Perturbation Theory in Quantum Electrodynamics

V. B. Berestetskii

The progress achieved in recent years in quantum electrodynamics[^1] is to a considerable extent connected with a new formulation of perturbation theory. The new perturbation theory has made it possible to represent results in a compact and relativistically invariant form, which has made it possible to develop methods for regularizing the divergent expressions encountered in the theory, expressions that previously made attempts to compute higher approximations meaningless. This is its chief merit. However, even for calculations of first approximations—problems that had been solved by the old methods—the use of the new perturbation theory offers significant advantages. Its essential features—the absence of any need to consider intermediate states and the absence of any need to separate the interactions of charges into Coulomb and “transverse” parts—make the application of the new methods more convenient.

In the present article the foundations and methods of applying the perturbation theory of modern quantum electrodynamics are set forth. The exposition of regularization methods is not included in the task of this article.

We shall use a system of units in which

\[ \hbar = c = 1 \]

(\(c\) is the velocity of light, \(\hbar\) the quantum constant).

1. Collision Matrix

We shall consider a system consisting of electrons (positive and negative) and photons. The general results will also apply to other particles (protons), if one restricts oneself to effects connected only with the presence of their electric charge. Let \(\Psi\) be the wave function of our system.

The equation determining the change of \(\Psi\) in time is

\[ i\,\frac{\partial \Psi}{\partial t}=\hat H\Psi, \tag{1} \]

where \(\hat H\) is the energy operator of the system. It is composed of the energy operators of the photons \(\hat W_\gamma\) and electrons \(\hat W_e\), and the interaction-energy operator \(\hat V\):

\[ \hat H=\hat H_0+\hat V;\qquad \hat H_0=\hat W_e+\hat W_\gamma . \tag{2} \]

Let \(\Psi_n\) be a complete system of eigenfunctions of the operator \(\hat H_0\),

\[ \left. \begin{aligned} i\,\frac{\partial \Psi_n}{\partial t}&=\hat H_0\Psi_n=E_n\Psi_n,\\ \Psi_n&=\Psi_n^{0}e^{-iE_n t}. \end{aligned} \right\} \tag{3} \]

Each function \(\Psi_n\) describes a definite state of the system of electrons and photons not interacting with one another. It can be represented in the form of a product

\[ \Psi_n=\Psi_{n_e}^{(e)}\cdot \Psi_{n_\gamma}^{(\gamma)}, \tag{3a} \]

where \(\Psi^{(e)}\) is the wave function of the system of electrons, and \(\Psi^{(\gamma)}\) that of the system of photons. \(E_n\) is the sum of the energies of all electrons and photons. As independent variables of the functions \(\Psi_n^{0}\) we choose the set of numbers of electrons \(n_f\) and photons \(n_q\) that are in definite individual states \(f, q\) (“occupation numbers”) (\(f,q\) denotes the complete set of quantum numbers of a particle, for example, momentum and polarization). In this representation

\[ \Psi_n^{0}=\prod_{q,f}\delta_{n_q n_q^{0}}\delta_{n_f n_f^{0}}, \tag{4} \]

where \(q, f\) run through all possible values; \(n\) now denotes a definite set of the numbers \(n_q^{0}, n_f^{0}\).

The functions \(\Psi_n\) (4) form an orthonormal system

\[ (\Psi_{n'}^{*},\Psi_n)=\sum_{n_q,n_f}\Psi_{n'}^{*}(n_f,n_q)\Psi_n(n_f,n_q)=\delta_{nn'} . \tag{4a} \]

The equality \(n=n'\) means equality of all occupation numbers \(n_q^{0}=n_q^{\prime 0};\ n_f^{0}=n_f^{\prime 0}\). The operators \(\hat W_e\) and \(\hat W_\gamma\) are expressed through the operators of the quantized wave functions of the particles.

A real system cannot be in a state described by a wave function \(\Psi_n\), since the latter is not an eigen-

...an eigenfunction of the interaction operator \(\hat V\). However, we can expand the wave function of the system satisfying equation (1) in the complete system of functions (3)

\[ \Psi=\sum_n \Phi(n)\Psi_n . \tag{5} \]

The coefficients \(\Phi(n)\) depend on time. If at some initial instant \(t_0\) the system is characterized by a definite distribution of the electron and photon occupation numbers (we shall denote their aggregate by \(n_0\)), then, owing to the interaction, this distribution changes with time. \(|\Phi(n,t)|^2\) determines the probability of some other aggregate of occupation numbers at the instant \(t\). Determining \(\Phi(n)\) is equivalent to finding \(\Psi\).

Let us find the equation that determines the change of \(\Phi(n)\) with time. To this end we substitute the expansion (5) into (1):

\[ \sum_n\left(i\,\frac{\partial\Psi_n}{\partial t}\,\Phi(n)+i\,\frac{\partial\Phi(n)}{\partial t}\,\Psi_n\right) = \sum_n\left[\Phi(n)\hat H_0\Psi_n+\Phi(n)\hat V\Psi_n\right]. \]

The first terms on the left- and right-hand sides of this equality cancel by virtue of (3). We multiply the remaining equation scalarly by \(\Psi_{n'}^{*}\). Then, by virtue of the orthonormality of the system of functions \(\Psi_n\) (4a), we obtain:

\[ i\,\frac{\partial\Phi(n)}{\partial t} = \sum_{n'} \langle n|V|n'\rangle\,\Phi(n'), \tag{6} \]

where \(\langle n|V|n'\rangle\) is the matrix element of the interaction-energy operator \(\hat V\):

\[ (n|V|n')=(\Psi_n^{*},\hat V\Psi_{n'}^{\phantom{*}}) = (\Psi_n^{0*},\hat V\Psi_{n'}^{0})e^{-i(E_{n'}-E_n)t}. \tag{7} \]

We can write (6) briefly as

\[ i\,\frac{\partial\Phi}{\partial t}=\hat V\Phi . \tag{8} \]

The function \(\Phi\), which is still the aggregate of the coefficients \(\Phi(n)\), may be regarded as the wave function of the system. Its energy operator is the matrix of the operator \(\hat V\) (7). (We note that it explicitly contains time.) In view of this, the description of the system by the function \(\Phi\) and equation (8) is sometimes called the interaction representation\(^2\).

The solution of equation (8) may be represented in the following form:

\[ \Phi(t)=\hat S(t)\Phi_0, \tag{9} \]

where \(\Phi_0\) is the value of \(\Phi\) at \(t=t_0\), and \(\hat S(t)\) is an operator. More explicitly,

\[ \Phi(n,t)=\sum_{n'}(n|\hat S(t)|n')\Phi_0(n'). \tag{9a} \]

Let at \(t=t_0\) the system be in the state \(n_0\), i.e.

\[ \Phi_0(n')=\delta_{n'n_0}. \tag{10} \]

Then, by (9a),

\[ \Phi(n,t)=(n|\hat S(t)|n_0)=(\Psi_n^*,\hat S(t)\Psi_{n_0}). \tag{11} \]

A typical physical problem is the collision problem. Its formulation is as follows. At the initial instant \(t_0=-\infty\), a state of motion of noninteracting particles*) is specified; it is required to determine the state of motion of the particles (again noninteracting) at \(t=\infty\). The problem is solved if the matrix \(\hat S(\infty)\) has been found, which therefore may be called the scattering matrix. In fact, as we shall see below, knowledge of the \(S\)-matrix makes it possible to solve also a broader class of problems going beyond the collision problem.

Substitution of (9) into (8) gives the equation for the matrix \(\hat S\)

\[ i\frac{\partial \hat S(t)}{\partial t}=\hat V(t)\hat S(t) \tag{12} \]

and the initial condition

\[ \hat S(t_0)=1. \tag{12a} \]

Equation (12) may serve as the fundamental equation of quantum electrodynamics.

2. PERTURBATION THEORY

To solve equation (12) we shall use the fact that the interaction energy of electrons with the electromagnetic field is proportional to the magnitude of the electric charge of the electron \(e\), which may be regarded as a small parameter. In the system of units chosen by us,

\[ e\approx \frac{1}{\sqrt{137}}. \]

Therefore one may solve (12) by the method of successive approximations, i.e. seek for \(\hat S(t)\) a solution in the form of a series in

*) The term “noninteracting particles” is understood here in the sense of the absence of interaction with one another, i.e. in the sense of the initial condition (10). Such a state may take into account the presence of an external field.

in powers of \(e\):

\[ \hat S(t)=\sum_{k=0}^{\infty}\hat S_k(t), \tag{13} \]

where \(\hat S_k\) is proportional to \((e)^k\). Substituting (13) into (12) and equating terms of the same order, we obtain (taking into account the initial condition (12a))

\[ \hat S_k(t)=-i\int_{t_0}^{t}\hat V(t')\hat S_{k-1}(t')\,dt' \tag{14} \]

or, successively substituting \(\hat S_{k-1}\) through \(\hat S_{k-2}\), etc., down to \(\hat S_0=1\),

\[ \hat S_k(t)=(-i)^k\int_{t_0}^{t}dt'\int_{t_0}^{t'}dt''\ldots\int_{t_0}^{t^{(k-1)}}dt^{(k)}\, \hat V(t')\times \]

\[ {}\times \hat V(t'')\ldots \hat V(t^{(k)}). \tag{15} \]

It is convenient to represent the multiple integral (15) in a somewhat different form.\(^3\) For simplicity, let us first consider the term \(k=2\)

\[ \hat S_2(t)=-\int_{t_0}^{t}dt'\int_{t_0}^{t'}dt''\,\hat V(t')\hat V(t''). \tag{16} \]

If \(t'\) and \(t''\) are represented as coordinates of a point in the plane (\(t'\)—abscissa, \(t''\)—ordinate), then the region of integration in (16) is a triangle lying below the bisector of the coordinate angle (Fig. 1). If in (16) we interchange the notations \(t'\) and \(t''\), then

\[ \hat S_2(t)=-\int_{t_0}^{t}dt''\int_{t_0}^{t''}dt'\,\hat V(t'')\hat V(t'). \tag{16a} \]

Fig. 1.

In the plane \(t'\), \(t''\) (still taking \(t'\) as the abscissa and \(t''\) as the ordinate) the region of integration is a triangle lying above the bisector of the coordinate angle. The integrand in (16a) differs from the integrand in (16) by a change in the order of the factors. If \(\hat V(t')\) and \(\hat V(t'')\)

commuted, then both integrands would coincide and \(\hat S_2\) could be represented as half the integral over the whole square of Fig. 1. We can do this also in the general case of noncommuting \(\hat V(t')\) and \(\hat V(t'')\), if we introduce the “chronologizing operator” \(P\):

\[ P\bigl[\hat V(t')\hat V(t'')\bigr] = P\bigl[\hat V(t'')\hat V(t')\bigr] = \begin{cases} \hat V(t')\hat V(t''), & \text{for } t'>t'',\\ \hat V(t'')\hat V(t'), & \text{for } t'<t''. \end{cases} \tag{17} \]

Then

\[ \hat S_2(t) = -\frac{1}{2} \int_{t_0}^{t} dt' \int_{t_0}^{t} dt''\, P\bigl[\hat V(t')\hat V(t'')\bigr]. \tag{18} \]

Similarly, in the general case (15) can be written as

\[ \hat S_k(t) = \frac{(-i)^k}{k!} \int_{t_0}^{t} dt' \int_{t_0}^{t} dt'' \ldots \int_{t_0}^{t} dt^{(k)} P\bigl[\hat V(t')\hat V(t'')\ldots \hat V(t^{(k)})\bigr], \tag{19} \]

where \(P\) is the operator that rearranges the factors so that, from left to right, the time arguments of the operators \(\hat V\) always decrease. Formula (19) is the basic starting formula for quantum-electrodynamic calculations.

In experiments the directly measured quantity is the probability of transitions per unit time from a state \(n_0\) to a state \(n\), which can be defined as

\[ w_{n_0 n} = \lim_{t\to\infty} \frac{d}{dt} \left|\left(n\middle|S(t)\middle|n_0\right)\right|^2 \tag{20} \]

(or simply the collision cross section associated with \(w_{n_0 n}\)). Let us find an expression for \(w_{n n_0}\) through the collision matrix \(\hat S(\infty)\). According to (19),

\[ S(\infty) = \frac{(-i)^k}{k!} \int_{-\infty}^{\infty} dt' \int_{-\infty}^{\infty} dt'' \ldots \int_{-\infty}^{\infty} dt^{(k)} \times \]

\[ \times P\bigl[\hat V(t') \ldots \hat V(t^{(k)})\bigr]. \tag{21} \]

Let \(k-1\) integrations be carried out here. Since the matrix \(V\) contains time only in the form \(e^{i(E_{n'}-E_n)t}\) [see (7)], then,

the resulting expression will have the form*)

\[ (n|S(\infty)|n_0)=-iU_{nn_0}\int_{-\infty}^{\infty} e^{i\nu t'}\,dt' =-2\pi i U_{nn_0}\delta(\nu), \tag{22} \]

where \(U_{nn_0}\) is a time-independent quantity. We shall call it the matrix of the effective energy of the perturbation. Let us now change the limiting procedure so as to find (20). Let

\[ (n|S'(t)|n_0)=-iU_{nn_0}\int_{t_0}^{t} e^{i\nu t'}\,dt' \tag{22a} \]

(\(S'(t)\) is not the true matrix \(S(t)\), but \(S'(\infty)=S(\infty)\)).

Further calculations coincide with those carried out in the usual quantum-mechanical nonstationary perturbation theory:

\[ (n|S'_k(t)|n_0) = -i\,\frac{e^{i\nu t}-e^{i\nu t_0}}{\nu}\,U_{nn_0}, \]

\[ |(n|S'(t)|n_0)|^2 = |U_{nn_0}|^2 \frac{\sin^2 \dfrac{\nu}{2}(t-t_0)} {\left(\dfrac{\nu}{2}\right)^2}; \]

for large \(t-t_0\),

\[ \frac{\sin^2 \dfrac{\nu}{2}(t-t_0)} {\left(\dfrac{\nu}{2}\right)^2} = 2\pi\delta(\nu)\,(t-t_0). \]

We obtain the usual quantum-mechanical formula

\[ w_{n_0n}=2\pi |U_{nn_0}|^2\delta(\nu). \tag{23} \]

The main result obtained here reduces to the connection of the collision matrix \(S(\infty)\) with the matrix \(U_{nn_0}\) (22).**

* \((n|S_k|n_0)\) may contain a number of terms of type (22), but, as we shall see below, the time dependence in all these terms is the same. In fact, \(\nu\) turns out to be nothing other than \(E_n-E_{n_0}\), and the factor \(\delta(\nu)\) ensures conservation of energy. Form (21) applies to \(S\) in any approximation.

