A SOLUTION TO THE SCHWINGERDYSON 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
21 The SchwingerDyson Equations . . . . . 9
22 Initial Approximations . . . . . . .. 14
23 Approximating the Vertex Equation with
Green's Generalized Ward Identity . . . ... 19
24 Converting the Integral Equations into
Differential Equations . ... . . ..... 23
III THE ELECTRON PROPAGATOR EQUATION . . . . .. 27
IV THE VERTEX EQUATION . . . . . . . .... .33
41 Introduction . . . . . . . . . 33
42 The Main Computer Program . . . . . . 37
43 The LeftHand Sides of the Eight Equations .... . 39
44 The RightHand Sides of the Eight Equations . . 44
V CHECKING THE ALGEBRA . . . . . . . . . 57
VI THE MASS SHELL SOLUTION . . . . . . . .. 75
61 An Approximation Solution . . . . . . 75
62 The F and I Functions . . . . . . . 80
63 The Go, G1 and G2 Functions . . . . .... 82
64 The Ho, H1 and H2 Functions . . . . .... 84
65 Summary of the Mass Shell Solution . . . ... 86
VII VERIFICATION OF THE MASS SHELL SOLUTION . . . ... 91
71 The Mass Shell Program . . . . . . .. 91
72 Contributions to Error in the Main Program .... . 93
73 The Mass Shell Program . .. .. . . .. . 95
74 Summary of Results of Mass Shell Program . . .. 104
iii
TABLE OF CONTENTS (Continued)
CHAPTER Page
VIII EXTENDING THE MASS SHELL SOLUTION . . ..... . 108
81 A Scaling Symmetry .. . .......... 108
82 The Large p2 Region . . . . . . 112
83 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 SCHWINGERDYSON EQUATIONS
OF QUANTUM ELECTRODYNAMICS
By
Joan F. Cartier
Chairman: Charles E. Reid
Major Department: Chemistry
A nonperturbative solution to the unrenormalized SchwingerDyson
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 NonAbelian 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
electroweak 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 selfconsistent and finite
approach to Q.E.D. In 1954, GellMann 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 nonperturbative 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 nonperturbative 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 SchwingerDyson 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 nonperturbative 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
thirdorder 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 SchwingerDyson 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
21 SchwingerDyson 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 spacetime is related
to its amplitude at a different point in spacetime 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 (21)
where W is a shorthand notation for yu and yW is a Dirac gamma
matrix.t The photon wave equation is
[] A' = 0 (22)
where ] is the D'ALambertian,  The propagator for the
ax axin
tSee Appendix A for representations of the y1.
electron satisfies a corresponding equation,
(imo)S (x' ,x) = 64(x'x). (23)
The solution to this equation in momentum space determines the Fourier
transform of the free electron propagator to be
S(p) m (24)
Similarly the photon propagator satisfies
Do (xx') =i64(xx'), (25)
so that the Fourier transform of the photon progagator is
D(q2) = (26)
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) (27)
or equivalently
S (p) = S (p) _(p) (28)
where
ie2
(p) = p(pq)S(q)D (pq)yd4q. (29)
S (2T)4
(b) D (k2) = D (k2) +Do(k2)~R B(k2)D (k2) (210)
where
ie 2
t (k2 0e Tr [yS(q) (qq+)_q+)] d4q. (211)
(27Tr)
(c) r'(pq) rl+A.:p,q) (212)
where
A(p,q) =ie2 fD )(k2) (p pp)S(pk)
r (pk,q)S( (qk,,q)d4k
+ ... f... f (p p pk) ... ...
rn (qn l ,qn)S(q )r. (qvq)
d4k ... d4k (2 )4n + ... (213)
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 SchwingerDyson 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 righthand 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 2m d k. (214)
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 fourdimensional 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 fourdimensional
integrals.
and in the photon propagator function. These are of the type (d)
category of divergences of Janch and Rohrlich8the socalled "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 SwingerDyson 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.
22 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. (28) relative to Eq. (27), it is of
interest to focus on the form of S(p),
S1(p) = A(p2) B(p2)O. (215)
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 (216)
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 = (217)
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 fourvector 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. (218)
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 (gm) (219)
X(p+kp) 1 X (220
D (k2) Z (221)
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[ (1b) 1 ]. (222)
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. (222).
