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R. P. Feynman, Phys. Rev. {\bf 76,} 769 \hfill {\large \bf 1949}\\
\vspace{2cm}
\begin{center}
{\Large \bf Space-Time Approach to Quantum Electrodynamics}\\
\end{center}
\vspace{0.5cm}
\begin{center}
R.P. Feynman\\
Department of Physics, Cornell University,\\
Ithaca, New York\\
(Received May 9, 1949)
\end{center}
\vspace{0.5cm}
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\noindent
Reprinted in ``Quantum Electrodynamics'', edited by Julian
Schwinger\\
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\vspace{0.5cm}
\begin{abstract}
In this paper two things are done. (1) It is shown that a considerable
simplification can be attained in writing down matrix elements for
complex processes in electrodynamics. Further, a physical point of
view is available which permits them to be written down directly for
any specific problem. Being simply a restatement of conventional
electrodynamics, however, the matrix elements diverge for complex
processes. (2) Electrodynamics is modified by altering the interaction
of electrons at short distances. All matrix elements are now finite,
with the exception of those relating to problems of vacuum
polarization. The latter are evaluated in a manner suggested by Pauli
and Bethe, which gives finite results for these matrices also. The only
effects sensitive to the modification are changes in mass and charge
of the electrons. Such changes could not be directly observed.
Phenomena directly observable, are insensitive to the details of the
modification used (except at extreme energies). For such phenomena,
a limit can be taken as the range of the modification goes to zero. The
results then agree with those of Schwinger. A complete,
unambiguous, and presumably consistent, method is therefore available for the
calculation of all processes involving electrons and photons.
The simplification in writing the expressions results from an
emphasis on the over-all space-time view resulting from a study of
the solution of the equations of electrodynamics. The relation of this
to the more conventional Hamiltonian point of view is discussed. It
would be very difficult Co make the modification which is proposed
if one insisted on having the equations in Hamiltonian form.
The methods apply as well to charges obeying the Klein-Gordon
equation, and to the various meson theories of nuclear forces.
Illustrative examples are given. Although a modification like that
used in electrodynamics can make all matrices finite for all of the
meson theories, for some of the theories it is no longer true that all
directly observable phenomena are insensitive to the details of the
modification used.
The actual evaluation of integrals appearing in the matrix elements
may be facilitated, in the simpler cases, by methods described in the
appendix.
\end{abstract}
This paper should be considered as a direct continuation of a preceding
one\footnote{R. P. Feynman, Phys. Rev. {\bf 76,} 749 (1949), hereafter called I.}
(I) in which the motion of electrons, neglecting interaction, was analyzed,
by dealing directly with the {\it solution} of the
Hamiltonian differential equations. Here the same
technique is applied to include interactions and in
that way to express in simple terms the solution of
problems in quantum electrodynamics.
For most practical calculations in quantum
electrodynamics the solution is ordinarily expressed
in terms of a matrix element. The matrix is worked
out as an expansion in powers of $e^2/ \hbar c$, the
successive terms corresponding to the inclusion of
an increasing number of virtual quanta. It appears
that a considerable simplification can be achieved in
writing down these matrix elements for complex
processes. Furthermore, each term in the expansion
can be written down and understood directly from a
physical point of view, similar to the space-time
view in I. It is the purpose of this paper to describe
how this may be done. We shall also discuss
methods of handling the divergent integrals which
appear in these matrix elements.
The simplification in the formulae results mainly
from the fact that previous methods unnecessarily
separated into individual terms processes that were
closely related physically. For example, in the
exchange of a quantum between two electrons there
were two terms depending on which electron
emitted and which absorbed the quantum. Yet, in
the virtual states considered, timing relations are not
significant. Only the order of operators in the
matrix must be maintained. We have seen (I), that
in addition, processes in which virtual pairs are
produced can be combined with others in which
only positive energy electrons are involved. Further, the
effects of longitudinal and transverse waves can be
combined together. The separations previously
made were on an unrelativistic basis (reflected in
the circumstance that apparently momentum but not
energy is conserved in intermediate states). When
the terms are combined and simplified, the
relativistic invariance of the result is self-evident.
We begin by discussing the solution in space and
time of the Schr\"odinger equation for particles
interacting instantaneously. The results are
immediately generalizable to delayed interactions
of relativistic electrons and we represent in that way
the laws of quantum electrodynamics. We can then
see how the matrix element for any process can be
written down directly. In particular, the self-energy
expression is written down.
So far, nothing has been done other than a restatement of
conventional electrodynamics in other
terms. Therefore, the self-energy diverges. A
modification\footnote{For a discussion of this modification in classical physics
see R. P. Feynman, Phys. Rev. {\bf 74} 939 (1948), hereafter referred to as A.}
in interaction between charges is next
made, and it is shown that the self-energy is made
convergent and corresponds to a correction to the
electron mass. After the mass correction is made,
other real processes are finite and-insensitive to the
``width'' of the cut-off in the interaction.\footnote{A brief summary of the methods
and results will be found in R. P. Feynman, Phys. Rev. {\bf 74,} 1430 (1948),
hereafter referred to as B.}
Unfortunately, the modification proposed is not
completely satisfactory theoretically (it leads to
some difficulties of conservation of energy). It does,
however, seem consistent and satisfactory to define
the matrix element for all real processes as the limit of that computed
here as the cut-off width goes to zero. A
similar technique suggested by Pauli and by Bethe
can be applied to problems of vacuum polarization
(resulting in a renormalization of charge) but again a
strict physical basis for the rules of convergence is
not known.
After mass and charge renormalization, the limit of
zero cut-off width can be taken for all real processes.
The results are then equivalent to those of
Schwinger\footnote{J. Schwinger, Phys. Rev. {\bf 74,} 1439 (1948), Phys.
Rev. {\bf 75,} 651 (1949). A proof of this equivalence is
given by F. J. Dyson, Phys. Rev. {\bf 75,} 486 (1949).}
who does not make explicit use of the
convergence factors. The method of Schwinger is to
identify the terms corresponding to corrections in
mass and charge and, previous to their evaluation, to
remove them from the expressions for real processes.
This has the advantage of showing that the results
can be strictly independent of particular cut-off
methods. On the other hand, many of the properties
of the integrals are analyzed using formal properties
of invariant propagation functions. But one of the
properties is that the integrals are infinite and it is not
clear to what extent this invalidates the
demonstrations. A practical advantage of the present
method is that ambiguities can be more easily
resolved; simply by direct calculation of the otherwise
divergent integrals. Nevertheless, it is not at all clear
that the convergence factors do not upset the physical
consistency of the theory. Although in the limit the
two methods agree, neither method appears to be
thoroughly satisfactory theoretically. Nevertheless, it
does appear that we now have available a complete
and definite method for the calculation of physical
processes to any order in quantum electrodynamics.
Since we can write down the solution to any
physical problem, we have a complete theory which
could stand by itself. It will be theoretically
incomplete, however, in two respects. First, although
each term of increasing order in $e^2/ \hbar c$ can be written
down it would be desirable to see some way of
expressing things in finite form to all orders in $e^2/ \hbar c$
at once. Second, although it will be physically
evident that the results obtained are equivalent to
those obtained by conventional electrodynamics the
mathematical proof of this is not included. Both of
these limitations will be removed in a subsequent
paper (see also Dyson$^5$).
Briefly the genesis of this theory was this. The
conventional electrodynamics was expressed in the
Lagrangian form of quantum mechanics described in
the Reviews of Modern Physics.\footnote{R. P. Feynman, Rev. Mod. Phys. {\bf
20,} 367 (1948). The application to electrodynamics is described in
detail by H. J. Groenewold, Koninklijke Nederlandsche Akademia van Weteschappen.
Proceedings Vol. LII, {\bf 3} (226) 1949.} The motion of the
field oscillators could be integrated out (as described
in Section 13 of that paper), the result being an
expression of the delayed interaction of the particles.
Next the modification of the delta-function
interaction could be made directly from the analogy
to the classical case.\footnote{For a discussion of this modification in classical
physics see R. P. Feynman, Phys. Rev. {\bf 74} 939 (1948), hereafter referred to as A.}
This was still not complete because the Lagrangian
method had been worked out in detail only for
particles obeying the non-relativistic Schr\"dinger
equation. It was then modified in accordance with the
requirements of the Dirac equation and the
phenomenon of pair creation. This was made easier
by the reinterpretation of the theory of holes (I).
Finally for practical calculations the expressions
were developed in a power series in $e^2/ \hbar c$. It was
apparent that each term in the series had a simple
physical interpretation. Since the result was easier to
understand than the derivation, it was thought best to
publish the results first in this paper. Considerable
time has been spent to make these first two papers as
complete and as physically plausible as possible
without relying on the Lagrangian method, because it
is not generally familiar. It is realized that such a
description cannot carry the conviction of truth
which would accompany the derivation. On the other
hand, in the interest of keeping simple things simple
the derivation will appear in a separate paper.
The possible application of these methods to the
various meson theories is discussed briefly. The
formulas corresponding to a charge particle of zero
spin moving in accordance with the Klein Gordon
equation are also given. In an Appendix a method is
given for calculating the integrals appearing in the
matrix elements for the simpler processes.
The point of view which is taken here of the interaction of charges differs
from the more usual point of view of field theory. Furthermore, the familiar
Hamiltonian form of quantum mechanics must be
compared to the over-all space-time view used here.
The first section is, therefore, devoted to a discussion
of the relations of these viewpoints.
\section{COMPARISON WITH THE HAMILTONIAN METHOD}
~~~~Electrodynamics can be looked upon in two
equivalent and complementary ways. One is as the
description of the behavior of a field (Maxwell's
equations). The other is as a description of a direct
interaction at a distance (albeit delayed in time)
between charges (the solutions of Lienard and
Wiechert). From the latter point of view light is
considered as an interaction of the charges in the
source with those in the absorber. This is an
impractical point of view because many kinds of
sources produce the same kind of effects. The field
point of view separates these aspects into two simpler
problems, production of light, and absorption of
light. On the other hand, the field point of view is
less practical when dealing with close collisions of
particles (or their action on themselves). For here the
source and absorber are not readily distinguishable,
there is an intimate exchange of quanta. The fields
are so closely determined by the motions of the
particles that it is just as well not to separate the
question into two problems but to consider the
process as a direct interaction. Roughly, the field
point of view is most practical for problems involving real quanta,
while the interaction view is best for
the discussion of the virtual quanta involved. We
shall emphasize the interaction viewpoint in this
paper, first because it is less familiar and therefore
requires more discussion, and second because the
important aspect in the problems with which we
shall deal is the effect of virtual quanta.
The Hamiltonian method is not well adapted to
represent the direct action at a distance between
charges because that action is delayed. The
Hamiltonian method represents the future as
developing out of the present. If the values of a
complete set of quantities are known now, their
values can be computed at the next instant in time. If
particles interact through a delayed interaction,
however, one cannot predict the future by simply
knowing the present motion of the particles. One
would also have to know what the motions of the
particles were in the past in view of the interaction
this may have on the future motions. This is done in
the Hamiltonian electrodynamics, of course, by
requiring that one specify besides the present motion
of the particles, the values of a host of new variables
(the coordinates of the field oscillators) to keep track
of that aspect of the past motions of the particles
which determines their future behavior. The use of
the Hamiltonian forces one to choose the field
viewpoint rather than the interaction viewpoint.
In many problems, for example, the close
collisions of particles, we are not interested in the
precise temporal sequence of events. It is not of
interest to be able to say how the situation would
look at each instant of time during a collision and
how it progresses from instant to instant. Such ideas
are only useful for events taking a long time and for
which we can readily obtain information during the
intervening period. For collisions it is much easier to
treat the process as a whole.\footnote{This is the viewpoint of the theory of the
$S$ matrix of Heisenberg.} The M$\varnothing$ller interaction
matrix for the the collision of two electrons is not
essentially more complicated than the nonrelativistic Rutherford formula,
yet the mathematical machinery used to obtain the former from quantum
electrodynamics is vastly more complicated than
Schr\"odinger's equation with the $e^2/r_{12}$ interaction
needed to obtain the latter. The difference is only
that in the latter the action is instantaneous so that
the Hamiltonian method requires no extra variables,
while in the former relativistic case it is delayed and
the Hamiltonian method is very cumbersome.
We shall be discussing the solutions of equations
rather than the time differential equations from
which they come. We shall discover that the
solutions, because of the over-all space-time view
that they permit, are as easy to understand when
interactions are delayed as when they are instantaneous.
As a further point, relativistic invariance will be
self-evident. The Hamiltonian form of the equations
develops the future from the instantaneous present.
But for different observers in relative motion the instantaneous
present is different, and corresponds to a
different 3-dimensional cut of space-time. Thus the
temporal analyses of different observers is different
and their Hamiltonian equations are developing the
process in different ways. These differences are
irrelevant, however, for the solution is the same in
any space time frame. By forsaking the Hamiltonian
method, the wedding of relativity and quantum
mechanics can be accomplished most naturally.
We illustrate these points in. the next section by
studying the solution of Schr\"odinger's equation for
non-relativistic particles interacting by an
instantaneous Coulomb potential (Eq. 2). When the
solution is modified to include the effects of delay in
the interaction and the relativistic properties of the
electrons we obtain an expression of the laws of
quantum electrodynamics (Eq. 4).
\section{THE INTERACTION BETWEEN CHARGES}
~~~~We study by the same methods as in I, the
interaction of two particles using the same notation
as I. We start by considering the non-relativistic case
described by the Schr\"dinger equation (I, Eq. 1). The
wave function at a given time is a function $\psi ({\bf x}_a, {\bf x}_b,
t)$ of the coordinates ${\bf x}_a$ and ${\bf x}_b$ of each particle.
Thus call $K({\bf x}_a, {\bf x}_b, t; {\bf x'}_a, {\bf x'}_b, t')$ the amplitude that
particle $a$ at ${\bf x'}_a$ at time $t'$ will get to ${\bf x}_a$ at $t$ while
particle $b$ at ${\bf x'}_b$ at $t'$ gets to ${\bf x}_b$ at $t$. If the particles
are free and do not interact this is
$$
K({\bf x}_a, {\bf x}_b, t; {\bf x}_a^{\prime}, {\bf x}_b^{\prime}, t') =
K_{0a}({\bf x}_a,
t; {\bf x}_a^{\prime}, t') K_{0b} ({\bf x}_b, t; {\bf x}_b^{\prime}, t')
$$
where $K_{0a}$ is the $K_0$ function for particle a
considered as free. In {\it this} case we can obviously
define a quantity like K, but for which the time $t$
need not be the same for particles $a$ and $b$ (likewise
for $t'$); e.g.,
\begin{equation}
K_0(3,4;1,2)=K_{0a}(3,1)K_{0b}(4,2)
\end{equation}
can be thought of as the amplitude that particle $a$
goes from ${\bf x}_1$ at $t_1$ to ${\bf x}_3$ at $t_3$ and that particle $b$ goes
from ${\bf x}_2$ at $t_2$ to ${\bf x}_4$ at $t_4$.
When the particles do interact, one can only define
the quantity $K(3,4;1,2)$ precisely if the interaction
vanishes between $t_1$ and $t_2$ and also between $t_3$ and
$t_4$. In a real physical system such is not the case.
There is such an enormous advantage, however, to
the concept that we shall continue to use it,
imagining that we can neglect the effect of
interactions between $t_1$ and $t_2$ and between $t_3$ and $t_4$.
For practical problems this means choosing such
long time intervals $t_3 - t_1$ and $t_4 - t_2$ that the extra
interactions near the end points have small relative
effects. As an example, in a scattering problem it
may well be that the particles are so well separated
initially and finally that the interaction at these times
is negligible. Again energy values can be defined by
the average rate of change of phase over such long
time intervals that errors initially and finally can be
neglected. Inasmuch as any physical problem can be
defined in terms of scattering processes we do not lose much in
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig1e.gif}}}
\caption{The fundamental interaction Eq. (4).
Exchange of one quantum between two electrons.}
\end{figure}
a general theoretical sense by this approximation. If
it is not made it is not easy to study interacting
particles relativistically, for there is nothing
significant in choosing $t_1 = t_3$ if ${\bf x}_1 \ne {\bf x}_3$ as absolute
simultaneity of events at a distance cannot be
defined invariantly. It is essentially to avoid this
approximation that the complicated structure of the
older quantum electrodynamics has been built up.
We wish to describe electrodynamics as a delayed
interaction between particles. If we can make the
approximation of assuming a meaning to $K(3,4;1,2)$
the results of this interaction can be expressed very simply.
To see how this may be done, imagine first that the
interaction is simply that given by a Coulomb
potential $e^2/r$ where $r$ is the distance between the
particles. If this be turned on only for a very short
time $\Delta t_0$ at time $t_0$ the first order correction to
$K(3,4;1,2)$ can be worked out exactly as was Eq.
(9) of I by an obvious generalization to two particles:
$$
\begin{array}{c}
K^{(1)}(3,4;1,2) = - ie^2 \int \int K_{0a}(3,5)K_{0b}(4,6)r_{56}^{-1} \\\\
\times K_{0a}(5,1)K_{0b}(6,2)d^3 {\bf x}_5 d^3 {\bf x}_6 \Delta t_0,
\end{array}
$$
where $t_5 = t_6 = t_0$. If now the potential were on at all
times (so that strictly $K$ is not defined unless $t_4 = t_3$
and $t_1 = t_2$), the first-order effect is obtained by
integrating on $t_0$, which we can write as an integral
over both $t_5$ and $t_6$ if we include a delta-function
$\delta(t_5 - t_6)$ to insure contribution only when $t_5 = t_6$
Hence, the first-order effect of interaction is (calling $t_5 - t_6 = t_{56}$):
\begin{equation}
\begin{array}{c}
K^{(1)}(3,4;1,2) = - ie^2 \int \int K_{0a}(3,5)K_{0b}(4,6)r_{56}^{-1}\\\\
\times \delta(t_{56})K_{0a}(5,1)K_{0b}(6,2)d \tau_5 d \tau_6,
\end{array}
\end{equation}
where $d \tau = d^3 {\bf x}dt.$
We know, however, in classical electrodynamics,
that the Coulomb potential does not act
instantaneously, but is delayed by a time $r_{56},$ taking
the speed of light as unity. This suggests simply
replacing $r_{56}^{-1} \delta(t_{56})$ in (2) by something like
$r_{56}^{-1}\delta(t_{56} - r_{56})$
to represent the delay in the effect of $b$ on $a$.
This turns out to be not quite right,\footnote{It and a like term for the effect
of a on b, leads to a theory which,
in the classical limit, exhibits interaction through half-advanced and
half-retarded potentials. Classically, this is equivalent to purely
retarded effects within a closed box from which no light escapes (e.g.,
see A, or J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. {\bf 17,} 157
(1945)). Analogous theorems exist in quantum mechanics but it would
lead us too far astray to discuss them now.} for when this
interaction is represented by photons they must be of
only positive energy, while the Fourier transform of
$\delta(t_{56} - r_{56})$ contains frequencies of both signs. It
should instead be replaced by $\delta_+(t_{56} - r_{56})$ where
\begin{equation}
\delta_+(x) = \int \limits_0^{\infty} e^{-i \omega x} d \omega/ \pi = \lim
\limits_{\epsilon \rightarrow 0} \frac{(\pi i)^{-1}}{x - i \epsilon} =
\delta(x) + (\pi i x)^{-1}.