** Let us note once again that \(\hat S(\infty)\) is expressed in the form (21) in any approximation, while \(U_{nn_0}\) enters expression (22), which has the form of the equation of first-order quantum-mechanical perturbation theory.

3. INTERACTION ENERGY. THE FIELD OF PHOTONS AND ELECTRONS

Up to now we have not specified the form of the interaction-energy operator \(\hat V\). It can be found by analogy with the corresponding classical expression.

The latter has the form

\[ V=\int j_i A_i (dr), \tag{24} \]

where \(j_i\) is the four-dimensional vector of the electron current density, \(A_i\) is the vector potential of the electromagnetic field \((i=1,2,3,0)\),

\[ j_i A_i = j_0 A_0-\mathbf{j}\mathbf{A}. \]

In the second-quantization representation all physical quantities pertaining to photons are expressed through the operators of quantized fields in the same way as the corresponding classical quantities.

It also follows from the Dirac equation for the motion of an electron in an external field that the mean interaction energy has the form (24), where \(j_i\) is the quantum-mechanical expression for the current density. We shall take for \(\hat V\) the expression (24), in which \(j_i\) and \(A_i\) are the corresponding operators expressed in terms of the operators of the quantized wave functions of electrons and photons. The same expression (24) follows from considerations of relativistic invariance. \(j_i A_i\) is an invariant. Then the integral \(\int \hat j_i \hat A_i (dr)\,dt\), which enters the matrix \(\hat S\), is also invariant.

The potential operator has the form

\[ \hat A_i=\sum_q \left(\hat C_q A_{qi}+\hat C_q^{+} A_{qi}^{*}\right), \tag{25} \]

where \(\hat C_q\) and \(\hat C_q^{+}\) are the absorption and emission operators of a photon in the state \(q\), and \(A_{qi}+A_{qi}^{*}\) is the vector potential of the field of the corresponding photon. Since by the operator \(\hat V\) in the interaction representation [equation (12)] we mean the matrix (7), \(\hat A_i\) contains factors \(e^{i(E_{n'}-E_n)t}\). But the matrix \((n|C_q|n')\) is different from zero only if \(n\) differs from \(n'\) by the absence of one photon in the state \(q\), i.e.

\[ E_n-E_{n'}=-k, \]

where \(k\) is the frequency of the absorbed photon.

Similarly, \((n|\hat C_q^\dagger|n')\) is different from zero when \(n\) differs from \(n'\) by one extra photon in the state \(q\), i.e.

\[ E_n-E_{n'}=k. \]

Thus, \(A_{qi}\) in (25) may be regarded as including the time dependence \(e^{-ikt}\).

If the quantum numbers \(q\) correspond to photon states with definite momentum and polarization, then

\[ \hat A_i=\sum_{\mathbf{k}\mu}(\mathbf e_{\mathbf{k}\mu})_i \left(\hat C_{\mathbf{k}\mu}\varphi_{\mathbf{k}\mu} +\hat C_{\mathbf{k}\mu}^{\dagger}\varphi_{\mathbf{k}\mu}^{*}\right) =\sum_{\mathbf{k}\mu}(\hat A_{\mathbf{k}\mu})_i, \tag{25a} \]

\[ \varphi_{\mathbf{k}\mu} = \frac{1}{2\pi}\sqrt{\frac{(d\mathbf{k})}{k}}\, e^{-iqx} \tag{25b} \]

(\(q\) is the four-dimensional wave vector, \(x\) is the set of \(\mathbf r\) and \(t\), \(-qx=\mathbf{k}\mathbf r-kt\)); \((\mathbf e_{\mathbf{k}\mu})\) is a unit polarization vector.

The normalization of the potentials (25b) is such that the energy of the electromagnetic field having the form of a wave packet of width \((d\mathbf{k})\) is

\[ \frac{1}{8\pi}\int \left(E_{\mathbf{k}\mu}^{2}+H_{\mathbf{k}\mu}^{2}\right)(d\mathbf r)=k. \]

For a given wave vector \(\mathbf k\), only two photon polarizations are possible \((\mu=1,2)\), those for which the vectors of the electric and magnetic fields are transverse. This could have been achieved by also taking the potentials to be transverse
\[ \left((\hat A_{\mathbf{k}\mu})_3=(\hat A_{\mathbf{k}\mu})_0=0, \ \text{if the axis } x_3 \text{ is directed along } \mathbf k\right). \]
However, such a formulation is not relativistically invariant. Therefore one must consider all four components of the vector-potential operator and, correspondingly, four polarizations (\(\mu=3\)—“longitudinal quanta” and \(\mu=0\)—“scalar quanta”).

For \(\mu=1,2,3\) the commutation relations hold:

\[ \hat C_{\mathbf{k}\mu}\hat C_{\mathbf{k}\mu'}^{\dagger} - \hat C_{\mathbf{k}\mu'}^{\dagger}\hat C_{\mathbf{k}\mu} = \delta_{\mu\mu'}, \quad (\mathbf e_{\mathbf{k}\mu})_i=\delta_{\mu i}. \tag{26} \]

For \(\mu=0\) two formulations are possible: either

\[ \hat C_{\mathbf{k}0}\hat C_{\mathbf{k}0}^{\dagger} - \hat C_{\mathbf{k}0}^{\dagger}\hat C_{\mathbf{k}0} = -1, \quad (\mathbf e_{\mathbf{k}0})_i=\delta_{0i}, \]

or

\[ \hat C_{\mathbf{k}0}\hat C_{\mathbf{k}0}^{\dagger} - \hat C_{\mathbf{k}0}^{\dagger}\hat C_{\mathbf{k}0} = 1, \quad (\mathbf e_{\mathbf{k}0})_i=-i\delta_{0i}. \tag{26a} \]

We shall use the latter formulation\(^4\); it is convenient

by the fact that, as for the transverse components, the operators \(\hat C^+_{k0}\) and \(\hat C_{k0}\) can be interpreted as emission and absorption operators. In this case, however, the operator \(\hat A_0\) turns out to be non-self-adjoint. The operator
\[ \hat A_4=i\hat A_0 \]
is now self-adjoint. In order that the mathematical expectation of \(\hat A_0\) be real, it is necessary to change its definition. Instead of the usual
\[ \overline{A_j}=(\Psi^*,\,\hat A_j\Psi) \]
let
\[ \overline{A_j}=(\Psi^+,\,\hat A_j\Psi), \tag{27} \]
where
\[ \Psi^+=\Psi^*\hat\eta, \tag{27a} \]
so that
\[ \overline{A_0}=-\,i\Psi^*\hat\eta\hat A_4\Psi, \]
\[ (\overline{A_0})^*=i\Psi^*\hat A_4\hat\eta\Psi \qquad (\hat A_4^+=\hat A_4;\ \hat\eta^+=\hat\eta). \]

The reality of \(\overline{A_0}\),
\[ (\overline{A_0})^*=\overline{A_0}, \]
will be satisfied if the operator \(\hat\eta\) is defined in the following way:
\[ \hat A_4\hat\eta=-\hat\eta\hat A_4 \tag{27б} \]
or
\[ \hat C_{k0}\hat\eta=-\hat\eta\hat C_{k0};\qquad \hat C^+_{k0}\hat\eta=-\hat\eta\hat C^+_{k0} \]
(\(\hat A_j,\hat\eta\) commute with the remaining components; it can be shown that these commutation rules are invariant).

From the explicit form of the matrices \(\hat C_0\) it is easy to find that
\[ \hat\eta=(-1)^{\nu_0}, \tag{27в} \]
where \(\nu_0\) is the number of scalar quanta.

The potentials \(\hat A_j\) satisfy the wave equation. In order for it to be equivalent to Maxwell’s equations, the Lorentz condition is usually added:

\[ \left(\operatorname{div}\hat{\mathbf A}+\frac{\partial \hat A_0}{\partial t}\right)\Psi=0. \]

It is sufficient, however, to impose a weaker condition ensuring the fulfillment of Maxwell’s equations for mathematical expectations,

\[ \Psi^{+}\left(\operatorname{div}\hat{\mathbf A}+\frac{\partial \hat A_0}{\partial t}\right)\Psi=0. \tag{28} \]

Let

\[ \hat{\mathbf A}=\hat{\mathbf A}^{(+)}+\hat{\mathbf A}^{(-)}, \]

\[ \hat A_4=\hat A_4^{(+)}+\hat A_4^{(-)}, \]

where

\[ \hat A_j^{(+)}=\sum_{\mathbf k\mu}(\mathbf e_{\mathbf k\mu})_j\,\hat C_{\mathbf k\mu}\varphi_{\mathbf k\mu}, \]

\[ \hat A_j^{(-)}=\sum_{\mathbf k\mu}(\mathbf e_{\mathbf k\mu})_j\,\hat C_{\mathbf k\mu}^{+}\varphi_{\mathbf k\mu}^{*}. \]

(Such a decomposition is relativistically invariant.) Suppose the condition

\[ \left[\operatorname{div}\hat{\mathbf A}^{(+)}-i\,\frac{\partial \hat A_4^{(+)}}{\partial t}\right]\Psi=0. \tag{28a} \]

is satisfied. The conjugate expression has the form

\[ \Psi^{*}\left(\operatorname{div}\hat{\mathbf A}^{(-)}+i\,\frac{\partial \hat A_4^{(-)}}{\partial t}\right)=0, \]

or, multiplying on the right by \(\eta\), by virtue of (27б):

\[ \Psi^{+}\left(\operatorname{div}\hat{\mathbf A}^{(-)}-i\,\frac{\partial \hat A_4^{(-)}}{\partial t}\right)=0, \]

i.e. condition (28a) is equivalent to (28). (28a) may be written as (the \(x_3\) axis is directed along \(\mathbf k\)):

\[ (\hat C_{\mathbf k3}+i\hat C_{\mathbf k0})\Psi_n(v_3v_0)=0, \tag{28б} \]

where \(\nu_3\) is the number of longitudinal quanta, \(\nu_0\) is the number of scalar quanta, and \(\Psi_{\parallel}\) is the wave function of the “longitudinal degrees of freedom” of the electromagnetic field.

The general solution of (286) is

\[ \Psi_{\parallel}=\sum_{s=0}^{\infty}\lambda_s \varphi_s(\nu_3,\nu_0), \tag{29} \]

where

\[ \varphi_s=\sum_{r=0}^{s}(i)^r \sqrt{\binom{s}{r}}\, \delta_{\nu_3,r}\,\delta_{\nu_0,s-r} \]

(\(\binom{s}{r}\) are the binomial coefficients).

It is noteworthy that

\[ (\Psi^{+}\Psi)=\lambda_0(\varphi_0^{+}\varphi), \tag{29a} \]

since for \(s\) or \(s'\ne 0\)

\[ (\varphi_s^{+}\varphi_{s'})=0. \]

Thus, the wave function \(\Psi_{\parallel}\) can be normalized:

\[ \lambda_0=1,\qquad \lambda_s\ne 0 \text{ arbitrary.} \tag{29б} \]

In particular, one may take

\[ \lambda_0=1,\qquad \lambda_s=0\quad (s\ne 0). \tag{29в} \]

One can verify that any other choice of \(\lambda_s\) leaves unchanged the gradient-invariant quantities (for example, mathematical expectations of the fields, energy, momentum, etc.).

This result means that there exists only one state of the longitudinal degrees of freedom. This is the state in which, under the simplest choice of the constants, longitudinal and scalar quanta are absent (the vacuum, or zero oscillations). Therefore the emission of these quanta is impossible in accordance with the transverse polarization of photons. Nevertheless, the presence of this “longitudinal vacuum” affects the interaction of particles.

The operator of the current-density vector has the form:

\[ \hat{j}^{\,i}=\frac{1}{2}\left(\hat{\bar{\psi}}\,\gamma_i\,\hat{\psi} -\hat{\bar{\psi}}\,\gamma_i\,\hat{\psi}\right), \tag{30} \]

where \(\gamma_i=\beta\alpha_i\;(\alpha_0=1)\) are the Dirac matrices and \(\bar{\psi}=\psi^{*}\beta\).

The operators of the quantized wave function of the electron are

\[ \hat{\psi}=\sum_f\left(\hat{a}_f\psi_f+\hat{b}_f^{+}\psi_{-f}\right), \tag{30a} \]

\[ \hat{\bar{\psi}}=\sum_f\left(\hat{a}_f^{+}\bar{\psi}_f+\hat{b}_f\bar{\psi}_{-f}\right), \]

where \(\hat{a}_f\) and \(\hat{a}_f^{+}\) are the operators of absorption and emission of an electron, and \(\hat{b}_f\) and \(\hat{b}_f^{+}\) are the same for a positron. \(\psi_f\) are solutions of the Dirac equation with positive frequencies, and \(\psi_{-f}\) are solutions with negative frequencies. As in the case of photon operators, in \(\psi_f\) one must include the time factor \(e^{-i\varepsilon_f t}\), and in \(\psi_{-f}\), \(e^{-i\varepsilon_{-f}t}=e^{i|\varepsilon_{-f}|t}\). Indeed, in accordance with (7), the matrix elements \(\hat{a}_f\) differ from 0 only when \(E_{n'}-E_n=\varepsilon_f\), and the matrix elements of \(\hat{b}_{-f}\) when \(E_{n'}-E_n=-\varepsilon_{-f}\) (\(|\varepsilon_{-f}|=-\varepsilon_{-f}\) is the positron energy; since \(\hat{b}_f\) is multiplied by \(\bar{\psi}_{-f}\), this is equivalent to saying that \(\psi_{-f}\) contains \(e^{-i\varepsilon_{-f}t}\)). Correspondingly the same is obtained for \(\hat{a}_f^{+}\) and \(\hat{b}_f^{+}\).