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 (+) S1p). (223)
In the limit as the photon momentum, k, grows small this can be expressed
in a differential form,
as'I (p)
_(p,p)  (224)
As illustrated by Eq. (214), 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)]. (225)
To solve the electron propagator equation a knowledge of the vertex
function r (p+k,p) is needed. Equation (225) represents a very good
approximation to the vertex function where Ik21 is small. In consider
ing the electron propagator Eqs. (27) and (29), 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)] (226)
is a reasonable first approximation for the vertex.
By using the approximations in equations (222) and (226) 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.
23 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(_pk)"(p_,p4__)
S(pqk)Fa(pqkp )d k
ie2 f 2
+ Jo DaB (k (p ja(p q)d4k. (227)
(211)
A new function has appeared, E"a, a function of three external
moment, which corresponds to four independent points in spacetime;
hence Ela is a fourpoint function. Just as the twopoint functions
S(p) and D (k2) were related to the threepoint function, r A ,p+R),
and as the threepoint vertex, r was related to the fourpoint func
tion Ea(pR,qqk), so the fourpoint function, El", can be related to
a fivepoint function and so forth. The greater the number of moment
involved the lower the contribution from such a cross section should be.
Each new npoint function is created by pulling a dressed photon and
dressed electron propagator out of an nl point diagram, creating in
this way a new vertex or point.
In Table 21 the equations for the twopoint electron propagator,
the threepoint vertex, and the fourpoint 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
wellknown Ward Identity which states
q A1(pq,p) = z(Pq) + E(p). (228)
A generalization of this for the four and fivepoint diagrams would be
q EUV(p+,p,k) = AV(p+q,pR,k) A"(p,pk,R) (229)
q F'uv(p+q,p,k,) = E'a(p+q,R+ipq,kl) E'(p,k+7p,k,L). (230)
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 npoint
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. (229).
In conclusion, it has been proposed the electron propagator can be
found by approximating the threepoint vertex by Ward's Identity.
TABLE 21. RELATIONSHIP BETWEEN NPOINT DIAGRAMS
Two point diagram
S1
S1(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,pq) = r + AP(p,pq)
where o = y
A = (pE,,pq)S(p )o D(k2 yvd4k
(2) TJ
Four point diagram
(p,,p = + E'l(pK,q,pq)
0
where D]a = yp(pqA()S(pqR)(pq,)
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 (pk_s,l,k,q,pq)S(ps)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 firstorder generalized
Ward Identity to determine an approximation for the fourpoint diagram.
This process has two very important aspects to it. One is that at all
levels in the solution to the SchwingerDyson equations the Ward Identity
is exactly preserved. The second is never are bare functions substituted
for dressed functionsa procedure which has always been associated with
divergences.
24 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 ]. (231)
k k
Because
2 g k k
S n (k2) 2 9 4 (232)
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) (233)
Notice also that the D'Alemberian operator,
 (234)
k oku Sk '
will operate on L to yield a delta function,
k
k = 4T2i5(k2). (235)
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. (236)
Substituting Eq. (233) into Eq. (236) results in
( k ) = i e 2 kb ) pl q k 
in((kq)2) (b+1) y F(q2)d4q (237)
nK q )J
Apply the D'Alembertian
nI kl(k2)= ie2 (lb) (k)2
0 4bD k) 4i k6 (kq )2
(b+1)g 4Ti6 (kq )I1" FXV(q2)d4q
= 2e2(b+l)i2y F (k2)
+ ie2f (1b)  Fv(q2)d4q. (238)
v (kq)
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 thirdorder differential equation
aka
lkIDk(k2) = 2e2 2(b+l ky FXv(k2)
+ 4e272(1b) F(k2). (239)
The second method is to identify the remaining integral as some func
tion GX(k2) such that
G (k2) = i (1b)  F (q2)q. (240)
v (kq)
Then the problem of solving Eq. (238) becomes the problem of solving
the pair of equations,
SkI (k2) = 2e2(b+1) 2 y F (k2) + e2(k2)
VGX(k2) = 42 (1b)  FX(k2). (241)
By either of these routes a solution to an integral equation of the
form (236) is equivalent to a solution to the differential Eq. (239)
or the pair of differential Eqs. (241) when the appropriate boundary
conditions are satisfied.
