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\begin{document}
F. J. Dyson, Phys. Rev. {\bf 75,} 1736 \hfill {\large \bf 1949}\\
\vspace{2cm}
\begin{center}
{\Large \bf The $S$ Matrix in Quantum Electrodynamics}\\
\end{center}
\vspace{0.5cm}
\begin{center}
F.J. Dyson\\
Institute for Advanced Study, Princeton, New Jersey \\
(Received February 24, 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}
The covariant quantum electrodynamics of Tomonaga, Schwinger, and Feynman
is used as the basis for a general treatment of scattering problems involving
electrons, positrons, and photons. Scattering processes, including the
creation and annihilation of particles, are completely described by the $S$ matrix
of Heisenberg. It is shown that the elements of this matrix can be calculated, by
a consistent use of perturbation theory, to any desired order in the fine-structure
constant. Detailed rules are given for carrying out such calculations, and it is shown that divergences
arising from higher order radiative corrections can be removed from the $S$ matrix
by a consistent use of the ideas of mass and charge renormalization.
Not considered in this paper are the problems of extending the treatment to include
bound-state phenomena, and of proving the convergence of the theory as the order
of perturbation itself tends to infinity.
\end{abstract}
\section{ INTRODUCTION}
~~~~In a previous paper\footnote{F. J. Dyson. Phys. Rev. {\bf 75}, 486 (1949).}
(to be referred to in what follows as I) the radiation theory of
Tomonaga\footnote{Sin-Itiro Tomonaga, Prog. Theor. Phys. {\bf 1,} 27 (1946);
Koba. Tati, and Tomonaga, Prog. Theor. Phys. {\bf 2,} 101 (1947) and {\bf
2,} 198 (1947); S. Kanesawa and S. Tomonaga, Prog. Theor. Phys. {\bf 3,} 1
(1948) and {\bf 3,} 101 (1948); Sin-Itiro Tomonaga, Phys. Rev. {\bf 74,} 224
(1948); Ito, Koba. and Tomonaga, Prog. Theor. Phys. {\bf 3,} 276 (1948);
Z. Koba and S. Tomonaga. Prog. Theor. Phys. {\bf 3,} 290 (1948).}
and Schwinger\footnote{Julian Schwinger, Phys. Rev. {\bf 73,} 416 (1948); {\bf
74,} 1439 (1948); {\bf 75,} 651 (1949).} was applied in detail to the problem of
the radiative corrections to the motion of a single
electron in a given external field. It was shown that
the rules of calculation for corrections of this kind
were identical with those which had been derived
by Feynman\footnote{Richard P. Feynman, Phys. Rev. {\bf 74,} 1430 (1948).}
from his own radiation theory. For the one-electron problem the radiative
corrections were fully described by an operator $H_T$ (Eq. (20) of
I) which appeared as the ``effective potential'' acting
upon the electron, after the interactions of the
electron with its own self-field had been eliminated
by a contact transformation. The difference between
the Schwinger and Feynman theories lay only in the
choice of a particular representation in which the
matrix elements of $H_T$ were calculated (Section V of I).
The present paper deals with the relation between
the Schwinger and Feynman theories when the
restriction to one-electron problems is removed. In
these more general circumstances the two theories
appear as complementary rather than identical. The
Feynman method is essentially a set of rules for the
calculation of the elements of the Heisenberg $S$
matrix corresponding to any physical process, and
can be applied with directness to all kinds of
scattering problems.\footnote{The idea of using standard electrodynamics as a starting point for
an explicit calculation of the $S$ matrix has been previously developed
by E. C. G. Stueckelberg, Helv. Phys. Acta, {\bf 14,} 51 (1941); {\bf 17,} 3
(1944); {\bf 18}, 195 (1945); {\bf 19,} 242 (1946); Nature, {\bf 153,} 143 (1944);
Phys. Soc. Cambridge Conference Report, 199 (1947); E. C. G. Stueckelberg and
D. Rivier, Phys. Rev. {\bf 74,} 218 (1948). Stueckelberg anticipated several features
of the Feynman theory, in particular the use of the function $D_F$ (in
Stuecketberg's notation $D^C$) to represent retarded (i.e., causally
transmitted) electromagnetic interactions. For a review of the earlier
part of this work, see Gregor Wentzel, Rev. Mod. Phys. {\bf 19,} 1 (1947).
The use of mass renormalization in scattering problems is due to
H. W Lewis, Phys. Rev {\bf 73,} 173 (1948).}
The Schwinger method evaluates radiative corrections by exhibiting them as
extra terms appearing in the Schrodinger equation
of a system of particles and is suited especially to
bound-state problems. In spite of the difference of
principle, the two methods in practice involve the
calculation of closely related expressions;
moreover, the theory underlying them is in all cases
the same. The systematic technique of Feynman, the
exposition of which occupied the second half of I
and occupies the major part of the present paper, is
therefore now available for the evaluation not only
of the $S$ matrix but also of most of the operators
occurring in the Schwinger theory.
The prominent part which the $S$ matrix plays in
this paper is due to its practical usefulness as the
connecting link between the Feynman technique of
calculation and the Hamiltonian formulation of
quantum electrodynamics. This practical usefulness
remains, whether or not one follows Heisenberg in
believing that the $S$ matrix may eventually replace
the Hamiltonian altogether. It is still an unanswered
question, whether the finiteness of the $S$ matrix
automatically implies the finiteness of all
observable quantities, such as bound-state energy
levels, optical transition probabilities, etc.,
occurring in electrodynamics. An affirmative
answer to the question is in no way essential to the
arguments of this paper. Even if a finite $S$ matrix
does not of itself imply finiteness of other
observable quantities, it is probable that all such
quantities will be finite; to verify this, it will be
necessary to repeat the analysis of the present paper,
keeping all the time closer to the original Schwinger
theory than has here been possible. There is no reason for
attributing a more fundamental significance to the $S$
matrix than to other observable quantities, nor was it
Heisenberg's intention to do so. In the last section of
this paper, tentative suggestions are made for a
synthesis of the Hamiltonian and Heisenberg philosophies.
\section{THE FEYNMAN THEORY AS AN $S$ MATRIX THEORY}
~~~~The $S$ matrix was originally defined by Heisenberg
in terms of the stationary solutions of a scattering
problem. A typical stationary solution is represented
by a time-independent wave function $\Psi'$ which has a
part representing ingoing waves which are
asymptotically of the form $\Psi_1'$, and a part
representing outgoing waves which are asymptotically of the form $\Psi_2'$.
The $S$ matrix is the transformation operator $S$ with the property that
\begin{equation}
\Psi_2' = S \Psi_1'
\end{equation}
for every stationary state $\Psi'$.
In Section III of I an operator $U(\infty)$ was defined
and stated to be identical with the $S$ matrix. Since
$U(\infty)$ was defined in terms of time-dependent wave
functions, a little care is needed in making the identification. In fact, the equation
\begin{equation}
\Psi_2 = U(\infty) \Psi_1
\end{equation}
held, where $\Psi_1$ and $\Psi_2$ were the asymptotic forms of
the ingoing and outgoing parts, of a wave function $\Psi$
in the $\Psi$--representation of I (the ``interaction representation'' of
Schwinger\footnote{Julian Schwinger, Phys. Rev. {\bf 73,} 416 (1948); {\bf
74,} 1439 (1948); {\bf 75,} 651 (1949).}). Now the time-independent wave function
$\Psi'$ corresponds to a time-dependent wave function
$$
\mbox{exp} [(-i/ \hbar) Et] \Psi'
$$
in the Schrodinger representation, where $E$ is the
total energy of the state; and this corresponds to a
wave function in the interaction representation
\begin{equation}
\Psi = \mbox{exp} [( + i/ \hbar)t(H_0 - E)] \Psi',
\end{equation}
where $H_0$ is the total free particle Hamiltonian.
However, the asymptotic parts of the wave function
$\Psi'$, both ingoing and outgoing, represent freely
traveling particles of total energy $E$, and are there-fore eigenfunctions of
$H_0$ with eigenvalue $E$. This implies, in virtue of (3), that the asymptotic
parts $\Psi_1$ and $\Psi_2$ of $\Psi$ are actually time-independent and
equal, respectively, to $\Psi_1'$ and $\Psi_2'$. Thus (1) and (2)
are identical, and $U(\infty)$ is indeed the $S$ matrix.
Incidentally, $U(\infty)$ is also the ``invariant collision
operator'' defined by Schwinger.\footnote{Julian Schwinger, Phys. Rev. {\bf 73,} 416 (1948); {\bf
74,} 1439 (1948); {\bf 75,} 651 (1949).}
There is a series expansion of $U(\infty)$ analogous to
(32) of I, namely,
\begin{equation}
U(\infty) = \sum \limits^{\infty}_{n=0} \left( \frac{-i}{\hbar c} \right)^n
\frac{1}{n!} \int \limits^{\infty}_{- \infty} dx_1 \ldots \int \limits^{\infty}_{-
\infty} dx_n \times P(H_1(x_1), \ldots , H_1(x_n)).
\end{equation}
Here the $P$ notation is as defined in Section V of I, and
\begin{equation}
H_1(x) = H^{I} (x) + H^e (x)
\end{equation}
is the sum of the interaction energies of the electron
field with the photon field and with the external
potentials. The Feynman radiation theory provides a
set of rules for the calculation of matrix elements of
(4), between states composed of any number of
ingoing and outgoing free particles. Also, quantities
contained in (4) are the only ones with which the
Feynman rules can deal directly. The Feynman
theory is thus correctly characterized as an $S$ matrix theory.
One particular way to analyze $U(\infty)$ is to use (5)
to expand (4) in a series of terms of ascending order
in $H^e$. Substitution from (5) into (4) gives
$$
U(\infty) = \sum \limits^{\infty}_{m=0} \sum \limits^{\infty}_{n=0} \left(
\frac{-i}{\hbar c} \right)^{m+n} \frac{1}{m!n!} \int \limits^{\infty}_{-
\infty} dx_1 \ldots
$$
\begin{equation}
\times \int \limits^{\infty}_{- \infty} dx_{m+n} P(H^e(x_1),
\ldots , H^e (x_m),
\times \int \limits^{\infty}_{- \infty} H^I (x_{m+1}), \ldots H^I (x_{m+n})).
\end{equation}
In this double series, the term of zero order in $H^e$ is
$S(\infty)$, given by (32) of I. The term of first order is
\begin{equation}
U_1 = (-i/ \hbar c) \int \limits^{\infty}_{- \infty} H_F (x) dx,
\end{equation}
where $H_F$ is given by (31) of I. Clearly, $S(\infty)$ is the $S$
matrix representing scattering of electrons and
photons by each other in the absence of an external
potential; $U_1$ is the $S$ matrix representing the
additional scattering produced by an external
potential, when the external potential is treated in the
first Born approximation; higher terms of the series
(6) would correspond to treating the external
potential in the second or higher Born approximation. The operator $H_F$ played
a prominent part in I, where it was in no way connected with a Born approximation;
however, it was there introduced in a somewhat unnatural manner, and its physical
meaning is made clearer by its appearance in (7). In
fact, $H_F$ may be defined by the statement that
$$
(-i / \hbar) (\delta t) (\delta \omega) H_F (x)
$$
is the contribution to the $S$ matrix that would be
produced by an external potential of strength $H^e$,
acting for a small duration $\delta t$ and over a small
volume $\delta \omega$ in the neighborhood of the space-time
point $x$.
The remainder of this section will be occupied
with a statement of the Feynman rules for evaluating
$U(\infty)$. Proofs will not be given, because the rules are
only trivial generalizations of the rules
which were given in I for the evaluation of matrix
elements of $H_F$ corresponding to one-electron transitions.
In evaluating $U(\infty)$ we shall not make any distinction between the
external and radiative parts of the electromagnetic field; this is physically
reasonable since it is to some extent a matter of convention how much of the field
in a given situation is to be regarded as ``external.'' The interaction
energy occurring in (4) is then
\begin{equation}
H_1(x) = - ie A_{\mu} (x) \tilde \psi (x) \gamma_{\mu} \psi (x) - \delta
mc^2 \tilde \psi (x) \psi (x),
\end{equation}
where $A_{\mu}$ is the total electromagnetic field, and the
term in $\delta m$ is included in order to allow for the fact
that the interaction representation is defined in terms
of the total mass of an electron including its ``electromagnetic mass'' $\delta
m$ (see Section IV of I).
The first step in the evaluation of $U(\infty)$ is to substitute from (8) into
(4), writing out in full the
suffixes of the operators $\tilde \psi_{\alpha}, \psi_{\beta}$ which are concealed
in the matrix product notation of (8). After such a substitution, (4) becomes
\begin{equation}
U(\infty) = \sum \limits^{\infty}_{n=0} J_n,
\end{equation}
where $J_n$, is an $n$--fold integral with an integrand
which is a polynomial in $\tilde \psi_{\alpha}, \psi_{\beta}$ and $A_{\mu}$, operators.
The most general matrix element of $J_n$ is obtained
by allowing some of the $\tilde \psi_{\alpha}, \psi_{\beta}$ and $A_{\mu}$
operators to annihilate particles in the initial state, some to create
particles in the final state, while others are
associated in pairs to perform a successive creation
and annihilation of intermediate particles. The
operators which are not associated in pairs, and
which are available for the real creation and annihilation of particles,
are called ``free''; a particular type of matrix element of $J_n$ is specified
by enumerating which of the operators in the integrand are
to be free and which are to be associated in pairs. As
described more fully in Section VII of I, each type of
matrix element of $J_n$. is uniquely represented by a
``graph'' $G$ consisting of $n$ points (bearing the labels
$x_1, \ldots, x_n$) and various lines terminating at these points.
The relation between a type of matrix element of
$J_n$. and its graph $G$ is as follows. For every associated
pair of operators $(\tilde \psi (x), \psi (y))$, there is a directed line
(electron line) joining $x$ to $y$ in $G$. For every
associated pair of operators $(A(x), A(y))$, there is
an undirected line (photon line) joining $x$ and $y$ in $G$.