The operator \(\hat{j}_i\), in accordance with (30) and (30a), contains the absorption and emission operators bilinearly in the following combinations:

\[ \hat{a}_f^{+}\hat{a}_{f'}-\hat{a}_{f'}\hat{a}_f^{+}; \qquad \hat{b}_f\hat{b}_{f'}^{+}-\hat{b}_{f'}^{+}\hat{b}_f, \]

or

\[ \hat{a}_f\hat{b}_{f'}-\hat{b}_{f'}\hat{a}_f; \qquad \hat{a}_f^{+}\hat{b}_{f'}^{+}-\hat{b}_{f'}^{+}\hat{a}_f^{+}. \]

The operators entering here anticommute, except for the operators \(\hat{a}_f\hat{a}_{f'}^{+}\) and \(\hat{b}_f\hat{b}_{f'}^{+}\) when \(f=f'\), when their anticommutator is equal to 1. Therefore, except in the last case, the difference in each bracket may be replaced by twice the first term. Thus the current operator may be written in the following form:

\[ \hat{j}_i= \sum_{f\ne f'} \left( \hat{a}_f^{+}\hat{a}_{f'}\bar{\psi}_f\gamma_i\psi_{f'} + \hat{b}_f\hat{b}_{f'}^{+}\bar{\psi}_{-f}\gamma_i\psi_{-f'} \right) + \]

\[ +\sum_{f\ne f'} \left( \hat{a}_f^{+}\hat{b}_{f'}^{+}\bar{\psi}_f\gamma_i\psi_{-f'} + \hat{b}_f\hat{a}_{f'}\bar{\psi}_{-f}\gamma_i\psi_{f'} \right) + \]

\[ +\frac{1}{2}\sum_f \left[ (\hat{a}_f^{+}\hat{a}_f-\hat{a}_f\hat{a}_f^{+})\bar{\psi}_f\gamma_i\psi_f - (\hat{b}_f^{+}\hat{b}_f-\hat{b}_f\hat{b}_f^{+})\bar{\psi}_{-f}\gamma_i\psi_{-f} \right]. \tag{31} \]

(The last sum becomes \(0\) for free particles, when the electron and positron states are identical.) The terms \(\hat a_f^+ \hat a_{f'}\) and \(\hat b_f^+ \hat b_{f'}\) correspond to invariance of the number of electrons and positrons separately (only the state may change), while the terms \(\hat a_f \hat b_{f'}\), \(\hat a_f^+ \hat b_{f'}^+\) correspond to the annihilation or creation of an electron–positron pair. Thus, the bilinear structure of \(\hat j_i\) expresses the law of charge conservation.

If \(\psi_f\) and \(\bar\psi_f\) are solutions of the Dirac equation without an external field, then they may be chosen as wave functions of states with definite momentum and polarization (spin projection). Then

\[ \hat\psi=\sum_{p\mu}\left(\hat a_{p\mu}\psi_{p\mu+}+\hat b_{p\mu}^{+}\psi_{-p,-\mu,-}\right), \]

\[ \hat{\bar\psi}=\sum_{p\mu}\left(\hat a_{p\mu}^{+}\bar\psi_{p\mu+}+\hat b_{p\mu}\bar\psi_{-p,-\mu,-}\right), \tag{32} \]

\[ \psi_{p\mu+}=u_{p\mu+}e^{i\left(pr-\sqrt{p^2+m^2}\,t\right)} \sqrt{\frac{(dp)}{(2\pi)^3}}, \]

\[ \psi_{-p,-\mu,-}=u_{-p,-\mu,-}e^{-i\left(pr-\sqrt{p^2+m^2}\,t\right)} \sqrt{\frac{(dp)}{(2\pi)^3}}. \tag{32a} \]

Here \(\psi_{p\mu+}\) is the wave function corresponding to the wave vector (momentum) \(p\), spin projection \(\mu\), and positive frequency; \(\psi_{p\mu-}\) is the function corresponding to the wave vector \(p\), polarization \(\mu\), and negative frequency, i.e. to the momentum \(-p\) and spin projection \(-\mu\) of the positron. \(u_{p\mu\pm}\) are the corresponding spinor amplitudes satisfying the relation following from the Dirac equation:

\[ f\gamma u=mu, \tag{33} \]

where

\[ f\gamma=f_i\gamma_i=\gamma_0\left(\pm\sqrt{p^2+m^2}\right)-p\gamma, \tag{33a} \]

and the sign at the root corresponds to the sign of the frequency.

The wave functions (32b) are normalized so that, for a wave packet of width \((dp)\),

\[ \int |\psi_{p\mu}|^2\,(dr)=1, \]

i.e.

\[ u_{p\mu}^{*}u_{p\mu}=1. \tag{33b} \]

The external field can be taken into account either exactly, by determining the wave functions \(\psi_f\), or approximately, as a perturbation. In the second case we may use wave functions without taking the external field into account (32a), and include the latter in the perturbation operator (24), which in this case has the form

\[ \hat V=e\int j_i\bigl(\hat A_i+A_i^{(e)}\bigr)(dr). \tag{34} \]

The external potential \(A_i^{(e)}\) is not restricted by the transversality condition. An intermediate method is of course also possible, in which part of the external field is taken into account exactly, while part enters into the perturbation. The potentials \(A_i^{(e)}\), not being operators, in (31) do not act on the photon variables and do not change the number of photons. Their time dependence may be contained explicitly. If it is represented in the form of a Fourier series or integral, then the \(S\)-matrix retains the form (22).

4. FIRST-ORDER PROCESSES

Let us now return to the basic formula for the collision matrix \(\hat S(\infty)\) (21). In what follows, for brevity, we shall denote it simply by \(\hat S\). Let us first consider the first-order collision matrix*)

\[ \hat S_1=-ie\int_{-\infty}^{\infty}\hat j_i(x)\bigl(\hat A_i(x)+A_i^{(e)}(x)\bigr)\,d^4x \tag{35} \]

and determine to what processes its elements correspond. According to (25), the matrix elements of \(\hat A_i\) correspond to the emission or absorption of one photon. Let the number of photons in some state \(q\) in the initial state be \(n_q\); then the following elements of the matrix \(\hat S_1\) are nonzero:

\[ (\ldots n_q+1\ldots|S_1|\ldots n_q\ldots)\quad \text{(emission)};\qquad (n_q+1|C_q^+|n_q)=\sqrt{n_q+1} \]

and

\[ (\ldots n_q-1\ldots|S_1|\ldots n_q\ldots)\quad \text{(absorption)};\qquad (n_q-1|C_q|n_q)=\sqrt{n_q}, \]

with the numbers of photons in the other states unchanged. In what follows we shall be interested in the case \(n_q=0\) or \(1\). The potential

*) In application to first-order processes, the new perturbation theory essentially coincides with the old one\({}^{7}\).

of the external field, since it contains no operators \(\hat C_q\) and \(\hat C_q^+\), is diagonal in the photon numbers. Therefore that term in (35) which contains \(A^{(e)}\) will enter the matrix elements

\[ (\ldots n_q \ldots |S|\ldots n_q \ldots). \]

The matrix \(\hat j_i\), according to (31), contains elements corresponding to the following processes.

I. Transition of an electron from the state \(f\) to the state \(f'\):

\[ (0_f 1_{f'}|a_{f'}^+ a_f|1_f 0_{f'})=1 \]

(by virtue of the Pauli principle \(n_f\) can take the values 0 or 1). This may correspond to:

a) Emission of a photon. The corresponding element of the matrix \(S\), according to (35), (31), and (25), is

\[ (f'q|S|f)=-ie\int \bar\psi_{f'}(x)\gamma_i A_{qi}^*(x)\psi_f(x)\,d^4x. \tag{36} \]

Here the notation is

\[ (f'q|S|f)=(\ldots 1_{f'}\ldots 0_f\ldots 1_q|S|\ldots 0_q\ldots 1_f\ldots 0_{f'}\ldots). \]

b) Absorption of a photon

\[ (f'|S|qf)=-ie\int \bar\psi_{f'}(x)\gamma_i A_{qi}(x)\psi_f(x)\,d^4x. \tag{36a} \]

c) Scattering of an electron by the external field

\[ (f'|S|f)=-ie\int \bar\psi_{f'}(x)\gamma_i A_i^{(e)}(x)\psi_f(x)\,d^4x. \tag{36b} \]

Let us note that (36) and (36a) vanish if \(f\) and \(f'\) are free states of the electron (without an external field). Indeed, then the matrix element contains the integral

\[ \int e^{i(f-f'-q)\cdot x}\,d^4x=(2\pi)^4\delta(f-f'-q), \tag{37} \]

where \(f_i\), \(f'_i\), \(q_i\) are the corresponding four-dimensional momenta of the electron and photon. Formula (37) expresses the laws of conservation of energy and momentum, which cannot be simultaneously satisfied. (A free electron cannot emit or absorb a photon.)

II. Transition of a positron from the state \(g\) to the state \(g'\). The corresponding matrix elements are:

\[ (g'q|S_1|g)=ie\int \bar\psi_{-g}(x)\gamma_i A_{qi}^*(x)\psi_{-g'}(x)\,d^4x, \tag{38} \]

\[ (g'|S_1|qg)=ie\int \bar\psi_{-g}\gamma_i A_{qi}\psi_{-g'}\,d^4x, \tag{38a} \]

\[ (g'|S_1|g)=ie\int \bar\psi_{-g}\gamma_i A_i^{(e)}\psi_{-g'}\,d^4x. \tag{38b} \]

Expression (37) differs from (36) by a sign (which corresponds to the opposite charge of the positron). In addition, in (37), as compared with (36), the initial and final wave functions have, as it were, changed places (this corresponds to the fact that the change of state of the positron, as a “hole,” is opposite to the change of the electron state with negative frequency).

III. Creation of a pair by a photon:

\[ (fg|S_1|q)=-ie\int \bar\psi_f A_{qi}\gamma_i\psi_{-g}\,d^4x \tag{39} \]

or by an external field:

\[ (fg|S_1|0)=-ie\int \bar\psi_f A_i^{(e)}\gamma_i\psi_{-g}\,d^4x . \tag{39a} \]

The matrix element (39a) vanishes for free electrons, for, analogously to (37), it contains

\[ \int e^{-i(f+g-q)x}d^4x=(2\pi)^4\delta(f+g-q), \]

where \(f\), \(g\), and \(q\) are the 4-momenta of the electron, positron, and photon, and again the requirements of the laws of conservation of energy and momentum turn out to be incompatible. By virtue of the law of conservation of energy, in any case the matrix element

\[ (fgq|S_1|0), \]

which contains

\[ \int e^{\,i(\varepsilon_f+|\varepsilon_{-g}|+k)t} \]

(each of these three quantities is positive), vanishes.

IV. Conversion of a pair into a photon:

\[ (q|S_1|fg)=-ie\int \bar\psi_{-g}\gamma_i A_{qi}\psi_f d^4x \tag{39b} \]

and absorption of a pair by an external field:

\[ (0|S_1|fg)=-ie\int \bar\psi_{-g}\gamma_i A_i^{(e)}\psi_f d^4x . \tag{39v} \]

Expression (39v) vanishes for free electrons. The matrix element

\[ (0|S_1|qfg) \]

is always equal to 0.

Let us also note that the creation or annihilation of a pair by an external field requires the presence in its Fourier expansion of high frequencies of order \(2m\), since (39a) or (39v) contain time integrals

\[ \int e^{\pm i(\varepsilon_f+|\varepsilon_{-g}|)t} A_i^{(e)}(t)\,dt . \]

The last terms in (31) are diagonal with respect to the electronic states. Therefore, by virtue of the law of conservation of energy, they lead neither to emission nor to absorption of a photon.

It is convenient to depict the matrix elements of the corresponding processes by diagrams constructed according to the following principle. Each of the matrix elements written above represents

Fig. 2. a)—emission of a photon by an electron (36a); б)—absorption of a photon by an electron (36a); в)—emission of a photon by a positron (37a); г)—absorption of a photon by a positron (38a); д)—scattering of an electron by an external field (36б); е)—scattering of a positron by an external field (38б); ж)—pair creation by a photon (39a); з)—conversion of a pair into a photon (39б); и)—pair creation in an external field (39a)

an integral of the product of three quantities that are functions of one and the same argument \(x\). Correspondingly to this, we draw three rays from one nodal point. The electronic functions are represented by solid lines, the photon function by a dashed line. To distinguish \(\psi\) and \(\bar{\psi}\), the line corresponding to \(\psi\) is provided with an arrow directed toward the nodal point, and the line \(\bar{\psi}\) with an arrow directed away from the nodal point. Lines corresponding to the state of the system at \(t=-\infty\) are placed to the right of the nodal point, and lines of states at \(t=\infty\) to the left (“the time axis is directed to the left”). Thus, the dashed line on the right always corresponds to \(A_q^{*}\), and on the left to \(A_q\). The external field is represented by a vertical dashed line. Electron lines on the right correspond to \(\psi\), and on the left to \(\bar{\psi}\).

(direction of the arrows along the “time axis”), and positrons, conversely (direction of the arrows against the “time axis”*). In Fig. 2 the diagrams of all the matrix elements of this paragraph are given.

5. THE INTERACTION FUNCTION OF TWO CHARGES

Let us proceed to consider the collision matrix of the second order. Substituting into (18) the expression \(\hat V\), we have:

\[ \hat S_2=-\frac{e^2}{2}\iint P\left[\hat j_i(x_1)\hat A_i(x_1)\hat j_j(x_2)\hat A_j(x_2)\right]\,d^4x_1\,d^4x_2 . \]

Since the electron and photon operators commute with one another, the order of the arrangement of \(\hat j\) relative to \(\hat A\) is immaterial, and

\[ \hat S_2=-\frac{e^2}{2}\iint P\left[\hat j_i(x_1)\hat j_j(x_2)\right]\,P\left[\hat A_i(x_1)\hat A_j(x_2)\right]. \tag{40} \]

Expression (40) contains the product of two photon operators. Therefore the matrix \(\hat S_2\) contains elements diagonal with respect to the number of photons. Of special significance for us will be the diagonal elements corresponding to the absence of photons. We shall therefore find the matrix element

\[ \left(0\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|0\right) = \left(\Psi_0^{(x)*},\,P[A_i(x_1)A_j(x_2)]\Psi_0^{(x)}\right), \tag{41} \]

which may be called the vacuum average of this operator.

