order vertex
equation is given by

V 2 1 A V (6-l-6)
V2 =- c[- y y + 1]. (6-6)

This is a far simpler equation than (6-2).
This equation can be decomposed into a set of four linearly inde-
pendent matrix equations by the following device. Take the product of
Eq. (6-6) with 1, y y and Y then take the trace of each product.
To facilitate this the following definitions weremade:

C = tr [rF ] (6-7a)

C = tr [rh y] (6-7b)

CA 1 tr [ryV ] (6-7c)


C = 1 tr [rF ] (6-7d)
Wvp 4 L vvp

D = tr [AA] (6-7e)

D = tr [A F] (6-7f)

D tr [AyI ] (6-7g)

DA 1 tr [Ay ] (6-7h)
Pvp 4 'vp








where Yv = [Y X'Y

and y { 'Yv (6-8)
and Yp 2 Y >YI*

Now it can be seen that the set of four equations below is equivalent
to Eq. (6-6).

2C = 3cDx (6-9a)

V2Cx = 2 E D 2 e V23DAX (6-9b)






S p pvo


The D D, DO and D tensors are evaluated by using the




= [(PI-'P2+A1A2) (A1Pe+A2P3 )C- p pp2C v]/DX (6-10a)


D = 4 tr [r(#2+A2)Y( 1+A1)]/D12
S[(A2~IP 2 OD) x + A2-P P2)C + (p DP2pp
2 D VC 2 Dx ( 6-19d)














d+ A2P 2 q. (-4)+ (6-10b) (6 ) (6 )



[- (p p 2v 2p +P I) C +(A2 P1 v-A1P2G)C (A2 VPU-A1P2P)C (Pl P2+AA2)C
S ( 02 2Vp 2 l p 1)2
+-APlVI2 PPlV P P)C(llA VI xApl VI .-10a)
(Ap VA ^c +ip PCA PI/D (6-1Oc)








DA = tr [FXA(2+A2)y (~p +Al)]/D12
=vv 4 02 2 2

= (Pl2-P2Pl) + (PlvP2p-PvPl)p + (PlpP2p-P2pP P


+ (A2PI +AIP2 )C + (A2Piv+AIP2v)C + (A2Plp+A P2)Cv

+(A1A2-P'P2)Cp + (Pip+P22pPo)Cpo + (plvP2+P2vP)Cpo

+ (PlP2+P2pP)Cu/ 2. (6-10d)

These definitions are somewhat intimidating in appearance but the
full effect of assuming the vertex function will be dominated by y in
the region of the mass shell has not yet been fully explored. If P is
well expressed by yX for some range of values of p2 and u, then this
means C is well expressed by 6 and the other tensors C C and
X A
CA have negligible effect relative to C Furthermore, the electron
propagator has the simple form Z2/0-m. Under these assumptions the
D D D and D tensors take on the greatly simplified appearance of

D = 2mpA (6-1la)
D12

D = [(m2 _-P 2)6A + (plp+P2p)pA]/Dl2 (6-11b)
DXA X -X ]/D (6-11c)
D = m[[(l-p 2)6 (PlpP2)6/D12 (6- )

Dp = [(plvP2-P2vPlv)6 + (P1vP2pP2vPl)p 6

+ X]/D (6-11d)
+ (PlpP2p-P2pP1p) ]/D12 (6-11d)

where D12 = (p-m 2(p2-m ).

5 6A is defined in Appendix A.
11









Green's work using these definitions of the D type tensors to solve
the four equations, (6-9a) through (6-9d), for the four tensors C C,
C and C is reproduced in Appendix B. In the next three sections
his expressions for C C C and C are used to determine the near-
the-mass-shell forms of the eight scalar functions F, Go, G1, G2, Ho, H1
H2 and I.

6-2 The F and I Functions
In Appendix B, Eq (B-32) gives the expression for the C tensor


where


S= Emp 1 [e In ( ) + ]



x = [N + (8+1) ]2 and u = m2- (1-82)k2.