To briefly summarize this chapter, a scheme has been laid out by
which the first two unrenormalized SchwingerDyson equations could be
solved. The first step of the scheme involved solving for the electron
propagator. To solve the electron equation required using Eq. (222)
and (226) 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. (27) where S the Fourier transform of the bare propagatator, is
given by
S = 1 (31)
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 (ptq)yVd4q. (32)
(2Tr)J
On the basis of rationalizations detailed in section 22 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 ) + (1b) p 1
(pq)2 (q)
and
rX p+q,p) a S (p). (34)
When these substitutions are made the electron propagator Eq. (31)
becomes
ie2 aSl1p)
S (p)= S(p) + (q)
0 (2)4Jf 3p
S (1b) ( 4 d] 4qyV. (35)
(pq) (pq)
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). (36)
So that
S (p)= 2pA'(p2) + y (p2) + 2pl B'(p2) (37)
P,
where
A' (p2) = and B' (p2) = d (38)
dp2 dp2
Substituting Eqs. (32), (37) and (38) into the electron propagator
Eq. (36) yields,
A(p2) + lB(p2) m + ie 2 J 2p4 A'(p2)
+ yvB(p2) + 2plB' (p2))(A(q2) + gB(q2))
 (1b) P d4q (39)
(pq) (pq)
The equation was converted into a differential equation by
application of the D'Alembertian operator, as described in section 24.
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 (310) 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 4b) AB (2+b)p (B'AA'B)
At p 2 2 4 22 
4 22 4 (A p2B2) 4 A p B
(310)
where
S 6 B'AA'BA (311)
The V equation is
B"p2 + 3B' e2 (2+b) (A'AB'B 2) (1b)B2 (312)
4n2 4 (A2p2B2) (A2_p2)j
where
A' = dA and B' =dB (313)
dp dp
SA" d2A and B" d2B (314)
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 (pq)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) = jlp21(P)p2 2 (315)
B(p2) = 1.0 (316)
where e = 1.74517 x 103
The tabulated values of A then predicted an asymptotic form of
A = Ip21E (317)
and an asymptotic B of
B = 1. (318)
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 + ... (319)
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 + ... (320)
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 + lp21 c(p2)/p2 (331)
where e = 1.74517 x 103.
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
41 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(pR)/(pkq')S(qk)r (qkq)d4k
(2n)4 a
2
+ le /DB(k)S(pK)E'A(pkqk,q)d4k. (41)
(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
2gpv k k\7
DU(k2) Z3 (42)
The second element is to recognize that the fourpoint diagram,
E which is defined in terms of the fivepoint 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 fourpoint diagram can be related to the vertex
through a Generalized Ward's Identity. There are two possible longi
tudinal components of the fourpoint 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(pk,q,pq) = A'(p,pq) _A(pk,pq) (43)
q E (pk,q,pq) = A (pk,p) A(p,pq). (44)
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. (44) and (45) in the differential form it can be
seen that
EX(p ,qpq) r ( ppq) (45)
ap V
EX(pk,0,p) = (pV,p). (46)
3p
Thus in this procedure the complete E V will be approximated by these
two parts,
E (pk,q,pq)  F (p,pq) + V(_pk,p). (47)
,v
For the purposes of simplification it is observed that the expression
r (pk,pq)S(pqk)r(Pqk,pq) r(p,pq) (48)
Pv
can be reduced to
a [(p k,pq)(pqR)] S(pq). (49)
aP,
To obtain this simplification use is made of the fact that
[(pq) S pq) S(pq) q), (410)
) 3ap
and the vertex function was again approximated by
v a 1 
r (pqk pq) S (pq). (411)
pv 
ap
By substituting the expressions in Eqs. (48) and (49) into the vertex
Eq. (41), one obtains a simplified vertex equation,
2
p,q) = + ie YD(k2)S(pk) rV (pk,p)d4k
(an7T) f
2
+ 4 yD (k2) S(pc) 3 [r(pk,pqk)S(pqk)]
(2n) 4p4 
S (pqk)d4k. (412)
The last step in preparing the vertex equation was to operate with
the D'Alembertian,
02 = (413)
'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 (pr) = 22i g, 6(pr)Z3
Z a 1 (414)
3 3(p~r) D(prV) (pr)2
Therefore,
2 2
l2 r(prl+ ie 24 f [2 2i g 6(pr)]S(r)
(27T
[ r(r,r+q)S(r+q) ]_S(F+q)d4r
ie2 _4 S()
(27T)4 a(p _r) (pvr ) (pr) 
[_ (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(pvr",) (pr)2
a [rA,(+q)S(Fr+)1] S(Fr+)d4r. (415)
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)] (416)
where
3= C, E = e2/47r2
and
F~ = S(p) [Fr(p,p+q) S(p+q) ] S(p+q). (417)
PV
42 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). (418)
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 (419)
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
(420)
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). (421)
In addition to the general expression for the vertex, the general
expression of the Fourier transform of the electron propagator, as given
in Eq. (216), is needed. So that
S(p) = A(p ) B(p 2 (422)
A2 (p2) p2B2(p2)
and
S(p+) = A((p+k)2) ( 2+ )B((p+k)2) (423)
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 (424)
D1 =1 2 = A2 2
Thus S(p) AI  (425)
D1
A2 ( +X)B2
S(p+k) = 2 (426)
D2
Equation (418) presents the differential vertex equation in a
straightforward and simple form. Unfortunately when substitution of
the scalar functions of Eqs. (421), (422) and (423) 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 43 the lefthand side of the eight equations will be explicitly
given, and in section 44 the righthand 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.