Substitute into this expression the expansion of the potential operators (25). The product \(\hat A_i\hat A_j\) will contain various products of emission and absorption operators, and

\[ \left(0\left|\hat C_q\hat C_{q'}\right|0\right) = \left(0\left|\hat C_q^{+}\hat C_{q'}^{+}\right|0\right) = \left(0\left|\hat C_q^{+}\hat C_{q'}\right|0\right) =0, \]

\[ \left(0\left|C_q C_{q'}^{+}\right|0\right)=\delta_{qq'}. \tag{42} \]

This applies both to the transverse and to the longitudinal and scalar components.

* The possibility of such a depiction of the motion of a positron was first pointed out by G. A. Zisman \({}^{5}\).

To take account of the chronological operator \(P\), let us write (41) separately for the cases \(t_1>t_2\) and \(t_1<t_2\):

\[ \left(0\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|0\right) = \left(0\left|\hat A_i(x_1)\hat A_j(x_2)\right|0\right) = \sum_q A_{qi}(x_1)A^*_{qj}(x_2) \qquad (t_1>t_2), \tag{43} \]

\[ \left(0\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|0\right) = \left(0\left|\hat A_j(x_2)\hat A_i(x_1)\right|0\right) = \sum_q A^*_{qi}(x_1)A_{qj}(x_2) \qquad (t_1<t_2). \tag{43a} \]

For \(A_q\) we shall use states with definite momentum and polarization (25б). Then

\[ \left(0\left|P\left[\hat A_i(x_1),\hat A_j(x_2)\right]\right|0\right) = \begin{cases} \dfrac{1}{(2\pi)^2}\displaystyle\int \dfrac{(dk)}{k}\, e^{i\mathbf{k}(\mathbf{r}_1-\mathbf{r}_2)-ik(t_1-t_2)} \displaystyle\sum_\mu (e_{\mathbf{k}\mu})_i(e_{\mathbf{k}\mu})_j, & (t_1>t_2),\\[1.2em] \dfrac{1}{(2\pi)^2}\displaystyle\int \dfrac{(dk)}{\kappa}\, e^{-i\mathbf{k}(\mathbf{r}_1-\mathbf{r}_2)+ik(t_1-t_2)} \displaystyle\sum_\mu (e_{\mathbf{k}\mu})_i(e_{\mathbf{k}\mu})_j, & (t_1<t_2). \end{cases} \tag{44} \]

According to (26), (26a)

\[ \sum_\mu (e_{\mathbf{k}\mu})_i(e_{\mathbf{k}\mu})_j=g_{ij}. \tag{45} \]

(Here \(g_{ij}=-1\) for \(i=j\ne0\); \(g_{00}=1\).)

Substituting this into (44), we obtain:

\[ \left(0\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|0\right) =-g_{ij}D(x_1x_2), \tag{46} \]

where

\[ D(x_1x_2)= \dfrac{1}{(2\pi)^2} \int e^{i\mathbf{k}(\mathbf{r}_1-\mathbf{r}_2)-ik|t_1-t_2|} \dfrac{(dk)}{k}. \tag{46a} \]

(In (44), \(e^{-i\mathbf{k}(\mathbf{r}_1-\mathbf{r}_2)}\), upon integration, after making the change of variable \(\mathbf{k}\to-\mathbf{k}\), may be replaced by \(e^{i\mathbf{k}(\mathbf{r}_1-\mathbf{r}_2)}\).)

The product of the components of the vector potential is not gradient-invariant. But for any arbitrary choice of the constants \(\lambda_s\) in (29), the expression for

\[ \left(0\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|0\right) \]

will differ from (46) by a term of the form \(\dfrac{\partial^2 \Lambda}{\partial x_i \partial x_j}\) (where \(\Lambda\) is some function of \(x_1\) and \(x_2\)), which will give in the matrix \(\hat S_2\) (40) the integral

\[ \int F_{ij}(x_1 x_2)\, \frac{\partial^2 \Lambda}{\partial x_{1i}\partial x_{2j}}\, d^4x_1\,d^4x_2, \]

where

\[ F_{ij}=\left(n\left|P\left[\hat j_i(x_1)\hat j_j(x_2)\right]\right|n'\right). \]

By integration by parts it can be reduced to an integral containing \(\dfrac{\partial F_{ij}}{\partial x_i}\). In view of the continuity equation for the current-density vector

\[ \frac{\partial \hat j_i}{\partial x_i}=0 \]

the same property will also be possessed by \(F_{ij}\). Then the second term in (46) may be discarded. We shall not prove in general form that

\[ \frac{\partial F_{ij}}{\partial x_i}=0;\qquad \frac{\partial F_{ij}}{\partial x_j}=0 \tag{47} \]

(it is sufficient that one of these conditions be fulfilled). In all concrete calculations it is easy to verify that (47) holds.

The expression for \(D\) (46a) can be put into the form of a four-dimensional integral, whence its invariance with respect to Lorentz transformations will be evident.

For this purpose let us consider the integral

\[ I(\tau)=\int \frac{e^{-i\omega\tau}}{\omega^2-k^2}\,d\omega, \tag{48} \]

taken in the plane of the complex variable \(\omega\) along a contour going along the real axis and passing around the poles of the integrand, \(\omega_1=k\) from above and \(\omega_2=-k\) from below (Fig. 3, a). For \(\tau>0\) the contour can be closed by an infinitely large semicircle below.

Fig. 3.

Then the value of \(I\) is determined by the residue at the point \(\omega_1=k\):

\[ I(\tau)=-\pi i\,\frac{e^{-ik\tau}}{k}\qquad (\tau>0). \]

For \(\tau<0\) the contour may be closed along an infinitely large semicircle from above, and the residue at the point \(\omega_2=-k\) will play the role:

\[ I(\tau)=-\pi i\,\frac{e^{+ik\tau}}{k}\qquad (\tau<0). \]

In both cases

\[ I(t)=-\pi i\,\frac{e^{-ik|t|}}{k}. \tag{48a} \]

Thus,

\[ \frac{e^{-ik|t_1-t_2|}}{k}=\frac{i}{\pi}\,I(t_1-t_2) \tag{48б} \]

and (46a) takes the following form:

\[ D(x_1x_2)=-\frac{i}{4\pi^3}\int e^{ik(r_1-r_2)-\omega(t_1-t_2)} \,\frac{(dk)\,d\omega}{k^2-\omega^2} \tag{49} \]

or\(^6\)

\[ D(x_1x_2)=\frac{i}{4\pi^3}\int e^{iq(x_2-x_1)}\,\frac{d^4q}{q^2}. \tag{49a} \]

In (49), \(qx=kr-\omega t\) and \(q_0=\omega\) is an independent variable (not connected with \(k\)). The condition for bypassing the poles may be formulated somewhat differently. We shall regard the denominator in the integral (49) as the limit of the expression \(q^2+i\delta\) as \(\delta\to0\), where \(\delta\) is a positive quantity. Then the poles of the integrand are displaced, the first \((q_0=k)\) downward, and the second \((q_0=-k)\) upward (Fig. 3, б), and the integration in (49) is performed over all four variables from \(-\infty\) to \(\infty\). From (49) it is evident that \(D\) is an invariant.

Let us denote the photon-vacuum-averaged matrix \(\hat S_2\) by

\[ \hat S_2^{(e)}=\left(\Psi_0^{(\gamma)*},\,\hat S_2\Psi_0^{(\gamma)}\right). \tag{50} \]

Then from (40) and (46), under the condition that (47) is satisfied,

\[ \hat S_2^{(e)} =-\frac{e^3}{2}\iint P\left[\hat j_i(x_1)\hat j_j(x_2)\right] D(x_1x_2)\,d^4x_1\,d^4x_2. \tag{50a} \]

In nonrelativistic quantum mechanics the interaction of particles is described by a potential energy depending on the differences of their coordinates. The operator of the potential energy of interac-

... in the representation of the method of second quantization has the form

\[ \iint \hat{\rho}(\mathbf r_1)\hat{\rho}(\mathbf r_2)\,U(\mathbf r_1\mathbf r_2)\,(d\mathbf r_1)(d\mathbf r_2). \]

In electrodynamics there is no such operator. Electrons interact not directly, but through the electromagnetic field. However, there exists a collision matrix, represented in the first approximation by the operator (50), which plays an analogous role. \(S_2^{(e)}\) contains only the coordinates of the electrons. The degrees of freedom of the electromagnetic field have been eliminated. The function \(D'\) may be called the interaction function of two charges. In what follows we shall see that, in the nonrelativistic approximation, it indeed contains the operator of the interaction energy of two charges (in particular, Coulomb’s law).

6. THE INTERACTION FUNCTION OF AN ELECTRON WITH A PHOTON

As a typical matrix element of \(\hat S_2\) that is nondiagonal with respect to photons, let us consider the matrix element for the collision of an electron with a photon:

\[ (f'q'|S_2|qf) = -\frac{e^2}{2} \iint \left(f'\left|P\left[\hat j_i(x_1)\hat j_j(x_2)\right]\right|f\right) \times \]

\[ {}\times \left(q'\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|q\right) \,d^4x_1\,d^4x_2 \tag{51} \]

(the photon passes from the state \(q\) to \(q'\), the electron—from \(f\) to \(f'\)).

From the operator \(\hat A_i(x_1)\hat A_j(x_2)\), entering into \(\hat S_2\), only those terms containing \(C_{q'}^{+}C_q\) will enter into (51).

Since \(C_{q'}^{+}\) commutes with \(C_q\) \((q\ne q')\), the operator \(P\) is immaterial, and

\[ \left(q'\left|P\left[\hat A_i(x_1)\hat A_j(x_2)\right]\right|q\right) = A_{qi}(x_1)A^{*}_{q'j}(x_2) + A^{*}_{q'i}(x_1)A_{qj}(x_2). \tag{52} \]

The operator of the product of currents

\[ \hat j_i(x_1)\hat j_j(x_2) = (\gamma_i)_{\alpha\beta}(\gamma_j)_{\alpha'\beta'} \hat\psi_{\alpha}(x_1)\hat\psi_{\beta}(x_1) \hat\psi_{\alpha'}(x_2)\hat\psi_{\beta'}(x_2) \]

(\(\alpha,\beta,\alpha',\beta'\) are spinor indices, over which summation is implied) consists of products of four operators of emission or absorption of electrons. Nonzero matrix elements will be given by those terms which contain \(\hat a_{f'}^{+}\) and \(\hat a_f\). Therefore we may replace one of the operators \(\hat\psi(x)\) by \(\hat\psi_f(x)=\hat a_f\psi_f(x)\) and one of the operators \(\hat\psi(x)\) by \(\hat\psi_{f'}(x)=\hat a_{f'}^{+}\bar\psi_{f'}(x)\).

In the remaining two operators \(\left(\hat{\psi}\ \text{and}\ \hat{\bar{\psi}}\right)\) one should retain only those terms which correspond to the emission and absorption of a particle in one and the same state,

\[ \hat{\psi}_{\alpha}(x_1)\hat{\bar{\psi}}_{\beta}(x_2) = \sum_f \hat{a}_f \hat{a}_f^{+}\psi_{f\alpha}(x_1)\bar{\psi}_{f\beta}(x_2). \]

Expressions containing these operators with coinciding arguments should be omitted, since the corresponding terms in (51) will contain:

\[ \int \bar{\psi}_{f\alpha}(x)\psi_{f\beta}(x)A_{qi}(x)\,dt \sim \int e^{-ikt}\,dt = 0. \]

Thus,

\[ P\left[\hat{j}_i(x_1)\hat{j}_j(x_2)\right] = \]

\[ = (\gamma_i)_{\alpha\beta}(\gamma_j)_{\alpha'\beta'}P \left[ \hat{\bar{\psi}}_{f'\alpha}(x_1)\hat{\psi}_{\beta}(x_1) \hat{\bar{\psi}}_{\alpha'}(x_2)\hat{\psi}_{f\beta'}(x_2) + \right. \]

\[ \left. + \hat{\bar{\psi}}_{\alpha}(x_1)\hat{\psi}_{f\beta}(x_1) \hat{\bar{\psi}}_{f'\alpha'}(x_2)\hat{\psi}_{\beta'}(x_2) \right]. \tag{53} \]

Equality (53) is to be understood in the sense that the operators on the left- and right-hand sides give identical matrix elements (51). The two terms in (53) differ from one another only by the replacement of the arguments \(x_1\) by \(x_2\) and conversely, which corresponds to an interchange of the variables of integration in the integral (51):

\[ \frac{1}{2}P\left[\hat{j}_i(x_1)\hat{j}_j(x_2)\right] = \]

\[ = (\gamma_i)_{\alpha\beta}(\gamma_j)_{\alpha'\beta'}P \left[ \hat{\bar{\psi}}_{f'\alpha}(x_1)\hat{\psi}_{\beta}(x_1) \hat{\bar{\psi}}_{\alpha'}(x_2)\hat{\psi}_{f\beta'}(x_2) \right]. \tag{53a} \]

This expression can be reduced to a simpler form. Let us note that for \(t_1<t_2\), owing to the action of the operator \(P\), the order of the factors in (53a) becomes the following:

\[ \hat{\bar{\psi}}_{\alpha'}(x_2)\hat{\psi}_{f\beta'}(x_2) \hat{\bar{\psi}}_{f'\alpha}(x_1)\hat{\psi}_{\beta}(x_1). \]