(6-12)



(6-13)


Furthermore, from the definition of C Eq. (6-7a), the scalar function F
is determined.


C tr (') = p F
S t .


(6-14)


Combining Eqs. (6-12) and (6-14) the scalar function is identified in
terms of an integration over beta.

3 cmp/I d u x
F = p [ In(I--6 ) + 1]. (6-15)
S-l 1 X B


Obtaining the I function is slightly more involved but in the end
it will be seen that the expression for I is simply related to the

expression for F. The defining equation for C0 is given in Appendix B,
Eq. (B-59) and it states,










V2 AP 1
( PipC ) k2 1 d (6-16)


Using the fact that when
U0 X
S (1 ) In (1--5) (6-17)
4 x u

then

V2= 1 (6-18)
x -uB

so that Eq. (B-16) can be re-expressed as

V2 (pP p C ) k2B V2 de. (6-19)

It follows that


Pp aC k2 J k dR. (6-20)

From the definition of CX the scalar function I is defined.
pvp

C =4 tr [y ]vp] = {6 [kp -k p ]

+ [k p -k p ]+ 6[k p-k p k]} k (6-21)


From the above it can be shown that it follows that

P p C =2 I (6-22)


From Eq. (6-17) it can be shown that
2
3 fD dB= 2 2p {- 1n (1 A) + }dB. (6-23)
x 25 X









By substituting Eqs. (6-22) and (6-23) into Eq. (6-20) the function I

is identified,
l _iI u x x
I = pk [ In (1 ) + ]dp. (6-24)
xu x

It is now apparent that a simple relationship has emerged. That is,

I k F. (6-25)

6-3 The G G1, and G2 Functions

The C tensor is defined in the appendix by the following statement.

C C*6 xC* (B-48)

where

C = [1 ] n (1- U)dB


{( In (1-p2) + (1 ) In (1- p2)} (B-40)
Pi P2
and where


Spk2 1 1


k ( + 21) (m2-u6 In (1 --B) d. (B-47)
4 2 B XU

For greater ease in manipulating these quantities, the following defin-

itions are made:


2d = 1- -i) n (1- -) d (6-26)
-l X 2x










fl3dB = u ) [(- 1) In (1- ) + 1]dB (6-27)


14dB = (1- u) n (1- ) (6-28)


CK= (1--) n (- p) + (1- ) In (1-p). (6-29)
Pl P2
Then
C5= 1 6J dSB + Ei x 3 dx

k k x
6 k E[p k + 4d k [p + - I4 dB

(6-30)
where
Sx apk pk
S2p ; Ix 31 ; = 0; and p = p -k
ax' A 92

Then
t k pk

2 3 2 3 2 4


and
C = J2d5 CK (6-32)

so that
Cx = C 6 6 C*

=h 6 2d CK /13d]









+ pXp[- f13ddg

+ p k[E (2 + 1)(JI3dB + I dB) fE I d]. (6-33)

From the definition of C Eq. (6-7b), the identification of G G1, and
G2 can be made.

C= 1 tr [FAy]

G1 A 62
+ pp 2+ p k (6-34)

From a comparison of Eqs. (6-33) and (6-34) the final expressions for the
three G functions are formed.

G = 12d - CK I 3d6 (6-35)
2
G1 = E- I 3d (6-36)

G2 = (2p2u + pk)( fI~ d + f/ d)

/Ix pkd6. (6-37)

6-4 The Ho, H1, H2 Functions

The defining equation for CX is shown in the appendix to be

CA EM [6Ak P-k D A)% (6 Ak Dk 3A) ]JfI5d
1cv 16 va v va app/

+ m (Ak -6 k )f/I d (B-55)

where
1 0 x6
d 1 [k2L ) + I x6) In (1 x-)]d, (6-38)
5 ) + (1(6-38)
-l 2D










I6de= 1 n (1 )]de (6-39)


L (z) = In(1-z) d (6-40)


In order to perform the implied operations first notice that

a = I(x) x- a (2p + (B+l)k )Ix (6-41)

where

Ix =_ I (6-42)
Dx
and
a23 I = 26a x + k ka(B+1)2 xx + pk 2(B+1)Ixx


+ 2p ka(B+1)Ixx + 4pnpaIXX. (6-43)

Completing the operations implied in Eq. (B-55) yields a new expression.