43 LeftHand Sides of the Eight Equations
The formation of the lefthand 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 41.
TABLE 41 LeftHand Side of the Eight Vertex Equations
Equation 1 Coefficient of p
** 2 2 ***
S2 3F + 10 u F u(lu2) 14
9F + 23u 4 44
p p p p p P
**I
18 u F + 2(1u2) F2 + 20 F" 4u F + 8 p 'F
p P
*
2 Ho u *** 2 2 2***
64 u H + (30 86 u2) 4 + 26 (1lu2)H* (1u2)2 H0
p p p p
H **
+64 Y4 H + (44 u216) H 4 (lu2)Ho 32 u H"
P P P
S(1u2)H 16 p2uHo
**
H1 H1 2 6H1 ***
+ 18 + 46 u + (20 u 6) 2 4(1u2)H
p p P p
H *' 4 2 **'
28  36 H1 + (lu )H1 + 40 H1 8 uH1
p P P
+ 16 pH1 p
(428)
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 + (1u2) 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
u24(1u2)] 
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 (1u 2
p p p
*
8 H0 + 4(1u2) + 8 uHo
p p
**
2 H1
12 uH1 + 4(1u2) + 20 H1
P
4 ,, 8u *
+ 16 p H1 2 1
P
**
2 H2 2 H
+ (18 u 6) 14 u(1u 2)
P P
***
22
 2u(1u ) 42
p
**'
+ 4(1u2 H22
P
2 *1
28 H2
P
(429)
8 F' + 4 p2F"
2 **u 2 H
+ 4(1u )H1 + 64 p Hl
; I I4pH
2 2 H2
+ 2(1u2) H2 2
p
628 2 + 28 p
+ (1628u )H2' + 4u(1u )H2 + 28 p uH2
+ p4 2 (2 *"
+ 16 up H2 + 8p (lu )H2 .
(430)
Equation 4 Coefficient of e Xa'5y yk pn
 8 H Ho + 2 (lu 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 u24(1u2) 4 +
p P
***
2u(1u2) 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 (1u ) 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(1u2) +20
P P P
***
u(1u2) G2
P
S *" 2
G24 uG2 + 8 p G2
** ***
+ 9 + 23 u [3 10 (u2) i (u
P P P P
 14 18 u +2(1u2 I +20 I 4 ul
p P P
+ 8 pI I].
(431)
(432)
Equation 6 Coefficient of ioa.p
** *** *1
Go G G Go
9 u (c7u2) uu2) G 14 u 
p p P P
G**' 2
+ 2(1u2) 2 + 24 GO 4 u GO + 8 p Go
P
**
G1 G1 G G1
+ 4 4 + 5 u (1u ) 4 4G
p P p P
**
9 u [23 (u2) 8] + 10(u2)u I4
p P P
***
(1u2)2 + 14 u  (818u2) 
P P P
2 I 2 (433
2u(lu2) 20 uI 4(1u2)1 pul (433)
P
p
Equation 7 Coefficient of io ka
** ***
G G G *' 2 *"
[3 + 5 u (1u2) G 4 p G
P P P
**
3 5 u + (1u2) + G2
P P P
2 **
+4 p2G 3 5 u + (1u2) 2
P P P
2 I + 10 u I 2(1u2) I 28 p21 8 p41". (434)
Equation 8 Coefficient of p
** *** *1
G 2 G0 G 2
9 u [2 7 (u2 ) u(lu2)  14 u 
P p p p
**I
2 Go 2 '
p
+ 2(1u2) ~2 + 24 G" 4 u Go + 8 p Go
P
** *I
G G G G' G1
12 15 u + 3(1u2)  + 12  10 u T
p P p p p
**'
G G2
+ 2(1u2) ~ + 36 G" + 8 p2 G"' + 9 u
P P
** ***
[8 23 u] 2 2 10 u(lu2) 2 + (1u2 2 2
*i 'l
G6 2 G2 2 G2
14 u +(818 u2) 6 + 2 u(1u ) 2
P P P
+ 20 u G" + 4(1u2)G2 + 8 p2u G". (435)
44 The RightHand Sides of the Eight Equations
The righthand side of the vertex equation,
y Fv + F ], (436)
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
(437)
(438)
(439)
(440)
(441)
(442)
The Z is a tensor with twentyeight different linearly independent
combinations of the available matrices. These twentyeight 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 (443)
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 (444)
a e12 a 13 el4
where
G H1
Z = G0BBD + + 2 ABD ul BBD (445)
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 (446)
p p
kG1 k Ho k
Z3 3 p2 BBD 2 ABD + I BBD (447)
3 p p P
4 = G2 + 2 H2 ABD + pk I BBD (448)
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 (449)