In view of the fact that the operators

\[ \hat{\psi}_f\ \text{with}\ \hat{\psi}_{f'}\ (f\ne f'), \qquad \hat{\psi}_f\ \text{with}\ \hat{\bar{\psi}} \quad\text{and}\quad \hat{\bar{\psi}}_{f'}\ \text{with}\ \hat{\psi} \]

anticommute, this expression is equal to

\[ - \hat{\bar{\psi}}_{f'\alpha}(x_1)\hat{\bar{\psi}}_{\alpha'}(x_2) \hat{\psi}_{\beta}(x_1)\hat{\psi}_{f\beta'}(x_2), \]

i.e., it differs from the corresponding expression for the case \(t_1>t_2\) by the sign and by the order of the internal factors. Thus,

\[ P\left[ \hat{\psi}_{f'a}(x_1)\hat{\psi}_{\beta}(x_1) \hat{\psi}_{a'}(x_2)\hat{\psi}_{f\beta'}(x_2) \right] = \hat{\psi}_{f'a}P'\left[ \hat{\psi}_{\beta}(x_1)\hat{\psi}_{a'}(x_2) \right]\hat{\psi}_{f\beta'}(x_2), \tag{54} \]

where

\[ P'\left[\hat{X}(t_1)\hat{Y}(t_2)\right]= \begin{cases} \hat{X}\hat{Y}, & \text{for } t_1>t_2,\\ -\hat{Y}\hat{X}, & \text{for } t_1<t_2. \end{cases} \tag{54a} \]

The operator \(\hat{\psi}_f\) has the single nonzero matrix element

\[ \left(0\left|\hat{\psi}_f\right|f\right)=\psi_f, \]

whereas the operator \(\hat{\psi}_{f'}\)

\[ \left(f'\left|\hat{\psi}_{f'}\right|0\right)=\overline{\psi}_{f'}. \]

Therefore

\[ \left(f'\left|P\left[ \hat{\psi}_{f'a}(x_1)\hat{\psi}_{\beta}(x_1) \hat{\psi}_{a'}(x_2)\hat{\psi}_{f\beta'}(x_2) \right]\right|f\right) = \]

\[ = \overline{\psi}_{f'a}(x_1)\psi_{f\beta'}(x_2) \left(0\left|P'\left[ \hat{\psi}_{\beta}(x_1)\hat{\psi}_{a}(x_2) \right]\right|0\right). \tag{55} \]

The problem is now reduced to finding the “vacuum average” of products of electron operators entering into (55). We introduce the notation

\[ K_{a_1a_2}(x_1x_2)= \left(0\left|P'\left[ \hat{\psi}_{a_1}(x_1)\hat{\psi}_{a_2}(x_2) \right]\right|0\right). \tag{56} \]

(We shall use the notation \(K(1,2)=K(x_1a_1;\,x_2a_2)\), with the spinor operators \(a_1,a_2\) understood to be included among the arguments.) The operator \(\hat{S}_2\) can now be replaced by the operator \(\hat{S}_2^{(\gamma)}\) (in the sense that

\[ \left(f'q'\left|\hat{S}_2\right|qf\right) = \left(f'q'\left|\hat{S}_2^{(\gamma)}\right|qf\right), \tag{57} \]

where

\[ \hat{S}_2^{(\gamma)} = -e^2\iint \hat{\bar{\psi}}(1)\hat{A}_i(1)\gamma_i K(1,2)\gamma_j\hat{A}_j(2)\hat{\psi}(2) \,d^4x_1\,d^4x_2). \tag{57a} \]

The function \(K(x_1 a_1; x_2 a_2)\) may be called the function of interaction of the electron with the photon.

It is easy to see that the matrix \(\hat S'_2\) pertains not only to scattering, but also to other processes in which two electronic and two photon states occur, namely: the emission or absorption of two photons

\[ (f' q' q \mid S'_2 \mid f);\qquad (f' \mid S'_2 \mid q q' f), \]

or the transformation of a pair into two photons

\[ (q q' \mid S'_2 \mid f g), \]

and the analogous processes.

Let us return to the expression for \(K(1,2)\) (56):

\[ K(1,2)= \begin{cases} (0|\hat\psi(1)\hat{\bar\psi}(2)|0), & \text{for } t_1>t_2,\\ -(0|\hat{\bar\psi}(2)\hat\psi(1)|0), & \text{for } t_1<t_2. \end{cases} \]

Let us substitute here the expansion of the operators \(\hat\psi\) and \(\hat{\bar\psi}\) (30a). The nonzero matrix elements will give those terms which contain the product of an absorption operator by the emission operator of a particle in the same state, the latter having to stand to the right of the former. Since \(\hat\psi\) contains the operators of electron absorption and positron emission, and \(\hat{\bar\psi}\), conversely, then for \(t_1>t_2\), \(K(1,2)\) will contain only electronic operators, while for \(t_1<t_2\) only positronic ones.

We obtain\(^6\)

\[ K(1,2)= \left\{ \begin{array}{ll} \displaystyle \sum_f \psi_f(1)\bar\psi_f(2) & (t_1>t_2),\\[6pt] \displaystyle -\sum_{-f}\psi_{-f}(1)\bar\psi_{-f}(2) & (t_1<t_2), \end{array} \right\} \tag{58} \]

where \(\displaystyle \sum_f\) denotes summation over states with positive frequencies, and \(\displaystyle \sum_{-f}\) over states with negative frequencies.

Expression (58) can be represented as the solution of an inhomogeneous Dirac equation. Let \(\hat L\) be the Dirac operator,

\[ \hat L=\gamma_i\left(\hat p_i-eA_i^{(e)}\right)-m =\beta\left(i\frac{\partial}{\partial t}-\hat H_D\right) \]

\[ \left(\hat H_D=(\alpha p+\beta m)\right), \tag{59} \]

acting on the variables \(x_1 a_1\) (we shall regard \(x_2, a_2\) as parameters). We shall prove that \(K(1,2)\) is a solution of the equation

\[ \hat L(1)K(1,2)=i\delta(x_1-x_2)\delta_{a_1a_2}. \tag{60} \]

The spatial part of the four-dimensional \(\delta\)-function

\[ \delta(x_1-x_2)=\delta(\mathbf r_1-\mathbf r_2)\delta(t_1-t_2) \]

will be represented, using the completeness property of the system of eigenfunctions of the operator \(\hat H_D\), in the form

\[ \delta(\mathbf r_1-\mathbf r_2)\delta_{a_1a_2} = \sum_f \psi_f^0(\mathbf r_1 a_1)\psi_f^{0*}(\mathbf r_2 a_2) = \sum_f \bar\psi_f^0(\mathbf r_2 a_2)\beta\psi_f^0(\mathbf r_1 a_1), \]

where

\[ \hat H_D\psi_f^0=\varepsilon_f\psi_f^0,\qquad \psi_f=\psi_f^0 e^{-i\varepsilon_f t}; \]

the summation is carried out over all states (both with positive and with negative frequencies). We write the time part of the \(\delta\)-function as

\[ \delta(t_1-t_2)=\frac{1}{2\pi}\int e^{-i\omega(t_1-t_2)}\,d\omega . \]

We shall seek the solution of equation (60) in the form of the expansion

\[ K(1,2)=\int d\omega\sum_f C_{f\omega}\psi_f^0(\mathbf r_1 a_1)e^{-i\omega t_1}. \]

Substituting all this into (60), we obtain:

\[ \beta(\omega-\varepsilon_f)C_{f\omega} = \frac{i}{2\pi}\bar\psi_f^0(\mathbf r_2 a_2)\beta e^{i\omega t_2}. \]

Since the operator \(\beta\) does not act on the variable \(a_2\), multiplying this equation by \(\beta\), we obtain:

\[ (\omega-\varepsilon_f)C_{f\omega} = \frac{i}{2\pi}\bar\psi_f^0(\mathbf r_2 a_2)e^{i\omega t_2} \]

or

\[ K(1,2)=\frac{i}{2\pi}\sum_f \bar\psi_f^0(\mathbf r_2 a_2)\psi_f(\mathbf r_1 a_1) \int \frac{e^{-i\omega(t_1-t_2)}}{\omega-\varepsilon_f}\,d\omega . \tag{61} \]

We establish the following rule for going around the poles situated on the real axis at the points \(\omega=\varepsilon_f\): on the positive axis the contour passes above the poles; on the negative axis—below the poles (Fig. 3, c). Then, closing the contour along a semicircle of infinitely large radius downward for \(t_1>t_2\) and upward for \(t_1<t_2\), we obtain (58).

For a free electron it is not difficult to obtain an explicit expression for \(K(1,2)\). To this end we seek the solution of equation (60) in the form of the Fourier integral

\[ K(1,2)=\int K_f e^{-i f x_1}\,d^4 f \tag{62} \]

and, since the right-hand side of the equation is

\[ i\delta(x_1-x_2)\delta_{a_1a_2} = \frac{i\delta_{a_1a_2}}{(2\pi)^4} \int e^{i f(x_2-x_1)}\,d^4 f, \]

we have

\[ (\gamma_i f_i-m)K_f = \frac{i\delta_{a_1a_2}}{(2\pi)^4}e^{i f x_2}. \tag{62a} \]

Multiplying this equality on the left by

\[ (\gamma f+m), \]

we obtain, in view of the fact that

\[ (f\gamma+m)(f\gamma-m)=f^2-m^2, \]

\[ K_f= \frac{i}{(2\pi)^4}\frac{\gamma f+m}{f^2-m^2}e^{i f x_2}, \tag{62b} \]

i.e.

\[ (K_f)_{a_1a_2} = \frac{i}{(2\pi)^4} \frac{e^{i f x_2}}{f^2-m^2} (\gamma f+m)_{a_1a_2}. \]

Substituting in (62), we obtain:

\[ K(1,2)= \frac{i}{(2\pi)^4} \int \frac{(\gamma f+m)}{f^2-m^2} e^{-i f(x_1-x_2)}\,d^4 f. \tag{63} \]

This may also be rewritten in the form

\[ K(1,2)= -i\frac{1}{4\pi} (\gamma_i p_i+m)\Delta(x_1x_2), \tag{64} \]

where

\[ \Delta(x_1x_2)= \frac{i}{4\pi^3} \int \frac{e^{-i f(x_1-x_2)}}{f^2-m^2}\,d^4 f \tag{64a} \]

passes into expression (49) for \(D(x_1x_2)\) when \(m=0\). The subintegral expressions (63) and (64a) contain two poles:

\[ f_0=\pm\sqrt{p^2-m^2}. \]

The contour of integration coincides with the contour in Fig. 3, \(a\). It may be taken as coinciding with the real axis if \(m^2\) is replaced by \(m^2-i\delta\). If one defines the inverse Dirac operator by

\[ (\gamma f-m)^{-1}=\frac{\gamma f+m}{f^2-m^2}, \tag{65} \]

PERTURBATION THEORY

then (63) can also be written in the form^6

\[ K(1,2)=\frac{i}{(2\pi)^4}\int(\gamma f-m)^{-1}e^{-if(x_1-x_2)}\,d^4f . \tag{66} \]

Formula (63) can also be arrived at directly from (58). After substituting plane waves for \(\psi_f\), (58) gives

\[ K(1,2)= \begin{cases} \displaystyle \frac{1}{(2\pi)^3}\int(dp)\, e^{i\mathbf p(\mathbf r_1-\mathbf r_2)-i\sqrt{p^2+m^2}(t_1-t_2)} \times \sum_{\mu+}(u_{\mathbf p\mu+})_{\alpha_1}(\bar u_{\mathbf p\mu+})_{\alpha_2}, & (t_1>t_2),\\[1.2em] \displaystyle -\frac{1}{(2\pi)^3}\int(dp)\, e^{i\mathbf p(\mathbf r_1-\mathbf r_2)+i\sqrt{p^2+m^2}(t_1-t_2)} \times \sum_{\mu-}(u_{\mathbf p\mu-})_{\alpha_1}(\bar u_{\mathbf p\mu-})_{\alpha_2}, & (t_1<t_2), \end{cases} \tag{67} \]

where \(u_{\mathbf p\mu}\) are unit spinor amplitudes, the sign \(\sum_{\mu+}\) denotes summation over the amplitudes of positive frequencies, and \(\sum_{\mu-}\) over those of negative frequencies. For the summation one may use the fact that, since

\[ (\gamma\mathbf p\mp\gamma_0\sqrt{p^2+m^2})\,u_{\mathbf p\mu\pm}=-m u_{\mathbf p\mu\pm}, \]

\[ \bar u_{\mathbf p\mu\pm}(\gamma\mathbf p\mp\gamma_0\sqrt{p^2+m^2})=-m\bar u_{\mathbf p\mu\pm}, \]

then one may, replacing \(u_{\mathbf p\mu}\) by

\[ u_{\mathbf p\mu}= \frac{m+\gamma_0\sqrt{p^2+m^2}-\gamma\mathbf p}{2m}\,u_{\mathbf p\mu}, \tag{68a} \]

carry out the summation over the complete system of eigenfunctions of the operator

\[ \gamma\mathbf p\mp\gamma_0\sqrt{p^2+m^2}, \]

which, besides the eigenvalue \(-m\), has also the eigenvalue \(+m\). This gives:

\[ \sum_{\mu\pm}(u_{\mathbf p\mu\pm})_{\alpha_1}(\bar u_{\mathbf p\mu\pm})_{\alpha_2} = \sum_{\mu\pm}\sum_{\alpha} (\bar u_{\mathbf p\mu\pm})_{\alpha_2}(u_{\mathbf p\mu\pm})_{\alpha} \frac{\left(m\pm\gamma_0\sqrt{p^2+m^2}\mp\gamma\mathbf p\right)_{\alpha,\alpha_1}}{2m}. \tag{68b} \]

Furthermore, since

\[ u^*u=\bar u\beta u=1, \]

then

\[ \sum_{\mu \pm}(\bar u_{\mathbf p\mu\pm})_\alpha (u_{\mathbf p\mu\pm})_\sigma = \pm \frac{m}{\sqrt{p^2+m^2}}\delta_{\alpha\sigma}. \tag{68в} \]

After this

\[ K(1,2)=\frac{1}{2(2\pi)^3}\int -\frac{\gamma p+m\mp \gamma_0\sqrt{p^2+m^2}}{\sqrt{p^2+m^2}} \times \]

\[ \times e^{i\mathbf p(\mathbf r_1-\mathbf r_2)\mp i\sqrt{p^2+m^2}(t_1-t_2)}(d\mathbf p) \qquad (t_1 \gtrless t_2). \tag{69} \]

Similarly to (48),

\[ -\frac{\gamma p+m\mp \gamma_0\sqrt{p^2+m^2}}{\sqrt{p^2+m^2}} e^{-i\sqrt{p^2+m^2}|t_1-t_2|} = \]

\[ = \frac{i}{\pi}\int \frac{-\gamma p+m+\gamma_0\omega}{f^2+m^2-\omega^2} e^{-i\omega(t_1-t_2)}\,d\omega, \tag{70} \]

where the integral is taken along the contour of Fig. 3, \(d\), which leads to (63).