For every free operator $\tilde \psi (x)$, there is a directed line
in $G$ leading from $x$ to the edge of the diagram. For
every free operator $\psi (x)$, there is a directed line in
$G$ leading to $x$ from the edge of the diagram. For
every free operator $A(x)$, there is an undirected line
in $G$ leading from $x$ to the edge of the diagram. Finally, for a particular type of
matrix element of $J_n$. it is specified that at each point
$x_i$, either the part of $H_1(x_i)$ containing $A_{\mu}(x_i)$ or the
part containing $\delta m$ is operating; correspondingly, at
each vertex $x_i$ of $G$ there are either two electron
lines (one ingoing and one outgoing) and one photon
line, or else two electron lines only. Lines joining
one point to itself are always forbidden.
In every graph $G$, the electron lines form a finite
number $m$ of open polygonal arcs with ends at the
edge of the diagram, and perhaps in addition a
number I of closed polygonal loops. The corresponding type of matrix element
of $J_n$, has $m$ free operators $\tilde \psi$ and $m$ free operators $\psi$ the
two end segments of any one open arc correspond to two free
operators, one $\tilde \psi$ and one $\psi$, which will be called a
``free pair.'' The matrix elements of $J_n$, are now to be
calculated by means of an operator $J(G)$, which is
defined for each graph $G$ of $n$ vertices, and which is
obtained from $J_n$ by making the following five
alterations.
First, at each point $x_i, H_1 (x_i)$ is to be.replaced by
either the first or the second term on the right of (8),
as indicated by the presence or absence of a photon
line at the vertex $x_i$ of $G$. Second, for every electron
line joining a vertex $x$ to a vertex $y$ in $G$, two
operators $\tilde \psi_{\alpha} (x)$ and $\psi_{\beta}(y)$ in $J_n$, regardless
of their positions, are to be replaced by the function
\begin{equation}
\frac{1}{2} S_{F \beta \alpha} (x - y),
\end{equation}
as defined by (44) and (45) of I. Third, for every
photon line joining two vertices $x$ and $y$ of $G$, two
operators $A_{\mu}(x)$ and $A_{\nu}(y)$ in $J_n$ regardless of their
positions, are to be replaced by the function
\begin{equation}
\frac{1}{2} \hbar c \delta_{\mu \nu} D_F (x - y),
\end{equation}
defined by (42) of I. Fourth, all free operators in $J_n$
are to be left unaltered, but the ordering by the $P$
notation is to be dropped, and the order of the free $\tilde \psi$
and $\psi$ operators is to be arranged so that the two
members of each free pair stand consecutively and
in the order $\tilde \psi \psi$; the order of the free pairs among
themselves, and of all free $A_{\mu}$, operators, is left
arbitrary. Fifth, the whole expression $J_n$ is to be multiplied by
\begin{equation}
(-1)^{n - l - m}.
\end{equation}
The Feynman rules for the evaluation of $U(\infty)$
are essentially contained in the above definition of
the operators $J(G)$. To each value of $n$ correspond
only a finite number of graphs $G$, and all possible
matrix elements of $U(\infty)$ are obtained by substituting into (9) for each
$J_n$ the sum of all the corresponding $J(G)$. It is necessary only to specify how
the matrix element of a given $J(G)$ corresponding to
a given scattering process may be written down.
The matrix element of $J(G)$ for a given process
may be obtained, broadly speaking, by replacing
each free operator in $J(G)$ by the wave function of
the particle which it is supposed to create or annihilate. More specifically,
each free $\tilde \psi$ operator may either create an electron in the final state or
annihilate a positron in the initial state, and the reverse
processes are performed by a free $\psi$ operator.
Therefore, for a transition from a state involving $A$
electrons and $B$ positrons to a state involving $C$
electrons and $D$ positrons, only operators $J(G)$
containing $(A+D)=(B+C)$ free pairs contribute
matrix elements. For each such $J(G)$, the $(A+D)$ free
$\psi$ operators are to be replaced in all possible
combinations by the $A$ initial electron wave functions and the $D$ final
positron wave functions, and the $(B+C)$ free $\tilde \psi$ operators are to be
similarly replaced by the initial positron and final electron wave
functions, and the results of all such replacements
added together, taking account of the antisymmetry
of the total wave functions of the system in the
individual particle wave functions. In the case of the
free $A_{\mu}$, operators, the situation is rather different,
since each such operator may either create a photon
in the final state, or annihilate, a photon in the initial
state, or represent merely the external potential.
Therefore, for a transition from a state with $A$
photons to a state with $B$ photons, any $J(G)$ with not
less than $(A+B)$ free $A_{\mu}$, operators may give a matrix
element. If the number of free $A_{\mu}$, operators in $J(G)$ is
$(A+B+C)$, these operators are to be replaced in all
possible combinations by the $(A+B)$ suitably
normalized potentials corresponding to the initial
and final photon states, and by the external potential
taken $C$ times, and the results of all such
replacements added together, taking account now of
the symmetry of the total wave functions in the individual photon states.
In practice cases are seldom likely to arise of
scattering problems in which more than two similar
particles are involved. The replacement of the free
operators in $J(G)$ by wave functions can usually be
carried out by inspection, and the enumeration of
matrix-elements of $U(\infty)$ is practically complete as
soon as the operators $J(G)$ have been written down.
The above rules for the calculation of $U(\infty)$
describe the state of affairs before any attempt has
been made to identify and remove the various
divergent parts of the expressions. In particular,
contributions are included from all graphs $G$, even
those which yield nothing but self-energy effects.
For this reason, the rules here formulated are
superficially different from those given for the one-electron problem in
Section IX of I, which described
the state of affairs after many divergencies had been
removed. Needless to say, the rules are not complete
until instructions have been supplied for the removal
of all infinite quantities from the theory; in Sections
V-VII of this paper it will be shown how the formal-structure of the $S$ matrix
makes such a complete removal of infinities appear
attainable.
Another essential limitation is introduced into the
$S$ matrix theory by the use of the expansion (4). All
quantities discussed in this paper are expansions of
this kind, in which it is assumed that not only the
radiation interaction but also the external potential is
small enough to be treated as a perturbation. It is well known that
an expansion in powers of the
external potential does not give a satisfactory
approximation, either in problems involving bound
states or in scattering problems at low energies. In
particular, whenever a scattering problem allows the
possibility of one of the incident particles being
captured into a bound state, the capture process will
not be represented in $U(\infty)$, since the initial and
final states for processes described by $U(\infty)$ are
always free-particle states. It is the expansion in
powers of the external potential which breaks down
when such a capture process is possible. Therefore it
must be emphasized that the perturbation theory of
this paper is applicable only to a restricted class of
problems, and that in other situations the Schwinger
theory will have to be used in its original form.
\section{THE $S$ MATRIX IN MOMENTUM SPACE}
~~~~Both for practical applications to specific problems, and for
general theoretical discussion, it is
convenient to express the $S$ matrix $U(\infty)$ in terms of
momentum variables. For this purpose, it is enough
to consider an expression which will be denoted by
$M$, and which is a typical example of the units out
of which all matrix elements of $U(\infty)$ are built up. A
particular integer $n$ and a particular graph $G$ of $n$
vertices being supposed fixed, the operator $J(G)$ is
constructed as in the previous section, and $M$ is
defined as the number obtained by substituting for
each of the free operators in $J(G)$ one particular
free-particle wave function. More specifically, for
each free operator $\psi(x)$ in $J(G)$ there is substituted
\begin{equation}
\psi(k) e^{ik_{\mu} x_{\mu}},
\end{equation}
where $k$, is some constant 4 vector representing the
momentum and energy of an electron, or minus the
momentum and energy of a positron, and where
$\psi(k)$ is a constant spinor. For each free operator
$\tilde \psi(x)$ there is substituted
\begin{equation}
\tilde \psi(k')e^{ik_{\mu'} x_{\mu}},
\end{equation}
where $\tilde \psi(k')$ is again a constant spinor. For each free
operator $A_{\mu}(x)$ there is substituted
\begin{equation}
A_{\mu} (k'') e^{ik_{\mu''} x_{\mu}},
\end{equation}
where $A_{\mu}(k'')$ is a constant 4 vector which may
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig1.gif}}}
\caption{}
\end{figure}
represent the polarization vector of a quantum whose
momentum-energy 4-vector is either plus or minus
$k_{\mu}''$; alternatively, $A_{\mu}(k'')$ may represent the Fourier
component of the external potential with a particular
wave number and frequency specified by the 4
vector $k''$. There is no loss of generality in splitting
up the external potential into Fourier components of
the form (15). When the substitutions (13), (14), (15)
are made in $J(G)$, the expression $M$ which is
obtained is still an $n$--fold integral over the whole of
space-time, and in addition depends parametrically
upon $E$ constant 4 vectors in momentum-space,
where $E$ is the number of free operators in $J(G)$.
The graph $G$ will contain $E$ external lines, i.e.,
lines with one end at a vertex and the other end at
the edge of the diagram. To each of these external
lines corresponds one constant 4 vector, which may
be denoted by $k_{\mu}^i, i=1, \ldots , E$, and one constant
spinor or polarization vector appearing in $M$, either
$\psi(k^i)$ or $\tilde \psi(k^i)$ or $A_{\mu}(k^i)$.
Suppose that $G$ contains $F$ internal lines, i.e., lines
with both ends at vertices. To each of these lines
corresponds a $D_F$ or an $S_F$ function in $M$, as
specified by (11) or (10). These functions have been
expressed by Feynman as 4-dimensional Fourier
integrals of very simple form, namely
\begin{equation}
D_F(x) = \frac{1}{4\pi^3} \int e^{-ip_{\mu} x_{\mu}} \delta_+ (p^2) dp,
\end{equation}
\begin{equation}
S_F(x) = \frac{1}{4 \pi^3} \int e^{-ip_{\mu} x_{\mu}} [+ip_{\mu} \gamma_{\mu}
- \kappa_0] \times \delta_+ (p^2 + \kappa_0^2) dp,
\end{equation}
where $\kappa_0$ is the electron reciprocal Compton wave-length,
\begin{equation}
p^2 = p_{\mu} p_{\mu} = p_1^2 + p_2^2 +p_3^2 - p_0^2,
\end{equation}
and the $\delta_+$ function is denned by
\begin{equation}
\delta_+(a) = \frac{1}{2} \delta(a) + \frac{1}{2 \pi ia} = \frac{1}{2 \pi}
\int \limits^{\infty}_a e^{iaz}dz.
\end{equation}
Substituting from (16) and (17) into $M$ will
introduce an $F$--fold integral over momentum space.
Corresponding to each internal line of $G$, there will
appear in $M$ a 4 vector variable of integration. which
may be denoted by $p_{\mu}^i, i=1, \ldots, F$. However, after
this substitution is made, the space-time variables ti,
$x_1, \ldots, x_n$, occur in $M$ only in the exponential factors,
and the integration over these variables can be
performed. The result of the integration over $x_j$, is to give
\begin{equation}
(2\pi)^4 \delta (q_j),
\end{equation}
where the $\delta$ represents a simple 4-dimensional Dirac
$\delta$--function, and $q_i$, is a 4 vector formed by taking an
algebraic sum of the $k_i$ and $p^i$ vectors
corresponding to those lines of $G$ which meet at $x_j$.
The factor (20) in the integrand of $M$ expresses the
conservation of energy and momentum in the interaction occurring at the point $x_j$.
The transformation of $M$ into terms of momentum
variables is now complete. To summarize the results,
$M$ now appears as an $F$--fold integral over the
variable 4 vectors $p_{\mu}^i$ in momentum space. In the
integrand there appear, besides numerical factors;
(i) a constant spinor or polarization-vector, $\psi(k^i)$
or $\tilde \psi(k^i)$ or $A_{\mu}(k^i)$, corresponding to each external
line of $G$;
(ii) a factor
\begin{equation}
D_F (p^i) = \delta_+ ((p^i)^2)
\end{equation}
corresponding to each internal photon line of $G$;
(iii) a factor
\begin{equation}
S_F (p^i) = [+ip_{\mu}^i \gamma_{\mu} - \kappa_0] \delta_+ ((p^i)^2) -
\kappa_0^2)
\end{equation}
corresponding to each internal electron line of $G$;
(iv) a factor
\begin{equation}
\delta(q_i)
\end{equation}
corresponding to each vertex of $G$;
(v) a $\gamma_{\mu}$ operator, surviving from Eq. (8), corresponding to each
vertex of $G$ at which there is a photon line.
The important feature of the above analysis is that
all the constituents of $M$ are now localized and
associated with individual lines and vertices in the
graph $G$. It therefore becomes possible in an
unambiguous manner to speak of ``adding'' or
``subtracting'' certain groups of factors in $M$, when $G$
is modified by the addition or subtraction of certain
lines and vertices. As an example of this method of
analysis, we shall briefly discuss the treatment in the
$S$ matrix formalism of the ``Lamb shift'' and associated phenomena.
Suppose that a graph $G$, of any degree of complication, has a vertex $x_1$
at which two electron lines and a photon line meet. These three lines may be
either internal or external, and the momentum 4
vectors associated with them in $M$ may be either $p^i$
or $k^i$; these 4 vectors are denoted by $t^1, t^2, t^3$ as
indicated in Fig. 1. The factors in the integrand of $M$ arising from the
vertex $x_1$ are
\begin{equation}
-ie \gamma_{\mu} (2 \pi)^4 \delta (t^1-t^2-t^3),
\end{equation}
the two spinor indices of the $\gamma-{\mu}$, being available for
matrix multiplication on both sides with the factors
in $M$ arising from the two electron lines at $x_1$.
Now suppose that $G'$ is a graph identical with $G$,
except that in the neighborhood of $x_1$ it is modified
by the addition of two new vertices and three new
lines, as indicated in Fig. 2. With the three new lines,
which are all internal, are associated three 4 vector
variables $p^1,p^2,p^3$, which occur as variables of
integration in the expression $M'$ formed from $G'$ as
$M$ is from $G$. It can be proved, in view of Eqs. (21),
(22), (23), that $M'$ may be obtained from $M$ simply
by replacing the factor (24) in $M$ by the expression
\begin{equation}
\begin{array}{c}
- \frac{ie^4}{\hbar c} (2 \pi)^3 \int \int \int dp^1 dp^2 dp^3 \delta (t^1
- p^2 + p^3) \delta (p^2 - p^1 - t^3) \delta (p^1 - p^3 - t^2)\\\\
\gamma_{\lambda} (+ ip_{\rho}^2 \gamma_{\rho} - \kappa_0) \gamma_{\mu}
(=ip_{\sigma}^1 \gamma_{\sigma} - \kappa_0)\gamma-{\lambda} \\\\
\delta_+((p^2)^2) + \kappa_0^2) \delta_+ ((p^1)^2 + \kappa_0^2)\delta_+ ((p^2)^2).