\end{equation}
This is to be averaged with $r_{56}^{-1} \delta_+(-t_{56} - r_{56})$ which
arises when $t_5 < t_6$ and corresponds to $a$ emitting the
quantum which $b$ receives. Since
$$
(2r)^{-1} (\delta_+ (t - r) + \delta_+(- t - r)) = \delta_+(t^2
- r^2),
$$
this means $r_{56}^{-1} \delta(t_{56})$ is replaced by $\delta_+(s_{56}^2)$
where $s_{56}^2 = t_{56}^2 - r_{56}^2$ is the square of the
relativistically invariant interval between points 5 and 6. Since in
classical electrodynamics there is also an interaction
through the vector potential, the complete interaction
(see A, Eq. (I)) should be \mbox{$(1 - ({\bf v}_5 \cdot {\bf v}_6)
\delta_+ (s_{56}^2)$,} or in the relativistic case,
$$
(1 - {\bf \alpha}_a \cdot {\bf \alpha}_b) \delta_+ (s_{56}^2) = \beta_a
\beta_b \gamma_{a \mu} \gamma_{b \mu} \delta_+ (s_{56} ^2).
$$
Hence we have for electrons obeying the Dirac equation,
\begin{equation}
\begin{array}{c}
K^{(1)}(3,4;1,2) = i e^2 \int \int K_{+a}(3,5)K_{+b}(4,6) \gamma_{a \mu}
\gamma_{b \mu} \\\\
\times \delta_+ (s_{56}^2)K_{+a}(5,1)K_{+b}(6,2) d \tau_5 d \tau_6,
\end{array}
\end{equation}
where $\gamma_{a \mu}$ and $\gamma_{b \mu}$, are the Dirac matrices applying
to the spinor corresponding to particles $a$ and $b$,
respectively (the factor $\beta_a \beta_b$ being absorbed in the
definition, I Eq. (17), of $K_+$).
This is our fundamental equation for
electrodynamics. It describes the effect of exchange
of one quantum (therefore first order in $e^2$) between
two electrons. It will serve as a prototype enabling
us to write down the corresponding quantities
involving the exchange of two or more quanta
between two electrons or the interaction of an
electron with itself. It is a consequence of conventional electrodynamics.
Relativistic invariance is clear. Since one sums over $\mu$ it contains the effects
of both longitudinal and transverse waves in a relativistically symmetrical way.
We shall now interpret Eq. (4) in a manner which
will permit us to write down the higher order terms.
It can be understood (see Fig. 1) as saying that the
amplitude for ``$a$'' to go from 1 to 3 and ``$b$'' to go
from 2 to 4 is altered to first order because they can
exchange a quantum. Thus, ``$a$'' can go to 5 (amplitude $(K_+(5,1))$
emit a quantum (longitudinal, transverse, or scalar
$\gamma_{a \mu}$) and then proceed to 3 $(K_+(3,5))$. Meantime ``$b$''
goes to 6 $(K_+(6,2))$, absorbs the quantum $(\gamma_{b \mu})$ and
proceeds to 4 $(K_+(4,6))$. The quantum meanwhile
proceeds from 5 to 6, which it does with amplitude
$\delta_+(s_{56}^2)$. We must sum over all the possible quantum
polarizations it and positions and times of emission
5, and of absorption 6. Actually if $t_5 > t_6$ it would be
better to say that ``$a$'' absorbs and ``$b$'' emits but no
attention need be paid to these matters, as all such
alternatives are automatically contained in (4).
The correct terms of higher order in $e^2$ or
involving larger numbers of electrons (interacting
with themselves or in pairs) can be written down by
the same kind of reasoning. They will be illustrated
by examples as we proceed. In a succeeding paper
they will all be deduced from conventional quantum
electrodynamics.
Calculation, from (4), of the transition element between positive
energy free electron states gives the
M$\varnothing$ller scattering of two electrons, when account is
taken of the Pauli principle.
The exclusion principle for interacting charges is
handled in exactly the same way as for noninteracting charges (I).
For example, for two charges
it requires only that one calculate $K(3,4;1,2) - K(4,3;1,2)$
to get the net amplitude for arrival of charges
at 3 and 4. It is disregarded in intermediate states.
The interference effects for scattering of electrons by
positrons discussed by Bhabha will be seen to result
directly in this formulation. The formulas are
interpreted to apply to positrons in the manner discussed in I.
As our primary concern will be for processes in
which the quanta are virtual we shall not include
here the detailed analysis of processes involving real
quanta in initial or final state, and shall content
ourselves by only stating the rules applying to
them.\footnote{Although in the expressions stemming from (4) the quanta are
virtual, this is not actually a theoretical limitation. One way to deduce
the correct rules for real quanta from (4) is to note that in a closed
system all quanta can be considered as virtual (i.e., they have a known
source and are eventually absorbed) so that in such a system the
present description is complete and equivalent to the conventional
one. In particular, the relation of the Einstein $A$ and $B$ coefficients can
be deduced. A more practical direct deduction of the expressions for
real quanta will be given in the subsequent paper. It might be noted
that (4) can be rewritten as describing the action on $a,~ K^{(1)}(3,1)
= i \int K_+(3,5) \times A(5) K_+(5,1) d \tau_5$ of the potential
$A_{\mu}(5) = e^2 \int K_+(4,6) \delta_+ (s_{56}^2) \gamma_{\mu} \times
K_+(6,2) d \tau_6$ arising from Maxwell's equations $- \square^2 A_{\mu}
= 4 \pi j_{\mu}$ from a ``current'' $j_{\mu}(6) = e^2 K_+ (4,6) \gamma_{\mu}
K_+(6,2)$ produced by particle $b$ in going from 2 to 4. This is virtue of the fact
that $\delta_+$ satisfies
\begin{equation}
- \square_2^2 \delta_+ (s_{21}^2) = 4 \pi \delta(2,1).
\end{equation}
}
The result of the analysis is, as expected, that they
can be included by the same line of reasoning as is
used in discussing the virtual processes, provided the
quantities are normalized in the usual manner to
represent single quanta. For example, the amplitude
that an electron in going from 1 to 2 absorbs a
quantum whose vector potential, suitably
normalized, is $c_{\mu} \mbox{exp} (-ik \cdot x) = C_{\mu}(x)$ is just the
expression (I, Eq. (13)) for scattering in a potential with ${\bf A}$ (3)
replaced by ${\bf C}$ (3). Each quantum interacts only
once (either in emission or in absorption), terms like
(I, Eq. (14)) occur only when there is more than one
quantum involved. The Bose statistics of the quanta
can, in all cases, be disregarded in intermediate
states. The only effect of the statistics is to change
the weight of initial or final states. If there are
among quanta, in the initial state, some a which are
identical then the weight of the state is $(1/n!)$ of what
it would be if these quanta were considered as
different (similarly for the final state).
\section{THE SELF--ENERGY PROBLEM}
~~~~Having a term representing the mutual interaction
of a pair of charges, we must include similar terms
to represent the interaction of a charge with itself.
For under some circumstances what appears to be
two distinct electrons may, according to I, be viewed
also as a single electron (namely in case one electron
was created in a pair with a positron destined to
annihilate the other electron). Thus to the interaction
between such electrons must correspond the
possibility of the action of an electron on itself.\footnote{These considerations
make it appear unlikely that the contention of
J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. {\bf 17,} 157 (1945),
that electrons do not act on themselves, will be a successful concept in
quantum electrodynamics.}
This interaction is the heart of the self energy
problem. Consider to first order in $e^2$ the action of an
electron on itself in an otherwise force free region.
The amplitude $K(2,1)$ for a single particle to get
from 1 to 2 differs from $K_+(2,1)$ to first order in $e^2$ by a term
\begin{equation}
\begin{array}{c}
K^{(1)}(2,1) = - ie^2 \int \int K_+(2,4) \gamma_{\mu} K_+(4,3) \gamma_{\mu}\\\\
\times K_+(3,1) d \tau_3 d \tau_4 \delta_+ (s_{43}^2).
\end{array}
\end{equation}
It arises because the electron instead of going from 1
directly to 2, may go (Fig. 2) first to 3, $(K_+(3,1))$,
emit a quantum $(\gamma_{\mu})$, proceed to 4, $(K_+(4,3))$, absorb
it $(\gamma_{\mu})$, and finally arrive at 2 $(K_+(2,4))$. The
quantum must go from 3 to 4 $(\delta_+(s_{43}^2))$.
This is related to the self-energy of a free electron
in the following manner. Suppose initially, time $t_1$,
we have an electron in state $f(1)$ which we imagine to
be a positive energy solution of Dirac's equation for
a free particle. After a long time $t_2 - t_1$ the
perturbation will alter
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig2e.gif}}}
\caption{Interaction of an electron with itself, Eq. (6).}
\end{figure}
the wave function, which can then be looked upon
as a superposition of free particle solutions (actually
it only contains $f$). The amplitude that $g(2)$ is
contained is calculated as in (I, Eq. (21)). The
diagonal element $(g = f)$ is therefore
\begin{equation}
\int \int \tilde f(2) \beta K^{(1)}(2,1)\beta f(1) d^3 {\bf x}_1 d^3 {\bf
x}_2.
\end{equation}
The time interval $T = t_2 - t_1$ (and the spatial
volume $V$ over which one integrates) must be taken
very large, for the expressions are only approximate
(analogous to the situation for two interacting
charges).\footnote{This is discussed in reference 5 in which it is pointed
out that the concept of a wave function loses accuracy if there are delayed
self-actions.} This is because, for example, we are
dealing incorrectly with quanta emitted just before
$t_2$ which would normally be reabsorbed at times after $t_2$.
If $K^{(1)}(2,1)$ from (6) is actually substituted into
(7) the surface integrals can be performed as was
done in obtaining I, Eq. (22) resulting in
\begin{equation}
-ie^2 \int \int \tilde f(4) \gamma_{\mu} K_+(4,3) \gamma_{\mu} f(3) \delta_+
(s_{43}^2) d \tau_3 d \tau_4.
\end{equation}
Putting for $f(1)$ the plane wave $u~ \mbox{exp}(-ip \cdot x_1)$ where $p_{\mu}$
is the energy $(p_4)$ and momentum of the electron
$({\bf p}^2 = m^2)$, and $u$ is a constant 4-index symbol, (8) becomes
$$
\begin{array}{cc}
~~~~~-ie^2 \int \int (\tilde u \gamma_{\mu} K_+(4,3) \gamma_{\mu} u)&\\\\
&\times \mbox{exp} (ip \cdot (x_4 - x_2)) \delta_+ (s_{43}^2) d \tau_3 d
\tau_4,
\end{array}
$$
the integrals extending over the volume $V$ and time
interval $T$. Since $K_+(4,3)$ depends only on the
difference of the coordinates of 4 and 3, $x_{43 \mu}$, the
integral on 4 gives a result (except near the surfaces
of the region) independent of 3. When integrated on
3, therefore, the result is of order $VT$. The effect is
proportional to $V$, for the wave functions have been normalized to unit
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig3e.gif}}}
\caption{Interaction of an electron with itself. Momentum space, Eq. (11).}
\end{figure}
volume. If normalized to volume $V$, the result would
simply be proportional to $T$. This is expected, for if
the effect were equivalent to a change in energy
$\Delta E$, the amplitude for arrival in $f$ at $t_2$ is altered by a
factor $\mbox{exp} (-i \Delta E(t_2 - t_1))$, or to first order by the
difference $-i(\Delta E) T$. Hence, we have
\begin{equation}
\Delta E = e^2 \int (\tilde u \gamma_{\mu} K_+(4,3) \gamma_{\mu} u) \mbox{exp}
(ip \cdot x_{43}) \delta_+ (s_{43}^2) d \tau_4,
\end{equation}
integrated over all space-time $d \tau_4$. This expression
will be simplified presently. In interpreting (9) we
have tacitly assumed that the wave functions are
normalized so that $(u^{\ast} u = (\tilde u \gamma_4 u) = 1$. The equation
may therefore be made independent of the
normalization by writing the left side as $(\Delta E) (\tilde u \gamma_4)
u)$, or since $(\tilde u \gamma_4 u) = (E/m) (\tilde u u)$ and $m \Delta
m = E \Delta E$, as $\Delta m (\tilde u u)$
where $\Delta m$ is an equivalent change in mass of
the electron. In this form invariance is obvious.
One can likewise obtain an expression for the
energy shift for an electron in a hydrogen atom.
Simply replace $K_+$ in (8), by $K_+^{(V)}$, the exact kernel
for an electron in the potential, ${\bf V} = \beta e^2/r$, of the
atom, and $f$ by a wave function (of space and time)
for an atomic state. In general the $\Delta E$ which results
is not real. The imaginary part is negative and in
$\mbox{exp} (-i \Delta ET)$ produces an exponentially decreasing
amplitude with time. This is because we are asking
for the amplitude that an atom initially with no
photon in the field, will still appear after time $T$ with
no photon. If the atom is in a state which can
radiate, this amplitude must decay with time. The
imaginary part of $\Delta E$ when calculated does indeed
give the correct rate of radiation from atomic states.
It is zero for the ground state and for a free electron.
In the non-relativistic region the expression for
$\Delta E$ can be worked out as has been done by
Bethe.\footnote{H. A. Bethe, Phys. Rev. {\bf 72,} 339 (1947).}
In the relativistic region (points 4 and 3 as close
together as a Compton wave-length) the $K_+^{(V)}$ which
should appear in (8) can be replaced to first order in
${\bf V}$ by $K_+$ plus $K_+^{(1)}(2,1)$ given in I, Eq. (13). The
problem is then very similar to the radiationless
scattering problem discussed below.
\section{EXPRESSION IN MOMENTUM AND\\ ENERGY SPACE}
~~~~The evaluation of (9), as well as all the other more
complicated expressions arising in these problems,
is very much simplified by working in the
momentum and energy variables, rather than space
and time. For this we shall need the Fourier
Transform of $\delta_+(s_{21}^2)$ which is
\begin{equation}
- \delta_+(s_{21}^2) = \pi^{-1} \int \mbox{exp}(-ik \cdot x_{21}) {\bf
k}^{-2}
d^4 k,
\end{equation}
which can be obtained from (3) and (5) or from I, Eq. (32)
noting that $I_+(2,1)$ for $m^2 = 0$ is $\delta_+(s_{21}^2)$ from
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig4e.gif}}}
\caption{Radiative correction to scattering, momentum space.}
\end{figure}
I, Eq. (34). The ${\bf k}^{-2}$ means $(k \cdot k)^{-1}$ or more precisely
the limit as $\delta \rightarrow 0$ of $(k \cdot k + i \delta)^{-1}$.
Further $d^4 k$ means $(2 \pi)^{-2} dk_1 dk_2 dk_3 dk_4$. If we imagine that quanta
are particles of zero mass, then we can make the
general rule that all poles are to be resolved by
considering the masses of the particles and quanta
to have infinitesimal negative imaginary parts.
Using these results we see that the self-energy (9)
is the matrix element between u and u of the matrix
\begin{equation}
(e^2/ \pi i) \int \gamma_{\mu} ({\bf p - k} - m)^{-1} \gamma_{\mu} {\bf k}^{-2} d^4
k,
\end{equation}
where we have used the expression (I, Eq. (31)) for
the Fourier transform of $K_+$. This form for the self-energy is easier
to work with than is (9).
The equation can be understood by imagining
(Fig. 3) that the electron of momentum ${\bf p}$ emits $(\gamma_{\mu})$
a quantum of momentum ${\bf k}$, and makes its way now
with momentum ${\bf p - k}$ to the next event (factor $({\bf p - k} - m)^{-1})$
which is to absorb the quantum (another $\gamma_{\mu}$). The amplitude of
propagation of quanta is ${\bf k}^{-2}$.
(There is a factor $e^2/ \pi i$ for each virtual quantum).
One integrates over all quanta. The reason an
electron of momentum ${\bf p}$ propagates as $1/({\bf p} - m)$ is
that this operator is the reciprocal of the Dirac
equation operator, and we are simply solving this
equation. Likewise light goes as $1/{\bf k}^2$, for this is the
reciprocal D'Alembertian operator of the wave
equation of light. The first $\gamma_{\mu}$, represents the current
which generates the vector potential, while the
second is the velocity operator by which this potential is multiplied
in the Dirac equation when an external field acts on an electron.
Using the same line of reasoning, other problems
may be set up directly in momentum space. For
example, consider the scattering in a potential
${\bf A} = A_{\mu} \gamma_{\mu}$ varying in space and time as ${\bf a}~
\mbox{exp}(-iq \cdot x)$. An electron initially in state of momentum
${\bf p}_1 = p_{1 \mu} \gamma_{\mu}$ will be deflected to state ${\bf p}_2$
where ${\bf p}_2 = {\bf p}_1 + {\bf q}$.
The zero-order answer is simply the matrix element
of ${\bf a}$ between states 1 and 2. We next ask for the
first order (in $e^2$) radiative correction due to virtual
radiation of one quantum. There are several ways
this can happen. First for the case illustrated in Fig. 4(a), find the matrix:
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig5e.gif}}}
\caption{Compton scattering, Eq. (15).}
\end{figure}
\begin{equation}
(e^2/ \pi i) \int \gamma_{\mu} ({\bf p}_2 - {\bf k} - m)^{-1} {\bf a} ({\bf
p}_1 - {\bf k} - m)^{-1} \gamma_{\mu} {\bf k}^{-2} d^4 k.
\end{equation}
For in this case, first\footnote{First, next, etc., here refer not to the order
in true time but to the succession of events along the trajectory of the electron.
That is, more precisely, to the order of appearance of the matrices in the
expressions.} a quantum of momentum ${\bf k}$ is
emitted $(\gamma_{\mu})$, the electron then having momentum
${\bf p}_1 - {\bf k}$ and hence propagating with factor $({\bf p}_1 -
{\bf k} - m)^{-1}$. Next it is scattered by the potential (matrix ${\bf
a}$) receiving additional momentum ${\bf q}$, propagating on
then (factor $({\bf p}_2 - {\bf k} - m)^{-1}$) with the new momentum
until the quantum is reabsorbed $(\gamma_{\mu})$. The quantum
propagates from emission to absorption $({\bf k}^{-2})$ and
we integrate over all quanta $(d^4 k)$, and sum on
polarization $\mu$. When this is integrated on $k_4$, the
result can be shown to be exactly equal to the
expressions (16) and (17) given in $B$ for the same
process, the various terms coming from residues of
the poles of the integrand (12).
Or again if the quantum is both emitted and reabsorbed before the scattering
takes place one finds (Fig. 4(b))
\begin{equation}
(e^2/ \pi i) \int {\bf a} ({\bf p}_1 - m)^{-1} \gamma_{\mu} ({\bf p}_1
- {\bf k} - m)^{-1} \gamma_{\mu} {\bf k}^{-2} d^4 k,
\end{equation}
or if both emission and absorption occur after the scattering, (Fig. 4(c))
\begin{equation}
(e^2/ \pi i) \int \gamma_{\mu} ({\bf p}_2 - {\bf k} - m)^{-1} \gamma_{\mu}
({\bf p}_2 - m)^{-1} {\bf ak}^{-2} d^4 k.
\end{equation}
These terms are discussed in detail below.
We have now achieved our simplification of the
form of writing matrix elements arising from virtual
processes. Processes in which a number of real
quanta is given initially and finally offer no problem
(assuming correct normalization). For example,
consider the Compton effect (Fig. 5(a)) in which an
electron in state ${\bf p}_1$ absorbs a quantum of momentum
${\bf q}_1$, polarization vector $e{1 \mu}$ so that its interaction is
$e_{1 \mu} \gamma_{\mu} = {\bf e}_1$, and emits a second quantum of
momentum $-{\bf q}_2$ polarization ${\bf e}_2$ to arrive in final
state of momentum ${\bf p}_2$. The matrix for
this process is ${\bf e}_2 ({\bf p}_1 + {\bf q}_1 - m)^{-1} {\bf e}_1$.
The total matrix for the Compton effect is, then,
\begin{equation}
{\bf e}_2 ({\bf p}_1 + {\bf q}_1 - m)^{-1} {\bf e}_1 + {\bf e}_1 ({\bf
p}_1 + {\bf q}_2 - m)^{-1} {\bf e}_3,
\end{equation}
the second term arising because the emission of ${\bf e}_2$,
may also precede the absorption of ${\bf e}_1$ (Fig. 5(b)).