7. SECOND-ORDER PROCESSES

The first-order collision matrix \(\hat S_1\) contains elements corresponding to processes in which one photon state and two electron states change (instead of a change of the photon state, the action of an external field may be considered). The second-order matrix contains elements of two types: those corresponding to the change of four electron states without a change of photon states, and to the change of two electron and two photon states. The former are contained in the matrix \(\hat S_2^e\) (50a), the latter—in the matrix \(\hat S_2^\gamma\) (57a).

The essence of the method set forth in Sections 5–6 is that the collision matrix is brought to such a form that only the initial and final states occur in it. The “intermediate states” are eliminated and replaced by interaction functions.

In (50a) there will enter those terms from the product \(\hat j_i(x_1)\hat j_i(x_2)\) which contain creation or annihilation operators belonging to different states. Therefore these terms anticommute with one another, while their pairwise products commute, so that for the processes under consideration

\[ \hat j_i(x_1)\hat j_i(x_2)=\hat j_i(x_2)\hat j_i(x_1) \]

and the operator \(P\) in (50a) may be omitted:

\[ \hat S^{e}_{2} = -\frac{e^{2}}{2}\iint \hat j_i(x_1)D(x_1x_2)\hat j_i(x_2)\,d^4x_1\,d^4x_2 = \]

\[ = -\frac{e^{2}}{2}\iint \hat{\bar\psi}(1)\gamma_i\hat\psi(1)D(x_1x_2) \hat{\bar\psi}(2)\gamma_i\hat\psi(2)\,d^4x_1\,d^4x_2 . \tag{71} \]

Let \(a,b,c,d\) be the quantum numbers of the four electron states participating in the process, with \(b\) and \(d\) referring to the initial (or final positron) states, and \(a\) and \(c\) to the final (or initial positron) states. Then

\[ \hat\psi_\alpha(x_1)\hat\psi_\beta(x_1)\hat\psi_\gamma(x_2)\hat\psi_\delta(x_2) = \hat\psi_{a\alpha}(x_1)\hat\psi_{b\beta}(x_1) \hat\psi_{c\gamma}(x_2)\hat\psi_{d\delta}(x_2) + \]

\[ + \hat\psi_{c\alpha}(x_1)\hat\psi_{d\beta}(x_1) \hat\psi_{a\gamma}(x_2)\hat\psi_{b\delta}(x_2) - \hat\psi_{a\alpha}(x_1)\hat\psi_{d\beta}(x_1) \hat\psi_{c\gamma}(x_2)\hat\psi_{b\delta}(x_2) + \]

\[ + \hat\psi_{c\alpha}(x_1)\hat\psi_{b\beta}(x_1) \hat\psi_{a\gamma}(x_2)\hat\psi_{d\delta}(x_2) \tag{72} \]

or, using the anticommutation of all factors and arranging in all terms the creation and annihilation operators in the same order,

\[ \hat\psi_\alpha(x_1)\hat\psi_\beta(x_1)\hat\psi_\gamma(x_2)\hat\psi_\delta(x_2) = \hat\psi_{a\alpha}(x_1)\hat\psi_{b\beta}(x_1) \hat\psi_{c\gamma}(x_2)\hat\psi_{d\delta}(x_2) + \]

\[ + \hat\psi_{a\gamma}(x_2)\hat\psi_{b\delta}(x_2) \hat\psi_{c\alpha}(x_1)\hat\psi_{d\beta}(x_1) - \hat\psi_{a\alpha}(x_1)\hat\psi_{b\delta}(x_2) \hat\psi_{c\gamma}(x_2)\hat\psi_{d\beta}(x_1) - \]

\[ - \hat\psi_{a\gamma}(x_2)\hat\psi_{b\beta}(x_1) \hat\psi_{c\alpha}(x_2)\hat\psi_{d\delta}(x_2). \tag{72a} \]

In the last expression the second term differs from the first, and the fourth from the third, only by the change of variables \(x_1 \leftrightarrows x_2\); \(\alpha \leftrightarrows \gamma\); \(\beta \leftrightarrows \delta\). Since the function \(D(x_1x_2)\) is symmetric with respect to its arguments, upon substitution into the integral (71) the corresponding terms will give identical results.

Thus,

\[ (\ldots|S^{e}_{2}|\ldots) = -e^{3}\iint d^4x_1\,d^4x_2 \{ \bar\psi_a(1)\gamma_i\psi_b(1)D(12)\bar\psi_c(2)\gamma_i\psi_d(2) - \]

\[ - \bar\psi_a(1)\gamma_i\psi_d(1)D(12)\bar\psi_c(2)\gamma_i\psi_b(2) \}. \tag{73} \]

The matrix elements (73) can be represented graphically according to the same principle as was done in Fig. 2 for \(\hat S_1\). Since now the integral contains two variables, 1 and 2, the diagram will include two nodal points. From each of them two electron lines depart. Instead of the potentials corresponding to a photon, (73) contains the interaction function \(D(12)\) of two variables. In the diagram it is represented by a dotted line connecting the nodal points. The two terms in (73) will correspond to two diagrams describing one process. In Fig. 4 are presented

Fig. 4. a) scattering of an electron by an electron (also the Auger effect); b) scattering of a positron by a positron; c) scattering of a positron by an electron; d) creation of a pair by an electron; e) creation of a pair by a positron; f) absorption of a pair without radiation.

diagrams representing the matrix elements of the various processes described by formula (73). The processes shown in Figs. 4, г, 4, е, by virtue of the conservation laws, are impossible for free particles.

The matrix (71) also describes the interaction of two charges of different nature, for example an electron and a proton. In this case \(\hat{\jmath}(1)\) refers to one particle, and \(\hat{\jmath}(2)\) to the other; operators with different arguments commute with one another. Suppose, for example, that \(a\) and \(b\) refer to the proton, while \(c\) and \(d\) refer to the electron. In (72) and (72a) only the first two terms remain. The second two, expressing “exchange effects,” are absent because the particles are not identical. In (73) there will now remain only the first term (with changed sign, corresponding to the difference in the signs of the charges of the electron and proton)

\[ (\ldots \mid S_{2}^{e}\mid \ldots)= \]

\[ = - e^{2}\iint d^{4}x_{1}\,d^{4}x_{2}\, (\psi_{a}(1)\gamma_{i}\psi_{b}(1))\,D(1,2)\, (\psi_{c}(2)\gamma_{i}\psi_{d}(2)). \tag{73a} \]

The process of transition of a proton in the nucleus from a higher energy state to a lower one, with transfer of energy to an electron of the atomic shell or with formation of an electron–positron pair, is called internal conversion of \(\gamma\)-rays. Figure 5 shows diagrams of the processes of interaction of an electron with a proton corresponding to the matrix (73a). Proton lines, in contrast to electron lines, are represented by heavier lines.

The matrix \(S_{2}^{\gamma}\) contains one operator \(\hat{\psi}\) and \(\hat{\psi}\) each, and two operators \(\hat{A}\). Let the quantum numbers of the initial state of the electron (or final positron) be \(a\), and of the final (or initial positron) be \(b\), while the quantum numbers of the photon states participating in the process are \(c\) and \(d\). Then from the expansion of the operator \(\hat{\psi}\) we must take the term \(\hat{\psi}_{a}\), and from \(\hat{\psi}\) the term \(\hat{\psi}_{b}\). Each of the two photon operators contains operators corresponding to absorption or emission of photons in the states \(c\) and \(d\): \(\hat{A}_{c}\) and \(\hat{A}_{d}\), and these operators commute.

Fig. 5. a)—1) internal conversion of \(\gamma\)-rays on the electron shell; 2)—scattering of an electron on a proton; б)—internal conversion with pair formation.

Therefore

\[ (\cdots|S_2'|\cdots)=-e^2\iint d^4x_1\,d^4x_2\left\{\bar\psi_a(1)A'_{ai}(1)\gamma_i K(1,2)\gamma_j A'_{dj}(2)\times\right. \]
\[ \left.\times\psi_b(2)+\bar\psi_a(1)A'_{di}(1)\gamma_iK(1,2)\gamma_j A'_{cj}(2)\psi_b(2)\right\}. \tag{74} \]

Here the vector potentials of the photon states are \(A'_c=A_c\), if \(c\) is the initial state, and \(A'_d=A^*_d\), if \(c\) is the final state (the same for \(A_d\)). Formula (74) is also valid for processes involving an external field. In this case \(A'_e\) or \(A'_d\) is the external potential \(A^{(e)}\).

The matrix elements (74) are likewise represented by diagrams\(^6\) containing two vertex points, corresponding to two variables. From each point there issues one electron line and one photon line (or a line of the external field). The interaction function \(K(12)\) is represented by a solid line connecting the two vertex points. The two terms in (74) correspond to two diagrams, the totality of which represents the given process. Figure 6 shows diagrams of the processes described by the matrix (74).

Let us note that the matrix elements with an external field (Fig. 6, \(в\)—\(д\)) refer to the same processes as were contained in the first-order matrix \(\hat S_1\) (Section 1, Fig. 2, \(a\), \(б\), \(ж\), \(з\)). The difference is that the external field is included in the electron wave functions (in the absence of an external field, as was noted in Section 4, these matrix elements vanish), whereas here it is considered as a perturbation. Let us compare, for example, the matrix elements of diagrams 6, \(в\), and 2, \(a\). For the first of (74) we obtain \((b=f;\ a=f';\ e=q)\):

\[ (qf'|S'_2|f)=-e^2\iint \bar\psi_{f'}(1)\left[\gamma_iA_{qi}(1)K(12)\gamma_jA^{(e)}_j(2)+\right. \]
\[ \left.+A^{(e)}_i(1)\gamma_iK(12)\gamma_jA^*_{qj}(2)\right]\psi_f(2)\,d^4x_1\,d^4x_2, \tag{75} \]

for the second (36)

\[ (qf'|S_1|f)=-ie\int \bar\psi'_{f'}(1)A^*_{qi}(1)\gamma_i\psi'_f(1)\,d^4x_1. \tag{75a} \]

In (75a) the electron wave functions are supplied with primes in order to mark the circumstance that they take the external field into account; in (75) the wave functions of a free electron enter. Let

\[ \psi'_f=\psi_f+\psi^{(e)}_f, \tag{76} \]

where \(\psi^{(e)}_f\) is that part of the electron wave function which differs

Fig. 6. a) scattering of a photon by an electron (Compton effect); б) scattering of a photon by a positron; в) emission of an electron in an external field (the same for a positron, with the direction of the arrows reversed and with \(a\) replaced by \(b\)); г) absorption of a photon by an electron in an external field; д) creation of a pair by a photon in an external field; е) transformation of a pair into one photon in an external field; ж) transformation of a pair into two photons; з) creation of a pair by two photons.

it from the wave function of a free particle. Then, substituting (76) into (75a) and comparing with (75), we can obtain \(\psi_f^{(e)}\) in the first approximation with respect to the external field. Taking into account that (75b) goes into 0 when \(\bar\psi_{f'}\) is replaced by \(\bar\psi_f\), and \(\psi_{f'}\) by \(\psi_f\), we find\(^6\):

\[ \psi_f^{(e)}(2)=-ie\int K(12)\gamma_i A_i^{(e)}(2)\psi_f(2)\,d^4x_2 \tag{77} \]

and, analogously,

\[ \bar\psi_f^{(e)}(1)=-ie\int \bar\psi_f(2)\gamma_i A_i^{(e)}(2)K(21)\,d^4x_2. \]

8. EXAMPLES

Let us briefly analyze, as an example of the application of formula (73), the collision matrix of two electrons. Consider the collision of two free electrons. Let \(f_b\), \(f_d\) be the 4-momenta of the initial states, and \(f_a\) and \(f_c\) the final ones.