CX = C ((p p k -pxpk) 2 fxx d5

+ (6k -6k )[2f 1dB + 2k2fI XB2dB

+ (2pku+2k2) fIx OdB + 2pkuJIxxdB + I6gdo]

+ (6 p-6 A )[4k2 /fxxBd + 4pku Ixxd6]}. (6-44)

The relation between the tensor function CA and the three scalar H
ui
functions is given by the definition

C = 1 tr [FA Y ]. (6-7c)









This can be expanded to reveal the relation between C and the three
H functions.

A A 2H
C =(pp k -ppk ) 0-


S2H
+ (6 k -6 k ) 2

2H
+ (6 -6X p ) 22 (6-46)

By comparison of Eqs. (6-44) and (6-46) the three H functions are
now expressible in the following way,

Ho kp2JfIxdB (6-47)


H1 = [- pk2 I xxBdB- 2(p2ku+ k)I xxd ] (6-48)


2 = d + 5 fI6dJ
2
mk [(pku ixxdB]
8 2 4%/55

+ k3J B 2d + (pku + -) JIxxda. (6-49)

6-5 Summary of the Mass Shell Solution
All of the preceding description of the eight scalar functions, F,
G Gl, G2, H H2 and I has established the transverse part of the
vertex function in any region which is typified by the satisfaction of
two constraints. One constraint is that the electron propagator be well
represented by


_- m










where m is the experimental mass of the electron. This means that the
2 2
function A(p2) or A(p2) must be essentially constant and equal to the

experimental mass. The second constraint is that r must be dominated

by the contribution from y so that the transverse vertex is expressible

in a series expansion,
a A ()a2 A 3k
A = / + ~- L + 2 + ) L + ... (6-50)
rl 1 T 2 IT 3

where a is the fine structure constant. If it is possible to assume

that the first term in the expansion dominates and if the electron propa-

gator is on the mass shell then it is expected that the functions F

through I will satisfy the eight differential equations which have been

written into the Main Program. These eight function solutions have been

collected together for ease of reference in Table 6-1.

By using y as the first order contribution to r a solution was

found to the vertex equation up to second order. In effect the solu-

tion which was determined represents an identification of the L[ term in

Eq. (6-50). If this new improved version of r = y + L were put into

the vertex equation the vertex could be determined up to third order.

There is a practical limit to how far this process should be carried

toward self consistency. The coupling constant, is a very small

number; 2.32 x 10-3. Where it is true that r is dominated by y the

third order corrections would make little difference to the solution.

An even more important consideration; the determination of the vertex

function is only one step in a larger iterative procedure which seeks

to re-evaluate the electron and photon propagators to self consistency.










TABLE 6-1, THE MASS SHELL FUNCTIONS

F = ap I ld
4 f j

I = pkfI d

Go 1 I 2dB - CK jI 3d




G2 = pk (,2Pku + 1 )[I 3d + xd ] -pkf IXdd

Ho kp2m If d1

H1 [- pk2/fIX xdB 2(p2ku + k)JIxXdB]

H = -- [Id + -1 d ] + (Pu + )[i d/+ I xdB

+ -i 1ixx 2dO.


1 in (1 ) +1]










2 2
S(1- k2) In (1 x,)








2 p 2 p2
C = (1 u-) -I (1 - ) In (1 X) 1

P m px m
14 x

2 2
2 p1 2 P2
CK C(1 L-) in (1 --2) + (I %) in (1 --)
P1 m P2 m











TABLE 6-1 (Continued)