One takes matrix elements of this between initial and
final electron states $({\bf p}_1 + {\bf q}_1 = {\bf p}_2 - {\bf q}_2)$,
to obtain the Klein Nishina formula. Pair annihilation with
emission of two quanta, etc., are given by the same
matrix, positron states being those with negative
time component of ${\bf p}$. Whether quanta are absorbed
or emitted depends on whether the time component of ${\bf q}$ is positive or negative.
\section{THE CONVERGENCE OF PROCESSES WITH VIRTUAL QUANTA}
~~~~These expressions are, as has been indicated, no
more than a re-expression of conventional quantum
electrodynamics. As a consequence, many of them
are meaningless. For example, the self-energy
expression (9) or (11) gives an infinite result when
evaluated. The infinity arises, apparently, from the
coincidence of the $\delta$--function singularities in $K_+(4,3)$
and $\delta_+(s_{43}^2)$. Only at this point is it necessary to
make a real departure from conventional
electrodynamics, a departure other than simply
rewriting expressions in a simpler form.
We desire to make a modification of quantum
electrodynamics analogous to the modification of
classical electrodynamics described in a previous
article, $A$. There the $\delta(s_{12}^2)$ appearing in the action of
interaction was replaced by $f(s_{12}^2)$ where $f(x)$ is a
function of small width and great height.
The obvious corresponding modification in the
quantum theory is to replace the $\delta_+(s^2)$ appearing the
quantum mechanical interaction by a new function
$f_=(s^2)$. We can postulate that if the Fourier transform
of the classical $f(s_{12}^2)$ is the integral over ail ${\bf k}$ of $F({\bf
k}^2) \mbox{exp}(-ik \cdot x_{12})d^4 k$, then the Fourier transform of $f_+(s^2)$
is the same integral taken over only positive frequencies $k_4$ for $t_2
> t_1$ and over only negative ones for $t_2 < t_1$
in analogy to the relation of $\delta_+(s^2)$ to $\delta(s^2)$. The
function $f(s^2) = f(x \cdot x)$ can be written \footnote{This relation is
given incorrectly in $A$, equation just preceding 16.} as
$$
\begin{array}{ll}
f(x \cdot x) = (2 \pi)^{-2} \int \limits^{\infty}_{k_4 = 0} \int \sin (k_4|x_4|)&\\\\
&\times \cos ({\bf K \cdot x}) dk_4 d^3 {\bf K} g(k \cdot k),
\end{array}
$$
where $g(k \cdot k)$ is $k_4^{-1}$ times the density of oscillators
and may be expressed for positive $k_4$ as (A, Eq. (16))
$$
g({\bf k}^2) = \int \limits^{\infty}_0 (\delta({\bf k}^2) - \delta ({\bf
k}^2 - \lambda^2)) G(\lambda) d\lambda,
$$
where $\int \limits^{\infty}_0 G(\lambda) d\lambda = 1$ and $G$ involves values
of $\lambda$ large compared to $m$. This simply means that the amplitude
for propagation of quanta of momentum ${\bf k}$ is
$$
-F_+ ({\bf k}^3) = \pi^{-1} \int^{\infty}_0 ({\bf k}^{-2} - ({\bf k}^2
- \lambda^2)^{-1})G (\lambda) d \lambda,
$$
rather than ${\bf k}^{-2}$. That is, writing $F_+({\bf k}^2) = - \pi^{-1}
{\bf k}^{-2} C({\bf k}^2)$,
\begin{equation}
-f_+ (s_{12}^2) = \pi^{-1} \int \mbox{exp} (-ik \cdot x_{12}){\bf k}^{-2}
C ({\bf k}^2) d^4 k.
\end{equation}
Every integral over an intermediate quantum which
previously involved a factor $d^4k/{\bf k}^2$ is now supplied
with a convergence factor $C({\bf k}^2)$ where
\begin{equation}
C({\bf k}^2) = \int^{\infty}_0 - \lambda^2 ({\bf k}^2 - \lambda^2)^{-1}
G(\lambda) d \lambda.
\end{equation}
The poles are defined by replacing ${\bf k}^2$ by ${\bf k}^2 + i \delta$ in the
limit $\delta \rightarrow 0$. That is $\lambda^2$ may be assumed to have an
infinitesimal negative imaginary part.
The function $f_+(s_{12}^2)$ may still have a discontinuity
in value on the light cone. This is of no influence for
the Dirac electron. For a particle satisfying the Klein
Gordon equation, however, the interaction involves
gradients of the potential which reinstates the $\delta$ function if
$f$ has discontinuities. The condition that $f$ is to
have no discontinuity in value on the light cone
implies ${\bf k}^2 C({\bf k}^2)$ approaches zero as ${\bf k}^2$ approaches
infinity. In terms of $G(\lambda)$ the condition is
\begin{equation}
\int \limits^{\infty}_0 \lambda^2 G(\lambda) d \lambda = 0.
\end{equation}
This condition will also be used in discussing the
convergence of vacuum polarization integrals.
The expression for the self-energy matrix is now
\begin{equation}
(e^2/ \pi i) \int \gamma_{\mu} ({\bf p - k} - m)^{-1} \gamma_{\mu} {\bf
k}^{-2} d^4 kC ({\bf k}^2),
\end{equation}
which, since $C({\bf k}^2)$ falls off at least as rapidly as $1/{\bf k}^2$,
converges. For practical purposes we shall suppose
hereafter that $C({\bf k}^2)$ is simply $- \lambda^2/ ({\bf k}^2 - \lambda^2)$
implying that some average (with weight $G(\lambda)d \lambda$)
over values of $\lambda$ may be taken afterwards. Since in
all processes the quantum momentum will be
contained in at least one extra factor of the form $({\bf p - k} - m)^{-1}$
representing propagation of an electron
while that quantum is in the field, we can expect all
such integrals with their convergence factors to
converge and that the result of ail such processes will
now be finite and definite (excepting the processes
with closed loops, discussed below, in which the
diverging integrals are over the momenta of the electrons rather than the quanta).
The integral of (19) with $C({\bf k}^2) = - \lambda^2 ({\bf k}^2 - \lambda^2)^{-1}$
noting that \mbox{${\bf p}^2 = m^2,$} $\lambda \gg m$ and dropping terms of
order $m/\lambda$ is (see Appendix A)
\begin{equation}
(e^2/ 2 \pi) [4m (\ln (\lambda /m) + \frac{1}{2}) - {\bf p} (\ln (\lambda
/m) + 5/4)].
\end{equation}
When applied to a state of an electron of
momentum ${\bf p}$ satisfying ${\bf p}u = mu$, it gives for the
change in mass (as in B, Eq. (9))
\begin{equation}
\Delta m = m (e^2/ 2 \pi) (3 \ln (\lambda/m)+ \frac{3}{4}).
\end{equation}
\section{RADIATIVE CORRECTIONS TO\\ SCATTERING}
~~~~We can now complete the discussion of the
radiative corrections to scattering. In the integrals
we include the convergence factor $C({\bf k}^2)$, so that
they converge for large ${\bf k}$. Integral (12) is also not
convergent because of the well-known infrared
catastrophy. For this reason we calculate (as
discussed in B) the value of the integral assuming
the photons to have a small mass $\lambda_{\mbox{min}} \ll m \ll \lambda$.
The integral (12) becomes
$$
\begin{array}{c}
(e^2 / \pi i) \int \gamma_{\mu} ({\bf p}_2 - {\bf k} - m)^{-1} {\bf a}
({\bf p}_1 - {\bf k} - m)^{-1}\\\\
~~~~~~~~~~~~\times \gamma_{\mu} ({\bf k}^2 - \gamma_{\mbox{min}}^2)^{-1} d^4 k C ({\bf
k}^2 - \lambda_{\mbox{min}}^2),
\end{array}
$$
which when integrated (see Appendix B) gives $(e^2/ 2 \pi)$ times
\begin{equation}
\begin{array}{c}
\left[ 2 \left( \ln \frac{\displaystyle m}{\displaystyle \lambda_{\mbox{min}}}
- 1 \right) \left( 1 - \frac{\displaystyle 2 \theta}{\displaystyle {\rm
tan} 2 \theta} \right) + \theta {\rm tan} \theta \right.\\\\
\left. + \frac{\displaystyle 4}{\displaystyle {\rm tan} 2 \theta} \int \limits^{\theta}_0
\alpha ~{\rm tan}\alpha d \alpha \right] {\bf a} + \frac{\displaystyle 1}{\displaystyle 4m}
({\bf qa - aq}) \frac{\displaystyle 2 \theta}{\displaystyle \sin 2 \theta}
+ r {\bf a},
\end{array}
\end{equation}
where $({\bf q}^2)^{1/2} = 2 m$ and we have assumed the
matrix to operate between states of momentum ${\bf p}_1$
and ${\bf p}_2 = {\bf p}_1 + {\bf q}$ and have neglected terms of order
$\lambda_{\mbox{min}}/m, m/ \lambda$, and ${\bf q}^2 / \lambda^2$.
Here the only dependence on the convergence factor is in the term ${\bf
ra}$, where
\begin{equation}
{\bf r} = \ln (\lambda / m) + 9/4 - 2 \ln (m/ \lambda_{\mbox{min}}).
\end{equation}
As we shall see in a moment, the other terms (13),
(14) give contributions which just cancel the ${\bf ra}$
term. The remaining terms give for small ${\bf q}$,
\begin{equation}
(e^2 / 4 \pi) \left( \frac{1}{2m} ({\bf qa - aq}) + \frac{4 {\bf q}^2}{3m^2}
{\bf a} \left( \ln \frac{m}{\lambda_{\mbox{min}}} - \frac{3}{8} \right)
\right),
\end{equation}
which shows the change in magnetic moment and
the Lamb shift as interpreted in more detail in
B.\footnote{That the result given in B in Eq. (19) was in error was repeatedly
pointed out to the author, in private communication, by V. F. Weisskopf
and J. B. French, as their calculation, completed simultaneously with the
author's early in 1948, gave a different result. French has finally
shown that although the expression for the radiationless scattering B,
Eq. (18) or (24) above is correct, it was incorrectly joined into
Bethe's non-relativistic result. He shows that the relation
$\ln 2 k_{\mbox{max}} - 1 = \ln \lambda_{\mbox{min}}$ used by the author
should have been $\ln 2 k_{\mbox{max}} - 5/6 = \ln \lambda_{\mbox{min}}$.
This results in adding a term $-(1/6)$ to the
logarithm in B, Eq. (19) so that the result now agrees with that of
J. B. French and V. F. Weisskopf, Phys. Rev. {\bf 75,} 1240 (1949) and
N. H. Kroll and W. E. Lamb, Phys. Rev. {\bf 75,} 388 (1949). The author
feels unhappily responsible for the very
considerable delay in the publication of French's
result occasioned by this error. This footnote is
appropriately numbered.}
We must now study the remaining terms (13) and
(14). The integral on ${\bf k}$ in (13) can be performed
(after multiplication by $C({\bf k}^2)$) since it involves
nothing but the integral (19) for the self-energy and
the result is allowed to operate on the initial state $u_1$,
(so that ${\bf p}_1 u_1 = mu_1$). Hence the factor following
${\bf a}({\bf p}_1 - m)^{-1}$ will be just $\Delta m$. But, if one now tries to
expand $1/({\bf p}_1 - m) = ({\bf p}_1 + m)/({\bf p}_1^2 - m^2)$ one obtains
an infinite result, since ${\bf p}_1^2 = m^2$. This is, however,
just what is expected physically. For the quantum
can be emitted and absorbed at any time previous to
the scattering. Such a process has the effect of a
change in mass of the electron in the state 1. It
therefore changes the energy by $\Delta E$ and the
amplitude to first order in $\Delta E$ by $- i \Delta E \cdot t$ where
$t$ is the time it is acting, which is infinite. That is, the
major effect of this term would be canceled by the
effect of change of mass $\Delta m$.
The situation can be analyzed in the following
manner. We suppose that the electron approaching
the scattering potential ${\bf a}$ has not been free for an
infinite time, but at some time far past suffered a
scattering by a potential ${\bf b}$. If we limit our
discussion to the effects of $\Delta m$ and of the virtual
radiation of one quantum between two such
scatterings each of the effects will be finite, though
large, and their difference is determinate. The
propagation from {\bf b} to ${\bf a}$ is represented by a matrix
\begin{equation}
{\bf a}({\bf p}' - m)^{-1} {\bf b},
\end{equation}
in which one is to integrate possibly over ${\bf p}'$
(depending on details of the situation). (If the time
is long between ${\bf b}$ and ${\bf a}$, the energy is very nearly
determined so that ${\bf p}^{'2}$ is very nearly $m^2$.)
We shall compare the effect on the matrix (25) of
the virtual quanta and of the change of mass $\Delta m$.
The effect of a virtual quantum is
\begin{equation}
(e^2/ \pi i) \int {\bf a} ({\bf p}' - m)^{-1} \gamma_{\mu} ({\bf p}' -
{\bf k} - m)^{-1} \times \gamma_{\mu} ({\bf p}' - m)^{-1} {\bf bk}^{-2}
d^4 kC ({\bf k}^2),
\end{equation}
while that of a change of mass can be written
\begin{equation}
{\bf a}({\bf p}' - m)^{-1} \Delta m ({\bf p}' - m)^{-1} {\bf b},
\end{equation}
and we are interested in the difference (26)-(27). A
simple and direct method of making this
comparison is just to evaluate the integral on ${\bf k}$ in
(26) and subtract from the result the expression (27)
where $\Delta m$ is given in (21). The remainder can be
expressed as a multiple $- r({\bf p}^{'2})$ of the unperturbed
amplitude (25);
\begin{equation}
- r ({\bf p}^{'2}) {\bf a} ({\bf p}' - m)^{-1} {\bf b}.
\end{equation}
This has the same result (to this order) as replacing
the potentials ${\bf a}$ and ${\bf b}$ in (25) by $( 1 - \frac{1}{2} r
({\bf p}^{'2})){\bf a}$ and
$(1 - \frac{1}{2} {\bf r} ({\bf p}^{'2})){\bf b}$.
In the limit, then, as ${\bf p}^{'2} \rightarrow m^2$ the net effect
on the scattering is $- \frac{1}{2} {\bf ra}$ where ${\bf r}$,
the limit of ${\bf r}({\bf p}^{'2})$
as ${\bf p}^{'2} \rightarrow m^2$ (assuming the integrals have an
infrared cut-off), turns out to be just equal to that given in (23).
An equal term $- \frac{1}{2} {\bf ra}$ arises from virtual transitions
after the scattering (14) so that the entire ra term in (22) is canceled.
The reason that ${\bf r}$ is just the value of (12) when ${\bf q}^2 = 0$
can also be seen without a direct calculation as follows:
Let us call ${\bf p}$ the vector of length $m$ in the direction
of ${\bf p}'$ so that if ${\bf p}^{'2} = m (1 + \epsilon)^2$ we have
${\bf p}' = (1 + \epsilon0{\bf p}$ and
we take $\epsilon$ as very small, being of order $T^{-1}$ where $T$
is the time between the scatterings ${\bf b}$ and ${\bf a}$. Since
$({\bf p}' - m)^{-1} = ({\bf p}' + m)/({\bf p}^{'2} - m^2) \approx ({\bf
p} + m)/2m^2 \epsilon$, the quantity (25) is of order $\epsilon^{-1}$
or $T$. We shall compute corrections to it only to its own order $(\epsilon^{-1})$
in the limit $\epsilon \rightarrow 0$. The term (27) can be written
approximately\footnote{The expression is not exact because the substitution
of $\Delta m$ by the integral in (19) is valid only if ${\bf p}$ operates on a
state such that ${\bf p}$ can be replaced by $m$. The error, however, is of
order ${\bf a}({\bf p}' - m)^{-1} ({\bf p} - m)({\bf p}' - m)^{-1} {\bf
b}$ which is ${\bf a}((1 + \epsilon) {\bf p} + m) ({\bf p} - m) \times
((1 + \epsilon) {\bf p} + m) {\bf p} (2 \epsilon + \epsilon^2)^{-2} m^{-4}$.
But since ${\bf}^2 = m^2$ we have ${\bf p}({\bf p} - m)= - m ({\bf p}
- m) = ({\bf p} - m){\bf p}$ so the net result is approximately
${\bf a}({\bf p} - m){\bf b}/4m^2$ and is not of order $1/\epsilon$ but smaller,
so that its effect drops out in the limit.} as
$$
(e^2/\pi i) \int {\bf a} ({\bf p}' - m)^{-1} \gamma_{\mu} ({\bf p- k} -
m)^{-1} \times \gamma_{\mu} ({\bf p}' - m)^{-1} {\bf bk}^{-2} d^4 kC ({\bf
k}^2),
$$
using the expression (19) for $\Delta m$. The net of the two
effects is therefore approximately\footnote{We have used, to first order,
the general expansion (valid for any operators $A, B$)
$$
(A + B0^{-1} = A^{-1} - A^{-1} B A^{-1} + A^{-1} BA^{-1} BA^{-1} - \ldots
$$
with $A = {\bf p - k} - m$ and $B = {\bf p' - p} = \epsilon{\bf p}$ to expand
the difference of $({\bf p' - k} - m)^{-1}$ and $({\bf p - k} - m)^{-1}$.}
$$
\begin{array}{c}
-(e^2 / \pi i) \int {\bf a} ({\bf p'} - m)^{-1} \gamma_{\mu} ({\bf p -
k} - m)^{-1} \epsilon {\bf p} ({\bf p - k} - m)^{-1}\\\\
\times \gamma_{\mu} ({\bf p'} - m)^{-1} {\bf bk}^{-2} d^4 kC ({\bf k^2}),
\end{array}
$$
a term now of order $1/ \epsilon$ (since $({\bf p'} - m)^{-1} \approx ({\bf
p} + m) \times (2m^2 \epsilon)^{-1}$) and therefore the one desired in the limit.
Comparison to (28) gives for ${\bf r}$ the expression
\begin{equation}
\begin{array}{c}
({\bf p}_1 + m/2m) \int \gamma_{\mu} ({\bf p}_1 - {\bf k} - m)^{-1} ({\bf
p}_1 m^{-1})({\bf p}_1 - {\bf k} - m)^{-1}\\\\
\times \gamma_{\mu} {\bf k}^{-2} d^4 kC ({\bf k}^2).
\end{array}
\end{equation}
The integral can be immediately evaluated, since it
is the same as the integral (12), but with ${\bf q} = 0$, for ${\bf a}$
replaced by ${\bf p}_1/m$. The result is therefore ${\bf r} \cdot ({\bf
p}_1/m)$ which when acting on the state $u_1$ is just ${\bf r}$, as ${\bf
p}_1 u_1 = mu_1$. For the same reason the term $({\bf p}_1 + m)/2m$ in (29)
is effectively 1 and we are left with $- {\bf r}$ of
(23).\footnote{The renormalization terms appearing $B$, Eqs. (14), (15) when
translated directly into the present notation do not give twice (29) but
give this expression with the central ${\bf p}_1 m^{-1}$ factor replaced
by $m \gamma_4/E_1$ where $E_1 = p_{1 \mu}$, for $\mu = 4$. When integrated
it therefore gives ${\bf ra}(({\bf p}_1 + m)/2m)(m \gamma_4/E_1)$ or
${\bf ra - ra}(m \gamma_4/E_1)({\bf p}_1 - m)/2m$. (Since ${\bf p}_1 \gamma_4
+ \gamma_4 {\bf p}_1 = 2E_1$) which gives just ${ra}$, since ${\bf p}_1
u_1 = mu_1$.}
In more complex problems starting with a free electron the same type
of term arises from the effects of a
virtual emission and absorption both previous to the
other processes. They, therefore, simply lead to the
same factor ${\bf r}$ so that the expression (23) may be used
directly and these renormalization integrals need not
be computed afresh for each problem.