\end{array}
\end{equation}
(The factorial coefficients appearing in (4) are just
compensated by the fact that the two new vertices of
$G'$ may be labelled $x_i, x_j$ in $(n+1)(n+2)$ ways, where $n$
is the number of vertices in $G$.) In (25), two of the 4-dimensional
$\delta$--functions can be eliminated at once
by integration over $p^1$ and $p^2$, and the third then
reduces to the $\delta$--function occurring in (24).
Therefore $M'$ can be obtained from $M$ by replacing
the operator $\gamma_{\mu}$, in (24) by an operator
\begin{equation}
\begin{array}{c}
L_{\mu} = L_{\mu} (t^1, t^2) = 2 \alpha \int dp [\gamma_{\lambda} (+ i(p_{\rho}+
t_{\rho}^1) \gamma_{\rho} - \kappa_0) \gamma_{\mu} \\\\
\times (+i (p_{\sigma}
+ t_{\sigma}^2) \gamma_{\sigma} - \kappa_0) \gamma_{\lambda}]
\times \delta_+ ((p+t^1)^2 + \kappa_0^2)\\\\
\times \delta_+ ((p+t^2)^2 + \kappa_0^2)\delta_+ (p^2).
\end{array}
\end{equation}
Here $\alpha$ is the fine-structure constant, $e^2/4 \pi \hbar c)$ in
Heaviside units. The operator $L_{\mu}$, can without great
difficulty be calculated explicitly as a function of the
4 vectors $t^1$ and $t^2$, by methods developed by Feynman.
In the special case when Fig. 1 represents the
graph $G$ in its entirety, $M$ is a matrix element for the
scattering of a single electron by an external potential. Figure 2 then
represents $G'$ in its entirety, and $M'$
is a second-order radiative correction to the
scattering of the electron. In this case then the
operator $L_{\mu}$, gives rise to what may be called ``Lamb
shift and associated phenomena.'' However, the
above analysis applies equally to an expression $M$
which may occur anywhere among the matrix elements of $U(\infty)$, and
may represent any physical process whatever involving electrons, positrons and
photons. There will always appear in $U(\infty)$,
together with $M$, terms $M'$ representing second-order
radiative corrections to the same process;
one term $M'$ arises from each vertex of $G$ at which a
photon line ends; and $M'$ is always to be obtained
from $M$ by substituting for an operator $\gamma_{\mu}$ the same
operator $L_{\mu}$. Furthermore, some higher radiative
corrections to $M$ will be obtained by substituting $L_{\mu}$
for $\gamma_{\mu}$, independently at two or more of the vertices of $G$.
By a ``vertex part'' of any graph will be meant a
connected part of the graph, consisting of vertices
and internal lines only, which touches precisely two
electron lines and one photon line belonging to the
remainder of the graph. The central triangle of Fig. 2
is an example of such a part. In other words, a vertex
part of a graph is a part which can be substituted for
the single vertex of Fig. 1 and give a physically
meaningful result. Now the argument, by which the
replacement of Fig. 1 by Fig. 2 was shown to be
equivalent to the replacement of $\gamma_{\mu}$ by $L_{\mu}$ can be
used also when a more complicated vertex part is
substituted for the vertex in Fig. 1. If $G$ is any graph
with a vertex $x_1$ as shown in Fig. 1, and $G'$ is
obtained from $G$ by substituting for $x_1$ any vertex
part $V$, and if $M$ and $M'$ are elements of $U(\infty)$
formed analogously from $G$ and $G'$, then $M'$ can be
obtained from $M$ by replacing an operator $\gamma_{\mu}$ by an operator
\begin{equation}
\Lambda_{\mu} = \Lambda_{\mu} (V,t^1,t^2),
\end{equation}
dependent only on $V$ and the 4 vectors $t^1,t^2$ and independent of $G$.
To summarize the results of this section, it has
been shown that the $S$ matrix formalism allows a
wide variety of higher order radiative processes to be
calculated in the form of operators in momentum
space. Such operators appear as radiative corrections
to the fundamental interaction between the photon
and electron-positron fields, and need only to be
calculated once to be applicable to the various
special problems of electrodynamics.
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig2.gif}}}
\caption{}
\end{figure}
\section{FURTHER REDUCTION OF THE $S$ MATRIX}
It was shown in Section VII of I that, for the one-electron processes
there considered, only connected graphs needed to be taken into account. In
constructing the $S$ matrix in general, this is no
longer the case; disconnected graphs give matrix
elements of $U(\infty)$ representing two or more collision
processes occurring simultaneously among separate
groups of particles, and such processes have
physical reality. It is only permissable to omit a
disconnected graph when one of its connected
components is entirely lacking in external lines;
such a component without external lines will give
rise only to a constant multiplicative phase factor in
every matrix element of $U(\infty)$ and is therefore
devoid of physical significance.
On the other hand, the treatment in Section VII of
I of graphs with ``self-energy parts'' applies almost
without change to the general $S$ matrix formalism. A
``self-energy part'' of a graph is a connected part,
consisting of vertices and internal lines only, which
can be inserted into the middle of a single line of a
graph $G$ so as to give a meaningful graph $G'$. In Fig.
3 is shown an example of such an insertion made in
one of the lines of Fig. 1. Let $M$ and $M'$ be
expressions derived in the manner of the previous
section from the graphs $G$ and $G'$ of which parts are
shown in Figs. 1 and 3. Suppose for definiteness
that the line labelled $t^1$ is an internal line of $G$; then
according to (22) it will contribute a factor $S_F(t^1)$ in
$M$. By an argument similar to that leading to (26), it
can be shown that $M'$ may now be obtained from $M$ by replacing $S_F(t^1)$ by
\begin{equation}
\begin{array}{c}
S_F (t^1) N(t^1)S_F(t^1) = S_F (t^1) 2 \alpha \int dp [\gamma_{\lambda}
(+ i \gamma_{\rho} (p_{\rho} + t_{\rho}^1) - \kappa_0) \gamma_{\lambda}]\\\\
\times \delta_+ ((p+t^1)^2 + \kappa_0^2) \delta_+ (p^2) S_F (t_1).
\end{array}
\end{equation}
In the same way, if $G'$ were obtained from $G$ by
inserting in the $t_1$ line any self-energy part $W$, then
$M'$ would be obtained from $M$ by replacing $S_F(t^1)$ by
\begin{equation}
S_F(t^1) \Sigma (W, t^1) S_F (t^1),
\end{equation}
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig3.gif}}}
\caption{}
\end{figure}
where $\Sigma$ is an operator dependent only on $W$ and $t^1$
and not on $G$. Moreover, if the $t^1$ line were an
external line of $G$, then $M'$ would be obtained from $M$ by replacing a
factor $\tilde \psi(t^1)$ by
\begin{equation}
\tilde \psi(t^1) \Sigma (W, t^1) S_F (t^1).
\end{equation}
As a special case, $W$ may consist of a single point;
then at this point it is the term in $\delta m$ of the interaction (8)
which is operating, and $\Sigma$ reduces to a constant,
\begin{equation}
\Sigma (W, t^1) = - 2 \pi i (\delta mc/ \hbar) = - 2 \pi i \delta \kappa_0.
\end{equation}
The operator $N(t^1)$ in (28) describes in a general way
the second-order contribution to the electron self-energy and to
the phenomenon called ``vacuum polarization of the second kind'' in Section VIII of I.
The self-energy contribution is supposed to be
cancelled by (31); the constant $\delta \kappa_0$ being a power
series in $\alpha$, the linear term only is required to cancel
the self-energy part of (28), and the higher terms are
available for the cancellation of self-energy effects
from operators $\Sigma(W, t^1)$ of higher order. The $S$ matrix
formalism makes clear the important fact that, since
the operators $\Sigma(W, t^1)$ are universal operators
independent of the graph $G$, the electron self-energy
effects will be formally cancelled by a constant $\delta \kappa_0$
independent of the physical situation in which the effects occur.
When a self-energy part $W'$ is inserted into a
photon line of a graph $G$, for example the line
labelled $t^3$ in Fig. 1, then the modification produced
in $M$ may be again described by a function $\Pi (W', t^2)$
independent of $G$. Specifically, if the $t^3$ line in $G$ is
internal, $M'$ is obtained from $M$ by replacing a factor $D_F(t^3)$ by
\begin{equation}
D_F (t^3) \Pi (W', t^3) D_F (t^3).
\end{equation}
If the $t^3$ line is external, the replacement is of a factor $A_{\mu}(t^3)$ by
\begin{equation}
A_{\mu}(t^3) \Pi (W', t^3) D_F (t^3).
\end{equation}
In addition to terms of the form (33), there will appear terms such as
\begin{equation}
A_{\nu} (t^3) t_{\nu}^3 t_{\mu}^3 \Pi' (W', t^3) D_F (t^3);
\end{equation}
these however are zero in consequence of the gauge
condition satisfied by $A_{\nu}$. Similar terms in $t_{\nu}^3 t_{\mu}^3$ will
also appear with the expression (32); in this case the
extra terms can be shown to vanish in consequence
of the equation of conservation of charge satisfied
by the electron-positron field. The functions $\Pi(W', t^3)$
describe the phenomenon of photon self-energy and
the ``vacuum polarization of the first kind'' of
Section VIII of I. Following Schwinger, one does
not explicitly subtract away the divergent photon
self-energy effects from the $\Pi(W', t^3)$, but one asserts
that these effects are zero as a consequence of the
gauge invariance of electrodynamics.
In Section VII of I, it was shown how self-energy
parts could be systematically eliminated from all
graphs, and their effects described by suitably
modifying the functions $D_F$ and $G_F$. The analysis was
carried out in configuration space, and was confined
to the one-electron problem. We are now in a
position to extend this method to the whole $S$ matrix
formalism, working in momentum space, and
furthermore to eliminate not only self-energy parts
but also the ``vertex parts'' denned in the last section.
Every graph $G$ has a uniquely defined ``skeleton''
$G_0$, which is the graph obtained by omitting all self-energy
parts and vertex parts from $G$. A graph which
is its own skeleton is called ``irreducible;'' all of its
vertices will be of the kind depicted in Fig. 1. From
every irreducible $G_0$ the $G$ of which it is the
skeleton can be built by inserting pieces in all
possible ways at all vertices and lines of$G_0$; these $G$
form a well-defined class $\Gamma$. The term ``proper vertex
part'' must here be introduced, denoting a vertex part
which is not divisible into two pieces joined only by
a single line; thus a vertex part which is not proper is
essentially redundant, being a proper vertex part plus
one or more self-energy parts. The graphs of $\Gamma$ are
then accurately enumerated by inserting at some or
all of the vertices of $G_0$ a proper vertex part, and in
some or all of the lines a self-energy part, these
insertions being made independently in all possible combinations.
Suppose that $M$ is a constituent of a matrix
element of $U(\infty)$, obtained from $G_0$ as described in
Section III. Then every graph $G$ in $\Gamma$ will yield
additional contributions to the same matrix element
of $U(\infty)$; the sum of all such contributions, including $M$, is denoted
by $M_S$. As a result of the analysis leading to (27), (29), and (32), and in view
of the statistical independence of the insertions made
at the different vertices and lines of $G_0$ the sum $M_S$
will be obtained from $M$ by the following
substitutions. For every internal electron line of $G_0$,
a factor $S_F(p^i)$ of $M$ is replaced by
\begin{equation}
S_F' (p^i) = S_F (p^i) + S_F (p^i) \Sigma (p^i) S_F (p^i),
\end{equation}
where $\Sigma(p^i)$ is the sum of the $\Sigma(W, p^i)$ over all
electron self-energy parts $W$. For every internal
photon line, a factor $D_F(p^i)$ is replaced by
\begin{equation}
D_F' (p^i) = D_F (p^i) + D_F (p^i) \Pi (p^i) D_F (p^i),
\end{equation}
where $\Pi(p^i)$ is the sum of the $\Pi(W', p^i)$ over all
photon self-energy parts $W'$. For every external line, a
factor $\psi(k^i)$ or $\tilde \psi(k^i)$ or $A_{\mu}(k^i)$ is replaced by
\begin{equation}
\begin{array}{c}
\psi' (k^i) = S_F(k^i) \Sigma (k^i) \psi(k^i) + \psi(k^i),\\
\tilde \psi(k^i) = \tilde \psi(k^i) \Sigma (k^i) S_F (k^i) + \tilde \psi(k^i),\\
A_{\mu}' (k^i) = A_{\mu} (k^i) \Pi (k^i) D_F (k^i) + A_{\mu} (k^i),
\end{array}
\end{equation}
respectively. For every vertex of $G_0$, where the
incident lines carry momentum variables as shown in
Fig. 1, an operator $\gamma_{\mu}$, is replaced by
\begin{equation}
\Gamma_{\mu} (t^1,t^2) = \gamma_{\mu} + \Lambda_{\mu} (t^1,t^2),
\end{equation}
where $\Lambda_{\mu}(t^1,t^2)$ is the sum of the $\Lambda_{\mu}(V, t^1,t^2)$
over all proper vertex parts $V$. The matrix elements of $U(\infty)$
will be correctly calculated, if one includes
contributions only from irreducible graphs, after
making in each contribution the replacements (35), (36), (37), (38).
To calculate the operators $\Lambda_{\mu}, \Sigma$ and $\Pi$, it is
necessary to write down explicitly the integrals in
momentum space, examples being (26) and (28),
corresponding to every self-energy part $W$ or proper
vertex part $V$. When considering effects of higher
order than the second, the parts $W$ and $V$ will
themselves often be reducible, containing in their
interior self-energy and vertex parts. It will again be
convenient to omit such reducible $V$ and $W$, and to
include their effects by making the substitutions (35)-(38) in the
integrals corresponding to irreducible $V$
and $W$. In this way one obtains in general not explicit
formulas, but integral equations, for $\Lambda_{\mu}, \Sigma$ and $\Pi$. For
example,
\begin{equation}
\Lambda_{\mu} = \alpha I_{\mu} (\Lambda, \Sigma, \Pi)
\end{equation}
where $I_{\mu}$ is an integral in which $\Lambda_{\mu}, \Sigma$ and $\Pi$ occur
explicitly. Fortunately, the appearance of $\alpha$ on the
right side of (39) makes it easy to solve such equations by a process
of successive substitution, the
forms of $\Lambda_{\mu}, \Sigma$, and $\Pi$ being obtained correct to order
$\alpha^n$ when values correct to order $\alpha^{n-1}$ are substituted into the integrals.
