Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction

R.P. Feynman

(Received June 8, 1950)

In addition, a relation is established between the amplitude for a given process in an arbitrary unquantized potential and in a quantum electrodynamical field. This relation permits a simple general statement of the laws of quantum electrodynamics.

A description, in Lagrangian quantum-mechanical form, of particles satisfying the Klein-Gordon equation is given in an Appendix. It involves the use of an extra parameter analogous to proper time to describe the trajectory of the particle in four dimensions.

A second Appendix discusses in the special case of photons, the problem of finding what real processes are implied by the formula for virtual processes.

Problems of the divergences of electrodynamics are not discussed.

Introduction

In paper II it was pointed out that for many problems in electrodynamics the Hamiltonian method is not advantageous, and might be replaced by the over-all space-time point of view of a direct particle interaction. It was also mentioned that the Lagrangian form of quantum mechanics was useful in this connection. The rules given in paper II were, in fact, first deduced in this form of quantum mechanics. We shall give this derivation here.

The advantage of a Lagrangian form of quantum mechanics is that in a system with interacting parts it permits a separation of the problem such that the motion of any part can be analyzed or solved first, and the results of this solution may then be used in the solution of the motion of the other parts. This separation is especially useful in quantum electrodynamics which represents the interaction of matter with the electromagnetic field. The electromagnetic field is an especially simple system and its behavior can be analyzed completely. What we shall show is that the net effect of the field is a delayed interaction of the particles. It is possible to do this easily only if it is not necessary at the same time to analyze completely the motion of the particles. The only advantage in our problems of the form of quantum mechanics in C is to permit one to separate these aspects of the problem. There are a number of disadvantages, however, such as a lack of familiarity, the apparent (but not real) necessity for dealing with matter in non-relativistic approximation, and at times a cumbersome mathematical notation and method, as well as the fact that a great deal of useful information that is known about operators cannot be directly applied.

It is also possible to separate the field and particle aspects of a problem in a manner which uses operators and Hamiltonians in a way that is much more familiar. One abandons the notation that the order of action of operators depends on their written position on the paper and substitutes some other convention (such that the order of operators is that of the time to which they refer). The increase in manipulative facility which accompanies this change in notation makes it easier to represent and to analyze the formal problems in electrodynamics. The method requires some discussion, however, and will be described in a succeeding paper. In this paper we shall give the derivations of the formulas of II by means of the form of quantum mechanics given in C.

The problem of interaction of matter and field will be analyzed by first solving for the behavior of the field in terms of the coordinates of the matter, and finally discussing the behavior of the matter (by matter is actually meant the electrons and positrons). That is to say, we shall first eliminate the field variables from the equations of motion of the electrons and then discuss the behavior of the electrons. In this way all of the rules given in the paper II will be derived.

Actually, the straightforward elimination of the field variables will lead at first to an expression for the behavior of an arbitrary number of Dirac electrons. Since the number of electrons might be infinite, this can be used directly to find the behavior of the electrons according to hole theory by imagining that nearly all the negative energy states are occupied by electrons. But, at least in the case of motion in a fixed potential, it has been shown that this hole theory picture is equivalent to one in which a positron is represented as an electron whose space-time trajectory has had its time direction reversed. To show that this same picture may be used in quantum electrodynamics when the potentials are not fixed, a special argument is made based on a study of the relationship of quantum electrodynamics to motion in a fixed potential. Finally, it is pointed out that this relationship is quite general and might be used for a general statement of the laws of quantum electrodynamics.

Charges obeying the Klein-Gordon equation can be analyzed by a special formalism given in Appendix . A fifth parameter is used to specify the four-dimensional trajectory so that the Lagrangian form of quantum mechanics can be used. Appendix  discusses in more detail the relation of real and virtual photon emission. An equation for the propagation of a self-interacting electron is given in Appendix .

In the demonstration which follows we shall restrict ourselves temporarily to cases in which the particle’s motion is non-relativistic, but the transition of the final formulas to the relativistic case is direct, and the proof could have been kept relativistic throughout.

The transverse part of the electromagnetic field will be represented as an assemblage of independent harmonic oscillators each interacting with the particles, as suggested by Fermi. We use the notation of Heitler.

Quantum electrodynamics in Lagrangian form

(A^(t r))(x)=(((8  π))^((1/2)))(∑_K)[(Subscript[e, 1])((Subscript[q, K]^((1)))cos(K·x)+(Subscript[q, K]^((3)))sin(K·x))+(Subscript[e, 2])((Subscript[q, K]^((2)))cos(K·x)+(Subscript[q, K]^((4)))sin(K·x))]

(Subscript[H, tr])=(1/2)(∑_K^( )) (∑_(r = 1)^4)(((((Subscript[𝓅, K]^((r)))))^2)+(((k^2)((Subscript[q, K]^((r)))))^2))

(Subscript[H, c]) = (1/2) (∑_n)(∑_m) (Subscript[e, n]) 
              (Subscript[e, m])/(Subscript[r, nm])

({{(Subscript[L, p]), =(1/2) (∑_n) (Subscript[m, n]) (Subscript[x, n]^2)}, {(Subscript[L, I]), =(∑_n)(Subscript[e, n])(Subscript[(x^(.)), n])·(A^(t r))((Subscript[x, n]))}, { (Subscript[L, tr]), =(1/2)(∑_K)(∑_r)(((((Subscript[\(q\& . \), K]^((r)))))^2)-(((k^2)((Subscript[q, K]^((r)))))^2))}, {(Subscript[L, c]), \(\(=\)\(\(-\(1\/2\)\) \(∑\_n\(∑\_m\( e\_n\) 
                        e\_m/\(\(r\_\(mn\)\)\(.\)\)\)\)\)\)}})

When this Lagrangian is used in the Lagrangian forms of quantum mechanics of C, what it leads to is, of course, mathematically equivalent to the result of using the Hamiltonian H in the ordinary way, and is therefore equivalent to the more usual forms of quantum electrodynamics (at least for non-relativistic particles). We may, therefore, proceed by using this Lagrangian form of quantum electrodynamics, with the assurance that the results obtained must agree with those obtained from the more usual Hamiltonian form.

The Lagrangian enters through the statement that the functional which carries the system from one state to another is exp(i S) where

S = ∫L dt  = 
        (Subscript[S, p]) + (Subscript[S, I]) + 
          (Subscript[S, c]) + (Subscript[S, tr]).

(Subscript[S, I])=(∑_n)∫(Subscript[e, n])(Subscript[(x^(.)), n])(t)·(A^(t r))((Subscript[x, n])(t)) dt

(Subscript[S, I])=(∑_n)(∑_i)(1/2)(Subscript[e, n])((Subscript[x, n, i + 1])-(Subscript[x, n, i]))·((A^(t r))((Subscript[x, n, i + 1]))+(A^(t r))((Subscript[x, n, i])))

(Subscript[(x^(.)), n, i])=(1/2)(ϵ^(-1))((Subscript[x, n, i + 1])-(Subscript[x, n, i]))+(1/2)(ϵ^(-1))((Subscript[x, n, i])-(Subscript[x, n, i - 1])).

In the Lagrangian form it is possible to eliminate the transverse oscillators as is discussed in C, Section 13. One must specify, however, the initial and final state of all oscillators. We shall first choose the special, simple case that all oscillators are in their ground states initially and finally, so that all photons are virtual. Later we do the more general case in which real quanta are present initially or finally. We ask, then, for the amplitude for finding no quanta present and the particles in state (Subscript[χ, (t^″)]), at time (t^″), if at time (t^′) the particles were in state (Subscript[ψ, (t^′)]) and no quanta were present.