In this problem of the radiative corrections to
scattering the net result is insensitive to the cut-off.
This means, of course, that by a simple
rearrangement of terms previous to the integration
we could have avoided the use of the convergence
factors completely (see for example Lewis\footnote{H.W. Lewis, Phys. Rev.
{\bf 73}, 173 (1948).}). The
problem was solved in the manner here in order to
illustrate how the use of such convergence factors,
even when they are actually unnecessary, may
facilitate analysis somewhat by removing the effort
and ambiguities that may be involved in trying to
rearrange the otherwise divergent terms.
The replacement of $\delta_+$ by $f_+$ given in (16), (17) is
not determined by the analogy with the classical
problem. In the classical limit only the real part of $\delta_+$
(i.e., just $\delta$) is easy to interpret. But by what should
the imaginary part, $1/(\pi s^2)$, of $\delta_+$ be replaced? The
choice we have made here (in denning, as we have,
the location of the poles of (17)) is arbitrary and
almost certainly incorrect. If the radiation resistance
is calculated for an atom, as the imaginary part of (8),
the result depends slightly on the function $f_+$. On the
other hand the light radiated at very large distances
from a source is independent of $f_+$. The total energy
absorbed by distant absorbers will not check with the
energy loss of the source. We are in a situation
analogous to that in the classical theory if the entire $f$
function is made to contain only retarded
contributions (see A, Appendix). One desires instead
the analogue of $\langle F \rangle_{\mbox{ret}}$ of A.
This problem is being studied.
One can say therefore, that this attempt to find a
consistent modification of quantum electrodynamics
is incomplete (see also the question of closed loops,
below). For it could turn out that any correct form of
$f_+$ which will guarantee energy conservation may at
the same time not be able to make the self-energy
integral finite. The desire to make the methods of
simplifying the calculation of quantum
electrodynamic processes more widely available has
prompted this publication before an analysis of the
correct form for $f_+$ is complete. One might try to take
the position that, since the energy discrepancies
discussed vanish in the limit $\lambda \rightarrow \infty$, the correct
physics might be considered to be that obtained by
letting $\lambda \rightarrow \infty$ after mass renormalization. I have no
proof of the mathematical consistency of this
procedure, but the presumption is very strong that it
is satisfactory. (It is also strong that a satisfactory
form for $f_+$ can be found.)
\section{THE PROBLEM OF VACUUM\\ POLARIZATION}
~~~~In the analysis of the radiative corrections to
scattering one type of term was not considered. The
potential which we can assume to vary as $a_{\mu} \mbox{exp} (- iq \cdot
x)$ creates a pair of electrons (see Fig. 6), momenta ${\bf p}_{a,} - {\bf
p}_b$. This pair then reannihilates, emitting a
quantum ${\bf q} = {\bf q}_b - {\bf q}_{a,}$ which quantum scatters the
original electron from state 1 to state 2. The matrix
element for this process (and the others which can
be obtained by rearranging the order in time of the
various events) is
\begin{equation}
\begin{array}{c}
-(e^2/ \pi i) (\tilde u_2 \gamma_{\mu} u_1) \int S p [({\bf p}_a + {\bf
q} - m)^{-1}\\\\
\times \gamma_{\mu} ({\bf p}_a - m)^{-1} \gamma_{\mu}] d^4 p_a {\bf q}^{-2}
C ({\bf q}^2) a_{\nu}.
\end{array}
\end{equation}
This is because the potential produces the pair with
amplitude proportional to $a_{\nu} \gamma_{\nu}$ the electrons
of momenta ${\bf p}_a$, and $-({\bf p}_a + {\bf q})$ proceed from there to
annihilate, producing a quantum (factor $\gamma_{\mu}$) which
propagates (factor ${\bf q}^{-2}C({\bf q}^2)$) over to the other
electron, by which it is absorbed (matrix element of
$\gamma_{\mu}$, between states 1 and 2 of the original electron
$(\tilde u_2 \gamma_{\mu} u_1)$). All momenta ${\bf p}_a$ and spin states of the
virtual electron are admitted, which means the spur
and the integral on $d^4p_a$ are calculated.
One can imagine that the closed loop path of the
positron-electron produces a current
\begin{equation}
4 \pi J_{\mu} a_{\nu},
\end{equation}
which is the source of the quanta which act on the
second electron. The quantity
\begin{equation}
\begin{array}{c}
J_{\mu \nu} = - (e^2/ \pi i) \int Sp[({\bf p + q} - m)^{-1}\\\\
\times \gamma_{\mu} ({\bf p} - m)^{-1} \gamma_{\mu}] d^4 p,
\end{array}
\end{equation}
is then characteristic for this problem of
polarization of the vacuum.
One sees at once that $J_{\mu \nu}$ diverges badly. The
modification of $\delta$ to $f$ alters the amplitude with
which the current $j_{\mu}$, will affect the scattered
electron, but it can do nothing to prevent the
divergence of the integral (32) and of its effects.
One way to avoid such difficulties is apparent.
From one point of view we are considering all
routes by which a given electron can get from one
region of space-time to another, i.e., from the source
of electrons to the apparatus which measures them.
From this point of view the closed loop path leading
to (32) is unnatural. It might be assumed that the
only paths of meaning are those which start from
the source and work their way in a continuous path
(possibly containing many time reversals) to the
detector. Closed loops would be excluded. We have
already found that this may be done for electrons moving in a fixed potential.
Such a suggestion must meet several questions,
however. The closed loops are a consequence of the
usual hole theory in electrodynamics. Among other
things, they are required to keep probability
conserved. The probability that no pair is produced
by a potential is
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig6e.gif}}}
\caption{Vacuum polarization effect on scattering, Eq. (30).}
\end{figure}
not unity and its deviation from unity arises from
the imaginary part of $J_{\mu \nu}$. Again, with closed loops
excluded, a pair of electrons once created cannot
annihilate one another again, the scattering of light
by light would be zero, etc. Although we are not
experimentally sure of these phenomena, this does
seem to indicate that the closed loops are necessary.
To be sure, it is always possible that these matters
of probability conservation, etc., will work
themselves out as simply in the case of interacting
particles as for those in a fixed potential. Lacking
such a demonstration the presumption is that the
difficulties of vacuum polarization are not so easily
circumvented.\footnote{It would be very interesting to calculate the Lamb shift
accurately enough to be sure that the 20 megacycles expected from vacuum
polarization are actually present.}
An alternative procedure discussed in $B$ is to
assume that the function $K_+(2,1)$ used above is
incorrect and is to be replaced by a modified
function $K'_+$ having no singularity on the light cone.
The effect of this is to provide a convergence factor
$C({\bf p}^2 - m^2)$ for every integral over electron
momenta.\footnote{This technique also makes self-energy and radiationless
scattering integrals finite even without the modification of $\delta_+$ to $f_+$
for the radiation (and the consequent convergence factor $C({\bf k}^2)$ for the
quanta). See B.} This will multiply the integrand of (32)
by $C({\bf p}^2 - m^2)C(({\bf p + q})^2 - m^2)$, since the integral was
originally $\delta ({\bf p}_a - {\bf p}_b + {\bf q}) d^4 p_a d^4 p_b$
and both ${\bf p}_a$ and ${\bf p}_b$ get convergence factors. The integral now
converges but the result is unsatisfactory.\footnote{Added to the terms given
below (33) there is a term $\frac{1}{4} (\lambda^3 - 2 \mu^2 + \frac{1}{3}
{\bf q}^2) \delta_{\mu \nu}$ for $C({\bf k}^2) = - \lambda^2 ({\bf k}^2
- \lambda^2)^{-1}$ , which is not gauge invariant. (In
addition the charge renormalization has $- 7/6$ added to the
logarithm.)}
One expects the current (31) to be conserved, that
is $q_{\mu} j_{\mu} = 0$ or \mbox{$q_{\mu} J_{\mu \nu} = 0$.} Also one expects no
current if $a$ is a gradient, or $a_{\nu} = q_{\nu}$, times a constant. This leads
to the condition $J_{\mu \nu} q_{\nu} = 0$ which is equivalent to
$q_{\mu} J_{\mu \nu} = 0$ since $J_{\mu \nu}$ is symmetrical. But when the
expression (32) is integrated with such convergence
factors it does not satisfy this condition. By altering
the kernel from $K$ to another, $K'$, which does not
satisfy the Dirac equation we have lost the gauge
invariance, its consequent current conservation and
the general consistency of the theory.
One can see this best by calculating $J_{\mu \nu} q_{\nu}$ directly
from (32). The expression within the spur becomes
$({\bf p + q} - m)^{-1} {\bf q} ({\bf p} - m)^{-1} \gamma_{\mu}$ which can be
written as the difference of two terms: $({\bf p} - m)^{-1} \gamma_{\mu}
- ({\bf p + q} - m)^{-1} \gamma_{\mu}$
Each of these terms would give the same result if
the integration $d^4 p$ were without a convergence factor, for
the first can be converted into the second by a shift
of the origin of ${\bf p}$, namely ${\bf p}' = {\bf p + q}$. This does not
result in cancelation in (32) however, for the
convergence factor is altered by the substitution.
A method of making (32) convergent without
spoiling the gauge invariance has been found by
Bethe and by Pauli. The convergence factor for light
can be looked upon as the result of superposition of
the effects of quanta of various masses (some
contributing negatively). Likewise if we take the
factor $C({\bf p}^2 - m^2) = - \lambda^2 ({\bf p}^2 - m^2 - \lambda^2)^{-1}$
so that $({\bf p}^2 - m^2)^{-1} C({\bf p}^2 - m^2) = ({\bf p}^2 - m^2)^{-1}
- ({\bf p}^2 - m^2 - \lambda^2)^{-1}$ we are taking the difference
of the result for electrons of mass $m$ and mass
$(\lambda^2 + m^2)^{1/2}$. But we have taken this difference for
{\it each} propagation between interactions with photons.
They suggest instead that once created with a certain
mass the electron should continue to propagate with
this mass through all the potential interactions until
it closes its loop. That is if the quantity (32),
integrated over some finite range of ${\bf p}$, is called
$J_{\mu \nu}(m^2)$ and the corresponding quantity over the
same range of ${\bf p}$, but with $m$ replaced by $(m^2 + \lambda^2)^{1/2}$ is
$J_{\mu \nu} (m^2 + \lambda^2)$ we should calculate
$$
J_{\mu \nu}^{P} = \int \limits^{\infty}_0 [J_{\mu \nu}(m^2) - J_{\mu \nu}
(m^2 + \lambda^2)]G(\lambda)d\lambda, \eqno(32')
$$
the function $G(\lambda)$ satisfying $\int^{\infty}_0 G(\lambda) d \lambda
= 1$ and $\int^{\infty}_0 G(\lambda) \lambda^2 d \lambda = 0$.
Then in the expression for $J_{\mu \nu}^{P}$ the
range of ${\bf p}$ integration can be extended to infinity as
the integral now converges. The result of the
integration using this method is the integral on $d \lambda$
over $G(\lambda)$ of (see Appendix C)
\begin{equation}
\begin{array}{c}
J_{\mu \nu}^{P} = - \frac{\displaystyle e^2}{\displaystyle \pi} (q_{\mu} q_{\nu} -
\delta_{\mu \nu}
{\bf q}^2) \left( - \frac{\displaystyle 1}{\displaystyle 3} \ln
\frac{\displaystyle \lambda^2}{\displaystyle m^2} \right.\\\\
\left. - \left[ \frac{\displaystyle 4m^2 + 2{\bf q}^2}{\displaystyle 3 {\bf q}2} \left( 1 -
\frac{\displaystyle \theta}{\displaystyle \mbox{tan}\theta} \right) -
\frac{\displaystyle 1}{\displaystyle 9} \right] \right),
\end{array}
\end{equation}
with ${\bf q}^2 = 4m^2 \sin^2 \theta$.
The gauge invariance is clear, since $q_{\mu}(q_{\mu} q_{\nu} - {\bf q}^2
\delta_{\mu \nu}) = 0.$ Operating (as it always will) on a potential
of zero divergence the $(q_{\mu} q_{\nu} - \delta_{\mu \nu} {\bf q}^2)
a_{\nu}$, is simply $-q^2 a_{\mu}$, the D'Alembertian of the potential, that is, the
current producing the potential. The term $- \frac{1}{3} (\ln (\lambda^2/m^2))(q_{\mu}
q_{\nu} - {\bf q}^2 \delta_{\mu \nu})$ therefore gives a current proportional to the
current producing the potential. This would have the
same effect as a change in charge, so that we would
have a difference $\Delta(e^2)$ between $e^2$ and the
experimentally observed charge, $e^2 + \Delta(e^2)$,
analogous to the difference between $m$ and the
observed mass. This charge depends logarithmically
on the cut-off, $\Delta(e^2)/e^2 = - (2e^2/3\pi) \ln (\lambda/m)$.
After this renormalization of charge is made, no effects will be sensitive to the
cut-off.
After this is done the final term remaining in (33),
contains the usual effects\footnote{E. A. Uehling, Phys. Rev. {\bf 48,} 55 (1935),
R. Serber, Phys. Rev. {\bf 48,} 49 (1935).}
of polarization of the vacuum. It is zero for a free light quantum $({\bf
q}^2 = 0)$. For small ${\bf q}^2$ it behaves as $(2/15){\bf q}^2$ (adding $-
\frac{1}{5}$ to the logarithm in the Lamb effect). For \mbox{${\bf q}^2 > (2m)^2$ }
it is complex, the imaginary part representing the loss in
amplitude required by the fact that the probability
that no quanta are produced by a potential able to
produce pairs \mbox{$(({\bf q}^2)^{1/2} > 2m)$} decreases with time. (To
make the necessary analytic continuation, imagine
$m$ to have a small negative imaginary part, so that
\mbox{$(1 - {\bf q}^2/4m^2 - 1)^{1/2}$} becomes
$-i({\bf q}^2/4m^2 - 1)^{1/2}$ as ${\bf q}^2$ goes from below to above $4m^2$. Then
$\theta = \pi/2 + iu$ where $\sin {\rm h} u = + ({\bf q}^2/4m^2 - 1)^{1/2}$, and
$-1/ \mbox{tan} \theta = i \mbox{tan} hu = + i({\bf q}^2 - 4m^2)^{1/2} ({\bf
q}^2)^{-1/2})$.
Closed loops containing a number of quanta or
potential interactions larger than two produce no
trouble. Any loop with an odd number of
interactions gives zero (I, reference 9). Four or more
potential interactions give integrals which are
convergent even without a convergence factor as is
well known. The situation is analogous to that for
self-energy. Once the simple problem of a single
closed loop is solved there are no further divergence
difficulties for more complex processes.\footnote{There are loops completely
without external interactions. For example, a pair is created virtually
along with a photon. Next they annihilate, absorbing
this photon. Such loops are disregarded on the
grounds that they do not interact with anything and
are thereby completely unobservable. Any indirect
effects they may have via the exclusion principle
have already been included.}
\section{LONGITUDINAL WAVES}
~~~~In the usual form of quantum electrodynamics the
longitudinal and transverse waves are given separate
treatment. Alternately the condition $(\partial A_{\mu} / \partial x_{\mu})
\Psi = 0$ is carried along as a supplementary condition. In the
present form no such special considerations are
necessary for we are dealing with the solutions of
the equation $-\square^2 A_{\mu} = 4 \pi j_{\mu}$, with a current $j_{\mu}$,
which is conserved $\partial j_{\mu}/ \partial x_{\mu} = 0$. That means at
least $\square^2 (\partial A_{\mu} / \partial x_{\mu})=0$ and in
fact our solution also satisfies $\partial A_{\mu}/\partial x_{\mu} = 0$.
To show that this is the case we consider the
amplitude for emission (real or virtual) of a photon
and show that the divergence of this amplitude
vanishes. The amplitude for emission for photons
polarized in the $\mu$ direction involves matrix
elements of $\gamma_{\mu}$. Therefore what we have to show is
that the corresponding matrix elements of $q_{\mu} \gamma_{\mu} = {\bf
q}$ vanish. For example, for a first order effect we
would require the matrix element of ${\bf q}$ between two
states ${\bf p}_1$ and ${\bf p}_2 = {\bf p}_1 + {\bf q}$. But since
${\bf q} = {\bf p}_2 - {\bf p}_1$ and $(\tilde u_2 {\bf p}_1 u_1) = m(\tilde
u_2 u_1) = (\tilde u_2 {\bf p}_2 u_1)$ the matrix element
vanishes, which proves the contention in this case. It
also vanishes in more complex situations (essentially because of relation
(34), below) (for example, try putting ${\bf e}_2 = {\bf q}_2$ in the matrix (15)
for the Compton Effect).
To prove this in general, suppose ${\bf a}_i, i = 1$ to $N$ are
a set of plane wave disturbing potentials carrying
momenta ${\bf q}_i$, (e.g., some may be emissions or
absorptions of the same or different quanta) and
consider a matrix for the transition from a state of
momentum ${\bf p}_0$ to ${\bf p}_N$ such
as ${\bf a}_N \Pi_{t = 1}^{N - 1} ({\bf p}_i - m)^{-1} {\bf a}_i$, where
${\bf p}_i = {\bf p}_{i - 1} + {\bf q}_i$ (and in
the product, terms with larger $i$ are written to the
left). The most general matrix element is simply a
linear combination of these. Next consider the
matrix between states ${\bf p}_0$ and ${\bf p}_N + {\bf q}$ in a
situation in which not only are the ${a}_i$, acting but also another
potential ${\bf a} ~\mbox{exp}(- iq \cdot x)$ where ${\bf a = q}$. This may act
previous to all ${\bf a}_i$ in which case it gives
${\bf a}_N \Pi({\bf p}_i + {\bf q} - m)^{-1} {\bf a}_i ({\bf p}_0 + {\bf
q} - m)^{-1} {\bf q}$ which is equivalent to $+ {\bf a}_N \Pi({\bf p}_i + {\bf q} -
m)^{-1} {\bf a}_i$ since $+({\bf p}_0 + {\bf q} - m)^{-1} {\bf q}$ is
equivalent to $({\bf p}_0 + {\bf q} - m)^{-1} \times ({\bf p}_0 + {\bf
q} - m)$ as ${\bf p}_0$ is equivalent to $m$ acting on the initial state. Likewise
if it acts after all the potentials it gives \mbox{${\bf q} ({\bf p}_N - m)^{-1}
{\bf a}_N \Pi({\bf p}_i - m)^{-1} {\bf a}_i$} which is equivalent to $-
{\bf a}_N \Pi ({\bf p}_i - m)^{-1} {\bf a}_i$ since ${\bf p}_N + {\bf q}
- m$ gives zero on the final state. Or again it may act between the potential
${\bf a}_k$ and ${\bf a}_{k + 1}$ for each $k$. This gives
$$
\begin{array}{c}
\sum \limits^{N - 1}_{k = 1} {\bf a}_N \operatornamewithlimits{\Pi}^{N -
1}_{i = k + 1} ({\bf p}_i + {\bf q} - m)^{-1} {\bf a}_i ({\bf p}_k + {\bf
q} - m)^{-1}\\
\times {\bf q} ({\bf p}_k - m)^{-1} {\bf a}_k \operatornamewithlimits{\Pi}^{k
- 1}_{j = 1} ({\bf p}_j - m)^{-1} {\bf a}_j.