p p
47
G,
Z 2 (450)
6 2
F G1
7 = ABD + G BBD + 8 + G BBD + uG2 BBD (451)
p
P
H2
Go
H1AD + 42P uH MPD + uI B8 (452)
Zg = k 2 H2 ABD pk I BBD (453)
10 = G BBD 2 ABD + I BBD (454)
S2p p
S= 4 H2 MPD I + 2 k I u 2 I PPD (455)
ll p2 3 p
p p p
H1
Z12 = G BBD 2 ABD + uI BBD (456)
p
13 = 2 H2 ABD + I + pk I BBD (457)
14 (458)
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 (459)
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
(460)
(461)
(462)
(463)
(464)
(465)
(466)
(467)
(468)
(469)
(470)
H1 k
S= 2 BBD + I ABD (471)
H
Ho k
e14 =2 2 2 H BBD k I ABD (472)
P
Since FXv = S(p2)ZX
A1 V B1 X
D1 Z D ( +K)ZV (473)
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 (474)
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 (475)
where
W1 = AD Zel + BD[ (p2+pk u)Z2 (1 + )Z5 Z7 Z] (476)
W2 = AD Ze2 + BD[ (p2+pk u)Z3 (k2+pk u)Z6 k2 Z7 Z9 (477)
W3 = AD Ze3 + BD[Z13 + k2Z12] (478)
W4 = AD Ze4 + DB[(p2+pku)Z2 (1 +P u)Z13 (479)
Wg = AD Ze5 + BD[ (p2+pku)Z (1 + P u)Z4] (480)
W6 = AD Ze6 + BD[k2Z2 Z5  Z 11] (481)
W7 = AD Ze7 + BD[Z Z6 Z10 14] (482)
W8 = AD Ze + BD[ Z8 + ( +R U)Z1 + Z12] (483)
W9 = AD Zeg + BD[Z9 + (p2+pku)Z14 + Z13] (484)
W10= AD Zel0 + BD[k2Z7 + (p2+pku)Z10 Z13] (485)
W11 = AD Zel + BD[ Z7 + (1 + u)Zlo Z12] (486)
W12 = AD Zel2 + BD[k2Z8 + (p2pku)Z1 + Z13] (487)
W13 = AD Z13 + BD[ Z9 + (k2+pku)Z14 + k2Z12] (488)
W14 = AD Z e4 + BD[k 2Z Z4] (489)
and where
V1 = AD Z + BD[ Ze5 + u e4+ Z4] (490)
e5 k e14+ e14
V = AD Z + BD[ Z + ( u+1)Z + Z + Z ] (491)
2 2 el k e6 e ell
V3 = AD Z3 + BD[ Z2 + pku Z + Ze + ekZ e+ kZ (492)
4 = AD Z4 + BD[ p2Ze4 k2Ze5 pku Ze l4 (493)
V = AD Z + BD[ p2Z Z Z k2Z pku Z ] (494)
V6 = AD Z6 + BD[ (p2+pku)Ze7 Ze9 Ze2 Zel] (495)
V7 = AD Z + BD[ Z + u Z (p2+pku)Ze + Zel (496)
V8 = AD Z + BD[Z (p2pku)Ze8 + (1 + )Zel2] (497)
V9 = AD Z9 + BD[(k2+pku)Ze9 (p2+pku)Ze13 + k2Ze4] (498)
V10 = AD Z10 + BD[Ze3 + 0 + k2Z ell (499)
V11 = AD Z11 + BD[Z2 Ze3 + k2Ze8 (4100)
V12 = AD Z12 + BD[(1 + u)Ze + Ze] (4101)
12 k e3 e4
V13 = AD Z13 + BD[ k2Z + (2pku)Ze3 (4102)
V14 = AD Z14 + BD[Ze9 + Ze3 Ze3 (4103)
Another way to write the righthand side of the vertex equation is
to notice
F X+
1 r yvXF, aFX
V 9p' V
= a{ [ g 3 av]F (4104)
The next layer of definition is to choose
S 3 i Xv (4105)
T = [ ( g a ]F (4105)
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
Te3 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
(4106)
(4107)
(4108)
(4109)
(4110)
(4111)
(4112)
(4113)
(4114)
(4115)
(4116)
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 =IV85 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
(4117)
(4118)
(4119)
(4120)
(4121)
(4122)
(4123)
(4124)
(4125)
(4126)
(4127)
(4128)
(4129)
(4130)
(4131)
(4132)
(4133)
1 12 1 1
T13 2 V3 V p + Ppk V11 + p V9 (4134)
To4 = V14 (4135)
The righthand side of the vertex equation,
E T (4136)
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 firstorder 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 righthand side of the vertex equation is
comprised of eight linearly independent matrix terms. The coefficients
of each of these terms are given in Table 42.