The method of eliminating field oscillators is described in Section 13 of C. We shall simply carry out the elimination here using the notation and equations of C. To do this, for simplicity, we first consider in the next section the case of a particle or a system of particles interacting with a single oscillator, rather than the entire assemblage of the electromagnetic field.

Force harmonic oscillator

\(TraditionalForm\`⟨\(χ\_\(t\^″\)\) \(φ\_m\) |1| \(ψ\_\(t\^′\)\) φ\_n⟩\_\(S\_p + S\_0 + S\_I\) = ∫\(∫\(\(χ\_\(t\^″\)\^*\)(
                x\_\(t\^″\))\) \(\(φ\_m\^*\)(
                q\_\(t\^″\))\) 
              exp\ \(i(
                S\_p + S\_0 + 
                  S\_I)\)\)\ 
\(\(\(·\)\(\(φ\_n\)(
                q\_\(t\^′\))\)\) \(\(ψ\_\(t\^′\)\)(
              x\_\(t\^′\))\) dx\_\(t\^″\) dx\_\(t\^′\) dq\_\(t\^″\) dq\_\(t\^′\) 𝒟\ \(x(
              t)\)\ 𝒟\ \(q(t)\)\)\)

(Subscript[S, I]) = ∫q(t) γ(
            t) dt,

The problem may be broken in two. The result can be written as an integral over all paths of the particles only, of (exp i (Subscript[S, p]))·
        (Subscript[G, mn]) :

\(TraditionalForm\`⟨\(χ\_\(t\^″\)\) \(φ\_m\) |1| \(ψ\_\(t\^′\)\) φ\_n⟩\_\(S\_p + S\_0 + S\_I\) = \ ⟨\(χ\_\(t\^″\)\) |G\_\(mn\)| ψ\_\(t\^′\)⟩\_\(S\_p\)\)

\(TraditionalForm\`G\_\(\(\(m\)\(\)\) \(n\)\(\ \)\) = \(⟨\(φ\_m\) |exp\ i\ \(∫\(q(t)\) \(γ(t)\) dt\)| φ\_n⟩\_\(S\_0\) = \(∫\(\(φ\_m\^*\)(q\_\(t\^″\))\) 
              exp\ i\ \((S\_0 + S\_I)\) \(\(φ\_n\)(
                q\_\(t\^′\))\) dq\_\(t\^′\) dq\_\(t\^″\) 𝒟\ \(q(
                t)\)\)
\(\(=\)\(∫\(\(φ\_m\^*\)(q\_j)\) 
                exp\ i\ ϵ\ \(\(∑\+\(i = 0\)\%\(j - 1\)\(\([\)\(\(1\/2\) \(\(ϵ\^\(-2\)\)(q\_\(i + 1\) - q\_i)\)\^2 - \(1\/2\) \(ω\^2\) 
                        q\_i\^2 + \(q\_i\) γ\_i\)\(]\)\)\)\ 
\(\(\(·\)\(\(φ\_n\)(
                        q\_0)\)\) dq\_0\ \(a\^\(-1\)\) dq\_1\ \(a\^\(-1\)\) dq\_2\ ⋯\ \(a\^\(-1\)\) dq\_j\)\)\)\)\)\)

\(TraditionalForm\`G\_\(\(\(m\)\(\)\) \(n\)\(\ \)\) = \ ∫\(\(φ\_m\^*\)(q\_j)\) \(k(q\_j, t\^″; q\_0, 
            t\^′)\) \(\(φ\_n\)(
            q\_0)\) dq\_0\ dq\_j\)

k((Subscript[q, j]), (t^″); (Subscript[q, 0]), (t^′))=(((2 π i (ω^(-1)) 
                    sin ω(
                      (t^″) - (t^′))))^(-(1/2))) 
            exp i Q((Subscript[q, j]), (t^″); (Subscript[q, 0]), (t^′))

Q=(ω/(2 sin ω(
                  (t^″) - (t^′))))[(((Subscript[q, j])^2) + ((Subscript[q, 0])^2)) 
              cos ω((t^″) - (t^′))-2  (Subscript[q, j]) (Subscript[q, 0])+\(\(\(2 
                     q\_j\)\/ω\) \(∫\_\(t\^′\)\%\(t\^″\)\(γ(t)\) 
                  sin\ \(ω(t - t\^′)\)\ dt\)\)
+\(\(\(2 
                     q\_0\)\/ω\) \(∫\_\(t\^′\)\%\(t\^″\)\(γ(t)\) 
                  sin\ \(ω(
                    t\^″ - t)\)\ dt\)\)-(2/(ω^2))(∫_((t^′))^((t^″)))\(∫\_\(t\^′\)\%t\( γ(t)\) \(γ(s)\) 
                      sin\ \(ω(
                        t\^″ - t)\)\ sin\ \(ω(
                        s - t\^′)\) ds\ dt\)]

The solution of the motion of the oscillator can now be completed by substituting () and () into () and performing the integrals. The simplest case is for m,  n=0 for which case

\(\(φ\_0\) \((q\_0)\) = \(\((ω/π)\)\^\(1\/2\)\) \(exp \((\(-\(1\/2\)\) ω\ q\_0\^2)\)\) 
        exp \((\(-\(1\/2\)\) i\ ω\ t\^′)\)\)

(Subscript[G, 00])= exp(-(1/(2 ω))(∫_((t^′))^((t^″)))(∫_((t^′))^t) 
                      exp(-i ω(t - s)) γ(
                      t) γ(
                      s) dt ds)

\(TraditionalForm\`G\_\(00\) = 
      exp(\(-\(1\/\(4\ ω\)\)\) \(∫\_\(-∞\)\%∞\(∫\_\(-∞\)\%∞\ \(exp(\(-i\)\ ω\ |t - s|)\) \(γ(
                  t)\) \(γ(
                  s)\)\ dt\ ds\)\))\)

Since (Subscript[G, 00]) is simply an exponential, we can write it as exp(i I), consider that the complete “action” for the system of particles is S = (Subscript[S, p]) + I and that one computes transition elements with this “action” instead of (Subscript[S, p]) (see C, Sec. 12). The functional I which is given by

\(TraditionalForm\`I = \(1\/4\) 
        i\ \(ω\^\(-1\)\) \(∫\(∫\(exp(\(-i\)\ ω\ |t - s|)\) \(γ(
                s)\) \(γ(
                t)\)\ ds\ dt\)\)\)

Virtual transitions in the electromagnetic field

The ((Subscript[γ, K])^((r))), corresponding to each oscillator ((Subscript[q, K])^((r))), is found from the interaction term (Subscript[L, I]) [Eq. (2b)], substituting the value of (A^(t r))(x) given in (). There results, for example,

({{((Subscript[γ, K])^((1))), =(((8 π))^((1/2)))(∑_n)(Subscript[e, n]) ((Subscript[e, 1])·(Subscript[x, n]^(.)))cos(K·(Subscript[x, n]))}, {((Subscript[γ, K])^((3))), =(((8 π))^((1/2)))(∑_n)(Subscript[e, n]) ((Subscript[e, 1])·(Subscript[x, n]^(.)))sin(K·(Subscript[x, n]))}})

The complex action resulting from oscillator of coordinate ((Subscript[q, K])^((1))), is therefore

((Subscript[I, K])^((1)))= ((8  π i)/(4 k))(∑_n)(∑_m)∫∫(Subscript[e, n])(Subscript[e, m]) exp(-i k |t - 
                            s|)×((Subscript[e, 1])·(Subscript[x, n]^(.))(t))((Subscript[e, 1])·(Subscript[x, m]^(.))(s))cos(K·(Subscript[x, n])(t))cos(K·(Subscript[x, m])(s)) ds dt

(Subscript[I, t r])= i(∑_n)(∑_m)(∫_((t^′))^((t^″)))dt\(∫\_\(t\^′\)\%\(t\^″\)\) ds∫(Subscript[e, n])(Subscript[e, m])\(exp(\(-i\)\ k\ |t - s|)\) ×[(Subscript[x, n]^(.))(t)·(Subscript[x, m]^(.))(s)-(k^(-2))(K·(Subscript[x, n]^(.))(t))(K·(Subscript[x, m]^(.))(s))]
·cos(K·((Subscript[x, n]^(.))(t)-(Subscript[x, m]^(.))(s)))(((d^3)K)/(8 (π^2) k)).