\end{array}
$$
However,
\begin{equation}
({\bf p}_k + {\bf q} - m)^{-1} {\bf q} ({\bf p}_k - m)^{-1} = ({\bf p}_k
- m)^{-1} - ({\bf p}_k + {\bf q} - m)^{-1},
\end{equation}
so that the sum breaks into the difference of two
sums, the first of which may be converted to the
other by the replacement of $k$ by $k - 1$. There
remain only the terms from the ends of the range of
summation,
$$
+ {\bf a}_N \operatornamewithlimits{\Pi}^{N - 1}_{i = 1} ({\bf p}_i - m)^{-1}
{\bf a}_i - {\bf a}_N \operatornamewithlimits{\Pi}^{N - 1}_{i = 1} ({\bf
p}_i + {\bf q} - m)^{-1} {\bf a}_i.
$$
These cancel the two terms originally discussed so
that the entire effect is zero. Hence any wave
emitted will satisfy $\partial A_{\mu}/ \partial x_{\mu} = 0$. Likewise longitudinal
waves (that is, waves for which $A_{\mu} = \partial \phi / \partial x_{\mu}$ or
${\bf a = q}$ cannot be absorbed and will have no effect, for
the matrix elements for emission and absorption are
similar. (We have said little more than that a
potential $A_{\mu} = \partial \varphi / \partial x_{\mu}$ has no effect on a Dirac
electron since a transformation $\psi' = \mbox{exp}(-i \phi) \psi$
removes it. It is also easy to see in coordinate
representation using integrations by parts.)
This has a useful practical consequence in that in
computing probabilities for transition for
unpolarized light one can sum the squared matrix
over all four directions rather than just the two
special polarization vectors. Thus suppose the
matrix element for some process for light polarized
in direction $e_{\mu}$, is $e_{\mu} M_{\mu}$. If the light has wave vector
$q_{\mu}$, we know from the argument above that $q_{\mu} M_{\mu} = 0$.
For unpolarized light progressing in the $z$ direction
we would ordinarily calculate $M_x^2 + M_y^2$. But we can as well sum
\mbox{$M_x^2 + M_y^2 + M_z^2 - M_t^2$} for
$q_{\mu} M_{\mu}$ implies $M_t = M_z$ since $q_t = q_z$ for free quanta.
This shows that unpolarized light is a relativistically
invariant concept, and permits some simplification
in computing cross sections for such light.
Incidentally, the virtual quanta interact through
terms like \mbox{$\gamma_{\mu} \ldots \gamma_{\mu} {\bf k}^{-2} d^4 k$.} Real processes
correspond to poles in the formulae for virtual processes. The
pole occurs when ${\bf k}^2 = 0$, but it looks at first as though
in the sum on all four values of $\mu$, of $\gamma_{\mu} \ldots \gamma_{\mu}$ we
would have four kinds of polarization instead of
two. Now it is clear that only two perpendicular to ${\bf k}$ are effective.
The usual elimination of longitudinal and scalar
virtual photons (leading to an instantaneous
Coulomb potential) can of course be performed here
too (although it is not particularly useful). A typical
term in a virtual transition is $\gamma_{\mu} \ldots \gamma_{\mu} {\bf
k}^{-2} d^4 k$ where
the $\ldots$ represent some intervening matrices. Let us choose for the
values of $\mu$, the time $t$, the direction
of vector part ${\bf K}$, of ${\bf k}$, and two perpendicular
directions 1, 2. We shall not change the expression
for these two 1, 2 for these are represented by
transverse quanta. But we must find $(\gamma_t \ldots \gamma_t)-(\gamma_{\bf
K} \ldots \gamma_{\bf K})$. Now ${\bf k} = k_4 \gamma_t - K \gamma_{\bf
K}$, where $K = ({\bf K \cdot K})^{1/2}$,
and we have shown above that ${\bf k}$ replacing the $\gamma_{\mu}$.
gives zero.\footnote{A little more care is required when both $\gamma_{\mu}$'s act
on the same particle. Define ${\bf x} = k_4 \gamma_t + K \gamma_{\bf K}$, and
consider $({\bf k} \ldots {\bf x}) + {\bf x} \ldots {\bf k})$.
Exactly this term would arise if a system, acted on by potential ${\bf
x}$ carrying momentum $-{\bf k}$, is disturbed by an added potential ${\bf
k}$ of momentum $+{\bf k}$ (the reversed sign of the momenta in the intermediate
factors in the second term ${\bf x} \ldots {\bf k}$ has no effect since we will
later integrate over all ${\bf k}$). Hence as shown above the result is zero, hut
since $({\bf k} \ldots {\bf x}) + ({\bf x} \ldots {\bf k}) = k_4^2 (\gamma_t
\ldots \gamma_t) - K^2 (\gamma_{\bf K} \ldots \gamma_{\bf K})$ we can still
conclude $(\gamma_{\bf K} \ldots \gamma_{\bf K}) = k_4^2 K^{-2} (\gamma_t
\ldots \gamma_t)$.} Hence $K \gamma_{\bf K}$ is equivalent to $k_4 \gamma_t$ and
$$
(\gamma_t \ldots \gamma_t) - (\gamma_{\bf K} \ldots \gamma_{\bf K}) = ((K^2
- k_4^2)/K^2)(\gamma_t \ldots \gamma_y),
$$
so that on multiplying by ${\bf k}^{-2} d^4 k = d^4 k(k_4^2 - K^2)^{-1}$ the net
effect is $-(\gamma_t \ldots \gamma_t) d^4 k/K^2$. The $\gamma_t$ means just scalar
waves, that is, potentials produced by charge
density. The fact that $1/K^2$ does not contain $k_4$
means that $k_4$ can be integrated first, resulting in an
instantaneous interaction, and the $d^3 {\bf K}/K^2$ is just the
momentum representation of the Coulomb potential, $1/{\bf r}$.
\section{KLEIN GORDON EQUATION}
~~~~The methods may be readily extended to particles
of spin zero satisfying the Klein Gordon
equation,\footnote{The equations discussed in this section were deduced from the
formulation of the Klein Gordon equation given in reference 5,
Section 14. The function $\psi$ in this section has only one component
and is not a spinor. An alternative formal method of making the
equations valid for spin zero and also for spin 1 is (presumably) by
use of the Kemmer-Duffin matrices $\beta_{\mu}$ satisfying the commutation
relation
$$
\beta_{\mu} \beta_{\nu} \beta_{\sigma} + \beta_{\sigma} \beta_{\nu} \beta_{\mu}
= \delta_{\mu \nu} \beta_{\sigma} + \delta_{\sigma \nu} \beta_{\mu}.
$$
If we interpret ${\bf a}$ to mean
$a_{\mu} \beta_{\mu}$, rather than $a_{\mu} \gamma_{\mu}$, for any $a_{\mu}$,
all of the equations in momentum
space will remain formally identical to those for the spin $1/2$; with the
exception of those in which a denominator $({\bf p} - m)^{-1}$ has been
rationalized to $({\bf p} + m)({\bf p}^2 - m^2)^{-1}$ since ${\bf p}^2$ is no
longer equal to a number, $p \cdot p$. But ${\bf p}^3$ does equal $(p \cdot
p){\bf p}$ that $({\bf p} - m)^{-1}$ may now be
interpreted as $(m {\bf p} + m^2 + {\bf p}^2 - p \cdot p)(p \cdot p - m^2)^{-1}$.
This implies that equations in coordinate space will be valid of the function
$K_+(2,1)$ is given as $K_+(2,1) = [(i \triangledown_2 + m) - m^{-1}(\triangledown_2
+ \square_2^2)]i I_+(2,1)$ with $\triangledown_2 = \beta_{\mu} \partial
/ \partial x_{2 \mu}$. This is all in virtue of the fact that the many component
wave function $\psi$ (5 components for spin 0, 10 for spin 1) satisfies $(i
\triangledown - m) \psi = {\bf a} \psi$ which is formally identical to the
Dirac Equation. See W. Pauli, Rev. Mod. Phys. {\bf 13,} 203 (1940).}
\begin{equation}
\square^2 \psi - m^2 \psi = i \partial(A_{\mu} \psi)/ \partial x_{\mu}
+ i A_{\mu} \partial \psi/ \partial x_{\mu} - A_{\mu} A_{\mu} \psi.
\end{equation}
The important kernel is now $I_+(2,1)$ denned in (I, Eq.
(32)). For a free particle, the wave function $\psi(20$
satisfies $+ \square^2 \psi - m^2 \psi = 0$. At a point, 2, inside a
space time region it is given by
$$
\psi(2) = \int [\psi(1) \partial I_+(2,1)/ \partial x_{1 \mu} - (\partial
\psi / \partial x_{1 \mu}) I_+(2,1)] N_{\mu}(1) d^3 V_1,
$$
(as is readily shown by the usual method of demonstrating Green's theorem)
the integral being over an entire 3-surface boundary of the region (with normal
vector $N_{\mu}$). Only the positive frequency components
of $\psi$ contribute from the surface preceding the time
corresponding to 2, and only negative frequencies
from the surface future to 2. These can be interpreted
as electrons and positrons in direct analogy to the Dirac case.
The right-hand side of (35) can be considered as a
source of new waves and a series of terms written
down to represent matrix elements for processes of
increasing order. There is only one new point here,
the term in $A_{\mu} A_{\mu}$ by which two quanta can act at the
same time. As an example, suppose three quanta or
potentials, $a_{\mu} \mbox{exp}(-iq_a \cdot x), b_{\mu} \mbox{exp}(-i q_b
\cdot x)$, and $c_{\mu} \mbox{exp}(iq_e \cdot x)$
are to act in that order on a particle of
original momentum $p_{0 \mu}$, so that ${\bf p}_a = {\bf p}_0 + {\bf q}_a$, and
${\bf p}_b = {\bf p}_a + {\bf q}_b$; the final momentum being ${\bf p}_c
= {\bf p}_b + {\bf q}_c$. The matrix element is the sum of three terms $({\bf
p}^2 = p_{\mu} p_{\mu})$ (illustrated in Fig. 7)
\begin{equation}
\begin{array}{c}
(p_c \cdot c + p_b \cdot c) ({\bf p}_b^2 - m^2)^{-1} (p_b \cdot b + p_a
\cdot b) \times ({\bf p}_a^2 - m^2)^{-1} (p_a \cdot a + p_0 \cdot a)\\\\
- (p_c \cdot c + p_b \cdot c)({\bf p}_b^2 - m^2)^{-1} (b \cdot a) - (c
\cdot b) ({\bf p}_a^2 - m^2)^{-1} (p_a \cdot a + p_0 \cdot a).
\end{array}
\end{equation}
The first comes when each potential acts through the
perturbation $i \partial (A_{\mu} \psi) / \partial x_{\mu} + i A_{\mu}
\partial \psi/ \partial x_{\mu}$. These gradient operators in momentum space mean
respectively the momentum after and before the
potential $A_{\mu}$ operates. The second term comes from
$b_{\mu}$ and $a_{\mu}$ acting at the same instant and arises from
the $A_{\mu} A_{\mu}$ term in (a). Together $b_{\mu}$ and $a_{\mu}$ carry
momentum $q_{b \mu} + q_{a \mu}$ so that after $b \cdot a$ operates the
momentum is ${\bf p}_0 + {\bf q}_a + {\bf q}_b$ or ${\bf p}_b$. The final term
comes from $c_{\mu}$ and $b_{\mu}$ operating together in a similar
manner. The term $A_{\mu} A_{\mu}$ thus permits a new type of
process in which two quanta can be emitted (or
absorbed, or one absorbed, one emitted) at the same
time. There is no $a \cdot c$ term for the order $a,b,c$ we
have assumed. In an actual problem there would be
other terms like (36) but with alterations in the order
in which the quanta $a, b, c$ act. In these terms $a \cdot c$
would appear.
As a further example the self-energy of a particle
of momentum $p_{\mu}$ is
$$
(e^2/2 \pi i m) \int [(2 p - k)_{\mu} (({\bf p - k})^2 - m^2)^{-1} \times
(2p - k)_{\mu} - \delta_{\mu \mu}] d^4 k{\bf k}^{-2} C({\bf k}^2),
$$
where the $\delta_{\mu \mu}$ comes from the $A_{\mu} A_{\mu}$ term
and represents the possibility of the simultaneous emission
and absorption of the same virtual quantum. This
integral without the $C({\bf k}^2)$ diverges quadratically and
would not converge if $C({\bf k}^2) = - \lambda^2/(k^2 - \lambda^2)$. Since the
interaction occurs through the gradients of the
potential, we must use a stronger convergence factor,
for example $C({\bf k}^2) = \lambda^4 (k^2 - \lambda^2)^{-2}$, or in general (17)
with $\int^{\infty}_0 \lambda^2 G(\lambda) d \lambda = 0$. In this case the
self-energy converges but depends quadratically on the cut-off $\lambda$
and is not necessarily small compared to $m$. The
radiative corrections to scattering after mass
renormalization are insensitive to the cut-off just as
for the Dirac equation.
When there are several particles one can obtain
Bose statistics by the rule that if two processes lead
to the same state but with two electrons exchanged,
their amplitudes are to be added (rather than
subtracted as for Fermi statistics). In this case
equivalence to the second quantization treatment of
Pauli and Weisskopf should be demonstrable in a
way very much like that given in $I$ (appendix) for
Dirac electrons. The Bose statistics mean that the
sign of contribution of a closed loop to the vacuum
polarization is the opposite of what it is for the Fermi
case (see I). It is $({\bf p}_b = {\bf p}_a + {\bf q})$
$$
\begin{array}{c}
J_{\mu \nu} = \frac{\displaystyle e^2}{\displaystyle 2 \pi i m} \int [(p_{b \mu} + p_{a \mu})(p_{b
\nu} + p_{a \nu}) ({\bf p}_a^2 - m^2)^{-1}\\\\
\times ({\bf p}_b^2 - m^2)^{-1} - \delta_{\mu \nu} ({\bf p}_a^2 - m^2)^{-1}
- \delta_{\mu \nu}({\bf p}_b^2 - m^2)^{-1}] d^4 p_a
\end{array}
$$
giving,
$$
J_{\mu \nu}^P = \frac{e^2}{\pi} (q_{\mu} q_{\nu} - \delta_{\mu \nu} {\bf
q}^2) \left[ \frac{1}{6} \ln \frac{\lambda^2}{m^2} + \frac{1}{9} - \frac{4m^2
- {\bf q}^2}{3 {\bf q}^2} \left( 1 - \frac{\theta}{{\rm tan} \theta} \right)
\right],
$$
the notation as in (33). The imaginary part for $({\bf q}^2)^{1/2} > 2m$
is again positive representing the loss in the
probability of finding the final state to be a vacuum,
associated with the possibilities of pair production.
Fermi statistics would give a gain in probability (and
also a charge renormalization of opposite sign to that
expected).
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig7e.gif}}}
\caption{Klein-Gordon particle in three potentials, Eq, (36). The
coupling to the electromagnetic field is now, for example, $p_0 \cdot a
+ p_a \cdot a$, and a new possibility arises, (b), of simultaneous interaction
with two quanta $a \cdot b$. The propagation factor is now $(p \cdot p
- m^2)^{-1}$ for a particle of momentum $p_{\mu}$.}
\end{figure}
\section{APPLICATION TO MESON THEORIES}
~~~~The theories which have been developed to
describe mesons and the interaction of nucleons can
be easily expressed in the language used here.
Calculations, to lowest order in the interactions can
be made very easily for the various theories, but
agreement with experimental results is not obtained.
Most likely all of our present formulations are
quantitatively unsatisfactory. We shall content
ourselves therefore with a brief summary of the
methods which can be used.
The nucleons are usually assumed to satisfy
Dirac's equation so that the factor for propagation of
a nucleon of momentum ${\bf p}$ is $({\bf p} - M)^{-1}$ where $M$ is
the mass of the nucleon (which implies that
nucleons can be created in pairs). The nucleon is
then assumed to interact with mesons, the various
theories differing in the form assumed for this
interaction.
First, we consider the case of neutral mesons. The
theory closest to electrodynamics is the theory of
vector mesons with vector coupling. Here the factor
for emission or absorption of a meson is $g \gamma_{\mu}$, when
this meson is ``polarized'' in the $\mu$ direction. The
factor $g$ the ``mesonic charge,'' replaces the electric
charge $e$. The amplitude for propagation of a meson
of momentum ${\bf q}$ in intermediate states is $({\bf q}^2 - \mu^2)^{-1}$
(rather than ${\bf q}^{-2}$ as it is for light) where $\mu$ is the
mass of the meson. The necessary integrals are
made finite by convergence factors $C({\bf q}^2 - \mu^2)$ as in
electrodynamics. For scalar mesons with scalar
coupling the only change is that one replaces the $\gamma_{\mu}$
by 1 in emission and absorption. There is no longer
a direction of polarization, $\mu$, to sum upon. For
pseudo-scalar mesons, pseudoscalar coupling
replace $\gamma_{\mu}$ by $\gamma_5 = i \gamma_x \gamma_y \gamma_z \gamma_t$.
For example, the self-energy matrix of a nucleon of momentum ${\bf p}$ in this
theory is
$$
(g^2/ \pi i) \int \gamma_5 ({\bf p - k} - M)^{-1} \gamma_5 d^4 k ({\bf
k}^2 - \mu^2)^{-1} C({\bf k}^2 - \mu^2).
$$
Other types of meson theory result from the replacement of $\gamma_{\mu}$,
by other expressions (for example by $\frac{1}{2} (\gamma_{\mu} \gamma_{\nu}
- \gamma_{\nu} \gamma_{\mu})$ with a subsequent sum over all $\mu$ and $\nu$
for virtual mesons). Scalar mesons with vector
coupling result from the replacement of $\gamma_{\mu}$ by $\mu^{-1} {\bf
q}$ where ${\bf q}$ is the final momentum of the nucleon
minus its initial momentum, that is, it is the
momentum of the meson if absorbed, or the
negative of the momentum of a meson emitted. As
is well known, this theory with neutral mesons
gives zero for all processes, as is proved by our
discussion on longitudinal waves in
electrodynamics. Pseudoscalar mesons with pseudo-vector coupling
corresponds to $\gamma_{\mu}$ being replaced by
$\mu^{-1} \gamma_5 {\bf q}$ white vector mesons with tensor coupling
correspond to using $(2 \mu)^{-1} (\gamma_{\mu} {\bf q} - {\bf q} \gamma_{\mu})$.
These extra gradients involve the danger of producing
higher divergencies for real processes. For example,
$\gamma_5 {\bf q}$ gives a logarithmically divergent interaction of
neutron and electron.\footnote{M. Slotnick and W. Heitler, Phys. Rev. {\bf
75}, 1645 (1949).} Although these divergencies
can be held by strong enough convergence
factors, the results then are sensitive to the method
used for convergence and the size of the cut-off
values of $\lambda$. For low order processes $\mu^{-1} \gamma_5 {\bf q}$ is
equivalent to the pseudoscalar interaction $2 M \mu^{-1} \gamma_5$
because if taken between free particle wave
functions of the nucleon of momenta ${\bf p}_1$ and ${\bf p}_2 = {\bf p}_1
+ {\bf q}$, we have
$$
\begin{array}{c}
(\tilde u_2 \gamma_5 {\bf q} u_1) = (\tilde u_2 \gamma_5 ({\bf p}_2 - {\bf
p}_1)u_1) = - (\tilde u_2 {\bf p}_2 \gamma_5 u_1)\\\\
- (\tilde u_2 \gamma_5 {\bf p}_1 u_1) = - 2M(\tilde u_2 \gamma_5 u_1)
\end{array}
$$
since $\gamma_5$ anticommutes with ${\bf p}_2$ and ${\bf p}_2$ operating on
the state 2 equivalent to $M$ as is ${\bf }_1$ on the state 1.
This shows that the $\gamma_5$ interaction is unusually
weak in the non-relativistic limit (for example the
expected value of $\gamma_5$ for a free nucleon is zero), but
since $\gamma_5^2 = 1$ is not small, pseudoscalar theory gives
a more important interaction in second order than it
does in first. Thus the pseudoscalar coupling
constant should be chosen to fit nuclear forces
including these important second order processes.\footnote{H. A. Bethe, Bull. Am.