TABLE 42 RightHand 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 + (1u ) T + T ]
06 P 06 P 07
Equation 3 Coefficient of yA
01 07 0o 08 k 09
+ (1u2) *
pk To]
Equation 4 Coefficient of Fasay 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 (1u2) T ].
p2 013 014 P 014
Equation 5 Coefficient of io3Ska PpA
2' k u2) *
E[6 T + 2 p2 T + 2 pku T + (1u ) 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' (1u2 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 (1u ) 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 lefthand 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
lefthand 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 righthand 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 righthand side was obtained.
One of the reasons Reduce was found to be less useful than expected
on the righthand 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 largescale sort
ing of terms.
The righthand 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 righthand side,as it
appeared in the previous chapter, was expressed in a hierarchy of defin
itions. The righthand 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 51 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 (51)
The abbreviations PPD, ABD, BBD and MPD, represent combinations of the
electron functions, A and B, which were defined in Eqs. (439), (440),
(441) and (442). Step 2 forms a new tensor,
WX = (0 + b)Z (52)
Step 3 forms yet another tensor out of the former two tensors,
A B
F =DZ D W (53)
where D1 = (A p2B2)
Step 4 forms the last tensor,
T [g Y + OYy]F. (54)
Finally, in Step 5, then index is contracted by a differentiation with
respect to pa. The rightside equals
c a TAa. (55)
apC0
TABLE 51 Formation of the RightHand 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. Righthand side = e TXa
apa
The abbreviations PPD, BBD, ABD, and MPD were defined in
Eqs. (439) through (442). The abbreviations Al and 01 were
defined in Eq. (424)
After the righthand side is separated into layers so that the full
length and breadth of the righthand 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 (56)
Table A1 was used to obtain the product y5 n in the second line.
Equation A12 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 A1. 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 (57)
Now a slightly altered form will be used.
px = p + 2 + PA3 + PA4
+ p'ioa0kaB B5 + iojapc 6 + ioykLa 7
+ ca y kaB 8 (58)
wF 2Ho
where pk 5 k2
2H1
2 = 6 pk
G1 2H2
3 p 7 k
P
(59)
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" (510)
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. (511)
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). (512)
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). (513)
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) (514)
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).
(515)
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). (516)
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"). (517)
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 (518)
Translating the bracket notation back to the original components it is
found that
C By5y kp p [pxi [ kap pku ioXpa
+ p2icAak k 1 (519)
This is the same result which was achieved in Eq. (56), yet it was
achieved without using Table A1. 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 52.
Returning to Table 51 it is easy to see that a knowledge of the
following operations is all that is needed to generate the righthand
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. (510)
and (511) 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)
(520)
TABLE 52 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)
(521)
(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) (522)
(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) (523)
(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)
+ FavtSy ka(tl3 + pku tl6) + X 5p k BY (t4)
+ p kv(tg+pku tl8 + k2t19) + pXkvO(t0k2t20)
+ p kv (tl7 + pku t20 tl) + p k i6 k (t12ti8)
+ ykv(tl3+pku t22 + k2t23) + k'iopa(t14k2t24)
+ kviToXk (tl5+t21+pku t24)+ kVE AB Y5kaPB(516t22)
+ g (pku t26 + k2t27) + g (k2t28)
+ gx(t25 + pku t28) + g AvioBk PB(t26) (524)
(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) (525)
(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) (526)
These seven equations complete the work necessary to execute a
check on the correctness of the righthand side as it was described
in Chapter IV.