The term in (K·(Subscript[x, n]^(.))(t)) (K·(Subscript[x, m]^(.))(s)) can be simplified by integration by parts with respect to t and with respect to s [note that exp(-i k |t - 
              s|) has a discontinuous slope at t = s, or break the integration up into two regions]. One finds

(Subscript[I, t r]) = 
      R - (Subscript[I, c]) + (Subscript[I, transient])

R=-i(∑_n)(∑_m)(∫_((t^′))^((t^″)))dt\(∫\_\(t\^′\)\%\(t\^″\)\)ds∫(Subscript[e, n])(Subscript[e, m]) ×\(exp(\(-i\)\ k\ |t - s|)\)(1-(Subscript[(x^(.)), n])(t)·(Subscript[(x^(.)), m])(s))
·cos K·((Subscript[x, n])(t)-(Subscript[x, m])(s))(d^3)K/8(π^2)k

(Subscript[I, c])=-(∑_n)(∑_m)(∫_((t^′))^((t^″)))dt∫(Subscript[e, n])(Subscript[e, m])cos K·((Subscript[x, n])(t)-(Subscript[x, m])(t))(d^3)K/4(π^2)(k^2)

∫cos(K·R)(d^3)K/4(π^2)(k^2)=\(∫\_0\%∞\(\((kr)\)\^\(-1\)\) sin \((kr)\) dk/π = \((2  r)\)\^\(-1\)\)

(Subscript[I, transient])=-(∑_n)(∑_m)(Subscript[e, n])(Subscript[e, m])∫(((d^3)K)/(4 (π^2) (k^2)))×{\(∫\_\(t\^′\)\%\(t\^″\)\)[\(exp(\(-i\)\ k\ \((t\^″ - t)\))\) cos K·((Subscript[x, n])((t^″))-(Subscript[x, m])(t))
+exp\((\(-i\)\ k\ \((t - t\^′)\))\) cos K·((Subscript[x, n])(t)-(Subscript[x, m])((t^′)))]dt+(((2  k))^(-1))i[ cos K·((Subscript[x, n])((t^″))-(Subscript[x, m])((t^″)))+cos K·((Subscript[x, n])((t^′))-(Subscript[x, m])((t^′)))-2 \(exp(\(-i\)\ k\ \((t\^″ - t\^′)\))\) cos K·((Subscript[x, n])((t^′))-(Subscript[x, m])((t^″)))]}

To simplify R we need the integral

J=∫\(exp(\(-i\)\ k\ |t|)\) cos(K·R)(d^3)K/8(π^2)k=\(∫\_0\%∞\( 
                  exp(\(-i\)\ k\ |t|)\)\ \(sin(
                  kr)\) dk/2  π\ r\)

\(TraditionalForm\`∫\_0\%∞\( 
            exp(\(-i\)\ k\ x)\) dk = \(lim\+\(ϵ → 0\)\((\(-\(i(x - i\ ϵ)\)\^\(-1\)\))\) = \(\(-i\)\ x\^\(-1\) + π\ \(δ(x)\) = π\ \(\(δ\_+\)(
              x)\)\)\)\)

\(TraditionalForm\`J = \(-\((4\ π\ r)\)\^\(-1\)\)\ \((\((|t| - r)\)\^\(-1\) - \((|t| + r)\)\^\(-1\))\) + \(\((4\ i\ r)\)\^\(-1\)\) \((δ(|t| - 
                  r) - δ(|t| + 
                  r))\)\)
\(TraditionalForm\`\(\(=\)\(\(-\((2\ π)\)\^\(-1\)\) \((t\^2 - r\^2)\)\^\(-1\) + \((2\ i)\)\^\(-1\)\ \(δ(t\^2 - r\^2)\)\)\)\)
\(TraditionalForm\`\(\(=\)\(\(-\(1\/2\)\) 
        i\ \(\(δ\_+\)(t\^2 - r\^2)\)\)\)\)

\(TraditionalForm\`δ(
        t\^2 - r\^2) = \(\((2  
                r)\)\^\(-1\)\) \((δ(|t| - 
                r) + δ(|t| + 
                r))\)\)

Substitution of these results into () gives finally,

R=-(1/2)(∑_n)(∑_m)(∫_(-∞)^(+∞))(∫_(-∞)^(+∞))(Subscript[e, n])(Subscript[e, m])(1-(Subscript[(x^(.)), n])(t)·(Subscript[(x^(.)), m])(s))× \(δ\_+\)((((t - s))^2)-((((Subscript[x, n])(t)-(Subscript[x, m])(s)))^2))dt\(\(ds\)\(.\)\)

The total complex action of the system is then (Subscript[S, p]) + R. Or, what amounts to the same thing; to obtain transition amplitudes including the effects of the field we must calculate the transition element of exp(i R):

\(TraditionalForm\`⟨\(χ\_\(t\^″\)\) |exp\ i\ R| ψ\_\(t\^′\)⟩\_\(S\_p\)\)

Expression (), with (), then contains all the effects of virtual quanta on a (at least non-relativistic) system according to quantum electrodynamics. It contains the effects to all orders in (e^2)/ℏ c in a single expression. If expanded in a power series in (e^2)/ℏ c, the various terms give the expressions to the corresponding order obtained by the diagrams and methods of II. We illustrate this by an example in the next section.

Example of application of expression ()

\(TraditionalForm\`⟨\(ψ\_\(t\^″\)\) |1 + i\ R| ψ\_\(t\^′\)⟩\_\(S\_p\) = ⟨\(ψ\_\(t\^″\)\) |1| ψ\_\(t\^′\)⟩\_\(S\_p\) + 
        i ⟨\(ψ\_\(t\^″\)\) |R| ψ\_\(t\^′\)⟩\_\(S\_p\)\)

\(⟨\(ψ\_\(t\^″\)\) |R| ψ\_\(t\^′\)⟩\_\(S\_p\)\)=-(1/2)(e^2)(∫_((t^′))^((t^″)))dt (∫_((t^′))^((t^″)))ds(Subscript[⟨(Subscript[ψ, (t^″)])|(1-(Subscript[(x^(.)), t])·(Subscript[(x^(.)), s])) \(δ\_+\)((((t - s))^2)-((((Subscript[x, t])-(Subscript[x, s])))^2))|(Subscript[ψ, (t^′)])⟩, (Subscript[S, p])]).

The expression within the brackets \(TraditionalForm\`⟨\ \ ⟩\_\(S\_p\)\)on the right-hand side of () can be evaluated by the methods described in C [Eq. (29)]. An expression such as () can also be evaluated directly in terms of the propagation kernel K(2, 1) [see I, Eq. (2)] for an electron moving in the given potential.

The term (Subscript[x, t]^(.))·(Subscript[x, s]^(.)) in the non-relativistic case produces an interesting complication which does not have an analog for the relativistic case with the Dirac equation. We discuss it below, but for a moment consider, in further detail expression () but with the factor (1-(Subscript[x, s]^(.))·(Subscript[x, t]^(.))) replaced simply by unity.