2u
13 = (m2-u )[- 2 In
x


x = (m2- u)- In (1
XB


x5
I5 =L2 u)
B


+ U
x


X
UB


1 1
UBXB X1
(1 )
u


"B
In (1 -
u


L2(z) = in (1-z)
2 Jo


I 1 In (1
6 Xg


u
I 1 i_ n


2



3B
2u
In (1
x


X
(1 )
UB

XB
(1 )
u
6


- B
UB


2
uB
+ In
XB


u
+ x 1
XB Xp
(1 x
x (1 ,- )


2 P
UB 1
-7 l
XB (1 A)
UB


XB =[p + ( 1 k]2


uB = m- B2)k2
a


a = fine structure constant.


x
(1 )


xI
In (1 )
u


2
-7]
x


X)
uB


x


Xp
x


where


2
us
x2
3
XB










This project involved an initial approximation for the photon propagator,

from that a calculation of the electron propagator, from that a calcula-

tion of the vertex. Future plans for the project looked toward using

the information gained by this work to calculate the photon propagator.

From this new photon propagator a new electron propagator could be cal-

culated. From these two a new vertex. Each cycle of calculation should

contribute a finer resolution of the exact solutions. There would be no

point in refining the vertex function much beyond the level to which

the electron propagator is known. The functional form of the electron

propagator is accurate to about 0.1% of the value of the function.

Where -y is dominant, corrections to order (a )2 in Fr would be expected

to amount to less than a 0.1% difference.

A direct consequence of solving the vertex equation only up to the

-L L1 term will be that the eighth equation, the coefficient of

Syxvp k p will not be solved. It happens that the right-hand side

of the eighth equation is second order in Therefore to this level

of solution the eighth equation should equal zero on the right.hand side.

For this reason equation eight will not be used as a criterionin assess-

ing the correctness of the solution.

In the next chapter a test of the viability of the mass shell

solution is made, and a description of the programming methods necessary

to enact it is given.













CHAPTER VII


VERIFICATION OF THE MASS SHELL SOLUTION



7-1 The Mass Shell Program

Chapter VI established the functional forms of the eight scalar

vertex functions. There is no single absolutely superior way to compile

these functions which were summarized in Table 6-1. The Mass Shell

Program--the Fortran Program--which computes the values of the functions

at given points, assumed many strategies. The decision of how to com-

pute the functions was influenced by concern for ease of assembly, the

demand for machine time, and the inherent error in each method. Three

basic categories of method were tested, and we became familiar with their

merits. These three categories are described in section 7-3.

Once the mass shell solution was computed by one of these methods,

it had to be interfaced with the Main Program by way of a matrix of

numbers. The data matrix contains the value of each of the eight scalar

functions and the nine possible partial derivatives of each function at

one or more points in the infinite plane of the variables p2 and u.

Also included as data are the simpler electron propagator functions A

and B (which were given in Eqs. (3-15) and (3-16)) and their derivatives.

The Main Program uses the data matrix to evaluate the left- and right-

hand sides (L.H.S. and R.H.S.) of each of the eight equations at each

point provided. The Main Program evaluates the relative error in each

equation at each point.











Relative Error of Equation i = (L.HS. -R.H.S. of Equation i (7-1)
L.H.S. of Equation i
for i = ...8.

The relative error is reported as a percentage error in the output. An

optimal solution will have a minimal error over the entire plane defined

by p2 and u. The question of what constitutes an acceptable minimal

error invites an analysis of what are the possible sources of error which

contribute to the Main Program and the Mass Shell Program.

The relative error in the eight differential equations is the effect

of a number of contributory causes. There is inherent error in the data

matrix just because the Mass Shell Solution is an approximate solution

to the eight differential equations. There is inherent error in the

electron propagator functions A and B because they represent only approx-

imate solutions to the electron equation. There are truncation errors

introduced by the numerical procedures used to perform integration

and differentiations. Finally there is roundoff error--the inevitable

outcome of any calculation which is carried out to a fixed finite number

of figures. All of these sources of error had to be either maintained

below a preset, tolerable level or, where they could not be controlled,

they at least had to be understood well enough so that we could recognize

when they were contributing to a significant loss of information. The

next section will consider what were the sources of error in the Main

Program and how these were controlled. The following section will discuss

the three principal methods used to evaluate the vertex functions, F,

G Gi, G2, H HI, H2 and I, and how each method affected the overall

level of uncertainty.