CHAPTER VI
THE MASS SHELL SOLUTION
61 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). (61)
The circumflex signifies the transverse component. The scalar functions
are shown as functions of the electron momentumsquared, 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 VFv]. (62)
The third order tensor equation, Eq. (416), is obtained from this equa
tion by the operation of 4. The tensor functions Green found to be
approximate solutions to Eq. (62) 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. (62) the tensor
F V= S(p) [r(pX 2)S(p2)]S1 (p2) (63)
Av
was approximated by A v where
A 1 S( )F (Pl,2)S(p2).
(64)
This amounts to neglecting
S(p1)[ r pA(pl2) (Pp2)S(P2)(P2)( S 2) y)]. (65)
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 (6l6)
V2 = c[ y y + 1]. (66)
This is a far simpler equation than (62).
This equation can be decomposed into a set of four linearly inde
pendent matrix equations by the following device. Take the product of
Eq. (66) with 1, y y and Y then take the trace of each product.
To facilitate this the following definitions weremade:
C = tr [rF ] (67a)
C = tr [rh y] (67b)
CA 1 tr [ryV ] (67c)
C = 1 tr [rF ] (67d)
Wvp 4 L vvp
D = tr [AA] (67e)
D = tr [A F] (67f)
D tr [AyI ] (67g)
DA 1 tr [Ay ] (67h)
Pvp 4 'vp
where Yv = [Y X'Y
and y { 'Yv (68)
and Yp 2 Y >YI*
Now it can be seen that the set of four equations below is equivalent
to Eq. (66).
2C = 3cDx (69a)
V2Cx = 2 E D 2 e V23DAX (69b)
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 (610a)
D = 4 tr [r(#2+A2)Y( 1+A1)]/D12
S[(A2~IP 2 OD) x + A2P P2)C + (p DP2pp
2 D VC 2 Dx ( 619d)
d+ A2P 2 q. (4)+ (610b) (6 ) (6 )
[ (p p 2v 2p +P I) C +(A2 P1 vA1P2G)C (A2 VPUA1P2P)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 (61Oc)
DA = tr [FXA(2+A2)y (~p +Al)]/D12
=vv 4 02 2 2
= (Pl2P2Pl) + (PlvP2pPvPl)p + (PlpP2pP2pP P
+ (A2PI +AIP2 )C + (A2Piv+AIP2v)C + (A2Plp+A P2)Cv
+(A1A2P'P2)Cp + (Pip+P22pPo)Cpo + (plvP2+P2vP)Cpo
+ (PlP2+P2pP)Cu/ 2. (610d)
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 Z2/0m. Under these assumptions the
D D D and D tensors take on the greatly simplified appearance of
D = 2mpA (61la)
D12
D = [(m2 _P 2)6A + (plp+P2p)pA]/Dl2 (611b)
DXA X X ]/D (611c)
D = m[[(lp 2)6 (PlpP2)6/D12 (6 )
Dp = [(plvP2P2vPlv)6 + (P1vP2pP2vPl)p 6
+ X]/D (611d)
+ (PlpP2pP2pP1p) ]/D12 (611d)
where D12 = (pm 2(p2m ).
5 6A is defined in Appendix A.
11
Green's work using these definitions of the D type tensors to solve
the four equations, (69a) through (69d), 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
themassshell forms of the eight scalar functions F, Go, G1, G2, Ho, H1
H2 and I.
62 The F and I Functions
In Appendix B, Eq (B32) gives the expression for the C tensor
where
S= Emp 1 [e In ( ) + ]
x = [N + (8+1) ]2 and u = m2 (182)k2.
(612)
(613)
Furthermore, from the definition of C Eq. (67a), the scalar function F
is determined.
C tr (') = p F
S t .
(614)
Combining Eqs. (612) and (614) the scalar function is identified in
terms of an integration over beta.