The kernel K(2, 1) is defined and discussed in I. From its definition as the amplitude that the electron be found at (Subscript[x, 2]) at time (Subscript[t, 2]), if at (Subscript[t, 1]) it was at (Subscript[x, 1]), we have

K((Subscript[x, 2]),(Subscript[t, 2]);(Subscript[x, 1]),(Subscript[t, 1]))=(Subscript[⟨δ(x-(Subscript[x, 2]))(Subscript[t, 2])\(|1|\)δ(x-(Subscript[x, 1]))(Subscript[t, 1])⟩, (Subscript[S, p])])

In the integrations over all paths implied by the symbol in () we can first integrate over all the (Subscript[x, i]) variables corresponding to times (Subscript[t, i]) from (t^′) to s, not inclusive, the result being a factor K((Subscript[x, s]),s;(Subscript[x, (t^′)]),(t^′)) according to (). Next we integrate on the variables between s and t not inclusive, giving a factor K((Subscript[x, t]),t;(Subscript[x, s]),s) and finally on those between t and (t^″) giving K((Subscript[x, (t^″)]),(t^″);(Subscript[x, t]),t). Hence the left-hand term in () excluding the (Subscript[x, t]^(.))·(Subscript[x, s]^(.)) factor is

-(e^2)∫dt∫ds∫\(ψ\^*\)((Subscript[x, (t^″)]),(t^″))K((Subscript[x, (t^″)]),(t^″);(Subscript[x, t]),t)\(δ\_+\)((((t - s))^2)-((((Subscript[x, t])-(Subscript[x, s])))^2))·K((Subscript[x, t]),t;(Subscript[x, s]),s)K((Subscript[x, s]),s;(Subscript[x, (t^′)]),(t^′))
×ψ((Subscript[x, (t^′)]),(t^′))(d^3)(Subscript[x, (t^″)])(d^3)(Subscript[x, t])(d^3)(Subscript[x, s])(d^3)(Subscript[x, (t^′)])

We have made use of a special case of a principle which may be stated more generally as

(Subscript[⟨\(χ\_\(\(t\^″\) \*""""\)\)|F((Subscript[x, 1]),(Subscript[t, 1]);(Subscript[x, 2]),(Subscript[t, 2]);⋯ (Subscript[x, k]),(Subscript[t, k]))|(Subscript[ψ, (t^′)])⟩, (Subscript[S, p])])=∫\(χ\^*\)((Subscript[x, (t^″)]))K((Subscript[x, (t^″)]),(t^″);(Subscript[x, 1]),(Subscript[t, 1]))·K((Subscript[x, 1]),(Subscript[t, 1]);(Subscript[x, 2]),(Subscript[t, 2]))⋯
×K((Subscript[x, k - 1]),(Subscript[t, k - 1]);(Subscript[x, k]),(Subscript[t, k]))K((Subscript[x, k]),(Subscript[t, k]);(Subscript[x, (t^′)]),(t^′))·F((Subscript[x, 1]),(Subscript[t, 1]);(Subscript[x, 2]),(Subscript[t, 2]);⋯ (Subscript[x, k]),(Subscript[t, k]))
×ψ((Subscript[x, (t^′)]))(d^3)(Subscript[x, (t^″)])(d^3)(Subscript[x, 1])(d^3)(Subscript[x, 2])⋯ (d^3)(Subscript[x, k])(d^3)(Subscript[x, (t^′)])

Expressions of higher order arising for example from (R^2) are more complicated as there are quantities referring to several different times mixed up, but they all can be interpreted readily. One simply breaks up the ranges of integrations of the time variables into parts such that in each the order of time of each variable is definite. One then interprets each part by formula ().

As a simple example we may refer to the problem of element of the transition element

⟨(Subscript[χ, (t^″)])|∫U(x(t),t)dt∫V(x(s),s)ds| (Subscript[ψ, (t^′)])⟩

⟨(Subscript[χ, (t^″)])|∫U((Subscript[x, t]),t)dt∫V((Subscript[x, s]),s)ds| (Subscript[ψ, (t^′)])⟩=(∫_((t^′))^((t^″)))dt(∫_((t^′))^t)ds∫\(χ\^*\)((Subscript[x, (t^″)]))K((Subscript[x, (t^″)]),(t^″);(Subscript[x, t]),t)U((Subscript[x, t]),t)K((Subscript[x, t]),t;(Subscript[x, s]),s)
×V((Subscript[x, s]),s)K((Subscript[x, s]),s;(Subscript[x, (t^′)]),(t^′))ψ((Subscript[x, (t^′)]))(d^3)(Subscript[x, (t^″)]) (d^3)(Subscript[x, t])(d^3)(Subscript[x, s])(d^3)(Subscript[x, (t^′)])+(∫_((t^′))^((t^″)))dt(∫_t^((t^″)))ds∫\(χ\^*\)((Subscript[x, (t^″)]))K((Subscript[x, (t^″)]),(t^″);(Subscript[x, s]),s)
×V((Subscript[x, s]),s)K((Subscript[x, s]),s;(Subscript[x, t]),t)U((Subscript[x, t]),t)K((Subscript[x, t]),t;(Subscript[x, (t^′)]),(t^′))ψ((Subscript[x, (t^′)]))(d^3)(Subscript[x, (t^″)])(d^3)(Subscript[x, s])(d^3)(Subscript[x, t])(d^3)(Subscript[x, (t^′)])

It remains to study in more detail the term in () arising from (x^(.))(t)·(x^(.))(s) in the interaction. The interpretation of such expressions is considered in detail in C, and we must refer to Eqs. (39) through (50) of that paper for a more thorough analysis. A similar type of term also arises in the Lagrangian formulation in simpler problems, for example the transition element

⟨(Subscript[χ, (t^″)])|∫(x^(.))(t)·A(x(t),t)dt∫(x^(.))(s)·B(x(s),s)ds|(Subscript[ψ, (t^′)])⟩

(∑_i)(∑_j)[(1/2)((Subscript[x, i + 1])-(Subscript[x, i]))+(1/2)((Subscript[x, i])-(Subscript[x, i - 1]))]·A((Subscript[x, i]),(Subscript[t, i]))× [(1/2)((Subscript[x, j + 1])-(Subscript[x, j]))+(1/2)((Subscript[x, j])-(Subscript[x, j - 1]))]·B((Subscript[x, j]),(Subscript[t, j]))

((Subscript[x, i + 1]) - 
          (Subscript[x, i]))/ϵ⟶i(
          Hx - xH)

(1/(2  m))((1/i)(∂/(∂(Subscript[x, t])))·A((Subscript[x, t]),t)+A((Subscript[x, t]),t)·(1/i)(∂/(∂(Subscript[x, t]))))

(1/(2  m))((1/i)(∂/(∂(Subscript[x, s])))·B((Subscript[x, s]),s)+(1/i)B((Subscript[x, s]),s)·(∂/(∂(Subscript[x, s]))))

This is nearly but not quite correct, however, as there is an additional term coming from the terms in the sum corresponding to the special values, j = i, j = i + 1 and j =  i - 1. We have tacitly assumed from the appearance of the expression () that, for a given i, the contribution from just three such special terms is of order (ϵ^2). But this is not true. Although the expected contribution of a term like ((Subscript[x, i + 1]) - (Subscript[x, i])) ((Subscript[x, j + 1]) - 
            (Subscript[x, j])) for j ≠ i is indeed of order (ϵ^2), the expected contribution of ((((Subscript[x, i + 1]) - (Subscript[x, i])))^2) is \(TraditionalForm\`\(+i\)ϵm\^\(\(-1\)\(\ \)\)\)[C, Eq. (50)], that is, of order ϵ. In nonrelativistic mechanics the velocities are unlimited and in very short times ϵ the amplitude diffuses a distance proportional to the square root of the time. Making use of this equation then we see that the additional contribution from these terms is essentially

i(m^(-1))ϵ(∑_i)A((Subscript[x, i]),(Subscript[t, i]))·B((Subscript[x, i]),(Subscript[t, i]))=i(m^(-1))∫A(x(t),t)·B(x(t),t) dt

(i/m)(∫_((t^′))^((t^″)))dt∫\(χ\^*\)((Subscript[x, (t^″)]))K((Subscript[x, (t^″)]),(t^″);(Subscript[x, t]),t)A((Subscript[x, t]),t)·B((Subscript[x, t]),t)·K((Subscript[x, t]),t;(Subscript[x, (t^′)]),(t^′))ψ((Subscript[x, (t^′)]))(d^3)(Subscript[x, (t^″)])(d^3)(Subscript[x, t])(d^3)(Subscript[x, (t^′)]).