The functions $D_F'$ and $S_F'$ of (35) and (36) are the
Fourier transforms of the corresponding functions in
I. The interpretation of these functions in Section
VIII of I can be extended in an obvious way to
include the operator $\Gamma_{\mu}$. Since $\bar \psi \gamma_{\mu} \psi$
is the charge-current 4 vector of an electron without radiative
corrections, $\tilde \psi \Gamma_{\mu} \psi$ may be interpreted
as an ``effective current'' carried by an electron, including the effects
of exchange interactions between the electron and the electron-positron field around it.
An additional reduction in the number of graphs
effectively contributing to $U(\infty)$ is obtained from a
theorem of Furry.\footnote{Wendell H. Furry, Phys. Rev. {\bf 51,} 125 (1937).}
The theorem was shown by Feynman to be an elegant consequence of his theory.
In any graph $G$, a ``closed loop'' is a closed electron
polygon, at the vertices of which a number $p$ of
photon lines originate; the loop is called odd or even
according to the parity of $p$. If $G$ contains a closed
loop, then there will be another graph $\tilde G$ also
contributing to $U(\infty)$, obtained from $G$ by reversing
the sense of the electron lines in the loop. Now if $M$
and $\tilde M$ are contributions from $G$ and $\tilde G, \tilde M$ is derived
from $M$ by interchanging the roles of electron and
positron states in each of the interactions at the
vertices of the loop; such an interchange is called
``charge conjugation.'' It was shown by Schwinger
that his theory is invariant under charge conjugation, provided that the sign
of $e$ is at the same time reversed (this is the well-known charge
symmetry of the Dirac hole theory). It is clear from
(8) that the constant $e$ appears once in $M$ for each of
the $p$ loop vertices at which there is a photon line; at
the remaining vertices only the constant $\delta m$ is
involved, and $\delta m$ is an even function of $e$. Therefore
the principle of charge-symmetry implies
\begin{equation}
\tilde M = (-1)^p M.
\end{equation}
Taking $p$ odd in (40) gives Furry's theorem; all
contributions to $U(\infty)$ from graphs with one or more
odd closed loops vanish identically.
By an ``odd part'' of a graph is meant any part,
consisting only of vertices and Internal lines, which
touches no electron lines, and only an odd number
of photon lines, belonging to the rest of the graph.
The simplest type of odd part which can occur is a
single odd closed loop. Conversely, it is easy to see
that every odd part must include within itself at
least one odd closed loop. Therefore, Furry's
theorem allows all graphs with odd parts to be
omitted from consideration in calculating $U(\infty)$.
\section{INVESTIGATION OF DIVERGENCES IN THE $S$ MATRIX}
The $\delta_+$ function defined by (19) has the property
that, if $b$ is real and $f(a)$ is any function analytic in
the neighborhood of $b$, then
\begin{equation}
\int f(a) \delta_+ (a - b) da = (1/2 \pi i) \int f(a) (1/(a-b))da,
\end{equation}
where the first integral is along a stretch of the real
axis including $b$, and the second integral is along the
same stretch of the real axis but with a small detour
into the complex plane passing underneath $b$. In the
matrix elements of $U(\infty)$ there appear integrals of the form
\begin{equation}
\int dp F (p) \delta_+ (p_1^2 + p_2^2 - p_0^2 + c^2),
\end{equation}
integrated over all real values of $p_1, p_2, p_3, p_0$. By
(41), one may write (42) in the form
\begin{equation}
\frac{1}{2 \pi i} \int dp \frac{F(p)}{(p_1^2 + p_2^2 + p_3^2 - p_0^2 +
c^2)},
\end{equation}
in which it is understood that the integration is
along the real axis for the variables $p_1, p_2, p_3$, and for
$p_0$ is along the real axis with two small detours, one
passing above the point $+(p_1^2 + p_2^2 + p_3^2 + c^2)^{1/2}$, and
one passing below the point $-(p_1^2 + p_2^2 + p_3^2 + c^2)^{1/2}$. To
equate (42) with (43) is certainly correct, when $F(p)$
is analytic at the critical values of $p_0$. In practice
one has to deal with integrals (42) in which $F(p)$
itself involves $\delta_+$ functions (see for example (26) and (28)); in
these cases it is legitimate to replace each $\delta_+$ function by a
reciprocal, making a separate detour in the $p_0$
integration for each pole in the integrand, provided
that no two poles coincide. Thus every constituent
part $M$ of $U(\infty)$ can be written as an integral of a
rational algebraic function of momentum variables, by using instead of (21) and (22)
\begin{equation}
D_F (p^i) = \frac{1}{2 \pi i (p^i)^2},
\end{equation}
\begin{equation}
S_F (p^i) = \frac{(ip_{\mu}^i \gamma_{\mu} - \kappa_0)}{2 \pi i ((p^i)^2
+ \kappa_0^2)}.
\end{equation}
This representation of $D_F$ and $S_F$ as rational functions in momentum-space
has been developed and extensively used by Feynman (unpublished).
There may appear in $M$ infinities of three distinct
kinds. These are (i) singularities caused by the
coincidence of two or more poles of the integrand,
(ii) divergences at small momenta caused by a
factor (44) in the integrand, (iii) divergences at
large momenta due to insufficiently rapid decrease
of the whole integrand at infinity.
In this paper no attempt will be made to explore
the singularities of type (i). Such singularities occur,
for example, when a many-particle scattering
process may for special values of the particle
momenta be divided into independent processes
involving separate groups of particles. It is probable
that all singularities of type (i) have a similarly clear
physical meaning; these singularities have long
been known in the form of vanishing energy denominators in ordinary
perturbation theory, and have never caused any serious trouble.
A divergence of type (ii) is the so-called ``infrared
catastrophe,'' and is well known to be caused by the
failure of an expansion in powers of a to describe
correctly the radiation of low momentum quanta. It
would presumably be possible to eliminate this
divergence from the theory by a suitable adaptation
of the standard Bloch-Nordsieck\footnote{F. Bloch and A. Nordsieck, Phys. Rev. {\bf
52,} 54 (1937).}
treatment; we shall not do this here. From a practical point of
view, one may avoid the difficulty by arbitrarily writing instead of (44)
\begin{equation}
D_F (p^i) = \frac{1}{2 \pi i ((p^i)^2 + \lambda^2)},
\end{equation}
where $\lambda$ is some non-zero momentum, smaller than
any of the quantum momenta which are significant
in the particular process under discussion.\footnote{The device
of introducing $\lambda$ in order to avoid infra-red divergences must be used
with circumspection. Schwinger (unpublished) has shown that a long
standing discrepancy between two alternative calculations of the
Lamb shift was due to careless use of $\lambda$ in one of them.}
It is the divergences of type (iii) which have
always been the main obstacle to the construction of
a consistent quantum electrodynamics, and which it
is the purpose of the present theory to eliminate. In
the following pages, attention will be confined to
type (iii) divergences; when the word ``convergent''
is used, the proviso ``except for possible singularities
of types (i) and (ii)'' should always be understood.
A divergent $M$ is called ``primitive'' if, whenever
one of the momentum 4 vectors in its integrand is
held fixed, the integration over the remaining
variables is convergent. Correspondingly, a primitive divergent graph
is a connected graph $G$ giving rise to divergent $M$, but such that, if any internal
line is removed and replaced by two external lines,
the modified $G$ gives convergent $M$. To analyze the
divergences of the theory, it is sufficient to enumerate the
primitive divergent $M$ and $G$ and to examine their properties.
Let $G$ be a primitive divergent graph, with $n$
vertices, $E$ external and $F$ internal lines. A corresponding $M$ will
be an integral over $F$ variable $p^i$
of a product of $F$ factors (44) and (45) and $n$ factors
(23). Since $G$ is connected, the $\delta$--functions (23) in
the integrand enable $(n-1)$ of the variables $p^i$ to be
expressed in terms of the remaining $(F - n + 1)~ p^i$ and
the constants $k^i$, leaving one $\delta$--function involving the
$k^i$ only and expressing conservation of momentum
and energy for the whole system. An example of
such integration over the $\delta$--functions was the
derivation of (26) from (25). After this, the
integrations in $M$ may be arranged as follows the
fourth components of the $(F - n + 1)$ independent $p^i$ are written
\begin{equation}
p_4^i = ip_0^i = i \alpha \pi_0^i,
\end{equation}
and the integration over $\alpha$ is performed first; subsequently, integration
is carried out over the $3(F - n + 1)$ independent $p_1^i, p_2^i, p_3^i$,
and over the $(F-n)$ ratios of the $\pi_0^i$. $M$ then has the form
\begin{equation}
M = \int dp_1^i dp_2^i dp_3^i d \pi_0^i \int \limits^{\infty}_{- \infty}
R \alpha^{F - n} d \alpha,
\end{equation}
where $R$ is a rational function of $\alpha$, the denominator
of which is a product of $F$ factors
\begin{equation}
(p_1^i)^2 + (p_2^i)^2 + (p_3^i)^2 + \mu^2 - (\alpha \pi_0^i + c^i)^2.
\end{equation}
Here the constants $\pi_0^i, c^i$ are defined by the condition that
\begin{equation}
p_4^i = ip_0^j = i(\alpha \pi_0^j + c^j), \quad j = 1, 2, \ldots, F.
\end{equation}
Thus the $c^i$ corresponding to the $(F - n + 1)$ independent $p^i$
are zero by (47), and the remainder are
linear combinations of the $k^i$; also $(n - 1)$ of the $\pi_0^i$
are linear combinations of the independent $\pi_0^i$
denned in (47). In view of (43), we take the integration variables in (48)
to be real variables, with the
exception of a which is to be integrated along a
contour $C$ deviating from the real axis at each of the
$2F$ poles of $R$. As a general rule, $C$ will detour above
the real axis for $\alpha > 0$, and below it for a $\alpha < 0$
the reverse will only occur at certain of the poles
corresponding to denominators (49) for which
\begin{equation}
(p_1^i)^2 + (p_2^i)^2 + (p_3^i)^2 + \mu^2 \leq (c^i)^2.
\end{equation}
Such poles will be called ``displaced.'' The integration
over $\alpha$ alone will always be absolutely convergent.
Therefore the contour $C$ may be rotated in a counter-clockwise direction
until it lies along the imaginary
axis, and the value of $M$ will be unchanged except for
residues at the displaced poles.
Regarded as a function of the parameters $k^i$
describing the incoming and outgoing particles, $M$
will have a complicated behavior; the behavior will
change abruptly whenever one of the $c^i$ has a critical
value for which (51) begins to be soluble for $p_1^i, p_2^i, p_3^i$,
and a new displaced pole comes into existence.
This behavior is explained by observing that
displaced poles appear whenever there is sufficient
energy available for one of the virtual particles
involved ill $M$ to be actually emitted as a real
particle. It is to be expected that the behavior of $M$
should change when the process described by $M$
begins to be in competition with other real processes.
It is a feature of standard perturbation theory, that
when a process $A$ involves an intermediate state $I$
which is variable over a continuous range, and in this
range occurs a state $II$ which is the final state of a
competing process, then the matrix element for $A$
involves an integral over $I$ which has a singularity at
the position $II$. In standard perturbation theory, this
improper integral is always to be evaluated as a
Cauchy principal value, and does not introduce any
real divergence into the matrix element. In the theory
of the present paper, the displaced poles give rise to
similar improper integrals; these come under the
heading of singularities of type (i) and will not be
discussed further.
If $p_1^i, p_2^i, p_3^i,$ satisfying (51) are held fixed, then the
value of $p_4^i$ at the corresponding displaced pole is
fixed by (50). The contribution to $M$ from the
displaced pole is just the expression obtained by
holding the 4-vector $p^i$ fixed in the original integral
$M$, apart from bounded factors; since $M$ is primitive
divergent, this expression is convergent. The total
contribution to $M$ from the $i'$th displaced pole is the
integral of this expression over the finite sphere (51)
and is therefore finite. Strictly speaking, this
argument requires not only the convergence of the
expression, but uniform convergence in a finite
region; however, it will be seen that the convergent
integrals in this theory are convergent for large
momenta by virtue of a sufficient preponderance of
large denominators, and convergence produced in
this way will always be uniform in a finite region.
$M$ is thus, apart from finite parts, equal to the
integral $M'$ obtained by replacing $\alpha$ by ia in (48) and
(49). Alternatively, $M'$ is obtained from the original
integral $M$ by substituting for each $p_0^i$
\begin{equation}
ip_4^i + (1 - i) c^i.
\end{equation}
and then treating the $4(F - n + 1)$ independent $p_{\mu}^i,~ \mu = 1,2,3,4,$
as ordinary real variables. In $M'$ the denominators of the integrand take the form
\begin{equation}
(p_1^i)^2 + (p_2^i)^2 + (p_3^i)^2 + \mu^2 + (p_4^i - (1 + i) c^i)^2,
\end{equation}
and are uniformly large for large values of $p_{\mu}^i$. The
convergence of $M'$ can now be estimated simply by
counting powers of $p_{\mu}^i$ in numerator and denominator of the
integrand. Since $M'$ is known to converge whenever one of the $p^i$ is held fixed and
integration is carried out over the others, the convergence of the whole
expression is assured provided that
\begin{equation}
K = 2F - F_e - 4[F - n + 1] \geq 1.
\end{equation}
Here $2F$ is the degree of the denominator, and $F_e$,
that of the numerator, which is by (44) and (45)
equal to the number of internal electron lines in $G$.
Let $E_e$ and $E_p$ be the numbers of external electron
and photon lines in $G$, and let $n_s$, be the number of
vertices without photon lines incident. It follows
from the structure of $G$ that
$$
2F = 3n - n_e - E_e - E_p,
$$
$$
F_e = n - \frac{1}{2} E_e,
$$
and so the convergence condition (52) is
\begin{equation}
K = \frac{3}{2} E_e + E_p + n_s - 4 \geq 1.