3 cmp/I d u x
F = p [ In(I6 ) + 1]. (615)
Sl 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. (B59) and it states,
V2 AP 1
( PipC ) k2 1 d (616)
Using the fact that when
U0 X
S (1 ) In (15) (617)
4 x u
then
V2= 1 (618)
x uB
so that Eq. (B16) can be reexpressed as
V2 (pP p C ) k2B V2 de. (619)
It follows that
Pp aC k2 J k dR. (620)
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 pk p k]} k (621)
From the above it can be shown that it follows that
P p C =2 I (622)
From Eq. (617) it can be shown that
2
3 fD dB= 2 2p { 1n (1 A) + }dB. (623)
x 25 X
By substituting Eqs. (622) and (623) into Eq. (620) the function I
is identified,
l _iI u x x
I = pk [ In (1 ) + ]dp. (624)
xu x
It is now apparent that a simple relationship has emerged. That is,
I k F. (625)
63 The G G1, and G2 Functions
The C tensor is defined in the appendix by the following statement.
C C*6 xC* (B48)
where
C = [1 ] n (1 U)dB
{( In (1p2) + (1 ) In (1 p2)} (B40)
Pi P2
and where
Spk2 1 1
k ( + 21) (m2u6 In (1 B) d. (B47)
4 2 B XU
For greater ease in manipulating these quantities, the following defin
itions are made:
2d = 1 i) n (1 ) d (626)
l X 2x
fl3dB = u ) [( 1) In (1 ) + 1]dB (627)
14dB = (1 u) n (1 ) (628)
CK= (1) n ( p) + (1 ) In (1p). (629)
Pl P2
Then
C5= 1 6J dSB + Ei x 3 dx
k k x
6 k E[p k + 4d k [p +  I4 dB
(630)
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 (632)
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]. (633)
From the definition of C Eq. (67b), the identification of G G1, and
G2 can be made.
C= 1 tr [FAy]
G1 A 62
+ pp 2+ p k (634)
From a comparison of Eqs. (633) and (634) the final expressions for the
three G functions are formed.
G = 12d  CK I 3d6 (635)
2
G1 = E I 3d (636)
G2 = (2p2u + pk)( fI~ d + f/ d)
/Ix pkd6. (637)
64 The Ho, H1, H2 Functions
The defining equation for CX is shown in the appendix to be
CA EM [6Ak Pk D A)% (6 Ak Dk 3A) ]JfI5d
1cv 16 va v va app/
+ m (Ak 6 k )f/I d (B55)
where
1 0 x6
d 1 [k2L ) + I x6) In (1 x)]d, (638)
5 ) + (1(638)
l 2D
I6de= 1 n (1 )]de (639)
L (z) = In(1z) d (640)
In order to perform the implied operations first notice that
a = I(x) x a (2p + (B+l)k )Ix (641)
where
Ix =_ I (642)
Dx
and
a23 I = 26a x + k ka(B+1)2 xx + pk 2(B+1)Ixx
+ 2p ka(B+1)Ixx + 4pnpaIXX. (643)
Completing the operations implied in Eq. (B55) 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 p6 A )[4k2 /fxxBd + 4pku Ixxd6]}. (644)
The relation between the tensor function CA and the three scalar H
ui
functions is given by the definition
C = 1 tr [FA Y ]. (67c)
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 (646)
By comparison of Eqs. (644) and (646) the three H functions are
now expressible in the following way,
Ho kp2JfIxdB (647)
H1 = [ pk2 I xxBdB 2(p2ku+ k)I xxd ] (648)
2 = d + 5 fI6dJ
2
mk [(pku ixxdB]
8 2 4%/55
+ k3J B 2d + (pku + ) JIxxda. (649)
65 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 + ... (650)
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 61.
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. (650). 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 103. 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 reevaluate the electron and photon propagators to self consistency.
TABLE 61, 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 61 (Continued)
2u
13 = (m2u )[ 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 (1z)
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 righthand 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
71 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 61. The Mass Shell
Programthe Fortran Programwhich 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 73.
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. (315) and (316)) 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 (71)
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 errorthe 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.
72 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 twentyeight TAa components by taking firstorder
differences. (This tensor appears in the R.H.S. and it was defined in
Eqs. (4 95) through (4116).) This saved writing the analytic expres
sions for the partial derivatives of the twentyeight components of T a
which, in turn, had to be expressed in terms of the partial derivatives of
twentyeight components of F v which, in turn, had to be expressed in
terms of the partial derivatives of the twentyeight 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 51 to be reminded of the
hierarchy of tensors which define the R.H.S.) The numerical procedure
for evaluating the partial derivatives of the twentyeight 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 16t+, 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 161
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.
73 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 (72)
pk (p2)(k2)1 "
Due to the indefinite metric the domain of p2 is from m to + m.
p2 = pp
= gpVp (73)
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