H=(((2  m))^(-1))(((p-A))^2)=(((2  m))^(-1))((p^2)-p·A-A·p+(A^2))

Extension to Dirac particles

\(⟨χ |1| ψ\ ⟩\)=(lim_(ϵ → 0))∫⋯∫\(χ\^*\)((Subscript[x, N])) (Subscript[Φ, p])((Subscript[x, N]),(Subscript[x, N - 1]),⋯ (Subscript[x, 0]))·ψ((Subscript[x, 0]))(d^3)(Subscript[x, 0])(d^3)(Subscript[x, 1])⋯ (d^3)(Subscript[x, N])

(Subscript[Φ, p])=(Subscript[Π, i])(A^(-1))exp iS((Subscript[x, i + 1]),(Subscript[x, i])).

(Subscript[( (Subscript[x, i + 1])|(Subscript[x, i])), ϵ])=(A^(-1))exp[iS((Subscript[x, i + 1]),(Subscript[x, i]))]

(Subscript[Φ, p])=(Subscript[Π, i])(Subscript[((Subscript[x, i + 1])|(Subscript[x, i])), ϵ])

For a Dirac electron the (Subscript[((Subscript[x, i + 1])|(Subscript[x, i])), ϵ]) is a 4×4 matrix (or (4^N)×(4^N) if we deal with N electrons) but the expression () with () is still correct (as it is in fact for any quantum-mechanical system with a sufficiently general definition of the coordinate (Subscript[x, i])). The product () now involves operators, the order in which the factors are to be taken is the order in which the terms appear in time.

For a Dirac particle in a vector and scalar potential (times the electron charge e) A(x,t), (Subscript[A, 4])(x,t), the quantity (Subscript[((Subscript[x, i + 1])|(Subscript[x, i])), ϵ]^((A))) is related to that of a free particle to the first order in ϵ as

(Subscript[((Subscript[x, i + 1])|(Subscript[x, i])), ϵ]^((A)))=(Subscript[((Subscript[x, i + 1])|(Subscript[x, i])), ϵ]^((0)))exp[-i(ϵ (Subscript[A, 4])((Subscript[x, i]),(Subscript[t, i]))-((Subscript[x, i + 1])-(Subscript[x, i]))·A((Subscript[x, i]),(Subscript[t, i])))].

Thus (Subscript[Φ, p]^((A))) differs from that for a free particle, (Subscript[Φ, p]^((0))), by a factor (Subscript[Π, i]) exp-i(ϵ (Subscript[A, 4])((Subscript[x, i]),(Subscript[t, i]))-((Subscript[x, i + 1])-(Subscript[x, i]))·A((Subscript[x, i]),(Subscript[t, i]))) which in the limit can be written as

exp{-i ∫[(Subscript[A, 4])(x(t),t)-(x^(.))(t)·A(x(t),t)]dt}

The case of a Dirac particle interacting with the quantum-mechanical oscillators representing the field may now be studied. Since the dependence of (Subscript[Φ, p]^((A))) on A, (Subscript[A, 4]) is through the same factor as in the non-relativistic case, when A, (Subscript[A, 4]) are expressed in terms of the oscillator coordinates q, the dependence of Φ on the oscillator coordinates q is unchanged. Hence the entire analysis of the preceding sections which concern the results of the integration over oscillator coordinates can be carried through unchanged and the results will be expression () with formula () for R. Expression () is now interpreted as

\(⟨\(χ\_\(t\^″\)\) |exp\ iR| ψ\_\(t\^′\)\ ⟩\)=(lim_(ϵ → 0)) ∫\(χ\^*\)((Subscript[x, (t^″)]^((1))),(Subscript[x, (t^″)]^((2)))⋯ )×(∏_n)((Subscript[Φ, p, n]^((0)))(d^3)(Subscript[x, (t^″)]^((n)))(d^3)(Subscript[x, (t^″) - ϵ]^(\((n)\)))⋯ (d^3)(Subscript[x, (t^′)]^((n)))) exp(iR)ψ((Subscript[x, (t^′)]^((1))),(Subscript[x, (t^′)]^((2)))⋯)

For calculational purposes we can, as before, expand R as a power series and evaluate the various terms in the same manner as for the non-relativistic case. In such an expansion the quantity (x^(.))(t) is replaced, as we have seen in (), by the operator i(Hx-xH), that is, in this case by α operating at the corresponding time. There is no further complicated term analogous to () arising in this case, for the expected value of ((((Subscript[x, i + 1]) - (Subscript[x, i])))^2) is now of order (ϵ^2) rather than ϵ.

For example, for self-energy one sees that expression () will be (with other terms coming from those with (x^(.))(t) replaced by α and with the usual β in back of each (Subscript[K, 0]) because of the definition of (Subscript[K, 0]) in relativity theory)

\(⟨\(ψ\_\(t\^″\)\) |R| ψ\_\(t\^′\)⟩\_\(S\_p\)\)=-(e^2)∫\(ψ\^*\)((Subscript[x, (t^″)]))(Subscript[K, 0])((Subscript[x, (t^″)]),(t^″);(Subscript[x, t]),t)(Subscript[(β  a), μ])·\(δ\_+\)((((t - s))^2)– ((((Subscript[x, t])– (Subscript[x, s])))^2))(Subscript[K, 0])((Subscript[x, t]),t;(Subscript[x, s]),s)β(Subscript[α, μ])
·(Subscript[K, 0])((Subscript[x, s]),s;(Subscript[x, (t^′)]),(t^′))βψ((Subscript[x, (t^′)]))(d^3)(Subscript[x, (t^″)])(d^3)(Subscript[x, t])(d^3)(Subscript[x, s])(d^3)(Subscript[x, (t^′)])dt\(ds\),

Extension to positron theory

On the other hand, in paper I on the theory of positrons, it was shown that the results of the hole theory in a system with a given external potential (Subscript[A, μ]) were equivalent to those of the Dirac one electron theory if one replaced the propagation kernel, (Subscript[K, 0]), by a different one, (SubPlus[K]), and multiplied the resultant amplitude by factor C\(TraditionalForm\`\_ν\) involving (Subscript[A, μ]). We must now see how this relation, derived in the case of external potentials, can also be carried over in electrodynamics to be useful in simplifying expressions involving the infinite sea of electrons.