\end{equation}
This gives the vital information that the only
possible primitive divergent graphs are those with
$E_e = 2,$ $E_p = 0, 1,$ and with $E_e = 0,$ $E_p = 1,2,3,4.$
Further, the cases $E_e = 0,$ $E_p = 1,3,$ do not arise, since these give
graphs with odd parts which were shown to be
harmless in Section IV. It should be observed that
the course of the argument has been ``if $E_e$ and $E_p$ do
not have certain small values, then the integral $M$ is
convergent at infinity;'' there is no objection to
changing the order of integrations in $M$ as was done
in (48), since the argument requires that this be done
only in cases when $M$ is, in fact, absolutely
convergent.
The possible primitive divergent graphs that have
been found are all of a kind familiar to physicists.
The case $E_e = 2,$ $E_p = 0$ describes self-energy effects
of a single electron; $E_e = 0,$ $E_p = 2$ self-energy effects of
a single photon; $E_e = 2,$ $E_p = 1$ the scattering of a
single electron in an electromagnetic field; and $E_e = 0,$ $E_p = 4$
the ``scattering of light by light'' or the mutual scattering of two
photons. Further, (55) shows that the divergence
will never be more than logarithmic in the third and
fourth cases, more than linear in the first, or more
than quadratic in the second. Thus it appears that,
however far quantum electrodynamics is developed
in the discussion of many-particle interactions and
higher order phenomena, no essentially new kinds
of divergence will be encountered. This gives strong
support to the view that ``subtraction physics,'' of the
kind used by Schwinger and Feynman, will be
enough to make quantum electrodynamics into a
consistent theory.
\section{SEPARATION OF DIVERGENCES IN THE $S$ MATRIX}
First it will be shown that the ``scattering of light
by light'' does not in fact introduce any divergence
into the theory. The possible primitive divergent $M$
in the case $E_e = 0,$ $E_p = 4$ will be of the form
\begin{equation}
\delta (k^1 + k^2 + k^3 + k^4) A_{\lambda} (k^1) A_{\mu} (k^2) A_{\nu} (k^3)
A_{\rho} (k^4) I_{\lambda \mu \nu \rho},
\end{equation}
where $I_{\lambda \mu \nu \rho}$ is an integral of the type
\begin{equation}
\int R_{\lambda \mu \nu \rho} (k^1, k^2, k^3, k^4, p^i) dp_i,
\end{equation}
at most logarithmically divergent, and $R$ is a certain
rational function of the constant $k^i$ and the variable
$p^i$. In any physical situation where, for example, the
$A(k)$ are the potentials corresponding to particular
incident and outgoing photons, there will appear in
$U(\infty)$ a matrix element which is the sum of (56) and
the 23 similar expressions obtained by permuting
the suffixes of $I_{\lambda \mu \nu \rho}$ in all possible ways. It may
therefore be supposed that at the start $R_{\lambda \mu \nu \rho},$ has been
symmetrized by summation over all permutations of
suffixes; (56) is then a sum of contributions from 24
or fewer (according to the degree of symmetry existing) graphs $G$.
If, under the sign of integration in (57), the value
$R(0)$ of $R$ for $k^1 = k^2 = k^3 = k^4 = 0$ is subtracted from $R$, the
integrand acquires one extra power of $|p_{\mu}^i|^{-1}$ for
large $|p_{\mu}^i|$, and the integral becomes absolutely
convergent at infinity. Therefore
\begin{equation}
I_{\lambda \mu \nu \rho} = I_{\lambda \mu \nu \rho}(0) + J_{\lambda \mu \nu \rho},
\end{equation}
where $J(0)$ is a possibly divergent integral independent of the
$k^i$ and $J$ is a convergent integral
vanishing when all $k^i$'s are zero. To interpret this
result physically, it is convenient to write (56) again
in terms of space-time variables; this gives
\begin{equation}
M = \int I_{\lambda \mu \nu \rho} (0) A_{\lambda} (x) A_{\mu} (x) A_{\nu}
(x) A_{\rho} (x) dx + N,
\end{equation}
where $N$ is a convergent expression involving derivatives of the
$A(x)$ with respect to space and time. Now the first term in (59) is physically
inadmissable; it is not gauge-invariant, and implies
for example a scattering of light by an electric field
depending on the absolute magnitude of the scalar
potential, which has no physical meaning. Therefore
$I(0)$ must vanish identically, and the whole expression (56) is convergent.
The fact that the scattering of light by light is
finite in the lowest order in which it occurs has long
been known.\footnote{H. Euler and B. Kockel, Naturwiss. {\bf 23,} 246 (1935);
H. Euler, Ann. d. Phys. {\bf 26}, 398 (1936). In these early calculations of
the scattering of light by light, the theory used is the Heisenberg
electrodynamics, in which certain singularities are eliminated at the
start by a procedure involving non-diagonal elements of the Dirac
density matrix. In Feynman's calculation, on the other hand, a finite
result is obtained without subtractions of any kind.}
It has also been verified by Feynman by direct calculation, using his own theory as
described in this paper. The graphs which give rise
to the lowest order scattering are shown in Fig. 4. It
is found that the divergent parts of the corresponding
$M$ exactly cancel when the three contributions are
added, or, what comes to the same thing, when the
function $R_{\lambda \mu \nu \rho}$ is symmetrized. It is probable that the
absence of divergence in the scattering of light by
light is in all cases due to a similar cancellation, and
it should not be difficult to prove this by calculation
and thus avoid making an appeal to gauge-invariance.
The three remaining types of primitive divergent
$M$ are, in fact, divergent. However, these are just the
expressions which have been studied in Sections III
and IV and shown to be completely described by the
operators $\Lambda_{\mu}, \Sigma$ and $\Pi$. More specifically, when
$E_e = 2,~ E_p = 0,~ M$ will be of the form
\begin{equation}
\tilde \psi(k^1) \Sigma (W, k^1) \psi (k^1),
\end{equation}
where $W'$ is some electron self-energy part of a graph.
When $E_e = 2,~ E_p = 1,~ M$ will be
\begin{equation}
A_{\mu} (k^1) \Pi (W', k^1) A_{\mu} (k^1),
\end{equation}
with $W'$ some photon self-energy part. When, $E_e = 2,$ $E_p = 1,~ M$
will be
\begin{equation}
\tilde \psi (k^1) \Lambda_{\mu} (V, k^1, k^2) \psi (k^2) A_{\mu} (k^1 -
k^2),
\end{equation}
with $V$ some vertex part. Therefore, if some means
can be found for isolating and removing the divergent parts from
$\Lambda_{\mu}, \Sigma,$ and $\Pi$ the ``irreducible'' graphs
defined in Section IV will not introduce any fresh
divergences into the theory, and the rules of Section
IV will lead to a divergence-free $S$ matrix.
In considering $\Lambda_{\mu}, \Sigma$ and $\Pi$ in Section IV it was
found convenient to divide vertex and self-energy
parts themselves into the categories reducible and
irreducible. An irreducible self-energy part $W$ is
required not only to have no vertex and self-energy
parts inside itself; it is also required to be ``proper,''
that is to say, it is not to be divisible into two
pieces joined by a single line. In Section IV it was
shown that to avoid redundancy the operator $\Lambda_{\mu}$
should be defined as a sum over proper vertex parts $V$
only. By the same argument, in order to make (35),
(36), (37) correct, it is essential to define $\Sigma$ and $\Pi$ as
sums over both proper and improper self-energy
parts. However, it is possible to define $S_f'$ and $D_F'$ in
terms of proper self-energy parts only, at the cost of
replacing the explicit definitions (35), (36) by
implicit definitions. Let $\Sigma^{\ast} (p^i)$ be defined as the sum
of the $\Sigma(W, p^i)$ over proper electron self-energy parts
$W$, and let $\Pi^{\ast} (p^i)$ be defined similarly. Every $W$ is
either proper, or else it is a proper $W$ joined by a
single electron line to another self-energy part which
may be proper or improper. Therefore, using (35), $S'_F$
may be expressed in the two equivalent forms
\begin{eqnarray}
S_F' (p^i) = S_F (p^i) + S_F (p^i) \Sigma^{\ast} (p^i) S_F' (o^i) \nonumber\\
= S_F (p^i) + S'_F (p^i) \Sigma^{\ast} (p^i) S_F (p^i).
\end{eqnarray}
Similarly,
\begin{eqnarray}
S'_F (p^i) = D_F (p^i) + D_F (p^i) \Pi^{\ast} (p^i) D'_F (p^i) \nonumber\\
= D_F (p^i) + D'_F (p^i) \Pi^{\ast} (p^i) D_F (p^i).
\end{eqnarray}
It is sometimes convenient to work with the $\Sigma$ and $\Pi$
in the starred form, and sometimes in the un-starred form.
Consider the contribution $\Sigma (W, t^1)$ to the operator
$\Sigma^{\ast}$, arising from an electron self-energy part $W$. It is
supposed that $W$ is irreducible, and the effects of
possible insertions of self-energy and vertex parts
inside $W$ are for the time being neglected. Also it is
supposed that $W$ is not a single point, of which the
contribution is given by (31). Then $W$ has an even
number $2l$ of vertices, at each of which a photon
line is incident; and $\Sigma (W, t^1)$ will be of the form
\begin{equation}
e^{2l} \int R(t^1, p^i) dp^i,
\end{equation}
where $R$ is a certain rational function of the $t^1$ and $p^i$
and the integral is at most linearly divergent. The
integrand in (65) is now written in the form
\begin{equation}
R(t^1, p^i) = R (0, p^i)+t_{\mu}^1 \left( \frac{\partial R}{\partial t_{\mu}^1}
(o,p^i) \right) + R_e (t^1, p^i),
\end{equation}
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig4.gif}}}
\caption{}
\end{figure}
and for large values of the $|p_{\mu}^i|$ the remainder term $R_e$
will tend to zero more rapidly by two powers of $|P_{\mu}^i|$
than $R$. Therefore, in complete analogy with (58),
\begin{equation}
\Sigma (W, t_1) = e^{2l} [A + B_{\mu} t_{\mu}^1 + \Sigma_e (W, t^1)],
\end{equation}
where $A$ and $B_{\mu}$, are constant divergent operators, and
$\Sigma_e(W, t^1)$ is defined by a covariant and absolutely
convergent integral. $\Sigma_e (W, t^1)$ must, on grounds of
covariance, be of the form
\begin{equation}
R_1 ((t^1)^2) + R_2 ((t_1)^2) t_{\mu}^1 \gamma_{\mu}
\end{equation}
with $R_1$ and $R_2$ particular functions of $(t^1)^2;$ for the
same reason, $B_{\mu}$, must be of the form $B \gamma_{\mu}$ with $B$ a
certain divergent integral. Now if $t^1$ happens to be
the momentum-energy 4 vector of a free electron,
\begin{equation}
(t^1)^2 = - \kappa_0^2, \quad t_{\mu}^1 \gamma_{\mu} = i \kappa_0.
\end{equation}
It is convenient to write
\begin{equation}
\Sigma_e (W, t^1) = A' + B'(t_{\mu}^1 \gamma_{\mu} - i \kappa_0)+(t_{\mu}^1
\gamma_{\mu} - i \kappa_0)S(W,t^1),
\end{equation}
where $S(W, t^1)$ is zero for $t^1$ satisfying (69), and to
include the first two terms in the constants $A$ and $B$
of (67); since all terms in (70) are finite, the
separation of $S(W, t^1)$ is without ambiguity. Thus an
equation of the form (67) is obtained, with
\begin{equation}
\Sigma_e (W, t^1) = (t_{\mu}^1 \gamma_{\mu} - i \kappa_0) S(w, t^1).
\end{equation}
Summing (67) over all irreducible $W$ and including
(31), gives for the operator $\Sigma^{\ast}$,
\begin{equation}
\Sigma^{\ast} (t^1) = A - 2 \pi i \delta \kappa_0 + B (t_{\mu}^1 \gamma_{\mu}
- i \kappa_0) + (t_{\mu}^1 \gamma_{\mu} - i \kappa_0) S_e (t^1).
\end{equation}
Hence by (63) and (45)
\begin{equation}
\begin{array}{c}
S_F' (t^1) = (A - 2 \pi i \delta \kappa_0) S_F (t^1)\\\\
+ \frac{\displaystyle 1}{\displaystyle 2 \pi} BS_F (t^1) + S_F (t^1) +
\frac{\displaystyle 1}{\displaystyle 2 \pi} S_e (t_1) S'_F
(t^1).
\end{array}
\end{equation}
In (72) and (73), $A$ and $B$ are infinite constants, and
$S_e$ a divergence-free operator which is zero when
(69) holds; $A,B$ and $S_e$ are power series in $e$ starting
with a term in $e^2$. In (72) and (73), however, effects
of higher order corrections to the $\Sigma(W, t^1)$ themselves
are not yet included.
A similar separation of divergent parts may be
made for the $\Pi(W', t^1)$, when $W'$ is an irreducible
photon self-energy part. The integral (65) may now
be quadratically divergent, and so it is necessary to
use instead of (66)
$$
\begin{array}{c}
R(t^1, p^i) = R(0, p^i) + t_{\mu}^1 \left( \frac{\displaystyle \partial R}
{\displaystyle \partial t_{\mu}^1}
(0, p^i) \right)\\\\
+ \frac{\displaystyle 1}{\displaystyle 2} t_{\mu}^1 t_{\nu}^1 \left(
\frac{\displaystyle \partial^2 R}{\displaystyle \partial t_{\mu}^1
t_{\nu}^1 (0, p^i)} \right) + R_e (t^1, p^i),
\end{array}
$$
and derive instead of (67)
\begin{equation}
\Pi (W', t^1) = e^{2l} [A + B_{\mu} t_{\mu}^1 + C_{\mu \nu} t_{\mu}^1 t_{\nu}^1
+ \Pi_e (W', t^1)].
\end{equation}
The $A, B_{\mu}, C_{\mu \nu}$ are absolute constant numbers (not
Dirac operators) and therefore covariance requires
that $B_{\mu} = 0,~ C_{\mu \nu} = C \delta_{\mu \nu}.$ $\Pi_e (W', t^1)$ is defined by an
absolutely convergent integral, and will be an
invariant function of $(t^1)^2$ of a form
\begin{equation}
\Pi_e (W', t^1) = (t^1)^2 D (W', t^1),
\end{equation}
where $D(W', t^1)$ is zero for $t^1$ satisfying
\begin{equation}
(t^1)^2 = 0
\end{equation}
instead of (69). Summing (74) over all irreducible $W'$'s will give
\begin{equation}
\Pi^{\ast} (t^1) = A' + C (t^1)^2 + (t^1)^2 D_e (t^1),
\end{equation}
and hence by (64) and (44)
\begin{equation}
\begin{array}{c}
D'_F (t^1) = A' D_F (t^1) D'_F (t^1) + \frac{\displaystyle 1}{\displaystyle 2 \pi i}
CD'_F (t^1)\\\\
+D_F (t^1) + \frac{\displaystyle 1}{\displaystyle 2 \pi i} D_e (t^1) D'_F (t^1).