Their wave functions are

\[ \psi_a=\frac{1}{(2\pi)^{3/2}}\,u_a e^{if_a x}, \]

where \(u_a\) is the spinor amplitude; analogously for all the other states. For the interaction function \(D\) we shall use expression (49). Then (73) gives:

\[ \begin{aligned} (f_a f_c|S_2^e|f_b f_d) &= \\ &= -\frac{ie^2}{4\pi^3(2\pi)^3}\int \frac{d^4q}{q^2} \Bigg\{ (\bar u_a\gamma_i u_b)(\bar u_c\gamma_i u_d) \int e^{-i(f_b-f_a+q)x_1}\,d^4x_1 \times \\ &\quad \times \int e^{-i(f_d-f_c+q)x_2}\,d^4x_2 -(\bar u_a\gamma_i u_d)(\bar u_c\gamma_i u_b)\times \\ &\quad \times \int e^{-i(f_d-f_a+q)x_1}\,d^4x_1 \int e^{-i(f_b-f_c-q)x_2}\,d^4x_2 \Bigg\}. \end{aligned} \tag{78} \]

We first integrate over the coordinates

\[ \int e^{-i(f_b-f_a+q)x_1}\,d^4x_1=(2\pi)^4\delta(f_b-f_a+q) \]

(and analogously the others), and then over \(q\). We obtain:

\[ (f_a f_c|S_2^e|f_b f_d) =\frac{ie^2}{\pi} \left\{ \frac{(\bar u_a\gamma_i u_b)(\bar u_c\gamma_i u_d)}{(f_a-f_b)^2} - \frac{(\bar u_a\gamma_i u_d)(\bar u_c\gamma_i u_b)}{(f_a-f_d)^2} \right\} \times \delta(f_a+f_c-f_b-f_d). \tag{79} \]

If the spatial component of the 4-momentum (momentum) is \(\mathbf p\), and the temporal one (energy) is \(\varepsilon\), then

\[ \delta(f_a+f_c-f_b-f_d) = \delta(\mathbf p_a+\mathbf p_c-\mathbf p_b-\mathbf p_d)\, \delta(\varepsilon_a+\varepsilon_c-\varepsilon_b-\varepsilon_d). \]

According to (22), the matrix element of the effective perturbation energy is

\[ \langle f_a f_c|U|f_b f_d\rangle = \frac{e^2}{2\pi^2} \left\{ \frac{(\bar u_a\gamma_i u_b)(\bar u_c\gamma_i u_d)} {(\mathbf p_a-\mathbf p_b)^2-(\varepsilon_a-\varepsilon_b)^2} - \frac{(\bar u_a\gamma_i u_d)(\bar u_c\gamma_i u_b)} {(\mathbf p_a-\mathbf p_d)^2-(\varepsilon_a-\varepsilon_d)^2} \right\} \times \delta(\mathbf p_a+\mathbf p_c-\mathbf p_b-\mathbf p_d). \tag{80} \]

Formula (80) makes it possible, by the usual rules, to determine the effective cross section for electron–electron scattering (Møller’s formula). The same formula determines the scattering of a positron by an electron, if, as before, \(\mathbf p_b\) is the initial and \(\mathbf p_a\) the final momentum of the electron, while \(-\mathbf p_c\) is the initial momentum of the positron, and \(-\mathbf p_d\) its final momentum (\(\mathbf p_c\) and \(\mathbf p_d\) are the momenta of the corresponding electron states with negative frequencies).

Let us now transform (73) into a somewhat different form. Let

\[ \psi_a(x)=\psi_a(\mathbf r)e^{-i\varepsilon_a t} \]

(and analogously for the other states). \(\psi_a\) may refer to the state of an electron in an arbitrary external field, \(\varepsilon_a\) being the energy of the corresponding stationary state.

\[ \begin{aligned} (ac\, S_2|bd) &= \frac{ie^2}{4\pi^3} \iint (\bar\psi_a(\mathbf r_1)\gamma_i\psi_b(\mathbf r_2)) (\bar\psi_c(\mathbf r_2)\gamma_i\psi_d(\mathbf r_2)) (d\mathbf r_1)(d\mathbf r_2) \times \\ &\quad \times \int(d\mathbf k)e^{i\mathbf k(\mathbf r_1-\mathbf r_2)} \int\frac{d\omega}{k^2-\omega^2} \int e^{i(\varepsilon_a-\varepsilon_b-\omega)t_1}\,dt_1 \int e^{i(\varepsilon_c-\varepsilon_d+\omega)t_2}\,dt_2 \\ &\quad - \frac{ie^2}{4\pi^3} \iint (\bar\psi_a(\mathbf r_1)\gamma_i\psi_d(\mathbf r_1)) (\bar\psi_c(\mathbf r_2)\gamma_i\psi_b(\mathbf r_2)) (d\mathbf r_1)(d\mathbf r_2) \int(d\mathbf k)e^{i\mathbf k(\mathbf r_1-\mathbf r_2)} \times \\ &\quad \times \int\frac{d\omega}{k^2-\omega^2} \int e^{i(\varepsilon_a-\varepsilon_d-\omega)t_1}\,dt_1 \int e^{i(\varepsilon_c-\varepsilon_b+\omega)t_2}\,dt_2 . \tag{81} \end{aligned} \]

Let us first carry out the integration over \(t_1\) and \(t_2\):

\[ \int e^{i(\varepsilon_a-\varepsilon_b-\omega)t_1}\,dt_1 = 2\pi\delta(\varepsilon_a-\varepsilon_b-\omega). \]

Then we integrate with respect to \(\omega\):

\[ \int \delta(\varepsilon_a-\varepsilon_b-\omega) \delta(\varepsilon_c-\varepsilon_d+\omega) \frac{d\omega}{k^2-\omega^2} = \]

\[ = \frac{1}{k^2-(\varepsilon_a-\varepsilon_b)^2} \delta(\varepsilon_a+\varepsilon_c-\varepsilon_b-\varepsilon_d). \]

V. B. Berestetskii

and, finally, over \(\mathbf{k}\):

\[ \int \frac{e^{i\mathbf{k}(\mathbf{r}_1-\mathbf{r}_2)}}{k^2-(\varepsilon_a-\varepsilon_b)^2}\,(d\mathbf{k}) = 4\pi \int_0^\infty \frac{\sin k|\mathbf{r}_1-\mathbf{r}_2|\cdot k\,dk}{k^2-(\varepsilon_a-\varepsilon_b)^2} = 2\pi^3 \frac{e^{\,i|\varepsilon_a-\varepsilon_b|\,|\mathbf{r}_1-\mathbf{r}_2|}}{|\mathbf{r}_1-\mathbf{r}_2|}. \]

In the last integral the pole of the integrand

\[ k=|\varepsilon_a-\varepsilon_b| \]

is bypassed according to the rule of Fig. 2. Substituting the results of these integrations and passing, according to (22), to the matrix element of the effective perturbation energy, we obtain:

\[ (ac|U|bd) = e^2 \iint (d\mathbf{r}_1)(d\mathbf{r}_2) \left\{ [\bar{\varphi}_a(\mathbf{r}_1)\gamma_i\varphi_b(\mathbf{r}_1)] [\bar{\varphi}_c(\mathbf{r}_2)\gamma_i\varphi_d(\mathbf{r}_2)] \frac{e^{\,i|\varepsilon_a-\varepsilon_b|\,|\mathbf{r}_1-\mathbf{r}_2|}}{|\mathbf{r}_1-\mathbf{r}_2|} - [\bar{\varphi}_a(\mathbf{r}_1)\gamma_i\varphi_d(\mathbf{r}_1)] [\bar{\varphi}_b(\mathbf{r}_2)\gamma_i\varphi_c(\mathbf{r}_2)] \frac{e^{\,i|\varepsilon_a-\varepsilon_d|\,|\mathbf{r}_1-\mathbf{r}_2|}}{|\mathbf{r}_1-\mathbf{r}_2|} \right\}. \tag{82} \]

Expression (82) has the simple meaning of the retarded interaction of two charges. The first term in (82) may be written as

\[ \int (j_i(\mathbf{r}_1))_{ab}\,(A_i(\mathbf{r}_1))_{cd}\,(d\mathbf{r}_1), \tag{82a} \]

where \((j_i)_{ab}=e\bar{\varphi}_a\gamma_i\varphi_b\) is the density of the “transition current,” and \((A_i)_{cd}\) are the retarded potentials created by the second electron:

\[ (A_i(\mathbf{r}_1))_{cd} = e\int \frac{(j_i(\mathbf{r}_2))_{cd}}{|\mathbf{r}_1-\mathbf{r}_2|} e^{ik|\mathbf{r}_1-\mathbf{r}_2|} (d\mathbf{r}_2), \tag{82б} \]

\[ (k=|\varepsilon_c-\varepsilon_d|). \]

The presence of the “retardation factor” \(e^{\,i|\varepsilon_a-\varepsilon_b|\,|\mathbf{r}_1-\mathbf{r}_2|}\), explicitly containing the initial and final states of the electrons, does not allow one to introduce the “true” operator of the interaction energy, i.e., an operator acting on the spatial and spin variables of the electron functions whose matrix element would be (82). For small particle velocities, however, such an operator exists with accuracy up to \(v^2/c^2\). It can be obtained by expanding in (82) the retardation factor into a series (Breit formula). The principal term of the operator is the Coulomb term

\[ \frac{e^2}{|\mathbf{r}_1-\mathbf{r}_2|}. \]

Formula (82) also applies to the interaction of an electron with a positron. In this case \(c\) is the final state, and \(d\) the initial state of the positron, while \(-\varepsilon_c = |\varepsilon_c|\) is its energy (the same for \(\varepsilon_d\)). (The change of sign of the interaction will be obtained automatically in view of the fact that now \(\hat{\psi}_c\) contains a creation operator, and \(\hat{\psi}_a\) an absorption operator, which anticommute.) The second term in (82) expresses their exchange (i.e. connected with the possibility of annihilation; see the second diagram, Fig. 4, в) interaction. In the approximation to accuracy up to \(\dfrac{v^2}{c^2}\)

\[ \varepsilon_a - \varepsilon_d = 2m \]

and the operator of the exchange interaction of an electron with a positron is

\[ \frac{e^{\,2 i |\mathbf r_1-\mathbf r_2|/\lambda_0}}{|\mathbf r_1-\mathbf r_2|} \]

\[ \left(\lambda_0=\frac{1}{m}\text{ is the Compton wavelength of the electron}\right). \]

Formula (82) (if in it, as indicated in Section 7, the second term is omitted, correspondingly (73a)) describes the interaction of an electron with a proton.

The matrix element

\[ (aA|U|bB)=e^2\int \bar{\psi}_A(\mathbf r_1)\gamma_i\psi_B(\mathbf r_1)(d\mathbf r_1)\times \]

\[ \times \int \bar{\psi}_a(\mathbf r_2)\gamma_i\psi_b(\mathbf r_2) \frac{e^{\,i|\varepsilon_a-\varepsilon_b|\,|\mathbf r_1-\mathbf r_2|}}{|\mathbf r_1-\mathbf r_2|}(d\mathbf r_2), \tag{82a} \]

where \(A\) and \(B\) refer to the proton, and \(a\) and \(b\) to the electron, is the starting point for the theory of internal conversion of \(\gamma\)-rays and the theory of excitation of nuclei by electrons.

To illustrate the application of the interaction function of the electron and the photon, let us consider the matrix element for the process of scattering of a photon by a free electron. Let the wave function of the initial state of the electron be:

\[ \frac{u^0}{(2\pi)^{3/2}} e^{i f^0 x} \]

and of the final state:

\[ \frac{u'}{(2\pi)^{3/2}} e^{i f' x}; \]

the vector potential of the initial state:

\[ \frac{e_i^0}{2\pi} e^{i q^0 x} \]

and the final one:

\[ \frac{e'_i}{2\pi} e^{iq'x}. \]

Then, according to (57a), (52), and (63),

\[ \begin{aligned} (f'q'|S'_2|f^0q^0) &= \frac{-ie^2}{(2\pi)^4(2\pi)^3(2\pi)^2} \int d^4 f \Biggl[ \frac{u'e'_j\gamma_i(\gamma f+m)\gamma_j e_j^0 u^0}{f^2-m^2} \\ &\qquad{}\times \int e^{i(f^0-q'+f)x_1}\,d^4x_1 \int e^{i(-f'+q^0-f)x_2}\,d^4x_2 \\ &\qquad{}+ \frac{u'e_i^0\gamma_i(\gamma f+m)\gamma_j e'_j u^0}{f^2-m^2} \int e^{i(f^0+q^0+f)x_1}\,d^4x_1 \\ &\qquad{}\times \int e^{i(-f'-q'-f)x_2}\,d^4x_2 \Biggr] \end{aligned} \tag{83} \]

or

\[ (f'q'|U|f^0q^0) = \frac{e^2}{(2\pi)^2} \left\{ \frac{ u'e'_i\gamma_i\,[\gamma(q'-f^0)+m]\gamma_j e_j^0 u^0 }{ (q'-f^0)^2-m^2 } + \frac{ u'e_i^0\gamma_i\,[\gamma(q'+f')+m]\gamma_j e'_j u^0 }{ (q'+f')^2-m^2 } \right\} \delta(p^0+k^0-p'-k'). \tag{84} \]

9. PROCESSES OF HIGHER ORDERS

Finding the matrix elements of higher orders for those processes for which the given order is the first one not vanishing in the zeroth approximation requires, in principle, nothing new in comparison with the preceding. The subintegral expression in the case of a matrix of the \(n\)-th order will contain, according to (19) and (24), \(2n\) electron operators (\(n\) operators \(\hat\psi\) and \(n\) operators \(\hat\psi^{+}\)) and \(n\) photon operators (some of them may be replaced by external potentials) depending on the variables \(x_1\ldots x_n\). Correspondingly, this matrix element is represented by diagrams consisting of \(n\) vertex points, at each of which two solid lines and one dotted line intersect.

Of the \(2n\) electron operators, some number \(k\) of pairs \((\hat\psi \text{ and } \hat\psi^{+})\) corresponds to the initial and final states (directed rays entering or leaving the edges of the diagram). Similarly, some number of photon operators corresponds to the initial or final states of photons (dotted lines at the con-

...of the diagram; the external potentials also belong to this category). The remaining electronic and photon operators may be combined into pairs corresponding to the emission and absorption of an electron (or photon) in arbitrary states, over which summation is performed (“vacuum operators”). These pairs are represented in the diagram by a solid (for electrons) or dashed (for photons) line connecting two vertices. Examples of diagrams of higher-order processes are given in Fig. 7. The structure of the diagram will always be such that it contains no “closed loops,” i.e., such sequences of pairs of electron operators that begin and end at one and the same variable. The presence of such a loop indicates that, for the process under consideration, there are nonzero elements in the collision matrix of lower order. An exception is the case where electrons are absent both in the initial and in the final states (for example, photon–photon scattering). A substantial simplification in calculating the matrix element arises from the fact that all electronic operators entering its expression, except for operators entering one pair, may be regarded as anticommuting. Indeed, all operators corresponding to initial or final states (the ends of the diagram) refer to different states and therefore anticommute. (We shall not consider the case where some electron does not change its state in the given process, so as not to burden the exposition; its treatment presents no fundamental differences.)

Next let us consider the products of an electron operator and a pair of “vacuum” operators

\[ \sum_f \hat{\bar{\psi}}_f \hat{\psi}_f: \]

\[ \hat{\psi}_a \sum_f \hat{\bar{\psi}}_f \hat{\psi}_f = -\sum_f \hat{\bar{\psi}}_f \hat{\psi}_a \hat{\psi}_f + \psi_a \bar{\psi}_a \hat{\psi}_a . \]

Here the term standing next to the sum should be neglected. (If the normalization of the wave functions is taken into account, the first term contains the integral \(\int \bar{\psi}_p \psi_p (dp)\), whereas the second is the differential without the integral \(\bar{\psi}_a\psi_a(dp_a)\to 0\).)