To do this we study in greater detail the relation between a problem involving virtual photons and one involving purely external potentials. In using () we shall assume in accordance with the hole theory that the number of electrons is infinite, but that they all have the same charge, e. Let the states \(TraditionalForm\`ψ\_\(t\^′\), χ\_\(t\^″\), \) represent the vacuum plus perhaps a number of real electrons in positive energy states and perhaps also some empty negative energy states. Let us call the amplitude for the transition in an external potential (Subscript[B, μ]), but excluding virtual photons, (Subscript[T, 0])[B], a functional of (Subscript[B, μ])(1). We have seen ()

(Subscript[T, 0])[B]=⟨(Subscript[χ, (t^″)])|expiP|(Subscript[ψ, (t^′)])⟩

P=-(∑_n)∫[(Subscript[B, 4])((x^((n)))(t),t)– (x^(.(n)))(t)·B((x^((n)))(t),t)]dt

P=-(∑_n)∫(Subscript[B, μ])(((Subscript[x, v])^((n)))(t))\(\(\(x\& . \)\_μ\^\((n)\)\)(
                  t)\)dt

(Subscript[T, (e^2)])[B]=⟨(Subscript[χ, (t^″)])|exp(iR) exp(iP)|(Subscript[ψ, (t^′)])⟩.

d (Subscript[T, (e^2)])[B]/d((e^2))= ⟨(Subscript[χ, (t^″)])|—(i/2)(∑_n)(∑_m)∫∫dtds((Subscript[(OverDot[x]), μ])^((n)))(t)\(\(\(x\& . \)\_μ\^\((m)\)\)(s)\)·\(δ\_+\)((((((Subscript[x, v])^((n)))(
                                        t) -  ((Subscript[x, v])^((m)))(s)))^2)) expi(R + P)|(Subscript[ψ, (t^′)])⟩.

\(TraditionalForm\`dT\_\(e\^2\)[
              B]/δ\ \(\(B\_μ\)(
          1)\) = \(-i\) ⟨\(χ\_\(t\^″\)\) |∑\+n∫dt \(\( 
                            x\& . \)\_μ\%\((n)\)\) \(\(δ\^4\)(\(x\_α\%\((n)\)\)(t) - 
                          x\_\(α, 1\))\)·exp\ \(i(
                      R + P)\)| ψ\_\(t\^′\)⟩\)

\(\(δ\^2\) 
        T\_\(e\^2\)[
            B]/δ\ \(B\_μ\) \((1)\) δ\ \(B\_ν\) \((2)\) = \(-\(\(⟨\(χ\_\(t\^″\)\) |\ ∑\+n\(∑\+m∫dt \(\(∫ds \(\( 
                                      x\& . \)\_μ\%\((n)\)\) \((t)\) \(\(x\& . \)\_ν\%\((m)\)\) \((s)\)\)
\(\(\(·\)\(δ\^4\)\) \((\(x\_α\%\((n)\)\) \((t)\) - 
                                x\_\(α, 1\))\) \(δ\^4\) \((\(x\_β\%\((n)\)\) \((s)\) - x\_\(β, 2\))\)\ exp\ i \((R + 
                                P)\)\)\)\)| ψ\_\(t\^′\)⟩\)\(.\)\)\)\)

\(TraditionalForm\`d\ T\_\(e\^2\)[B]/\(d(e\^2)\) = \(1\/2\) 
        i \(∫\(∫\(\((\(δ\^2\) 
                    T\_\(e\^2\)[B]/δ\ \(\(B\_μ\)(
                      1)\) δ\ \(\(B\_μ\)(
                      2)\))\)×\(\(δ\_+\)(
                  s\_12\%2)\)\) dτ\_1 dτ\_2\)\)\)

We now proceed to use this equation to prove the validity of the rules given in II for electrodynamics. This we do by the following argument. The equation can be looked upon as a differential equation for (Subscript[T, (e^2)])[B]. It determines (Subscript[T, (e^2)])[B] uniquely if (Subscript[T, o])[B] is known. We have shown it is valid for the hole theory of positrons. But in I we have given formulas for calculating (Subscript[T, o])[B] whose correctness relative to the hole theory we have there demonstrated. Hence we have shown that the (Subscript[T, (e^2)])[B] obtained by solving () with the initial condition (Subscript[T, o])[B] as given by the rules in I will be equal to that given for the same problem by the second quantization theory of the Dirac matter field coupled with the quantized electromagnetic field. But it is evident (the argument is given in the next paragraph) that the rules given in II constitute a solution in power series in (e^2) of the Eq. () [which for (e^2) = 0 reduce to the (Subscript[T, 0])[B] given in I]. Hence the rules in II must give, to each order in (e^2), the matrix element for any process that would be calculated by the usual theory of second quantization of the matter and electromagnetic fields. This is what we aimed to prove.

That the rules of II represent, in a power series expansion, a solution of () is clear. For the rules there given may be stated as follows: Suppose that we have a process to order k in (e^2) (i.e., having k virtual photons) and order n in the external potential (Subscript[B, μ]). Then, the matrix element for the process with one more virtual photon and two less potentials is that obtained from the previous matrix by choosing from the n potentials a pair, say (Subscript[B, μ])(1) acting at 1 and (Subscript[B, ν])(2) acting at 2, replacing them by \(TraditionalForm\`i\ \(e\^2\) \(δ\_μν\) \(\(δ\_+\)(s\_12\%2)\)\), adding the results for each way of choosing the pair, and dividing by k + 1, the present number of photons. The matrix with no virtual photons (k = 0) being given to any n by the rules of I, this permits terms to all orders in (e^2) to be derived by recursion. It is evident that the rule in italics is that of II, and equally evident that it is a word expression of Eq. (). [The factor (1/2) in () arises since in integrating over all d(Subscript[τ, 1])and d(Subscript[τ, 2]) we count each pair twice.The division by k + 1 is required by the rules of II for, there, each diagram is to be taken only once, while in the rule given above we say what to do to add one extra virtual photon to k others. But which one of the k + 1 is to be identified at the last photon added is irrelevant. It agrees with () of course for it is canceled on differentiating with respect to (e^2) the factor ((((e^2)))^(k + 1)) for the (k + 1) photons.]

Generalized formulation of quantum electrodynamics

In earlier sections we pointed out that as a consequence of the Lagrangian form of quantum mechanics the aspects of the particles’ motions and the behavior of the field could be analyzed separately. What we did was to integrate over the field oscillator coordinates first. We could, in principle, have integrated over the particle variables first. That is, we first solve the problem with the action of the particles and their interaction with the field and then multiply by the exponential of the action of the field and integrate over all the field oscillator coordinates. (For simplicity of discussion let us put aside from detailed special consideration the questions involving the separation of the longitudinal and transverse parts of the field.) Now the integral over the particle coordinates for a given process is precisely the integral required for the analysis of the motion of the particles in an unquantized potential. With this observation we may suggest a generalization to all types of systems.

Let us suppose the formal solution for the amplitude for some given process with matter in an external potential (Subscript[B, μ])(1) is some numerical quantity (Subscript[T, 0]). We mean matter in a more general sense now, for the motion of the matter may be described by the Dirac equation, or by the Klein-Gordon equation, or may involve charged or neutral particles other than electrons and positrons in any manner whatsoever. The quantity (Subscript[T, 0]) depends of course on the potential function (Subscript[B, μ])(1); that is, it is a functional (Subscript[T, 0])[B] of this potential. We assume we have some expression for it in terms of (Subscript[B, μ]) (exact, or to some desired degree of approximation in the strength of the potential).

Then the answer (Subscript[T, (e^2)])[B] to the corresponding problem in quantum electrodynamics is (Subscript[T, 0])[(Subscript[A, μ])(1) + (Subscript[B, μ])(1)] exp(
          i (Subscript[S, 0])) summed over all possible distributions of field (Subscript[A, μ])(1), wherein (Subscript[S, 0]) is the action for the field (Subscript[S, 0])=-(((8 π (e^2)))^(-1))(∑_μ)∫\((\((∂A\_μ/∂t)\)\^2 - \((∇A\_μ)\)\^2)\)(d^3)x d t the sum on μ carrying the usual minus sign for space components.

If F[A] is any functional of (Subscript[A, μ])(1) we shall represent by \(TraditionalForm\`\(\_0\)\(|\ F[A]\ |\_0\)\) this superposition of F[A] exp(i (Subscript[S, 0])) over distributions of (Subscript[A, μ]) for the case in which there are no photons in initial or final state. That is, we have

(Subscript[T, (e^2)])[
        B] = (Subscript[, 0])| 
                (Subscript[T, 0])[A + 
                    B] (Subscript[|, 0]).