\end{array}
\end{equation}
In (77) and (78), $D_e$ is zero for $t^1$ satisfying (76), and
is divergence free.
The constant $A'$ in (77) is the quadratically
divergent photon self-energy. It will give rise to
matrix elements in $U(\infty)$ of the form
\begin{equation}
M = A' \int A_{\mu} (x) A_{\mu} (x) dx,
\end{equation}
which are non-gauge invariant and inadmissable.
Such matrix elements must be eliminated from the
theory, as the first term of (59) was eliminated, by
the statement that $A'$ is zero. The verification of this
statement, by direct calculation of the lowest order
contribution to $A'$, has been given by Schwinger.\footnote{Julian Schwinger, Phys. Rev. {\bf 73,} 416 (1948); {\bf
74,} 1439 (1948); {\bf 75,} 651 (1949).}\footnote{Gregor Wentzel, Phys. Rev. {\bf
74,} 1070 (1948), presents the case against Schwinger's treatment of the photon
self-energy.}
The separation of the divergent part of $\Lambda_{\mu}$ again
follows the lines laid down for $\Sigma^{\ast}$. Since the integral
analogous to (65) is now only logarithmically
divergent, no derivative term is required in (66), and the analog of (67) is
\begin{equation}
\Lambda_{\mu} (V, t^1, t^2) = e^{2l} [L_{\mu} + \Lambda_{\mu e} (V, t^1,
t^2)],
\end{equation}
where $L_{\mu}$ is a constant divergent operator, and $\Lambda_{\mu e}$ is
convergent and zero for $t^1 = t^2 = 0$. In (80), $L{\mu}$ can only
be of the form $L \gamma_{\mu}$. Also, if $t^1 = t^2$ and $t^1$ satisfies
(69), $\Lambda_{\mu e}$ will reduce to a finite multiple of $\gamma_{\mu}$ which
can be included in the term $L \gamma_{\mu}$. Therefore it may be
supposed that $\Lambda_{\mu e}$ in (80) is zero not for $t^1 = t^2 = 0$
but for $t^1 = t^2$ satisfying (69). The meaning of this
physically is that $\Lambda_{\mu e}$ now gives zero contribution to
the energy of a single electron in a constant electromagnetic potential, so
that the whole measured static charge on an electron is included in the term
$L \gamma_{\mu}$. Summing (80) over all irreducible vertex parts $V$,
and using (38),
\begin{equation}
\Lambda_{\mu} (t^1, t^2) = L \gamma_{\mu} + \Lambda_{\mu e} (t^1, t^2),
\end{equation}
\begin{equation}
\Gamma_{\mu} (t^1, t^2) = (1 + L) \gamma_{\mu} + \Lambda_{\mu e} (t^1,
t^2).
\end{equation}
In (81) and (82), effects of higher order corrections to
the $\Lambda_{\mu} (V, t^1, t^2)$ are again not yet included. Formally,
(82) differs from (73) and (78) in not containing the
unknown operator $\Gamma_{\mu}$ on both sides of the equation.
\section{REMOVAL OF DIVERGENCES FROM THE $S$ MATRIX}
~~~~The task remaining is to complete the formulas
(73), (78), and (82), which show how the infinite
parts can be separated from the operators $\Gamma_{\mu}, S'_F$, and
$D'_F$, and to include the corrections introduced into
these operators by the radiative reactions which they
themselves describe. In other words, we have to
include radiative corrections to radiative corrections,
and renormalizations of renormalizations, and so on
{\it ad, infinitum.} This task is not so formidable as it
appears.
First, we observe that $\Lambda_{\mu}, \Sigma^{\ast}$, and $\Pi^{\ast}$ are defined
by integral equations of the form (39), which will be
referred to in the following pages as ``the integral
equations.'' More specifically, consider the contribution $\Lambda_{\mu}
(V, t^1, t^2)$ to $\Lambda_{\mu}$, represented by (80), arising
from a vertex part $V$ with $(2l + 1)$ vertices, $l$ photon
lines, and $2l$ electron lines. This contribution is
defined by an integral analogous to (65), with an
integrand which is a product of $(2l + 1)$ operators $\gamma_{\mu}, l$
functions $D_F$, and $2l$ operators $S_F$. The exact $\Lambda_{\mu} (V,
t^1, t^2)$ is to be obtained by replacing these factors,
respectively, by $\Gamma_{\mu}, D'_F, S_F$, as described in Section
IV. Now suppose that $S'_F$ in the integrand is
represented, to order $e^{2n}$ say, by the sum of $S_F$ and of
a finite number of finite products of $S_F$ with
absolutely convergent operators $S(\tilde W, t^1)$ such as
appear in (71); similarly, let $D'_F$ be represented by $D_F$
plus a finite sum of finite products of $D_F$ with
functions $D(\tilde W', t^1)$ appearing in (75); and let $\Gamma_{\mu}$ be
represented by the sum of $\gamma_{\mu}$ and of a finite set of
$\Lambda-{\mu e} (\tilde V, t^1, t^2)$ from (80). Then the integral $\Lambda_{\mu}(V,
t^1, t^2)$ will be determined to order $e^{2n + 2l}$; and since the
operators $S(\tilde W, t^1), D(\tilde W', t^1), \Lambda_{\mu e} (\tilde
V, t^1, t^2)$ always have a sufficiency of denominators for convergence,
the theory of Section V can be applied to prove that
this $\Lambda_{\mu} (V, t^1, t^2)$ will not be more than
logarithmically divergent. Therefore the new $\Lambda_{\mu} (V, t^1, t^2)$
can be again separated into the form (80). The
sum of these $\Lambda_{\mu} (V, t^1, t^2)$ will then be a $\Lambda_{\mu}
(t^1, t^2)$ of the form (81), with constant $L$ and convergent operator $\Lambda_{\mu
e}$ determined to order $e^{2n + 2}$. Thus (82) provides a new expression $\Gamma_{\mu}$
determined to order $e^{2n + 2}$.
\begin{figure}[h]
\centerline{\resizebox{7cm}{!}{\includegraphics{fig5.gif}}}
\caption{}
\end{figure}
The above procedure describes the general method
for separating out the finite part from the
contribution to $\Gamma_{\mu}$ arising from a reducible vertex
part $V_R$. First, $V_R$ is broken down into an irreducible
vertex part $V$ plus various inserted parts $\tilde W, \tilde W', \tilde
V;$ the contribution to $\Gamma_{\mu}$ from $V_R$ is an integral $M(V_R)$
which is not only divergent as a whole, but also
diverges when integrated over the variables
belonging to one of the insertions $\tilde W, \tilde W', \tilde V$ the
remaining variables being held fixed. The divergences are to be removed from
$M(V_R)$ in succession, beginning with those arising from the inserted parts,
and ending with those arising from $V$ itself. This
successive removal of divergences is a well-defined
procedure, because any two of the insertions made in
$V$ are either completely non-overlapping or else
arranged so that one is completely contained in the
other.
In calculating the contribution to $\Sigma^{\ast}$ or $\Pi^{\ast}$ from
reducible self-energy parts, additional complications
arise. There is in fact only one irreducible photon
self-energy part, the one denoted by $W'$ in Fig. 5; and
there is, besides the self-energy part consisting of a
single point, just one irreducible electron self-energy
part, denoted by $W$ in Fig. 5. All other self-energy
parts may be obtained by making various insertions
in $W$ or $W'$. However, reducible self-energy parts are
to be enumerated by inserting vertex parts at only
one, and not both, of the vertices of $W$ or $W'$;
otherwise the same self-energy part would appear
more than once in the enumeration. And the
contribution $M(W_R)$ to $\Sigma^{\ast}$ arising from a reducible
part $W_R$ will be, in general, an integral which
involves simultaneously divergences corresponding
to each of the ways in which $W_R$ might have been
built up by insertions of vertex parts at either or both
vertices of $W$. This complication arises because, in
the special case when two vertex parts are both
contained in a self-energy part and each contains one
end-vertex of the self-energy part (and in no other
case), it is possible for the two vertex parts to
overlap without either being completely contained in
the other.
The finite part of $M(W_R)$ is to be separated out as
follows. In a unique way, $W_R$ is obtained from $W$ by
inserting a vertex part $\tilde V_a$ at $a$ and self-energy parts
$\tilde W_a$ and $\tilde W'_a$ in the two lines of $W$. From $M(W_R)$ there
are subtracted all divergences arising from $\tilde V_a, \tilde W_a, \tilde
W'_a$ let the remainder after this subtraction be $M' (W_R)$. Next, $W_R$
is considered as built up from $W$ by inserting some vertex part $\tilde
V_b$ at $b$, and self-energy parts $\tilde W_b$, and $\tilde W'_b$ in the two
lines of $W$. The integral $M'(W_R)$ will still contain
divergences arising from $\tilde V_b$, (but none from $\tilde W_b$ and
$\tilde W'_b$), and these divergences are to be subtracted,
leaving a remainder $M''(W_R)$. The finite part of
$M''(W_R)$ can finally be separated by applying to the
whole integral the method of Section VI, which
gives for $M''(W_R)$ an expression of the form (67),
with $\Sigma_e$, given by (71). Therefore the finite part of
$M(W_R)$ is a well-determined quantity, and is an
operator of the form (71).
The behavior of the higher order contributions to
$\Sigma^{\ast}$ and $\Pi^{\ast}$ having now been qualitatively explained,
we may describe the precise rules for the calculation
of $\Sigma^{\ast}$ and $\Pi^{\ast}$ by the same kind of inductive scheme
as was given for $\Lambda_{\mu}$ in the second paragraph of this
Section. Apart from the constant term $(-2 \pi i \delta \kappa_0), \Sigma^{\ast}$
is just the contribution $\Sigma(W, t^1)$ from the $W$ of
Fig. 5; and $\Sigma(W, t^1)$ is represented by an integral of
the form (65) with $l=1$. The integrand in (65) was a
product of two operators $\gamma_{\mu}$, one operator $D_F$, and
one operator $S_F$. The exact $\Sigma(W, t^1)$ is to be obtained
by replacing $D_F$ by $D'_F, S_F$ by $S'_F$ and one only of the
factors $\gamma_{\mu}$ by $\Gamma_{\mu}$ say the $\gamma_{\mu}$,
corresponding to the vertex $a$ of $W$. Suppose that $S'_F$ in the integrand is
represented, to order $e^{2n}$, by the sum of $S_F$ and of a
finite number of finite products of $S_F$ with operators
$S(\tilde W, t^1)$ such as appear in (71); and suppose that $D'_F$
and $\Gamma_{\mu}$ are similarly represented. Then $\Sigma(W, t^1)$ will
be determined to order $e^{2n + 2}$. The new $\Sigma(W, t^1)$ will
be a sum of integrals like the $M'(W_R)$ of the previous
paragraph, still containing divergences arising from
vertex parts at the vertex $b$ of $W$, in addition to
divergences arising from the graph $W_R$ as a whole.
When all these divergences are dropped, we have a
$\Sigma_e(W, t^1)$ which is finite; substituting this $\Sigma_e(W, t^1)$
for $\Sigma^{\ast}$ in (63) gives an $S'_F$ which is also finite and
determined to the order $e^{2n + 2}$.
The above procedures start from given $S'_F, D'_F$
and $\Gamma_{\mu}$ represented to order $e^{2n}$ by, respectively, $S_F$
plus $S_F$ multiplied by a finite sum of products of
$S(\tilde W, t^1), D_F$ plus $D_F$ multiplied by a finite sum of
products of $D(\tilde W, t^1)$ and $\gamma_{\mu}$ plus a finite sum of
$\Lambda_{\mu e}(\tilde V, t^1, t^2)$. From these there are obtained new
expressions for $S'_F, D'_F, \Gamma_{\mu}$. In the new expressions
there appear new convergent operators $S(W, t^1), D(W', t^1), \Lambda_{\mu
e} (V, t^1, t^2)$, determined to order $e^{2n +2}$;
in the divergent terms which are separated out and
dropped from the new expressions, there appear
divergent coefficients $A,B,C,L$, such as occur in
(73), (78), (82), also now determined to order $e^{2n + 2}$.
After the dropping of the divergent terms, the new
$\Gamma_{\mu}$ by (82) is a sum of $\gamma_{\mu}$ and a finite set of
$\Lambda_{\mu e} (V, t^1, t^2)$; the new $S'_F$ by (73) is $S_F$ plus $S_F$
multiplied by a finite sum of products of $S(W, t^1)$;
and the new $D'_F$ by (78) is $D_F$ plus $D_F$ multiplied by
a finite sum of products of $D(W, t^1)$. That is to say,
the new $\Gamma_{\mu}, S'_F, D'_F$ can be substituted back into the
integrals of the form (65), and so a third set of
operators $\Gamma_{\mu}, S'_F, D'_F$ is obtained, determined to
order $2^{2n + 2}$, and again with finite and divergent
parts separated. In this way, always dropping the
divergent terms before substituting back into the
integral equations, the finite parts of $\Gamma_{\mu}, S'_F, D'_F,$ may
be calculated by a process of successive approximation, starting with
the zero-order values $\gamma_{\mu}, S_F, D_F$.
After $n$ substitutions, the finite parts of $\Gamma_{\mu}, S'_F, D'_F$
will be determined to order $e^{2n}$.
It is necessary finally to justify the dropping of
the divergent terms. This will be done by showing
that the ``true'' $\Gamma_{\mu}, S'_F, D'_F,$ which are obtained if the
divergent terms are not dropped, are only numerical
multiples of those obtained by dropping
divergences, and that the numerical multiples can
themselves be eliminated from the theory by a
consistent use of the ideas of mass and charge
renormalization. Let $\Gamma_{\mu 1}(e), S_{F1}(e), D_{F1}(e)$ be the
operators obtained by the process of substitution
dropping divergent terms; these operators are power
series in $e$ with finite operator coefficients (to avoid
raising the question of the convergence of these
power-series, all quantities are supposed defined
only up to some finite order $e^{2N}$). Then we shall
show that the true operators $\Gamma_{\mu}, S'_F, D'_F$ are of the form
\begin{equation}
\Gamma_{\mu} = Z_1^{-1} \Gamma_{\mu 1}(e_1),
\end{equation}
\begin{equation}
S'_F = Z_2S'_{F1}(e_1),
\end{equation}
\begin{equation}
D'_F = Z_3 D_{F1}(e_1),
\end{equation}
where $Z_1, Z_2, Z_3$ are constants to be determined, and $e_1$ is given by
\begin{equation}
e_1 = Z_1^{-1} Z_2 Z_3^{1/2} e.