Let us consider electron operators arranged in the following order:

\[ \left\{ \hat{\psi}_a(1) \left[ \hat{\psi}(1)\hat{\bar{\psi}}(2) \right] \left[ \hat{\psi}(2)\cdots \right]\cdots \left[ \cdots \hat{\bar{\psi}}(s) \right] \hat{\psi}_b(s) \right\} \left\{ \hat{\bar{\psi}}_c(s+1)\times \right. \]

\[ \left. \times \hat{\psi}(s+1)\ldots \psi_d(\ldots) \right\}\ldots, \tag{85} \]

Fig. 7. a) transformation of one photon into two upon collision with an electron; b) transformation of a pair into three photons; c) radiation upon collision of an electron with an electron; d) pair production upon collision of a photon with an electron; e) pair production upon collision of two electrons; f) scattering of a photon by a photon.

where \(a, b, c, d \ldots\) are the quantum numbers of the initial or final states, and the pairs of operators enclosed in square brackets,

\[ [\hat{\psi}(1)\hat{\psi}(2)] = \sum_f \hat{\psi}_f(1)\hat{\psi}_f(2) \]

and similarly for the other pairs (spinor indices are omitted everywhere). To each “chain” of operators placed in braces in (85)

\[ \left( \text{from } \hat{\psi}_a \text{ to } \hat{\psi}_b,\ \text{from } \hat{\psi}_c \text{ to } \hat{\psi}_d,\ \text{etc.} \right), \]

there corresponds a continuous solid line joining the “entrance” and “exit” of the diagram. The given matrix element may contain a number of terms corresponding to different numbers of pairs of “vacuum” operators between the operators \(\hat{\psi}_a\) and \(\hat{\psi}_b\), etc. (different numbers of links in the chains for a given total number of chains and links). Moreover, alongside each term of type (78) there are terms differing by the interchange of \(\hat{\psi}_b\) with \(\hat{\psi}_a\), etc.

Each chain consists of an even number of operators, and operators belonging to different chains anticommute. Therefore different chains commute with one another:

\[ P\!\left[\left(\hat{\psi}_a(1)\ldots \hat{\psi}_b(s)\right) \left(\hat{\psi}_c(s+1)\ldots \hat{\psi}_d(\ldots)\right)\ldots\right] = \]

\[ = P[\hat{\psi}_a(1)\ldots \hat{\psi}_b(s)]\, P[\hat{\psi}_c(s+1)\ldots \hat{\psi}_d(\ldots)]\,P[\ldots]. \tag{86} \]

Next, considering the various possible sequences of times, it is not hard to see that

\[ P\{\hat{\psi}_a(1)\hat{\psi}(1)\hat{\psi}(2)\ldots \hat{\psi}_b(s)\} = \]

\[ = \hat{\psi}_a(1)\, P'[\hat{\psi}(1)\hat{\psi}(2)]\, P'[\hat{\psi}(2)\hat{\psi}(3)]\ldots]\, \hat{\psi}_b(s). \tag{87} \]

In calculating the matrix element, from each factor \(P'[\ldots]\) there will enter its vacuum expectation value (56). Thus we arrive at the conclusion that to every solid line joining two vertex points of the diagram one must assign a factor \(K(1,2)\) in the corresponding term of the integrand expression of the matrix element.

Similarly it is easy to obtain that to every dotted line joining two points of the diagram there corresponds a factor \(D(1,2)\).

Indeed, all photon operators, except those entering in pairs, may be regarded as commuting (analogously to what was said above about the anticommutation of electron operators). Therefore, if

\(a'\), \(b'\), \(c'\), … are the quantum numbers of the initial or final states, then

\[ P\left\{ A_{a'} A_{b'} \ldots \left[\hat A \hat A\right]\left[\hat A \hat A\right]\right\} = A_{a'} A_{b'} \ldots P\left[\hat A \hat A\right] P\left[\hat A \hat A\right]\ldots \]

(all factors \(A(x_k)\) have different arguments). The preliminary construction of the diagram is practically very useful when considering processes of high order.

For illustration let us consider the transformation of one photon into two in a collision with a free electron (Fig. 7, \(a\)). This is a third-order process. According to (21),

\[ \hat S_3 = \frac{i e^3}{6} \int P\left[\hat j_i(1)\hat j_j(2)\hat j_k(3)\right] P\left[\hat A_i(1)\hat A_j(2)\hat A_k(3)\right] \times d^4x_1\, d^4x_2\, d^4x_3 . \tag{88} \]

We have one initial and two final photon states. To each of them there corresponds one of the three photon operators. (The diagram has no dotted lines connecting its points. Let us note that such lines can connect only two different electron chains if the matrix element under consideration is a first approximation.) Therefore the operator \(P\) before the photon operators may be omitted. Next, we have one initial (\(f\)) and one final (\(f'\)) electron state; the diagram consists of one chain. The operator \(\hat{\psi}_f\) may have any one of the three arguments \((x_1x_2x_3)\). For a given argument, \(\hat{\psi}_{f'}\) may have one of two arguments not coinciding with the argument of \(\hat{\psi}_f\) (if the arguments of \(\hat{\psi}_f\) and \(\hat{\psi}_{f'}\) are the same, then the integral

\[ \int \bar\psi_{f'}(1)\psi_f(1) A(1)\, d^4x_1, \]

which is equal to 0 according to Section 4, arises). Thus, we have 6 terms, differing only in the naming of the arguments. (81) can be rewritten in the following form:

\[ \hat S_3 = i e^3 \int P\left[ \hat{\bar\psi}_{f'}(1)\gamma_i\hat\psi(1) \hat{\bar\psi}(2)\gamma_j\hat\psi(2) \hat{\bar\psi}(3)\gamma_k\hat\psi_f(3) \right] \times \hat A_i(1)\hat A_j(2)\hat A_k(3)\, d^4x_1\, d^4x_2\, d^4x_2 \tag{88a} \]

or, according to (87) and (56),

\[ (q_1 q_2 f'|S_3|fq) = i e^3 \sum \int \bar\psi_{f'}(1) A'_{a i}(1)\gamma_i K(12) \gamma_j A'_{b j}(2)K(23) \times \gamma_k A'_{c k}(3)\psi_f(3)\, d^4x_1\, d^4x_2\, d^4x_3 . \tag{88б} \]

Here \(a\), \(b\), and \(c\) are one of the three quantum numbers \(q, q_1, q_2\), summed over in their various combinations; \(A'_q = A_q\); \(A'_{q_1}=A^*_{q_1}\); \(A'_{q_2}=A^*_{q_2}\).

The structure of the matrix element of an \(n\)-th order process is in the general case of a similar character. The matrix element consists of a series of terms, each of which corresponds to a definite diagram and is a \(4n\)-fold integral. The integrand is the product of wave functions of the initial and final electron states, potentials of the initial and final photon states, and the functions \(K\) and \(D\). Since in the case of free electron states \(K\) and \(D\) are explicitly expressed in the form of Fourier integrals, the integration over coordinates can be carried out (transition to the momentum representation). Then each term is a \(4n\)-fold integral over momenta. Each wave function of the initial and final electron states (the “entrances” and “exits” of the diagram) must then be replaced by their spinor amplitudes, respectively the photon ones by polarization vectors, the function \(K\) by \((\gamma f-m)^{-1}\), the function \(D\) by \(\dfrac{1}{q^2}\), and for each nodal point one must introduce the factor \(\delta(f_1-f_2-q)\), where \(f_1\), \(f_2\), and \(q\) are the 4-momenta corresponding to the lines converging at the given node. Owing to these \(\delta\)-functions, the matrix element in fact assumes a very simple form.

10. SECOND APPROXIMATION

The methods set forth in the preceding sections make it possible to write expressions for matrix elements in the following approximations (radiative corrections). Since the transition from the matrix \(S_n\) to \(S_{n+1}\) adds one operator \(\hat A\), the second approximation differs from the first by two orders (for the numbers of photons in the initial and final states remain the same).

The motion of a free electron is a process of zeroth order, to which the unit matrix corresponds

\[ \hat S_0 = 1. \]

However, the electron interacts with the electromagnetic field, and the matrix elements corresponding to the motion of a free electron are also contained in the second-order matrix

\[ \hat S_2 = -\frac{e^2}{2} \int \hat P\!\left[\hat j_i(1)\hat j_j(2)\right] P\!\left[\hat A_i(1)\hat{\bar A}_j(2)\right] \, d^4x_1\, d^4x_2 . \]

Let \(f\) and \(f'\) be the initial and final 4-momenta of the electron (there is no external field).

Then

\[ (f' \mid S_2 \mid f)=(f' \mid S_2^{\mathrm{self}} \mid f), \]

where

\[ \hat S_2^{\mathrm{self}} = -e^2 \int \hat{\bar\psi}_{f'}(1)\,D(12)\,\gamma_i K(12)\gamma_i\,\hat\psi_f(2)\,d^4x_1\,d^4x_2 . \tag{89} \]

\(D(12)\gamma_i K(12)\gamma_i\) is the function of the interaction of the electron with itself. Matrix (89) corresponds to the diagram shown in Fig. 8, a). Substituting the explicit expressions for \(\psi\), \(D\), and \(K\), we obtain (integrating over the coordinates)\(^{6}\):

Fig. 8. a) electron self-energy;
b) corrections to scattering in an external field;
c) corrections to scattering. The effect of “vacuum polarization.”

\[ (f' \mid U \mid f) = \frac{i e^2}{4\pi^3}\,\delta(p-p')\,\bar u_{f'} \int \left\{ \gamma_i[\gamma(f-q)-m]^{-1}\gamma_i \frac{d^4q}{q^2} \right\} u_f , \tag{90} \]

where the matrix \(\hat U\) is related to \(\hat S_2\) by relation (22). It is diagonal in the electron momenta and represents the electron’s self-energy. Expression (90) contains everything that, in the old formulations of the theory, was separated into the classical electromagnetic self-energy of the electron, the “transverse” energy associated with the virtual emission and absorption of photons, and the corrections to the latter connected with allowance for the “background” of negative-energy electrons.

A most important question is that of radiative corrections to the scattering of an electron in an external field. The first approximation is a first-order process (formula (36b), Fig. 2,d). The second approximation will be a matrix of third order. The corresponding diagrams are shown in Figs. 7,b and 7,c. The diagram in Fig. 7,c contains a “closed loop”; this part of the matrix element is connected with the “polarization of the vacuum,” i.e., with the virtual creation of pairs by the external field. To the diagram with a loop there corresponds the following arrangement of operators in the matrix \(\hat S_3\):

\[ P\left[ \hat{\bar\psi}_{f'}(1)\hat\psi_f(1)\hat{\bar\psi}(2)\hat\psi(2)\hat{\bar\psi}(3)\hat\psi(3) \right] P\left[ \hat A(1)\hat A(2) \right]A^{(e)}(3) = \]

\[ = -\hat{\bar\psi}_{f'}(1)\hat\psi_f(1) P'\left(\hat{\bar\psi}(2)\hat\psi(3)\right) P'\left[ \hat{\bar\psi}(3)\hat\psi(2) \right] \times \]

\[ \times P\left[ \hat A(1)\hat A(2) \right]A^{(e)}(3), \]

which gives the following term in the matrix element \((f'|S_3|f)\):

\[ -\int \bar\psi_f(1)\gamma_i\psi_f(1)D(12)\, Sp\!\left[K(2,3)\gamma_jK(3,2)\gamma_i\right] A_j^{(e)}(3)\,d^4x_1d^4x_2d^4x_3. \]

The remaining diagrams have the usual character (one chain).

For the matrix element of the effective perturbation energy we obtain\(^6\):

\[ (f'|U_3|f)= \frac{e^2}{4\pi^3} \left\{ \bar u_{f'} \int \frac{d^4q}{q^2} \left[ \gamma_i(f'\gamma-q\gamma-m)^{-1} \times \right. \right. \]

\[ \left. \times a_j\gamma_j(f\gamma-q\gamma-m)^{-1}\gamma_i + a_j\gamma_j(f\gamma-m)^{-1} \times \right. \]

\[ \left. \times \gamma_i(f\gamma-q\gamma-m)^{-1}\gamma_i + \gamma_i(f'\gamma-q\gamma-m)^{-1} \times \right. \]

\[ \left. \times \gamma_i(f'\gamma-m)^{-1}a_j\gamma_j \right]u_f - \frac{(\bar u_{f'}\gamma_i u_f)a_j}{(f'-f)^2} \times \]

\[ \left. \times \int Sp\!\left[(q\gamma+f'\gamma-f\gamma-m)^{-1} \gamma_j(q\gamma-m)^{-1}\gamma_i \right]d^4q \right\}, \tag{91} \]

where

\[ a_j=\frac{1}{(2\pi)^4}\int A_j^e(x)e^{i(f'-f)x}\,d^4x . \tag{91a} \]

The integrals (89) and (91) are divergent. Their regularization is not part of the problem of the present paper (see ¹).

References

1. The shift of atomic-electron levels and the additional magnetic moment of the electron according to the newest quantum electrodynamics. Collection of articles, IL, Moscow, 1950; Ya. A. Smorodinskii, UFN 39, 325 (1949).

2. J. Schwinger, Phys. Rev. 74, 1439 (1948).

3. F. J. Dyson, Phys. Rev. 75, 486 (1949) (contained in ¹).

4. S. N. Gupta, Proc. Phys. Soc. A63, 682 (1950); K. Bleuler, Helv. Phys. Acta 23, 567 (1950).

5. G. A. Zisman, ZhETF 10, 1163 (1940); 11, 631 (1941).

6. R. P. Feynman, Phys. Rev. 76, 749, 769 (1949); (abridged translations are contained in the collection Problems of Modern Physics, ser. 3, issue II, IL, Moscow, 1951).

7. W. Heitler, The Quantum Theory of Radiation, GTTI, 1940.