The evaluation of \(TraditionalForm\`\(\_0\)\(|\ F[A]\ |\_0\)\) directly from the definition of the operation \(TraditionalForm\`\(\_0\)\(|\ |\_0\)\) is not necessary. We can give the result in another way. We first note that the operation is linear,

\(TraditionalForm\`\(\_0\)\(|\ 
              F\_1[A] + 
                F\_2[A]\ |\_0\) = \(\_0\)\(|\ 
                F\_1[A]|\_0\) + \(\_0\)\(|\ 
                F\_2[A]\ |\_0\)\)

(Subscript[, 0])(Subscript[|exp(-i ∫(Subscript[j, μ])(1) (Subscript[A, μ])(
                          1) d(Subscript[τ, 1])) |, 0])=exp(-(1/2) i(e^2) ∫\(∫\(\(j\_μ\)(1)\)\ \(\(j\_μ\)(
                          2)\)\ \(\(δ\_+\)(
                          s\_\(1\ 2\)\%2)\) dτ\_1\)d(Subscript[τ, 2]))

Although this gives the evaluation of \(TraditionalForm\`\(\_0\)\(|\ \ |\_0\)\) for only a particular functional of (Subscript[A, μ]) the appearance of the arbitrary function (Subscript[j, μ])(1) makes it sufficiently general to permit the evaluation for any other functional. For it is to be expected that any functional can be represented as a superposition of exponentials with different functions (Subscript[j, μ])(1) (by analogy with the principle of Fourier integrals for ordinary functions). Then, by (), the result of the operation is the corresponding superposition of expressions equal to the right-hand side of () with the various j’s substituted for (Subscript[j, μ]).

In many applications F[A] can be given as a power series in (Subscript[A, μ]):

F[A]=(Subscript[f, 0])+∫(Subscript[f, μ])(1) (Subscript[A, μ])(
                    1) d(Subscript[τ, 1])+∫∫(Subscript[f, μ ν])(1, 
                              2) (Subscript[A, μ])(1) (Subscript[A, ν])(
                              2) d(Subscript[τ, 1])d(Subscript[τ, 2])+⋯

(Subscript[, 0])| 
                F[A] (Subscript[|, 0])=(Subscript[f, 0])+\(∫\(\(f\_μ\)(
                    1)\)\ \(\_0\)\(|\ \(A\_μ\)(
                            1)\ |\_0\) dτ\_1\)+∫\(∫\(\(f\_\(μ\ ν\)\)(1, 
                              2)\)\ \(\_0\)\(|\ \(\(A\_μ\)(1)\)\ \(\(A\_ν\)(
                                        2)\)\ |\_0\) dτ\_1\)d(Subscript[τ, 2])+⋯

(Subscript[, 0])(Subscript[| -i (Subscript[A, μ])(3) exp(-i ∫(Subscript[j, ν])(1) (Subscript[A, ν])(
                            1) d(Subscript[τ, 1]))|, 0])=-i(e^2) \(∫\(\(δ\_+\)(
                      s\_\(3\ 4\)\%2)\)\ \(\(j\_μ\)(
                      4)\) dτ\_4\)
×exp(-(1/2) i(e^2)∫\(∫\(\(j\_ν\)(
                                    1)\)\ \(\(j\_ν\)(
                                    2)\)\ \(\(δ\_+\)(
                                    s\_\(1\ 2\)\%2)\) dτ\_1\)d(Subscript[τ, 2])).

\(TraditionalForm\`\(\_0\)\ \(|\ \(A\_μ\)(3)\ |\_0\) = 
      0. \)

\(TraditionalForm\`\(\_0\)\(|\ \(\(A\_μ\)(3)\)\ \(\(A\_ν\)(
                  4)\)\ |\_0\) = 
      ie\^2\ δ\_\(μ\ ν\)\ \(\(δ\_+\)(
          s\_\(3\ 4\)\%2)\)\)

It is evident that (), (), () imply that (Subscript[T, (e^2)])[B] satisfies the differential equation () and conversely () with the definition () implies () and (). For if (Subscript[T, 0])[B] is an exponential

(Subscript[T, 0])[B]=exp(-i ∫(Subscript[j, μ])(1) (Subscript[B, μ])(
                    1) d(Subscript[τ, 1]))

(Subscript[T, (e^2)])[B]=exp[-(1/2) i(e^2) ∫\(∫\(\(j\_μ\)(1)\)\ \(\(j\_μ\)(
                            2)\)\ \(\(δ\_+\)(
                            s\_\(1\ 2\)\%2)\) dτ\_1\)d(Subscript[τ, 2])] ·exp[-i ∫(Subscript[j, ν])(1) (Subscript[B, ν])(
                        1) d(Subscript[τ, 1])].

Many of the formal representations of the matter system (such as that of second quantization of Dirac electrons) represent the interaction with a fixed potential in a formal exponential form such as the left-[hand] side of (), except that (Subscript[j, μ])(1) is an operator instead of a numerical function, Equation () may still be used if care is exercised in defining the order of the operators on the right-hand side. The succeeding paper will discuss this in more detail.

Equation () or its solution (), (), () constitutes a very general and convenient formulation of the laws of quantum electrodynamics for virtual processes. Its relativistic invariance is evident if it is assumed that the unquantized theory giving (Subscript[T, 0])[B] is invariant. It has been proved to be equivalent to the usual formulation for Dirac electrons and positrons (for Klein-Gordon particles see Appendix A). It is suggested that it is of wide generality. It is expressed in a form which has meaning even if it is impossible to express the matter system in Hamiltonian form; in fact, it only requires the existence of an amplitude for fixed potentials which obeys the principle of superposition of amplitudes. If (Subscript[T, 0])[B] is known in power series in B, calculations of ((Subscript[T, e])^2)[B] in a power series of (e^2) can be made directly using the italicized rule of Sec. 7. The limitation to virtual quanta is removed in the next section.

On the other hand, the formulation is unsatisfactory because for situations of importance it gives divergent results, even if (Subscript[T, 0])[B] is finite. The modification proposed in II of replacing \(TraditionalForm\`\(δ\_+\)(s\_12\^2)\) in (), () by \(TraditionalForm\`\(f\_+\)(s\_12\^2)\) is not satisfactory owing to the loss of the theorems of conservation of energy or probability discussed in II at the end of Sec. 6. There is the additional difficulty in positron theory that even (Subscript[T, 0])[B] is infinite to begin with (vacuum polarization). Computational ways of avoiding these troubles are given in II and in the references of footnote 2.