\end{equation}
This $e_1$ will turn out to be the ``true'' electronic
charge. It has to be proved that the result of substituting (83), (84), (85)
into the integral equations defining $\Gamma_{\mu}, S'_F, D'_F,$ is to reproduce
these expressions exactly, when $Z_1, Z_2, Z_3,$ and $\delta \kappa_0$ are
suitably chosen.
Concerning the $\Gamma_{\mu 1} (e), S'_{F1}(e), D'_{F1}(e),$ it is known
that, when these operators are substituted into the
integral equations, they reproduce themselves with
the addition of certain divergent terms. The
additional divergent terms consist partly of the
terms involving $A,B,C,L$, which are displayed in
(73), (78), (82), and partly of terms arising (in the
case of $S'_F$ and $D'_F$ only) from the peculiar behavior
of the vertices $b, b'$ in Fig. 5. The terms arising from
$b$ and $b'$ have been discussed earlier; they may be
called for brevity $b$--divergences. Originally, of
course, there is no asymmetry between the divergences arising in
$\Sigma^{\ast}$ from vertex parts inserted at
the two ends $a$ and $b$ of $W$; we have manufactured
an asymmetry by including the divergences arising
at a in the coefficient $Z_1^{-1}$ of (83), while at $b$ the
operator $\gamma_{\mu}$, has not been replaced by $\Gamma_{\mu}$ and so the $b$
divergences have not been so absorbed. It is thus to
be expected that the effect of the $b$ divergences, like
that of the a divergences, will be merely to multiply
all contributions to $\Gamma^{\ast}$ by the constant $Z_1^{-1}$.
Similarly, we expect that divergences at $b'$ will
multiply $\Pi^{\ast}$ by the constant $Z_1^{-1}$. It can be shown,
by a detailed argument too long to be given here,
that these expectations are justified. (The interested
reader is recommended to see for himself, by
considering contributions to $\Sigma^{\ast}$ arising from various
self-energy parts, how it is that the finite terms of a
given order are always reappearing in higher order
multiplied by the same divergent coefficients.)
Therefore, the complete expressions obtained by
substituting $\Gamma_{\mu 1}(e), S_{F1}(e), D_{F1}(e),$ into the integral
equations defining $\Lambda_{\mu}, \Sigma^{\ast}, \Pi^{\ast}$, are
\begin{equation}
\Lambda_{\mu 1} (e) = \Lambda_{\mu e} (e) + L (e) \gamma_{\mu},
\end{equation}
\begin{equation}
\begin{array}{c}
S_F \Sigma_1^{\ast} (e) = - 2 \pi i \delta \kappa_0 S_F\\\\
+ Z_1^{-1} \left( A(e) S_F + \frac{1}{2 \pi} B(e) + \frac{1}{2 \pi} S_e
(e) \right),
\end{array}
\end{equation}
\begin{equation}
D_F \Pi_1^{\ast} (e) = Z_1^{-1} \left( \frac{1}{2 \pi i} C(e) + \frac{1}{2
\pi i} D_e (e) \right).
\end{equation}
Here $A(e), B(e), C(e), L(e)$ are well-defined power
series in $e$, with coefficients which diverge never
more strongly than as a power of a logarithm. The
finite operators $\Lambda_{\mu e} (e), S_e(e), D_e(e), D_e(e),$ will, when all
divergent terms are dropped, lead back to the $\Gamma_{\mu1}(e), S'_{F1}(e),
D'_{F1}(e),$ from which the substitution started; thus, according to (38), (63), (64),
$$
\Gamma_{\mu1}(e) = \gamma_{\mu} + \Lambda_{\mu e} (e), \eqno(87')
$$
$$
S'_{F1} (e) = S_F + \frac{1}{2 \pi} S_e (e) S'_{F1} (e), \eqno(88')
$$
$$
D'_{F1}(e) = D_F + \frac{1}{2 \pi i} D_e (e) D'_{F1}(e).
\eqno(89')
$$
Equations (87)-(89), (87$'$)-(89$'$), describe precisely
the way in which the $\Gamma_{\mu 1} (e), S'_{F1}(e), D'_{F1}(e),$ when
substituted into the integral equations, reproduce
themselves with the addition of divergent terms.
And from these results it is easy to deduce the self-reproducing property of
the operators (83 )-(85), when substituted into the same equations.
Consider for example the effect of substituting
from (83)-(85) into the term $\Sigma(W, t^1)$, given by (65)
with $l=1$. The integrand of (65) is a product of one
factor $\Gamma_{\mu}$ one $\gamma_{\mu}$, one $S'_F$, and one $D'_F$.
Therefore the substitution gives
\begin{equation}
Z_1^{-1} Z_2 Z_3 Z_0 (W),
\end{equation}
where $\Sigma_0 (W)$ is the expression (65) obtained by
substituting $\Gamma_{\mu 1}(e_1),$ $S'_{F1}(e_1),$ $D'_{F1}(e_1)$, without the $Z$
factors. Now the $Z$ factors in (90) combine with the $e^2$ of (65) to give
$$
Z_1 Z_2^{-1} e_1^2,
$$
and the remaining factor of $\Sigma_0(W)$ is explicitly a
function of $e_1$ and not of $e$. Therefore (90) is
$$
Z_1 Z_2^{-1} Z_1 (W, e_1),
$$
where $\Sigma_1 (W,e)$ is the expression obtained by substituting the
operators $\Gamma_{\mu 1}(e), S'_{F1}(e), D'_{F1}(e)$ into
$\Sigma(W, t^1)$. Thus the $\Sigma^{\ast}(t^1)$, obtained by substituting
from (83)-(85) into (65), is identical with the result
of substituting the operators $\Gamma_{\mu 1}(e), S'_{F1}(e), D'_{F1}(e),$
and afterwards changing $e$ to $e_1$ and multiplying the
whole expression (except for the constant term in
$\delta \kappa_0$) by $Z_1 Z_2^{-1}$. More exactly, using (88), one can say
that the $\Sigma^{\ast}$ obtained by substituting from (83)-(85) is given by
\begin{equation}
S_F \Sigma^{\ast} = - 2 \pi i \delta \kappa_0 S_F + Z_2^{-1} \left( A(e_1)
S_F + \frac{1}{2 \pi} B(e_1) + \frac{1}{2 \pi} S_e (e_1) \right).
\end{equation}
Further, the $S'_F$ obtained by substituting from (83)-(85) into the integral
equations is given by (91) and
\begin{equation}
S'_f = S_F + S_F \Sigma^{\ast} S'_F.
\end{equation}
It is now easy to verify, using (88$'$), that $S'_F$ given by
(91) and (92) will be identical with (84), provided that
\begin{equation}
Z_2 + 1 + \frac{1}{2 \pi} B(e_1),
\end{equation}
\begin{equation}
\delta \kappa_0 = \frac{1}{2 \pi i} Z_2^{-1} A(e_1).
\end{equation}
In a similar way, the $D'_F$ obtained by substituting
from (83)-(85) into the integral equations can be
related with the $\Pi_1^{\ast}(e)$ of (89). This $D'_F$ will be
identical with (85) provided that
\begin{equation}
Z_3 = 1 + \frac{1}{2 \pi i} C(e_1).
\end{equation}
Finally, the $\Gamma_{\mu}$ obtained by substituting from (83)-(85) can be shown to be
$$
\Gamma_{\mu} = \gamma_{\mu} + Z_1^{-1} \Lambda_{\mu 1} (e_1),
$$
with $\Lambda_{\mu 1}(e)$ given by (87). Using (87$'$), this $\Gamma_{\mu}$ will
be identical with (83) provided that
\begin{equation}
Z_1 = 1 - L(e_1).
\end{equation}
Therefore, if $Z_1, Z_2, Z_3, \delta \kappa_0$ are defined by (96), (93),
(95), (94). it is established that (83)-(85) give the
correct forms of the operators $\Gamma_{\mu}, S'_F, D'_F$, including
all the effects of the radiative corrections which
these operators introduce into themselves and into
each other. The exact Eqs. (83)-(85) give a much
simpler separation of the infinite from the finite
parts of these operators than the approximate equations (73), (78), (82).
Consider now the result of using the exact
operators (83)-(85) in calculating a constituent $M$ of
$U(\infty)$, where $M$ is constructed from a certain
irreducible graph $G_0$ according to the rules of
Section IV. $G_0$ will have, say, $F_e$ internal and $E_e$
external electron lines, $F_p$ internal and $E_p$ external
photon lines, and
\begin{equation}
n = F_e + \frac{1}{2} E_e = 2F_p + E_p
\end{equation}
vertices. In $M$ there will be $\frac{1}{2} E_e$ factors $\psi'(k^i), \frac{1}{2}
E_e,$ factrs $\tilde \psi'(k^i)$ and $E_p$ factors $A'_{\mu}(k^i)$ given
by (37). In $\psi'(k^i), k^i$ is the momentum-energy 4 vector of an
electron, which satisfies (69), and the $S_e(k^i)$ in (73)
are zero at every stage of the inductive definition of
$S'_{F1}(e)$. Therefore (84), (35), (37) give in turn
\begin{equation}
\begin{array}{l}
S'_F (k^i) = Z_2 S_F (k^i),\\
\Sigma(k^i) = 2 \pi (Z_2 - 1) (k_{\mu}^i \gamma_{\mu} - i \kappa_0),\\
\psi' (k^i) = \psi(k^i) + 2 \pi (Z_2 - 1) S_F (k^i) (k_{\mu}^i \gamma_{\mu}
- i \kappa_0) \psi(k^i).
\end{array}
\end{equation}
The expression (98) is indeterminate, since
$(k_{\mu}^i \gamma_{\mu} - i \kappa_0)$ operating on $\psi(k^i)$ gives zero, while
operating on $S_F(k^i)$ it gives the constant $(1/ 2 \pi)$.
Thus, according to the order in which the factors are
evaluated, (98) will give for $\psi'(k^i)$ either the
value $\psi(k^i)$ or the value $Z_2 \psi(k^i)$. Similarly, $\tilde \psi(k^i)$ is
indeterminate between $\tilde \psi(k^i)$ and $Z_2 \tilde \psi(k^i)$, and,
excluding for the moment $A_{\mu}(k^i)$ which are Fourier
components of the external potential, $A'_{\mu}(k^i)$ is
indeterminate between $A_{\mu}(k^i)$ and $Z_3A_{\mu}(k^i)$. In any
case, considerations of covariance show that the
$\psi'(k^i), \tilde \psi'(k^i), A'_{\mu}(k^i)$ are numerical multiples of the
$\psi(k^i), \tilde \psi(k^i), A_{mu}(k^i)$; thus the indeterminacy lies only
in a constant factor multiplying the whole expression $M$.
There cannot be any indeterminacy in the
magnitude of the matrix elements of $U(\infty)$, so long
as this operator is restricted to be unitary. The
indeterminacy in fact lies only in the normalization
of the electron and photon wave functions $\psi(k^i), \tilde \psi(k^i),
A_{\mu}(k^i),$ which may or may not be regarded as
altered by the continual interactions of these particles with the
vacuum-fields around them. It can be
shown that, if the wave functions are everywhere
normalized in the usual way, the apparent indeterminacy is removed, and one must take
\begin{equation}
\begin{array}{c}
\psi'(k^i)=Z_2^{1/2} \psi(k^i),\\
\tilde \psi(k^i) = Z_2^{1/2} \tilde \psi(k^i),\\
A'_{\mu}(k^i) = Z_3^{1/2} A_{\mu}(k^i).
\end{array}
\end{equation}
It will be seen that (99) gives just the geometric
mean of the two alternative values of $\psi'(k^i)$ obtained from (98).
When $A_{\mu}(k^i)$ is a Fourier component of the external potential,
then in general $(k^i) \ne 0$ and $A'_{\mu} (k^i)$ is
not indeterminate but is given by (37) and (85) in the form
\begin{equation}
A'_{\mu}(k^i) = 2 \pi i Z_3 D'_{F1}(e_1)(k^i)^2 A_{\mu} (k^i).
\end{equation}
However, the unit in which external potentials are
measured is defined by the dynamical effects which
the potentials produce on known charges; and these
dynamical effects are just the matrix elements of
$U(\infty)$ in which (100) appears. Therefore the factor $Z_3$
in (100) has no physical significance, and will be
changed when $A_{\mu}$, is measured in practical units. The
correct constant which appears when practical units
are used is $Z_3^{1/2}$; this is because the photon potentials
$A_{\mu}$ in (99) were normalized in terms of practical.
units; and (100) should reduce to (99) when $(k^i) \rightarrow 0$,
if the external $A_{\mu}$ and the photon $A_{\mu}$ are measured
in the same units. Therefore the correct formula for
$A'_{\mu}$ covering the cases both of photon and of external potentials, is
\begin{equation}
\left. \begin{array}{l}
A'_{\mu}(k^i) = 2 \pi i Z_3^{1/2} D'_{F1}(e_1)(k^i)^2 A_{\mu}(k^i), \quad
(k^i) \ne 0,\\
A_{\mu}(k^i) = Z_3^{1/2} A_{\mu}(k^i), \quad (k^i)^2 = 0.
\end{array} \right\}
\end{equation}
In $M$ there will appear $E_e$ factors $S'_F, F_p$ factors
$D'_F$, and $n$ factors $\Gamma_{\mu}$, in addition to the factors of the
type (99), (101). Hence by (97) the $Z$ factors will occur in $M$ only as the
constant multiplier
$$
Z_1^{-n} Z_2^n Z_3^{1/2n}.
$$
By (86), this multiplier is exactly sufficient to
convert the factor $e^n$, remaining in $M$ from the
original interaction (8), into a factor $e_1^n$. Thereby,
both $e$ and $Z$ factors disappear from $M$, leaving only
their combination $e_1$ in the operators $\Gamma_{\mu 1}(e_1), S'_{F1}(e_1),
D'_{F1}(e_1),$ and in the factor $e_1^n$. If now $e_1$ is identified
with the finite observed electronic charge, there no
longer appear any divergent expressions in $M$. And
since $M$ is a completely general constituent of
$U(\infty)$, the elimination of divergences from the $S$ matrix is accomplished.