7-2 Contributions to Error in the Main Program

For the purpose of analyzing the kind of error that is being

generated within the Main Program alone, we will assume for the time being

that the data matrix of the eight scalar functions, their derivatives, the

two scalar electron functions and their derivatives, could be supplied to

the Main Program without error. If this could be done what would be the

remaining sources of error and how would they propagate through the

Main Program?

One of the earliest versions of the Main Program defined the partial

derivatives of the twenty-eight TAa components by taking first-order

differences. (This tensor appears in the R.H.S. and it was defined in

Eqs. (4- 95) through (4-116).) This saved writing the analytic expres-

sions for the partial derivatives of the twenty-eight components of T a

which, in turn, had to be expressed in terms of the partial derivatives of

twenty-eight components of F v which, in turn, had to be expressed in

terms of the partial derivatives of the twenty-eight components of Z

and W which, in turn, were at last expressed in terms of the partial

derivatives of the eight vertex functions and two electron functions

provided by the data matrix. (See Figure 5-1 to be reminded of the

hierarchy of tensors which define the R.H.S.) The numerical procedure

for evaluating the partial derivatives of the twenty-eight components

of TAa was quickly abandoned because the Tha components are very often

extremely large and slowly changing, and roundoff error eliminated most

useful details. It is a characteristic of the R.H.S. that, not only

are the T h components large, but that the R.H.S. is very much smaller











than its components. The R.H.S. is very sensitive to roundoff error

because it involves the difference of many large and almost equal terms.

In order to effectively calculate the R.H.S. analytic expressions of the

partial derivatives of the components of the T", F Z7 and W

tensors had to be supplied and the data matrix had to provide the func-

tions and their derivatives to more than six figures.

Once the numerical means of calculating the derivatives of the

components of TXa was discarded, the only remaining source of error in

the Main Program is the accumulative effects of roundoff error. In

hexadecimal based arithmetic the roundoff error for a single operation

will be proportional to the 16-t+, where t is the number of digits in

the mantissa when the number is expressed as a fraction times the base

raised the power of the exponent. For a calculation of standard pre-

cision, the number of figures in the mantissa is only 8. This is not

enough to provide an accurate evaluation of the R.H.S. It was neces-

sary to increase this precision to 16 significant figures. This meant
-15
the roundoff error for a single operation was proportional to 16-1

a very small number. However as the hundreds of thousands of single

operations of multiplication and addition take place this error will

grow systematically through the program. As mentioned before, the

problem grows particularly severe on the R.H.S. of the vertex equation

when the differences of large and nearly equal quantities are taken.

It would be a huge task to draw a process graph to follow the propaga-

tion of the approximate error throughout the program. Even if this was

done the projected error would be an upperbound with a large possible










deviation since the error would be assumed maximal at each individual

step. As an alternative measure to estimate the intrinsic roundoff error,

both the Main Program and the Mass Shell Program were converted to qua-

druple precision. The stability of the program results under the change

from double precision to quadruple precision was taken to indicate that

the data matrix was being supplied with numbers of sufficient accuracy

that roundoff error was not a matter of principal concern. In this way

the Main Program was established to be working satisfactorily. The

really significant problem of error management lay in the design of the

Mass Shell Program.

7-3 The Mass Shell Program

The Mass Shell Program takes the functional form of the F, G Gi,

G2, Ho, H1, H2 and I functions and computes the value of the functions
and all the derivatives at selected points. The vertex functions are

parametrized by k2 and are dependent on the variables p2 and u, where

p k" p ka.
u P= a a (7-2)
pk (p2)(k2)1 "


Due to the indefinite metric the domain of p2 is from m to + m.

p2 = pp
= gpVp (7-3)


Therefore both (p2) and (k2)+ can take on imaginary values. This leaves

a dilemma in the interpretation of the meanings of the symbols p and k
used in the definition of u and in the definitions of the functions in




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