Phys. Soc. {\bf 24,} 3, Z3 (Washington, 1949).}
The equivalence of pseudoscalar and pseudo-vector
coupling which holds for low order processes
therefore does not hold when the pseudoscalar
theory is giving its most important effects. These
theories will therefore give quite different results in
the majority of practical problems.
In calculating the corrections to scattering of a
nucleon by a neutral vector meson field $(\gamma_{\mu})$ due to
the effects of virtual mesons, the situation is just as
in electrodynamics, in that the result converges
without need for a cut-off and depends only on
gradients of the meson potential. With scalar (1) or
pseudoscalar $(\gamma_{\mu})$ neutral mesons the result
diverges logarithmically and so must be cut off. The
part sensitive to the cut-off, however, is directly
proportional to the meson potential. It may thereby
be removed by a renormalization of mesonic charge
$g$. After this renormalization the results depend only
on gradients of the meson potential and are
essentially independent of cut-off. This is in
addition to the mesonic charge renormalization
coming from the production of virtual nucleon pairs
by a meson, analogous to the vacuum polarization
in electrodynamics. But here there is a further
difference from electrodynamics for scalar or
pseudoscalar mesons in that the polarization also
gives a term in the induced current proportional to
the meson potential representing therefore an
additional renormalization of the {\it mass of the meson }
which usually depends quadratically on the cut-off.
Next consider charged mesons in the absence of
an electromagnetic field. One can introduce isotopic
spin operators in an obvious way. (Specifically
replace the neutral $\gamma_5$, say, by $\tau_i \gamma-5$ and sum over
$i = 1,2$, where $\tau_1 = \tau_+ + \tau_-,~ \tau_2 = i(\tau_+ - \tau_-)$ and $\tau_+$
changes neutron to proton ($\tau_+$ on proton $=0$) and $\tau_-$
changes proton to neutron.) It is just as easy for
practical problems simply to keep track of whether
the particle is a proton or a neutron on a diagram
drawn to help write down the
matrix element. This excludes certain processes. For
example in the scattering of a negative meson from
${\bf q}_1$ to ${\bf q}_2$ by a neutron, the meson ${\bf q}_2$ must be emitted
first (in order of operators, not time) for the neutron
cannot absorb the negative meson ${\bf q}_1$ until it becomes
a proton. That is, in comparison to the Klein Nishina
formula (15), only the analogue of second term (see
Fig. 5(b)) would appear in the scattering of negative
mesons by neutrons, and only the first term (Fig. 5
(a)) in the neutron scattering of positive mesons.
The source of mesons of a given charge is not
conserved, for a neutron capable of emitting
negative mesons may (on emitting one, say) become
a proton no longer able to do so. The proof that a
perturbation ${\bf q}$ gives zero, discussed for longitudinal
electromagnetic waves, fails. This has the
consequence that vector mesons, if represented by
the interaction $\gamma_{\mu}$, would not satisfy the condition
that the divergence of the potential is zero. The
interaction is to be taken\footnote{The vector meson field potentials $\varphi_{\mu}$
satisfy
$$
- \partial/ \partial x_{\nu} (\partial \varphi_{\mu}/ \partial x_{\nu}
- \partial \varphi_{\nu} / \partial x_{\mu}) - \mu^2 \varphi_{\mu} = -
4 \pi s_{\mu},
$$
where $s_{\mu}$, the source for such mesons, is the matrix
element of $\gamma_{\mu}$ between states of neutron and proton.
By taking the divergence $\partial / \partial x_{\mu}$ of both sides,
conclude that $\partial \varphi_{\nu} / \partial x_{\nu} = 4 \pi \mu^{-2}
\partial s_{\nu} / \partial x_{\nu}$, so that the original
equation can lie rewritten as
$$
\square^2 \varphi_{\mu} - \mu^2 \varphi_{\mu} = - 4 \pi (s_{\mu} + \mu^{-2}
\partial/ \partial x_{\mu} (\partial s_{\nu} / \partial x_{\nu})).
$$
The right hand side gives in momentum representation $\gamma_{\mu} - \mu^{-2}
q_{\mu} q_{\nu} \gamma_{\nu}$ the left yields the $({\bf q}^2 - \mu^2)^{-1}$
and finally the interaction $s_{\mu} \varphi_{\mu}$ in the
Lagrangian gives the $\gamma_{\mu}$ on absorption.
Proceeding in this way find generally that particles of spin one can
be represented by a four-vector $u_{\mu}$ (which, for a free particle of
momentum $q$ satisfies $q \cdot u = 0$). The propagation of virtual particles of
momentum $q$ from state $\nu$ to $\mu$ is represented by multiplication by the
4-4 matrix (or tensor) $P_{\mu \nu} = (\delta_{\mu \nu} - \mu^2 q_{\mu}
q_{\nu}) \times (q^2 - \mu^2)^{-1}$. The first-order
interaction (from the Proca equation) with an electromagnetic
potential $a ~\mbox{exp}(ik \cdot x)$ corresponds to multiplication by the matrix
$E_{\mu \nu} = (q_2 \cdot a + q_1 \cdot a) \delta_{\mu \nu} - q_{2\nu} a_{\mu}
- q_{1 \nu} a_{\nu}$, where $q_1$ and $q_2 = q_1 + k$ are the
momenta before and after the interaction. Finally, two potentials $a,b$
may act simultaneously, with matrix $E'_{\mu \nu} = -(a \cdot b)\delta_{\mu
\nu}+b_{\mu}a_{\nu}$} as $\gamma_{\mu}-\mu^{-2} q_{\mu}{\bf q}$
in emission and as $\gamma_{\mu}$ in absorption if the real emission of mesons
with a non-zero divergence of potential is to be
avoided. (The correction term $\mu^{-2} q_{\mu} {\bf q}$ gives zero in
the neutral case.) The asymmetry in emission and
absorption is only apparent, as this is clearly the
same thing as subtracting from the original $\gamma_{\mu} \ldots \gamma_{\mu}$,
a term $\mu^{-2} {\bf q} \ldots {\bf q}$. That is, if the term $- \mu^{-2}
q_{\mu} {\bf q}$ is omitted the resulting theory describes a combination
of mesons of spin one and spin zero. The spin zero
mesons, coupled by vector coupling ${\bf q}$, are removed
by subtracting the term $\mu^{-2} {\bf q} \ldots {\bf q}$.
The two extra gradients ${\bf q} \ldots {\bf q}$ make the problem
of diverging integrals still more serious (for example
the interaction between two protons corresponding
to the exchange of two charged vector mesons
depends quadratically on the cut-off if calculated in
a straightforward way). One is tempted in this
formulation to choose simply $\gamma_{\mu} \ldots \gamma_{\mu}$ and accept
the admixture of spin zero mesons. But it appears
that this leads in the conventional formalism to
negative energies for the spin zero component. This
shows one of the advantages of the
method of second quantization of meson fields over
the present formulation. There such errors of sign
are obvious while here we seem to be able to write
seemingly innocent expressions which can give
absurd results. Pseudovector mesons with
pseudovector coupling correspond to using $\gamma_5(\gamma_{\mu} - \mu^{-2}
q_{\mu} {\bf q})$ for absorption and $\gamma_5 \gamma_{\mu}$ for emission for
both charged and neutral mesons.
In the presence of an electromagnetic field,
whenever the nucleon is a proton it interacts with the
field in the way described for electrons. The meson
interacts in the scalar or pseudoscalar case as a
particle obeying the Klein-Gordon equation. It is
important here to use the method of calculation of
Bethe and Pauli, that is, a virtual meson is assumed
to have the same ``mass'' during all its interactions
with the electromagnetic field. The result for mass $\mu$
and for $(\mu^2 + \lambda^2)^{1/2}$ are subtracted and the difference
integrated over the function $G(\lambda) d \lambda$. A separate
convergence factor is not provided for each meson
propagation between electromagnetic interactions,
otherwise gauge invariance is not insured. When the
coupling involves a gradient, such as $\gamma-5 {\bf q}$ where ${\bf q}$ is
the final minus the initial momentum of the nucleon,
the vector potential ${\bf A}$ must be subtracted from the
momentum of the proton. That is, there is an
additional coupling $\pm \gamma_5 {\bf A}$ (plus when going from
proton to neutron, minus for the reverse)
representing the new possibility of a simultaneous
emission (or absorption) of meson and photon.
Emission of positive or absorption of negative
virtual mesons are represented in the same term, the
sign of the charge being determined by temporal
relations as for electrons and positrons.
Calculations are very easily carried out in this way
to lowest order in $g^2$ for the various theories for
nucleon interaction, scattering of mesons by
nucleons, meson production by nuclear collisions
and by gamma-rays, nuclear magnetic moments,
neutron electron scattering, etc., However, no good
agreement with experiment results, when these are
available, is obtained. Probably all of the
formulations are incorrect. An uncertainty arises
since the calculations are only to first order in $g^2$, and
are not valid if $g^2/ \hbar c$ is large.
The author is particularly indebted to Professor H. A. Bethe for his
explanation of a method of
obtaining finite and gauge invariant results for the
problem of vacuum polarization. He is also grateful
for Professor Bethe's criticisms of the manuscript,
and for innumerable discussions during the
development of this work. He wishes to thank
Professor J. Ashkin for his careful reading of the manuscript.
\section*{\bf APPENDIX}
{\small
~~~~In this appendix a method will be illustrated by which the simpler
integrals appearing in problems in electrodynamics can be directly
evaluated. The integrals arising in more complex processes lead to
rather complicated functions, but the study of the relations of one
integral to another and their expression in terms of simpler integrals
may be facilitated by the methods given here.
As a typical problem consider the integral (12) appearing in the
first order radiationless scattering problem:
$$
\int \gamma_{\mu} ({\bf p}_2 - {\bf k} - m)^{-1} {\bf a} ({\bf p}_1 - {\bf
k} - m)^{-1} \gamma_{\mu} {\bf k}^{-2} d^4 kC({\bf k}^2), \eqno(1a)
$$
where we shall take $C({\bf k}^2)$ to be typically $- \lambda^2 ({\bf k}^2
- \lambda^2)^{-1}$ and $d^4k$ means $(2 \pi)^{-2} dk_1 dk_2 dk_3 dk_4$.
We first rationalize the factors $({\bf p - k} - m)^{-1} = ({\bf p - k}
+ m)\\(({\bf p - k})^2 - m^2)^{-1}$ obtaining,
$$
\int \gamma_{\mu} ({\bf p}_2 - {\bf k} + m) {\bf a} ({\bf p}_1 - {\bf k}
+ m) \gamma_{\mu} {\bf k}^{-2} d^4 kC ({\bf k}^2)
$$
$$
\times (({\bf p}_1 - {\bf k})^2 - m^2)^{-1} (({\bf p}_2 - {\bf k})^2 -
m^2)^{-1}. \eqno(2a)
$$
The matrix expression may be simplified. It appears to be best to do
so after the integrations are performed. Since ${\bf AB} = 2A \cdot B -
{\bf BA}$ where $A \cdot B = A_{\mu} B_{\mu}$ is a number commuting with all
matrices, find, if $R$ is any expression, and ${\bf A}$ a vector, since
$\gamma_{\mu} {\bf A} = - {\bf A} \gamma_{\mu} + 2 A_{\mu}$,
$$
\gamma_{\mu} {\bf A} R \gamma_{\mu} = - {\bf A} \gamma_{\mu} R \gamma_{\mu}
+ 2 R {\bf A}. \eqno(3a)
$$
Expressions between two $\gamma_{\mu}$'s can be thereby reduced by induction.
Particularly useful are
$$
\gamma_{\mu} \gamma_{\mu} = 4
$$
$$
\gamma_{\mu} {\bf A} \gamma_{\mu} = - 2 {\bf A}
$$
$$
\gamma_{\mu} {\bf AB} \gamma_{\mu} = 2 ({\bf AB + BA}) = 4A \cdot B
$$
$$
\gamma_{\mu} {\bf ABC} \gamma_{\mu} = - 2 {\bf CBA} \eqno(4a)
$$
where ${\bf A,B,C}$ are any three vector-matrices (i.e., linear combinations
of the four $\gamma_{\mu}$s),
In order to calculate the integral in (2a) the integral may be
written as the sum of three terms (since ${\bf k} = k_{\sigma} \gamma_{\sigma}$),
$$
\gamma_{\mu} ({\bf p}_2 + m) {\bf a} ({\bf p}_1 + m) \gamma_{\mu} J_1 -
[ \gamma_{\mu} \gamma_{\sigma} {\bf a} ({\bf p}_1 + m) \gamma_{\mu}
$$
$$
+ \gamma_{\mu} ({\bf p}_2 + m) {\bf a} \gamma_{\sigma} \gamma_{\mu}] J_2
+ \gamma_{\mu} \gamma_{\sigma} {\bf a} \gamma_{\tau} \gamma_{\mu} J_3,
\eqno(5a)
$$
where
$$
J(1;2;3) = \int (1; k_{\sigma}; k_{\sigma} k_{\tau}){\bf k}^{-2} d^4 kC
({\bf k}^2)
$$
$$
\times (({\bf p}_2 - {\bf k})^2 - m^2)^{-1} (({\bf p}_1 - {\bf k})^2 -
m^2)^{-1}. \eqno(6a)
$$
That is for $J_1$ the $(1; k_{\sigma}; k_{\sigma} k_{\tau})$ is replaced by
1, for $J_2$ by $k_{\sigma}$ and for $J_3$ by $k_{\sigma} k_{\tau}$.
More complex processes of the first order involve more factors
like $(({\bf p}_3 - {\bf k})^2 - m^2)^{-1}$ and a corresponding increase
in the number of $k$'s which may appear in the numerator, as $k_{\sigma}
k_{\tau} k_{\nu} \ldots$. Higher order
processes involving two or more virtual quanta involve similar
integrals but with factors possibly involving ${\bf k + k}'$ instead of just ${\bf
k}$, and the integral extending on ${\bf k}^{-2} d^4 kC ({\bf k}^2) {\bf
k}^{' -2} d^4 kC ({\bf}^{'2})$. They can be simplified
by methods analogous to those used on the first order integrals.
The factors $({\bf p - k})^2 - m^2$ may be written
$$
({\bf p - k})^2 - m^2 = {\bf k}^2 - 2p \cdot k - \Delta, \eqno(7a)
$$
where $\Delta = m^2 - {\bf p}^2,$ $\Delta_1 = m_1^2 - {\bf p}_1^2$, etc., and
we can consider dealing
with cases of greater generality in that the different denominators
need not have the same value of the mass $m$. In our specific problem
(6a) ${\bf p}_1^2 = m^2$ that $\Delta_1 = 0$, but we desire to work with greater
generality.
Now for the factor $C({\bf k}^2)/{\bf k}^2$ we shall use $- \lambda^2 ({\bf
k}^2 - \lambda^2)^{-1} {\bf k}^{-2}$. This can be written as
$$
- \lambda^2/({\bf k}^2 - \lambda^2){\bf k}^2 = {\bf k}^{-2} C({\bf k}^2)
= - \int \limits^{\lambda^2}_0 dL ({\bf k}^2 - L)^{-2}. \eqno(8a)
$$
Thus we can replace ${\bf k}^{-2} C({\bf k}^2)$ by $({\bf k}^2 - L)^{-2}$ and
at the end integrate the result with respect to $L$ from zero to $\lambda^2$.
We can for many practical purposes consider $\lambda^2$ very large relative
to $m^2$ or $p^2$. When the
original integrai converges even without the convergence factor, it
will be obvious since the $L$ integration will then be convergent to
infinity. If an infra-red catastrophe exists in the integral one can
simply assume quanta have a small mass $\lambda_{\mbox{min}}$ and extend the
integral on $L$ from $\lambda^2_{\mbox{min}}$ to $\lambda^2$, rather than from
zero to $\lambda^2$.
We then have to do integrals of the form
$$
\int (1; k_{\sigma}; k_{\sigma} k_{\tau} d^4 k ({\bf k}^2 - L)^{-2} ({\bf
k}^2 - 2p_1 \cdot k - \Delta_1)^{-1}
\times ({\bf k}^2 - 2p_2 \cdot k - \Delta_2)^{-1}, \eqno(9a)
$$
where by $(1; k_{\sigma}; k_{\sigma} k_{\tau})$ we mean that in the place
of this symbol either 1, or $k_{\sigma}$ or $k_{\sigma} k_{\tau}$ may stand
in different cases. In more
complicated problems there may be more factors $({\bf k}^2 - 2p_i \cdot
k - \Delta_i)^{-1}$ or other powers of these factors (the $({\bf k}^2 -
L)^{-2}$ may be considered as a special case of such a factor with ${\bf
p}_i = 0,$ $\Delta_i = L$) and further factors like $k_{\sigma} k_{\tau}
k_{\rho} \ldots$ in the numerator. The poles in all the
factors are made definite by the assumption that $L$, and the $\Delta$'s have
infinitesimal negative imaginary parts.
We shall do the integrals of successive complexity by induction.
We start with the simplest convergent one, and show
$$
\int d^4 k({\bf k}^2 - L)^{-3} = (8iL)^{-1}. \eqno(10a)
$$
For this integral is $\int (2\pi)^{-2} dk_4 d^3 {\bf K} (k_4^2 - {\bf K}
\cdot {\bf K} - L)^{-3}$ where the vector ${\bf K}$, of
magnitude $K = ({\bf K \cdot K})^{1/2}$ is $k_1, k_2, k_3$. The integral
on $k_4$ shows third order poles at $k_4 = + (K^2 + L)^{1/2}$ and $k_4
= - (K^2 + L)^{1/2}$. Imagining, in
accordance with our definitions, that $L$ has a small negative
imaginary part only the first is below the real axis. The contour can
be closed by an infinite semi-circle below this axis, without change
of the value of the integral since the contribution from the
semi-circle vanishes in the limit. Thus the contour can be shrunk about
the pole {\mbox{$k_4 = + (K^2 + L)^{1/2}$} and the resulting $k_4$, integral is $-
2 \pi i$ times the residue at this pole. Writing $k_4 = (k^2 + L)^{1/2}
+ \epsilon$ and expanding $(k_4^2 - K^2 - L)^{-3} = \epsilon^{-3} (\epsilon
+ 2(K^2 + L)^{1/2})^{-3}$ in powers of $\epsilon$, the residue, being the coefficient
of the term $\epsilon^{-1}$, is seen to be $6(2(K^2 +L)^{1/2})^{-5}$ so our integral is
$$
-(3i/32 \pi) \int \limits^{\infty}_0 4 \pi K^2 dK (K^2 + L)^{-5/2} = (3/8i)(1/3L)
$$
establishing (10a).
We also have $\int k_{\sigma} d^4 k({\bf k}^2 - L)^{-3} = 0$ from the symmetry
in the $k$ space. We write these results as
$$
(8i) \int (1; k_{\sigma}) d^4 k({\bf k}^2 - L)^{-3} = (1;0)L^{-1}, \eqno(11a)
$$
where in the brackets $(1;k_{\sigma})$ and $(1; 0)$ corresponding entries are
to he used.
Substituting ${\bf k} = {\bf k}' - {\bf p}$ in (11a) and calling $L - p^2
= \Delta$ shows that
$$
(8i) \int (1;k_{\sigma})d^4k ({\bf k}^2 - 2p \cdot k - \Delta)^{-3} = (1;
p_{\sigma})(p^2 + \Delta)^{-1}. \eqno(12a)
$$
By differentiating both sides of (12a) with respect to $\Delta$ or with
respect to $p_{\tau}$ there follows directly
$$
(24i) \int (1;k_{\sigma}; k_{\sigma} k_{\tau})d^4 k({\bf k}^2 - 2p \cdot
k - \Delta)^{-4}
$$
$$
= - (1; p_{\sigma}; p_{\sigma} p_{\tau} - \frac{\displaystyle 1}{\displaystyle
2} \delta_{\sigma \tau} ({\bf p}^2 + \Delta))({\bf p}^2 + \Delta)^{-2}.