It hardly needs to be pointed out that the arguments of this
section have involved extensive
manipulations of infinite quantities. These manipulations have
only a formal validity, and must be
justified $a$ {\it posteriori} by the fact that they ultimately
lead to a clear separation of finite from infinite
expressions. Such an $a$ {\it posteriori} justification of
dubious manipulations is an inevitable feature of
any theory which aims to extract meaningful results
from not completely consistent premises.
We conclude with two disconnected remarks.
First, it is probable that $Z_1 = Z_2$ identically, though
this has been proved so far only up to the order $e^2$. If
this conjecture is correct, then all charge-renormalization effects
arise according to (86) from the
coefficient $Z_3$ alone, and the arguments of this paper
can be somewhat simplified. Second, Eqs. (88$'$),
(89$'$), which define the fundamental operators $S'_{F1}, D'_{F1}$,
may be solved for these operators. Thus
$$
S'_{F1}(e0 = \left[ 1 - \frac{1}{2 \pi} S_e (e) \right]^{-1} S_F, \eqno(88'')
$$
$$
D'_{F1}(e) = \left[ 1 - \frac{1}{2 \pi i} D_e (e) \right]^{-1} D_F. \eqno(89'')
$$
In electrodynamics, the $S_e$ and $D_e$ are small radiative
corrections, and it will always be legitimate and
convenient to expand (88$''$) and (89$''$) by the
binomial theorem. If, however, the methods of the
present paper are to be applied to meson fields, with
coupling constants which are not small, then it will
be desirable not to expand these expressions;
in this way one may hope to escape partially from
the limitations which the use of weak-coupling
approximations imposes on the theory.
\section{SUMMARY OF RESULTS}
The results of the preceding sections divide
themselves into two groups. On the one hand, there
is a set of rules by which the element of the $S$ matrix
corresponding to any given scattering process may
be calculated, without mentioning the divergent
expressions occurring in the theory. On the other
hand, there is the specification of the divergent
expressions, and the interpretation of these expressions as mass
and charge renormalization factors.
The first group of results may be summarized as
follows. Given a particular scattering problem, with
specified initial and final states, the corresponding
matrix element of $U(\infty)$ is a sum of contributions
from various graphs $G$ as described in Section II. A
particular contribution $M$ from a particular $G$ is to be
written down as an integral over momentum
variables according to the rules of Section III; the
integrand is a product of factors $\psi(k^i), \tilde \psi(k^i), A_{\mu}(k^i),
S_F(p^i), D_F(p^i), \delta(q_j), \gamma_{\mu}$, the factors corresponding in
a prescribed way to the lines and vertices of $G$.
According to Section IV, contributions $M$ are only to
be admitted from irreducible $G$; the effects of
reducible graphs are included by replacing in $M$ the
factors $\psi, \tilde \psi, A_{\mu} S_F, D_F, \gamma_{\mu}$, by the corresponding
expressions (37), (35), (36), (38). These
replacements are then shown in Section VII to be
equivalent to the following: each factor $S_F$ in $M$ is
replaced by $S'_{F1}(e)$, each factor $D_F$, by $D'_{F1}(e)$, each
factor $\gamma_{\mu}$, by $\Gamma_{\mu 1}(e)$, each factor $A_{\mu}$, when it represents
an external potential is replaced by
\begin{equation}
A_{\mu 1}(k^i) = 2 \pi i D'_{F1}(e) (k^i)^2 A_{\mu}(k^i),
\end{equation}
factors $\psi, \tilde \psi, A_{\mu}$ representing particle wave-functions
are left unchanged, and finally $e$ wherever it occurs
in $M$ is replaced by $e_1$. The definition of $M$ is completed by the
specification of $S'_{F1}(e), D'_{F1}(e), \Gamma_{\mu 1}(e)$;
it is in the calculation of these operators that the
main difficulty of the theory lies. The method of
obtaining these operators is the process of successive
substitution and integration explained in the first part
of Section VII; the operators so calculated are
divergence-free, the divergent parts at every stage of
the calculation being explicitly dropped after being
separated from the finite parts by the method of Section VI.
The above rules determine each contribution $M$ to
$U(\infty)$ as a divergence-free expression, which is a
function of the observed mass â and the observed
charge $e_1$ of the electron, both of which quantities
are taken to have their empirical values. The divergent parts of the
theory are irrelevant to the calculation of $U(\infty)$, being absorbed into
the unobservable constants $\delta m$ and $e$ occurring in (8). A place
where some ambiguity might appear in $M$ is in the
calculation of the operators $S'_{F1}(e), D'_{F1}(e), \Gamma_{\mu 1}(e)$,
when the method of Section VI is used to separate
out the finite parts $S(W, t^1), D(W', t^1), \Lambda_{\mu e}(V, t^1, t^2),$ from
the expressions (67), (74), (80). Even in this place
the rules of Section VI give unambiguous directions
for making the separation; only there is a question
whether some alternative directions might be equally
reasonable. For example, it is possible to separate
out a finite part from $\Sigma(W, t^1)$ according to (67), and
not to make the further step of using (70) to separate
out a finite part $S(W, t^1)$ which vanishes when (69)
holds. Actually it is easy to verify that such an
alternative procedure will not change the value of $M$,
but will only make its evaluation more complicated;
it will lead to an expression for $M$ in which one
(infinite) part of the mass and charge
renormalizations is absorbed into the constants $\delta m$
and $e$, while other finite mass and charge
renormalizations are left explicitly in the formulas. It
is just these finite renormalization effects which the
second step in the separation of $S(W, t^1)$ and $\Lambda_{\mu e}(V, t^1,
t^2)$ is designed to avoid. Therefore it may be
concluded that the rules of calculation of $U(\infty)$ are
not only divergence-free but unambiguous.
As anyone acquainted with the history of the Lamb
shift\footnote{H. A. Bethe, {\it Electromagnetic Shift of Energy Levels,} Report to
Solvay Conference, Brussels (1948).} knows, the utmost care is required
before it can be said that any particular rule of
calculation is unambiguous. The rules given in this
paper are unambiguous, in the sense that each
quantity to be calculated is an integral in momentum-space which is
absolutely convergent at infinity; such an integral has always a well-defined
value. However, the rules would not be unambiguous
if it were allowed to split the integrand into several
parts and to evaluate the integral by integrating the
parts separately and then adding the results;
ambiguities would arise if ever the partial integrals
were not absolutely convergent. A splitting of the
integrals into conditionally convergent parts may
seem unnatural in the context of the present paper,
but occurs in a natural way when calculations are
based upon a perturbation theory in which electron
and positron states are considered separately from
each other. The absolute convergence of the integrals
in the present theory is essentially connected with
the fact that the electron and positron parts of the
electron-positron field are never separated; this finds
its algebraic expression in the statement that the
quadratic denominator in (45) is never to be
separated into partial fractions. Therefore the
absence of ambiguity in the rules of calculation of
$U(\infty)$ is achieved by introducing into the theory
what is really a new physical hypothesis, namely that
the electron-positron field always acts as a unit and
not as a combination of two separate fields. A
similar hypothesis is made for the electromagnetic
field, namely that this field also acts as a unit and not
as a sum of one part representing photon emission
and another part representing photon absorption.
Finally, it must be said that the proof of the
finiteness and unambiguity of $U(\infty)$ given in this
paper makes no pretence of being complete and
rigorous. It is most desirable that these general
arguments should as soon as possible be supplemented by an explicit
calculation of at least one fourth-order radiative effect, to make sure that no
unforeseen difficulties arise in that order.
The second group of results of the theory is the
identification of $\delta m$ and $e$ by (94) and (86). Although
these two equations are strictly meaningless, both
sides being infinite, yet it is a satisfactory feature of
the theory that it determines the unobservable
constants $\delta m$ and $e$ formally as power series in the
observable $e_1$, and not vice versa. There is thus no
objection in principle to identifying <$e_1$ with the
observed electronic charge and writing
\begin{equation}
(e_1^2/4 \pi \hbar c) = \alpha = 1/137.
\end{equation}
The constants appearing in (8) are then, by (94) and
\begin{equation}
\delta m = m(A_1 \alpha + A_2 \alpha^2 + \ldots ),
\end{equation}
\begin{equation}
e = e_1 (1 + B_1 \alpha + B_2 \alpha^2 + \ldots ),
\end{equation}
where the $A_i$ and $B_i$ are logarithmically divergent
numerical coefficients, independent of $m$ and $e_1$.
\section{DISCUSSION OF FURTHER OUTLOOK}
The surprising feature of the $S$ matrix theory, as
outlined in this paper, is its success in avoiding
difficulties. Starting from the methods of Tomonaga,
Schwinger and Feynman, and using no new ideas or
techniques, one arrives at an $S$ matrix from which the
well-known divergences seem to have conspired to
eliminate themselves. This automatic disappearance
of divergences is an empirical fact, which must be
given due weight in considering the future prospects
of electrodynamics. Paradoxically opposed to the
finiteness of the $S$ matrix is the second fact. that the
whole theory is built upon a Hamiltonian formalism
with an interaction-function (8) which is infinite and
therefore physically meaningless.
The arguments of this paper have been essentially
mathematical in character, being concerned with the
consequences of a particular mathematical
formalism. In attempting to assess their significance
for the future, one must pass from the language of
mathematics to the language of physics. One must
assume provisionally that the mathematical formalism corresponds
to something existing in nature,
and then enquire to what extent the paradoxical
results of the formalism can be reconciled with such
an assumption. In accordance with this program, we
interpret the contrast between the divergent
Hamiltonian formalism and the finite $S$ matrix as a
contrast between two pictures of the world, seen by
two observers having a different choice of measuring
equipment at their disposal. The first picture is of a
collection of quantized fields with localizable
interactions, and is seen by a fictitious observer
whose apparatus has no atomic structure and whose
measurements are limited in accuracy only by the
existence of the fundamental constants $c$ and $\hbar$. This
observer is able to make with complete freedom on a
sub-microscopic scale the kind of observations
which Bohr and Rosenfeld\footnote{N. Bohr and L. Rosenfeld, Kgl. Dansk. Vid. Sels.
Math.-Phys. Medd. {\bf 12}, No. 8 (1933). A second paper by Bohr and Rosenfeld is to
be published later, and is abstracted in a booklet by A. Pais.
{\it Developments in the Theory of the Electron} (Princeton University Press,
Princeton, 1948).} employ in a more
restricted domain in their classic discussion of the
measurability of field-quantities;
and he will be referred to in what follows as the
``ideal'' observer. The second picture is of a collection of observable
quantities (in the terminology
of Heisenberg), and is the picture seen by a real
observer, whose apparatus consists of atoms and
elementary particles and whose measurements are
limited in accuracy not only by $c$ and $\hbar$ but also by
other constants such as $\alpha$ and $m$. The real observer
makes spectroscopic observations, and performs
experiments involving bombardments of atomic
systems with various types of mutually interacting
subatomic projectiles, but to the best of our knowledge he cannot
measure the strength of a single field
undisturbed by the interaction of that field with
others. The ideal observer, utilizing his apparatus in
the manner described in the analysis of the
Hamiltonian formalism by Bohr and Rosenfeld,\footnote{N. Bohr and L. Rosenfeld, Kgl. Dansk. Vid. Sels.
Math.-Phys. Medd. {\bf 12}, No. 8 (1933). A second paper by Bohr and Rosenfeld is to
be published later, and is abstracted in a booklet by A. Pais.
{\it Developments in the Theory of the Electron} (Princeton University Press,
Princeton, 1948).} makes measurements of precisely this last kind, and
it is in terms of such measurements that the
commutation-relations of the fields are interpreted.
The interaction-function (8) will presumably always
remain unobservable to the real observer, who is
able to determine positions of particles only with
limited accuracy, and who must always obtain finite
results from his measurements. The ideal observer,
however, using non-atomic apparatus whose
location in space and time is known with infinite
precision, is imagined to be able to disentangle a
single field from its interactions with others, and to
measure the interaction (8). In conformity with the
Heisenberg uncertainty principle, it can perhaps be
considered a physical consequence of the infinitely
precise knowledge of location allowed to the ideal
observer, that the value obtained by him when he
measures (8) is infinite.
If the above analysis is correct, the divergences of
electrodynamics are directly attributable to the fact
that the Hamiltonian formalism is based upon an
idealized conception of measurability. The
paradoxical feature of the present situation does not
then lie in the mere coexistence of a finite $S$ matrix
with an infinite interaction-function. The
empirically found correlation, between expressions
which are unobservable to a real observer and
expressions which are infinite, is a physically intelligible and
acceptable feature of the theory. The
paradox is the fact that it is necessary in the
present paper to start from the infinite expressions in
order to deduce the finite ones. Accordingly, what is
to be looked for in a future theory is not so much a
modification of the present theory which will make
all infinite quantities finite, but rather a turning-round of the theory
so that the finite quantities shall become primary and the infinite quantities
secondary.
One may expect that in the future a consistent
formulation of electrodynamics will be possible,
itself free from infinities and involving only the
physical constants $m$ and $e_1$, and such that a
Hamiltonian formalism with interaction (8), with
divergent coefficients $\delta m$ and $e$, may in suitably
idealized circumstances be deduced from it. The
Hamiltonian formalism should appear as a limiting
form of a description of the world as seen by a
certain type of observer, the limit being approached
more and more closely as the precision of measurement allowed to the
observer tends to infinity.
The nature of a future theory is not a profitable
subject for theoretical speculation. The future theory
will be built, first of all upon the results of future
experiments, and secondly upon an understanding of
the interrelations between electrodynamics and
mesonic and nucleonic phenomena. The purpose of
the foregoing remarks is merely to point out that
there is now no longer, as there has seemed to be in
the past, a compelling necessity for a future theory
to abandon some essential features of the present
electrodynamics. The present electrodynamics is
certainly incomplete, but is no longer certainly incorrect.
In conclusion, the author would like to express his
profound indebtedness to Professor Feynman for
many of the ideas upon which this paper is built, to
Professor Oppenheimer for valuable discussions,
and to the Commonwealth Fund of New York for
financial support.
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