\eqno(13a)
$$
Further differentiations give directly successive integrals including
more $k$ factors in the numerator and higher powers of $({\bf k}^2 - 2p
\cdot k - \Delta)$ in the denominator.
The integrals so far only contain one factor in the denominator.
To obtain results for two factors we make use of the identity
$$
a^{-1} b^{-1} = \int^1_0 dx (ax + b(1 - x))^{-2}, \eqno(14a)
$$
(suggested by some work of Schwinger's involving Gaussian integrals). This
represents the product of two reciprocals as a parametric integral over one
and will therefore permit integrals with two
factors to be expressed in terms of one. For other powers of $a,b$ we
make use of all of the identities, such as
$$
a^{-2} b^{-1} = \int^1_0 2xdx (ax + b(1 - x))^{-3}, \eqno(15a)
$$
deducible from (14a) by successive differentiations with respect to $a$
or $b$. To perform an integral, such as
$$
(8i) \int (1; k_{\sigma})d^4 k({\bf k}^2 - 2p_1 \cdot k - \Delta_1)^{-2}
({\bf k}^2 - 2p_2 \cdot k - \Delta_2)^{-1}, \eqno(16a)
$$
write, using (15a),
$$
({\bf k}^2 - 2p_1 \cdot k - \Delta_1)^{-2} ({\bf k}^2 - 2p_2 \cdot k -
\Delta_2)^{-1} = \int^1_0 2xdx ({\bf k}^2 - 2p_x \cdot k - \Delta_x)^{-3},
$$
where
$$
{\bf p}_x = x{\bf p}_1 + (1 - x){\bf p}_2 \quad \mbox{and} \quad \Delta_x
= x \Delta_1 + (1 - x) \Delta_2, \eqno(17a)
$$
(note that $\Delta_x$ is {\it not} equal to $m^2 - {\bf p}_x^2$) so that the
expression (l6a) is $(8i) \int^1_0 2xdx (1; k_{\sigma}) d^4 k({\bf k}^2
- 2p_x \cdot k \Delta_x)^{-3}$ which may now be evaluated by (12a)
and is
$$
(16a) = \int^1_0 (1; p_{x \sigma}) 2xdx ({\bf p}_x^2 + \Delta_x)^{-1},
\eqno(18a)
$$
where, ${\bf p}_x,~ \Delta_x$ are given in (17a). The integral in (18a) is elementary,
being the integral of ratio of polynomials, the denominator of second
degree in $x$. The general expression although readily obtained is a
rather complicated combination of roots and logarithms.
Other integrals can be obtained again by parametric differentiation. For example
differentiation of (16a), (18a) with respect to $\Delta_2$ or $p_{2 \tau}$ gives
$$
(8i) \int(1; k_{\sigma}; k_{\sigma} k_{\tau})d^4 k({\bf k}^2 - 2p_1 \cdot
k - \Delta_1)^{-2} ({\bf k}^2 - 2p_2 \cdot k - \Delta_2)^{-2}
$$
$$
= - \int^1_0 (1; p_{x \sigma}; p_{x \sigma} p_{x \tau} - \frac{\displaystyle
1}{\displaystyle 2} \delta_{\sigma \tau} (x^2 {\bf p}^2 + \Delta_x))
\times 2x (1 - x) dx ({\bf p}_x^2 + \Delta_x)^{-2}, \eqno(19a)
$$
again leading to elementary integrals.
As an example, consider the case that the second factor is just $(k^2 -
L)^{-2}$ and in the first put ${\bf p}_1 = {\bf p},$ $\Delta_1 = \Delta$.
Then ${\bf p}_x = x{\bf p}$, $\Delta_x - x\Delta + (1 - x) L$. There results
$$
(8i) \int (1; k_{\sigma}; k_{\sigma} k_{\tau})d^4 k({\bf k}^2 - L)^{-2}
({\bf k}^2 - 2p \cdot k - \Delta)^{-2}
$$
$$
= - \int^1_0 (1; xp_{\sigma}; x^2 p_{\sigma} p_{\tau} - \frac{\displaystyle
1}{\displaystyle 2} \delta_{\sigma \tau} (x^2 {\bf p}^2 + \Delta_x))
\times 2x (1 - x) dx (x^2 {\bf p}^2 + \Delta_x)^{-2}. \eqno(20a)
$$
Integrals with three factors can be reduced to those involving two
by using (14a) again. They, therefore, lead to integrals with two
parameters (e.g., see application to radiative correction to scattering
below).
The methods of calculation given in this paper are deceptively
simple when applied to the lower order processes. For processes of
increasingly higher orders the complexity and difficulty increases
rapidly, and these methods soon become impractical in their present
form.}
\section*{\bf A. Self-Energy}
{\small
~~~~The self-energy integral (19) is
$$
(e^2/ \pi i) \int \gamma_{\mu} ({\bf p - k} - m)^{-1} \gamma_{\mu} {\bf
k}^{-2} d^4 kC ({\bf k}^2), \eqno(19)
$$
so that it requires that we find (using the principle of (8a)) the integral
on $L$ from 0 to $\lambda^2$ of
$$
\int \gamma_{\mu} ({\bf p - k} + m) \gamma_{\mu} d^4 k ({\bf k}^2 - L)^{-2}
({\bf k}^2 - 2p \cdot k)^{-1},
$$
since $({\bf p - k})^2 - m^2 = {\bf k}^2 - 2p \cdot k$, as ${\bf p}^2 =
m^2$. This is of the form (16a) with $\Delta_1 = L,$ ${\bf p}_1 = 0$, $\Delta_2
= 0,$ ${\bf p}_2 = {\bf p}$ so that (18a) gives, since ${\bf p}_x = (1 -
x){\bf p},$ $\Delta_x = xL$,
$$
(8i) \int (1; k_{\sigma}) d^4 k ({\bf k}^2 - L)^{-2} ({\bf k}^2 - 2p \cdot
k)^{-1}
= \int^1_0 (1;(1 - x) p_{\sigma}0 2xdx ((1 - x)^2 m^2 _ xL)^{-1},
$$
or performing the integral on $L$, as in (8),
$$
(8i) \int (1; k_{\sigma}) d^4 k {\bf k}^{-2} C({\bf k}^2) ({\bf k}^2 -
2p \cdot k)^{-1}
= \int^1_0 (1; (1 - x) p_{\sigma}) 2dx \ln \frac{\displaystyle x \lambda^2
+(1 - x)^2 m2}{\displaystyle (1 - x)^2 m^2}.
$$
Assuming now that $\lambda^2 \gg m^2$ we neglect $(1 - x)^2 m^2$ relative to
$x \lambda^2$ in the argument of the logarithm, which then becomes $(\lambda^2/m^2)(x/(1
- x)^2)$. Then since $\int^1_0 dx \ln (x(1 - x)^{-2}) = 1$ and
$ \int^1_0 (1 - x) dx \ln (x(1 - x)^{-2}) = - (1/4)$ find
$$
(8i) \int (1; k_{\sigma}){\bf k}^{-2} C ({\bf k}^2) d^4 k({\bf k}^2 - 2p
\cdot k)^{-1}
= \left( 2 \ln \frac{\displaystyle \lambda^2}{\displaystyle m^2} + 2; p_{\sigma}
\left( \ln \frac{\displaystyle \lambda^2}{\displaystyle m^2} -
\frac{\displaystyle 1}{\displaystyle 2} \right) \right),
$$
so that substitution into (19) (after the $({\bf p - k} - m)^{-1}$ in (19) is
replaced by $({\bf p - k} + m)({\bf k}^2 - 2p \cdot k)^{-1})$ gives
$$
(19) = (e^2/ 8 \pi) \gamma_{\mu} [({\bf p} + m)(2 \ln (\lambda^2/m^2)+2)
- {\bf p}(\ln (\lambda^2/m^2) - \frac{\displaystyle 1}{\displaystyle 2})]
\gamma_{\mu}
$$
$$
= (e^2/8 \pi) [8m(\ln (\lambda^2/m^2)+1) - {\bf p} (2 \ln (\lambda^2 m^2)+5)],
\eqno(20)
$$
using (4a) to remove the $\gamma_{\mu}$'s. This agrees with Eq. (20) of the text,
and gives the self-energy (21) when ${\bf p}$ is replaced by $m$.
\section*{\bf B. Corrections to Scattering}
{\small
~~~~The term (12) in the radiationless scattering, after rationalizing the
matrix denominators and using $p_1^2 = p_2^2 = m^2$ requires the integrals (9a),
as we have discussed. This is an integral with three denominators
which we do in two stages. First the factors $({\bf k}^2 - 2p_1 \cdot k)$
and $({\bf k}^2 - 2p_2 \cdot k)$ are combined by a parameter $y$;
$$
({\bf k}^2 - 2p_1 \cdot k)^{-1} ({\bf k}^2 - 2p_2 \cdot k)^{-1} = \int^1_0
dy ({\bf k}^2 - 2p_y \cdot k)^{-2},
$$
from (14a) where
$$
p_y = yp_1 + (1 - y)p_2. \eqno(21a)
$$
We therefore need the integrals
$$
(8i) \int (1; k_{\sigma}; k_{\sigma} k_{\tau}) d^4 k ({\bf k}^2 - L)^{-2}
({\bf k}^2 - 2p_y \cdot k)^{-2}, \eqno(22a)
$$
which we will then integrate with respect to $y$ from 0 to 1. Next we do
the integrals (22a) immediately from (20a) with $p = p_y,$ $\Delta = 0$:
$$
\begin{array}{c}
(22a) = - \int^1_0 \int^1_0 (1; xp_{y \sigma}; x^2 p_{y \sigma} p_{y \tau}\\\\
- \frac{\displaystyle 1}{\displaystyle 2} \delta_{ \sigma \tau} (x^2 p_y^2
+ (1 - x) L)) 2x(1 - x) dx (x^2 p_y^2 + L(1 - x))^{-2} dy.
\end{array}
$$
We now turn to the integrals on $L$ as required in (8a). The first term,
(1), in $(1; k_{\sigma}; k_{\sigma} k_{\tau})$ gives no trouble for large
$L$, but if $L$ is put equal to
zero there results $x^{-2} p_y^{-2}$ which leads to a diverging integral on $x$ as
$x \rightarrow 0$. This infra-red catastrophe is analyzed by using
$\lambda_{\mbox{min}}^2$ for the lower limit of the $L$ integral. For the last
term the upper limit of $L$ must be kept as $\lambda^2$. Assuming $\lambda_{\mbox{min}}^2
\ll p_y^2 \ll \lambda^2$ the $x$ integrals which
remain are trivial, as in the self-energy case. One finds
$$
- (8i) \int ({\bf k}^2 - \lambda_{\mbox{min}}^2)^{-1} d^4 kC ({\bf k}^2
- \lambda_{\mbox{min}}^2) ({\bf k}^2 - 2p_1 \cdot k)^{-1} ({\bf k}^2 -
2p_2 \cdot k)^{-1}
$$
$$
= \int^1_0 p_y^{-2} dy \ln (p_y^2 / \lambda_{\mbox{min}}^2) \eqno(23a)
$$
$$
-(8i) \int k_{\sigma} {\bf k}^{-2} d^4 kC ({\bf k}^2) ({\bf k}^2 - 2p_1
\cdot k)^{-1} ({\bf k}^2 - 2p_2 \cdot k)^{-1}
$$
$$
= 2 \int^1_0 p_{y \sigma} p_y^{-2} dy, \eqno(24a)
$$
$$
-(8i) \int k_{\sigma} k_{\tau} {\bf k}^{-2} d^4 kC ({\bf k}^2) ({\bf k}^2
- 2p_1 \cdot k)^{-1} ({\bf k}^2 - 2p_2 \cdot k)^{-1}
$$
$$
= \int^1_0 p_{y \sigma} p_{y \tau} p_y^{-2} dy - \frac{\displaystyle 1}{\displaystyle 2}
\delta_{\sigma \tau} \int^1_0 dy \ln (\lambda^2 p_y^{-2}) + \frac{\displaystyle 1}{\displaystyle 4}
\delta_{\sigma \tau}. \eqno(25a)
$$
The integrals on $y$ give,
$$
\int^1_0 p_y^{-2} dy \ln (p_y^2 \lambda_{\mbox{min}}^{-2}) = 4(m^2 \sin
2 \theta)^{-1} \left[ \theta \ln (m \lambda_{\mbox{min}}^{-1})
- \int^{\theta}_0 \alpha {\rm tan} \alpha d \alpha \right], \eqno(26a)
$$
$$
\int^1_0 p_{y \sigma} p_y^{-2} dy = \theta(m^2 \sin 2 \theta)^{-1} (p_{1
\sigma} + P_{2 \sigma}), \eqno(27a)
$$
$$
\int^1_0 p_{y \sigma} p_{y \tau} p_y^{-2} dy = \theta (2m^2 \sin 2 \theta)^{-1}
(p_{1 \sigma} + p_{1 \tau} (p_{2 \sigma} + p_{2 \tau})
$$
$$
+ {\bf q}^{-2} q_{\sigma} q_{\tau} (1 - \theta {\rm ctn} \theta), \eqno(28a)
$$
$$
\int^1_0 dy \ln (\lambda^2 p_y^{-2}) = \ln (\lambda^2/m^2) + 2 (1 - \theta
{\rm ctn} \theta). \eqno(29a)
$$
These integrals on $y$ were performed as follows. Since $p_2 = p_1 + {\bf
q}$ where
${\bf q}$ is the momentum carried by the potential, it follows from ${\bf
p}_2^2 ={\bf p}_1^2 = m^2$ that $2p_1 \cdot q = - {\bf q}^2$ so that since ${\bf
p}_y = {\bf p}_1 + {\bf q} (1- y),$ ${\bf p}_y^2 = m^2 - {\bf q}^2 y(1 - y)$. The
substitution $2y - 1 = {\rm tan} \theta$ where $\theta$ is defined by
$4m^2 \sin^2 \theta = {\bf q}^2$ is useful for it means ${\bf p}_y^2 = m^2
{\rm sec}^2 \alpha/{\rm sec}^2 \theta$ and ${\bf p}_y^{-2} dy = (m^2 \sin 2 \theta))^{-1}
d \alpha$ where $\alpha$ goes from $- \theta$ to $+ \theta$.
These results are substituted into the original scattering formula
(2a), giving (22). It has been simplified by frequent use of the fact
that ${\bf p}_1$ operating on the initial state is $m$ and likewise ${\bf
p}_2$ when it
appears at the left is replacable by $m$. (Thus, to simplify:
$$
\gamma_{\mu} {\bf p}_2 {\bf ap}_1 \gamma_{\mu} = - 2{\bf p}_1 {\bf ap}_2
\quad \mbox{by} \quad (4a),
$$
$$
= - 2 ({\bf p}_2 - {\bf q}){\bf a} ({\bf p}_1 + {\bf q}) = - 2(m - {\bf
q}){\bf a} (m + {\bf q}).
$$
A term like ${\bf qaq} = - q^2 {\bf a} + 2 (a \cdot q){\bf q}$ is equivalent
to just $-q^2{\bf a}$ since \mbox{${\bf q} = {\bf p}_2 - {\bf p}_1 = m - m$}
has zero matrix element.) The renormalization term
requires the corresponding integrals for the special case ${\bf q} = 0$.}
\section*{\bf C. Vacuum Polarization}
{\small
~~~~The expressions (32) and (32$'$) for $J_{\mu \nu}$ in the vacuum polarization
problem require the calculation of the integral
$$
J_{\mu \nu}(m^2) = \frac{\displaystyle e^2}{\displaystyle \pi i} \int Sp[\gamma_{\mu}
({\bf p} - \frac{\displaystyle 1}{\displaystyle 2} {\bf q} + m) \gamma_{\nu}
{\bf p} + \frac{\displaystyle 1}{\displaystyle 2} {\bf q} + m)]d^4 p
$$
$$
\times (({\bf p} - \frac{\displaystyle 1}{\displaystyle 2} {\bf q})^2 -
m^2)^{-1} (({\bf p} + \frac{\displaystyle 1}{\displaystyle 2} {\bf q})^2
- m^2)^{-1}, \eqno(32)
$$
where we have replaced ${\bf p}$ by ${\bf p} - \frac{1}{2} {\bf q}$ to simplify
the calculation somewhat. We
shall indicate the method of calculation by studying the integral,
$$
I(m^2) = \int p_{\sigma} p_{\tau}d^4 p(({\bf p} - \frac{1}{2} {\bf q})^2 -
m^2)^{-1} (({\bf p} + \frac{1}{2}{\bf q})^2 - m^2)^{-1}.
$$
The factors in the denominator, ${\bf p}^2 - p \cdot q - m^2 + \frac{1}{4}
{\bf q}^2$ and ${\bf p}^2 + p \cdot q - m^2 + \frac{1}{4} {\bf q}^2$
are combined as usual by (8a) but for symmetry we
substitute $x = \frac{1}{2} (1 + \eta),$ $(1 - x) = \frac{1}{2} (1 - \eta)$ and
integrate $\eta$ from $- 1$ to $+1$:
$$
I(m^2) = \int \limits^{+1}_{-1} p_{\sigma} p_{\tau} d^4 p ({\bf p}^2 -
\eta p \cdot q - m^2 + \frac{1}{4} {\bf q}^2)^{-2} d \eta /2. \eqno(30a)
$$
But the integral on ${\bf p}$ will not be found in our list for it is badly
divergent. However, as discussed in Section 7, Eq. (34$'$) we do not
wish $I(m^2)$ but rather \mbox{$\int \limits^{\infty}_0 [I(m^2) - I(m^2 + \lambda^2)]
G(\lambda) d \lambda$.} We can calculate
the difference $I(m^2) - I(m^2 + \lambda^2)$ by first calculating the derivative
$I'(m^2 + L)$ of $I$ with respect to $m^2$ at $m^2 + L$ and later integrating $L$
from zero to $\lambda^2$. By differentiating (30a), with respect to $m^2$ find,
$$
I'(m^2 + L) = \int \limits^{+1}_{-1} p_{\sigma} p_{\tau} d^4 p({\bf p}^2
- \eta p \cdot q - m^2 - L + \frac{1}{4} {\bf q}^2)^{-3} d \eta.
$$
This still diverges, but we can differentiate again to get
$$
I''(m^2 + L) = 3 \int \limits^{+1}_{-1} p_{\sigma} p_{\tau} d^4 p ({\bf
p}^2 - \eta p \cdot q - m^2 - L + \frac{\displaystyle 1}{\displaystyle 4}
{\bf q}^2)^{-4} d \eta
$$
$$
= - (8i)^{-1} \int \limits^{+1}_{-1} (\frac{\displaystyle 1}{\displaystyle 4}
\eta^2 q_{\sigma} q_{\tau} D^{-2} - \frac{\displaystyle 1}{\displaystyle 2}
\delta_{\sigma \tau} D^{-1}) d \eta
\eqno(31a)
$$
(where $D = \frac{1}{4} (\eta^2 - 1) {\bf q}^2 + m^2 + L)$, which now converges
and has been evaluated by (13a) with ${\bf p} = \frac{1}{2} \eta{\bf q}$
and $\Delta = m^2 + L - \frac{1}{4} {\bf q}^2$. Now to get $I'$ we may
integrate $I''$ with respect to $L$ as an indefinite integral and {\it
we may choose any convenient arbitrary constant.} This is because a constant
$C$ in $I'$ will mean a term $-C \lambda^2$ in $I(m^2) - I (m^2 + \lambda^2)$
which vanishes since we will integrate the results times $G(\lambda) d
\lambda$ and $\int \limits^{\infty}_0 \lambda^2 G (\lambda) d \lambda =
0$. This means that the logarithm appearing on integrating $L$ in (31a) presents
no problem. We may take
$$
I'(m^2 + L) = (8i)^{-1} \int \limits^{+1}_{-1} [ \frac{1}{4} \eta^2 q_{\sigma}
q_{\tau} D^{-1} + \frac{1}{2} \delta_{\sigma \tau} \ln D] d \eta + C \delta_{\sigma
\tau},
$$
a subsequent integral on $L$ and finally on $\eta$ presents no new
problems. There results
$$
-(8i) \int p_{\sigma} p_{\tau} d^4 p (({\bf p} - \frac{\displaystyle 1}{\displaystyle 2}
{\bf q})^2 - m^2)^{-1} (({\bf p} + \frac{\displaystyle 1}{\displaystyle 2}
{\bf q})^2 - m^2)^{-1}
$$
$$
= (q_{\sigma} q_{\tau} - \delta_{\sigma \tau} {\bf q}^2) \left[ \frac{1}{9}
- \frac{4m^2 - {\bf q}^2}{3 {\bf q}^2} \left( 1 - \frac{\theta}{\mbox{tan}
\theta} \right) + \frac{\displaystyle 1}{\displaystyle 6} \ln \frac{\lambda^2}{m^2}
\right]
$$
$$
+ \delta_{\sigma \tau} [(\lambda^2 + m^2) \ln (\lambda^2 m^{-2} + 1) -
C' \lambda^2] \eqno(32a)
$$
where we assume $\lambda^2 \gg m^2$ and have put some terms into the arbitrary
constant $C'$ which is independent of $\lambda^2$ (but in principle could depend
on ${\bf q}^2$ and which drops out in the integral on $G(\lambda) d \lambda$. We
have set ${\bf q}^2 = 4m^2 \sin^2 \theta$.
In a very similar way the integral with $m^2$ in the numerator can be
worked out. It is, of course, necessary to differentiate this $m^2$ also
when calculating $I'$ and $I''$. There results
$$
-(8i) \int m^2 d^4 p(({\bf p} - \frac{\displaystyle 1}{\displaystyle 2}
{\bf q})^2 - m^2)^{-1} (({\bf p} + \frac{\displaystyle 1}{\displaystyle 2}
{\bf q})^2 - m^2)^{-1}
$$
$$
= 4m^2(1 - \theta \mbox{ctn} \theta) - {\bf q}^2/3 + 2 (\lambda^2 + m^2)
\ln (\lambda^2 m^{-2} + 1) - C'' \lambda^2), \eqno(33a)
$$
with another unimportant constant $C''$. The complete problem requires the
further integral,
$$
-(8i) \int (1; p_{\sigma})d^4 p(({\bf p} - \frac{\displaystyle 1}{\displaystyle 2}
{\bf q})^2 - m^2)^{-1} (({\bf p} + \frac{\displaystyle 1}{\displaystyle 2}
{\bf q})^2 - m^2)^{-1}
$$
$$
= (1, 0)(4(1 - \theta \mbox{ctn} \theta) + 2 \ln (\lambda^2 m^{-2})). \eqno(34a)
$$
The value of the integral (34a) times $m^2$ differs from (33a), of course,
because the results on the right are not actually the integrals on the
left, hut rather equal their actual value minus their value for $m^2 =
m^2 + \lambda^2$.
Combining these quantities, as required by (32), dropping the
constants $C', C''$ and evaluating the spur gives (33). The spurs are
evaluated in the usual way, noting that the spur of any odd number of
$\gamma$ matrices vanishes and $S_p(AB)=S_p(BA)$ for arbitrary $A,B$. The
$S_p(1) = 4$ and we also have
$$
\frac{1}{2} S_p [(p_1 + m_1)({\bf p}_2 - m_2)] = p_1 \cdot p_2 - m_1 m_2,
\eqno(35a)
$$
$$
\frac{\displaystyle 1}{\displaystyle 2} S_p [({\bf p}_1 + m_1)({\bf p}_2
- m_2)({\bf p}_4 - m_4)] = (p_1 \cdot p_2 - m_1 m_2)(p_3 \cdot p_4 - m_3
m_4)
$$
$$
-(p_1 \cdot p_3 - m_1 m_3)(p_2 \cdot p_4 - m_2 m_4) + (P-1 \cdot p_4 -
m_1 m_4)(p_2 \cdot p_3 - m_2 m_3), \eqno(36a)
$$
where ${\bf p}_i, m_i$ are arbitrary four-vectors and constants.
It is interesting that the terms of order $\lambda^2 \ln \lambda^2$ go out,
so that the charge renormalization depends only logarithmically on $\lambda^2$.
This is not true for some of the meson theories. Electrodynamics is
suspiciously unique in the mildness of its divergence.
\section*{\bf D. More Complex Problems}
{\small
~~~~Matrix elements for complex problems can be set up in a manner
analogous to that used for the simpler cases. We give three
illustrations; higher order corrections to the M$\varnothing$ller scattering,
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig8e.gif}}}
\caption{The interaction between two electrons to order $(e^2/ \hbar c)^2$. One
adds the contribution of every figure involving two virtual quanta,
Appendix D.}
\end{figure}
to the Compton scattering, and the interaction of a neutron with an
electromagnetic field.
For the M$\varnothing$ller scattering, consider two electrons, one in state $u_1$
of momentum ${\bf p}_1$ and the other in state $u_2$ of momentum ${\bf
p}_2$. Later they are found in states $u_3, {\bf p}_3$ and $u_4, {\bf p}_4$.
This may happen (first order in $e^2/ \hbar c$)
because they exchange a quantum of momentum ${\bf q} = {\bf p}_1 - {\bf
p}_3 = {\bf p}_4 - {\bf p}_2$ in the
manner of Eq. (4) and Fig. 1. The matrix element for this process is
proportional to (translating (4) to momentum space)
$$
(\tilde u_4 \gamma_{\mu} u_2)(\tilde u_3 \gamma_{\mu} u_1) {\bf q}^{-2}.
\eqno(37a)
$$
We shall discuss corrections to (37a) to the next order in $e^2/ \hbar
c$. (There is also the possibility that it is the electron at 2 which finally
arrives at 3, the electron at 1 going to 4 through the exchange of quantum of
momentum ${\bf p}_3 - {\bf p}_2$. The amplitude for this process,
$(\tilde u_4 \gamma_{\mu} u_1) (\tilde u_3 \gamma_{\mu} u_2) ({\bf p}_3
- {\bf p}_2)^{-2}$ must be subtracted from (37a) in
accordance with the exclusion principle. A similar situation exists to
each order so that we need consider in detail only the corrections to
(37a), reserving to the last the subtraction of the same terms with 3, 4
exchanged.)
One reason that (37a) is modified is that two quanta may be
exchanged, in the manner of Fig. 8a. The total matrix element for all
exchanges of this type is
$$
(e^2/ \pi i) \int (\tilde u_3 \gamma_{\nu}({\bf p}_1 - {\bf k} - m)^{-1}
\gamma_{\mu} u_1)(\tilde u_4 \gamma_{\nu} ({\bf p}_2 + {\bf k} - m)^{-1}
\gamma_{\mu} u_2)
$$
$$
\cdot {\bf k}^{-2} ({\bf q - k})^{-2} d^4 k, \eqno(38a)
$$
as is clear from the figure and the general rule that electrons of
momentum ${\bf p}$ contribute in amplitude $({\bf p} - m)^{-1}$ between interactions
$\gamma_{\mu}$ and that quanta of momentum ${\bf k}$ contribute ${\bf k}^{-2}$.
In integrating on $d^4 k$ and summing over $\mu$ and $\nu$, we add all alternatives
of the type of Fig. 8a. If the time of absorption, $\gamma_{\mu}$, of the quantum
${\bf k}$ by electron 2 is later than the absorption, $\gamma_{\mu}$, of ${\bf
q - k}$, this corresponds to the
virtual state ${\bf p}_2 + {\bf k}$ being a positron (so that (38a)
contains over thirty terms of the conventional method of analysis).
In integrating over all these alternatives we have considered all
possible distortions of Fig. 8a which preserve the order of events along
the trajectories. We have not included the possibilities corresponding
to Fig. 8b, however. Their contribution is
$$
(e^2/ \pi i) \int (\tilde u_3 \gamma_{\nu} ({\bf p}_1 - {\bf k} - m)^{-1}
\gamma_{\nu} u_1)
$$
$$
\times (\tilde u_4 \gamma_{\mu} ({\bf p}_2 + {\bf q - k} - m)^{-1} \gamma_{\nu}
u_2) {\bf k}^{-2} ({\bf q - k})^{-2} d^4 k, \eqno(39a)
$$
as is readily verified by labeling the diagram. The contributions of all
possible ways that an event can occur are to be added. This
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig9e.gif}}}
\caption{Radiative correction to the Compton scattering term (a)
of Fig. 5. Appendix D.}
\end{figure}
means that one adds with equal weight the integrals corresponding to
each topologically distinct figure.
To this same order there are also the possibilities of Fig. 8d which
give
$$
(e^2/ \pi i) \int (\tilde u_3 \gamma_{\nu} ({\bf p}_2 - {\bf k} - m)^{-1}
\gamma_{\mu} ({\bf p}_1 - {\bf k} - m)^{-1} \gamma_{\nu} u_1)
\times (\tilde u_4 \gamma_{\mu} u_2) {\bf k}^{-2} {\bf q}^{-2} d^4 k.
$$
This integral on ${\bf k}$ will be seen to be precisely the integral (12) for the
radiative corrections to scattering, which we have worked out. The
term may be combined with the renormalization terms resulting from
the difference of the effects of mass change and the terms, Figs. 8f and
8g. Figures 8e, 8h, and 8i are similarly analyzed.
Finally the term Fig. 8c is clearly related to our vacuum polarization
problem, and when integrated gives a term proportional to
$(\tilde u_4 \gamma_{\mu} u_2) (\tilde u_3 \gamma_{\nu} u_1) J_{\mu \nu}
{\bf q}^{-4}$. If the charge is renormalized the term
$\ln (\lambda/m)$ in $J_{\mu \nu}$ in (33) is omitted so there is no remaining dependence
on the cut-off.
The only new integrals we require are the convergent integrals
(38a) and (39a). They can be simplified by rationalizing the denominators
and combining them by (14a). For example (38a) involves the factors
\mbox{$({\bf k}^2 - 2p_1 \cdot k)^{-1} ({\bf k}^2 + 2p_2 \cdot k)^{-1} {\bf k}^{-2}
({\bf q}^2 + {\bf k}^2 - 2q \cdot k)^{-2}$.} The first two may
be combined by (14a) with a parameter $x$, and the second pair by an
expression obtained by differentation (l5a) with respect to $b$ and
calling the parameter $y$. There results a factor $({\bf k}^2 - 2p_x \cdot
k)^{-2} ({\bf k}^2 + y {\bf q}^2 - 2yq \cdot k)^{-4}$
so that the integrals on $d^4 k$ now involve two factors and can be
performed by the methods given earlier in the appendix. The
subsequent integrals on the parameters $x$ and $y$ are complicated and
have not been worked out in detail.
Working with charged mesons there is often a considerable reduction of
the number of terms. For example, for the interaction
between protons resulting from the exchange of two mesons only the
term corresponding to Fig. 8h remains. Term 8a, for example, is
impossible, for if the first proton emits a positive meson the second
cannot absorb it directly for only neutrons can absorb positive
mesons.
As a second example, consider the radiative correction to the
Compton scattering. As seen from Eq. (15) and Fig. 5 this scattering
is represented by two terms, so that we can consider the corrections
to each one separately. Figure 9 shows the types of terms arising
from corrections to the term of Fig. 5a. Calling ${\bf k}$ the momentum of
the virtual quantum, Fig. 9a gives an integral
$$
\int \gamma_{\mu} ({\bf p}_2 - {\bf k} - m)^{-1} {\bf e}_2 ({\bf p}_1 +
{\bf q}_1 - {\bf k} - m)^{-1} {\bf e}_1 ({\bf p}_1 - {\bf k} - m)^{-1}
\gamma_{\mu} {\bf k}^{-2} d^4 k,
$$
convergent without cut-off and reducible by the methods outlined in
this appendix.
The other terms are relatively easy to evaluate. Terms $b$ and $c$ of
Fig. 9 are closely related to radiative corrections (although somewhat
more difficult to evaluate, for one of the states is not that of a free
electron, $({\bf p}_1 + {\bf q})^2 \ne m^2)$. Terms $e, f$, are renormalization
terms. From term $d$ must be subtracted explicitly the effect of mass $\Delta
m$, as analyzed in Eqs. (26) and (27) leading to (28) with $p' = {\bf p}_1
+ {\bf q},$ ${\bf a} = {\bf e}_2$, ${\bf b} = {\bf e}_1$. Terms $g, h$
give zero since the vacuum polarization has zero effect on free light
quanta, ${\bf q}_1^2 = 0,$ ${\bf q}_2^2 = 0$. The total is insensitive to
the cut-off $\lambda$.
The result shows an infra-red catastrophe, the largest part of the
effect. When cut-off at $\lambda{\mbox{min}}$, the effect proportional to
$\ln (m/\lambda_{\mbox{min}})$ goes as
$$
(e^2/ \pi) \ln (m/\lambda_{\mbox{min}})(1 - 2 \theta ~\mbox{ctn} 2 \theta),
\eqno(40a)
$$
times the uncorrected amplitude, where $({\bf p}_2 - {\bf p}_1)^2 = 4m^2
\sin^2 \theta$. This is the same as for the radiative correction to scattering for
a deflection ${\bf p}_2 - {\bf p}_1$. This is physically clear since the long
wave quanta are not effected by short-lived intermediate states. The infra-red
effects arise\footnote{F. Bloch and A. Nordsieck, Phys. Rev. {\bf 52,} 54 (1937).}
from a final adjustment of the field from the asymptotic coulomb field
characteristic of the electron of
momentum ${\bf p}_1$ before the collision to that characteristic of an electron
moving in a new direction ${\bf p}_2$ after the collision.
The complete expression for the correction is a very complicated
expression involving transcendental integrals.
As a final example we consider the interaction of a neutron with an
electromagnetic field in virtue of the fact that the neutron may emit a
virtual negative meson. We choose the example of pseudoscalar
mesons with pseudovector coupling. The change in amplitude due to
an electromagnetic field ${\bf A} = {\bf a}~ \mbox{exp}(- iq \cdot x)$
determines the scattering of
a neutron by such a field. In the limit of small ${\bf q}$ it wilt vary as
${\bf qa - aq}$ which represents the interaction of a particle possessing a
magnetic moment. The first-order interaction between an electron
and a neutron is given by the same calculation by considering the
exchange of a quantum between the electron and the nucleon. In this
case $a_{\mu}$ is ${\bf q}^{-2}$ times the matrix element of $\gamma_{\mu}$
between the initial and final states of the electron, the states differing in
momentum by ${\bf q}$.
The interaction may occur because the neutron of momentum ${\bf p}_1$
emits a negative meson becoming a proton which proton interacts
with the field and then reabsorbs the meson (Fig. 10a). The matrix for
this process is $({\bf p}_2 = {\bf p}_1 + {\bf q}),$
$$
\int (\gamma_5 {\bf k} ({\bf p}_2 - {\bf k} - M)^{-1} {\bf a} ({\bf p}_1
- {\bf k} - M)^{-1} (\gamma_5 {\bf k}) ({\bf k}^2 - \mu^2)^{-1} d^4 k.
\eqno(41a)
$$
Alternatively it may be the meson which interacts with the field. We
assume that it does this in the manner of a scalar potential satisfying
the Klein Gordon Eq. (35), (Fig. l0b)
$$
- \int (\gamma_5 {\bf k}_2) ({\bf p}_1 - {\bf k}_1 - M)^{-1} (\gamma_5
{\bf k}_1) ({\bf k}_2^2 - \mu^2)^{-1}
$$
$$
\times (k_2 \cdot a + k_1 \cdot a) ({\bf k}_1^2 - \mu^2)^{-1} d^4 k_1,
\eqno(42a)
$$
where we have put ${\bf k}_2 = {\bf k}_1 + {\bf q}$. The change in sign arises
because the virtual meson is negative. Finally there are two terms arising from
the $\gamma_5 {\bf a}$ part of the pseudovector coupling (Figs. 10c, 10d)
$$
\int (\gamma_5 {\bf k}) ({\bf p}_2 - {\bf k} - M)^{-1} (\gamma_5 {\bf a})
({\bf k}^2 - \mu^2)^{-1} d^4 k, \eqno(43a)
$$
and
$$
\int (\gamma_5 {\bf a}) ({\bf p}_1 - {\bf k} - M)^{-1} (\gamma_5 {\bf k})
({\bf k}^2 - \mu^2)^{-1} d^4 k, \eqno(44a)
$$
Using convergence factors in the manner discussed in the section on
meson theories each integral can be evaluated and the results
combined. Expanded in powers of ${\bf q}$ the first term gives the magnetic
moment of the neutron and is insensitive to the cut-off, the next gives
the scattering amplitude of stow electrons on neutrons, and depends
logarithmically on the cut-off.
The expressions may be simplified and combined somewhat
before integration. This makes the integrals a little easier and also
shows the relation to the case of pseudoscalar coupling. For example
in (41a) the final $\gamma_5 {\bf k}$ can be written as $\gamma_5({\bf
k - p}_1 + M)$ since ${\bf p}_1 = M$
when operating on the initial neutron state. This is
\begin{figure}[h]
\centerline{\resizebox{7.5cm}{!}{\includegraphics{fig10e.gif}}}
\caption{According to the meson theory a neutron interacts with an
electromagnetic potential ${\bf a}$ by first emitting a virtual charged meson.
The figure illustrates the case for a pseudoscalar meson with
pseudovector coupling. Appendix D.}
\end{figure}
$({\bf p}_1 - {\bf k} - M) \gamma_5 + 2m \gamma_5$ since $\gamma_5$
anticommutes with ${\bf p}_1$ and ${\bf k}$. The
first term cancels the $({\bf p}_1 - {\bf k} - M)^{-1}$ and gives a term which
just cancels (43a). In a like manner the leading factor $\gamma_5 {\bf
k}$ in (41a) is written as $- 2 M \gamma-5 - \gamma_5 ({\bf p}_2 - {\bf
k} - M)$, the second term leading to a
simpler term containing no $({\bf p}_2 - k - M)^{-1}$ factor and combining with
a similar one from (44a). One simplifies the $\gamma_5 {\bf k}_1$
and $\gamma_5 {\bf k}_2$ in (42a) in
an analogous way. There finally results terms like (41a), (42a) but
with pseudoscalar coupling $2 M \gamma_5$ instead of $\gamma_5 {\bf k}$, no terms like
(43a) or (44a) and a remainder, representing the difference in effects
of pseudovector and pseudoscalar coupling. The pseudoscalar terms
do not depend sensitively on the cut-off, but the difference term
depends on it logarithmically. The difference term affects the
electron-neutron interaction but not the magnetic moment of the
neutron.
Interaction of a proton with an electromagnetic potential can be
similarly analyzed. There is an effect of virtual mesons on the
electromagnetic properties of the proton even in the case that the
mesons are neutral. It is analogous to the radiative corrections to the
scattering of electrons due to virtual photons. The sum of the
magnetic moments of neutron and proton for charged mesons is the
same as the proton moment calculated for the corresponding
neutral mesons. In fact it is readily seen by comparing diagrams.
that for arbitrary ${\bf q}$, the scattering matrix to {\it first order in the
electromagnetic potential} for a proton according to neutral meson
theory is equal, if the mesons were charged, to the sum of the matrix
for a neutron and the matrix for a proton. This is true, for any type
or mixtures of meson coupling, to all orders in the coupling
(neglecting the mass difference of neutron and proton).}
\end{document}
%ENCODE September 2002 BY NIS;