Case of real photons

\(TraditionalForm\`G\_\(\(m\)\(\)\(n\)\(\ \)\) = \ ∫\(\(φ\_m\^*\)(q\_j)\) \(k(q\_j, t\^″; q\_0, 
            t\^′)\) \(\(φ\_n\)(
            q\_0)\) dq\_0\ dq\_j\)

\(TraditionalForm\`g(X, 
        Y) = ∑\+m\(∑\+n\( 
                G\_\(m\ n\)\) \(X\^m\) \(\(Y\^n\)(\(m!\) \(n!\))\)\^\(-\(1\/2\)\)\)\)

\(TextForm\`∑\+n\( φ\_n\) \((q\_0)\) \(Y\^n\) \((\(n!\))\)\^\(-\(1\/2\)\) = \(\(\(\((ω/π)\)\^\(1\/2\)\) 
            exp \((\(-\(1\/2\)\) 
                iωt\^′)\)\)×exp\)\ \(1\/2\)[ω\ q\_0\^2 - \((Y\ exp[\(-i\)ωt\^′] - \(\((2  ω)\)\^\(1\/2\)\) q\_0)\)\^2]\)

\(TraditionalForm\`g(X, 
        Y) = \(G\_\(0\ 0\)\) \(exp(
          X\ Y + i\(β\^*\) X + 
            iβY)\)\)

\(TraditionalForm\`G\_\(m\ n\) = \ \(G\_\(0\ 0\)\) \(\((\(m!\)\ \(n!\))\)\^\(-\(1\/2\)\)\) \(∑\+r\(\(\(\( 
                      m!\)\/\(\(\((m - 
                            r)\)!\) \(r!\)\)\) \(n!\)\/\(\(\((n - r)\)!\) \(r!\)\)\)×\(r!\)\) \(\((i\(β\^*\))\)\^\(m - 
                  r\)\) \((iβ)\)\^\(n - r\)\)\)

({{β, \(\(\(=\)\(\(\((2  ω)\)\^\(-\(1\/2\)\)\) \(∫\_\(t\^′\)\%\(t\^″\)\(γ(
                        t)\) \(exp(\(-i\)ωt)\) dt\)\)\)\(,\)\)}, {\(β\^*\), \(\(\(=\)\(\(\((2  ω)\)\^\(-\(1\/2\)\)\) \(∫\_\(t\^′\)\%\(t\^″\)\(γ(
                        t)\) \(exp(\(+i\)ωt)\) dt\)\)\)\(,\)\)}})

Formula () is readily understandable. Consider first a simple case of absorption of one photon. Initially we have one photon and finally none. The amplitude for this is the transition element of (Subscript[G, 0 1]) = 
        iβ(Subscript[G, 0 0]) or \(TraditionalForm\`⟨\(χ\_\(t\^″\)\) |iβG\_\(0\ 0\)| ψ\_\(t\^′\)⟩\). This is the same as would result if we asked for the transition element for a problem in which all photons are virtual but there was present a perturbing potential -(((2  ω))^(-(1/2))) γ(
          t) exp(iωt) and we required the first-order effect of this potential. Hence photon absorption is like the first order action of a potential varying in time as γ(t) exp(
          iωt) that is with a positive frequency (i.e., the sign of the coefficient of t in the exponential corresponds to positive energy). The amplitude for emission of one photon involves \(TraditionalForm\`G\_\(1\ 0\) = 
        i\(β\^*\) G\_\(0\ 0\)\), which is the same result except that the potential has negative frequency. Thus we begin by interpreting \(TraditionalForm\`i\(β\^*\)\) as the amplitude for emission of one photon iβ as the amplitude for absorption of one.

Next for the general case of n photons initially and finally we may understand () as follows. We first neglect Bose statistics and imagine the photons as individual distinct particles. If we start with n and end with m this process may occur in several different ways. The particle may absorb in total n - r of the photons and the final m photons will represent r of the photons which were present originally plus m - r new photons emitted by the particle. In this case the n - r which are to be absorbed may be chosen from among the original n in n!/(n - r)! r! different ways, and each contributes a factor iβ, the amplitude for absorption of a photon. Which of the m - r photons from among the m arc emitted can be chosen in m!/(m - r)! r! different ways and each photon contributes a factor \(TraditionalForm\`i\ \(β\^*\)\) in amplitude. The initial r photons which do not interact with the particle can be re-arranged among the final r in r! ways. We must sum over the alternatives corresponding to different values of r. Thus the form of (Subscript[G, m n]) can be understood. The remaining factor \(TraditionalForm\`\(\((\(m!\))\)\^\(-\(1\/2\)\)\) \((\(n!\))\)\^\(-\(1\/2\)\)\) may be interpreted as saying that in computing probabilities (which therefore involves the square of (Subscript[G, m n])) the photons may be considered as independent but that if m are actually equal the statistical weight of each of the states which can be made by rearranging the m equal photons is only 1/m!. This is the content of Bose statistics; that m equal particles in a given state represents just one state, i.e., has statistical weight unity, rather than the m! statistical weight which would result if it is imagined that the particles and states can be identified and rearranged in m! different ways. This holds for both the initial and final states of course. From this rule about the statistical weights of states the derivation of the blackbody distribution law follows.

The actual electromagnetic field is represented as a host of oscillators each of which behaves independently and produces its own factor such as (Subscript[G, m n]). Initial or final states may also be linear combinations of states in which one or another oscillator is excited. The results for this case are of course the corresponding linear combination of transition elements.

For photons of a given direction of polarization and for sin or cos waves the explicit expression for β can be obtained directly from () by substituting the formulas () for the γ’s for the corresponding oscillator. It is more convenient to use the linear combination corresponding to running waves. Thus we find the amplitude for absorption of a photon of momentum K, frequency k=(((K·K))^((1/2))) polarized in direction e is given by including a factor i times

(Subscript[β, K,e])=(((4 π))^((1/2)))(((2 k))^(-(1/2)))(∑_n)(Subscript[e, n])(∫_((t^′))^((t^″)))exp(-ikt)×exp(i K·(Subscript[x, n])(t))e·(Subscript[x, n]^(.))(t) dt

(A^(P H))= (((2 π/k))^((1/2)))e exp(-i (kt-K·x))

We can express this directly in terms of the quantity (Subscript[T, (e^2)])[B], the amplitude for a given transition without emission of a photon. What we have said is that the amplitude for absorption of just one photon whose classical wave form is ((Subscript[A, μ])^(P H))(1) (time variation exp(-ik(Subscript[t, 1])) corresponding to positive energy k) is proportional to the first order (in ϵ) change produced in (Subscript[T, (e^2)])[B] on changing B to B + ϵ(A^(P H)). That is, more exactly,

∫(δ(Subscript[T, (e^2)])[
                B]/δ (Subscript[B, μ])(1)) ((Subscript[A, μ])^(P H))(
          1) d(Subscript[τ, 1])

\(TextForm\`∫\(∫\((\(δ\^2\) 
              T\_\(e\^2\)[
                  B]/δ\ \(B\_μ\) \((1)\) δ\ \(B\_ν\) \((2)\))\) \(A\_μ\^\(P\ H\_1\)\) \((1)\) \(A\_ν\^\(P\ H\_2\)\) \((2)\) dτ\_1 dτ\_2\)\)

It is possible, of course, to deal with situations in which the electromagnetic field is not in a definite state after the interaction. For example, we might ask for the total probability of a given process, such as scattering, without regard for the number of photons emitted. This is done of course by squaring the amplitude for the emission of m photons of a given kind and summing on all m. Actually the sums and integrations over the oscillator momenta can usually easily be performed analytically. For example, the amplitude, starting from vacuum and ending with m photons of a given kind, is by () just

\(TraditionalForm\`G\_\(m\ 0\) = \(\((\(m!\))\)\^\(-\(1\/2\)\)\) \(\(\(\(G\_\(0\ 0\)\)(\ 
                i\(β\^*\))\)\^m\)\(.\)\)\)

\(TraditionalForm\`∑\_m\( G\_\(m\ 0\)\^*\) 
          G\_\(m\ 0\)\^′ = \(∑\_m\( G\_\(0\ 0\)\^*\) \(\(\(G\_\(0\ 0\)\^′\)(\(m!\))\)\^\(-1\)\) \(\(β\^m\)(β\^\(\(′\)\(*\)\))\)\^m = \(G\_\(0\ 0\)\^*\) \(G\_\(0\ 0\)\^′\) \(exp(ββ\^\(\(′\)\(*\)\))\)\)\)

The author appreciates his opportunities to discuss these matters with Professor H. A. Bethe and Professor J. Ashkin, and the help of Mr. M. Baranger with the manuscript.

We start with the Klein-Gordon equation

Since the potentials (Subscript[A, μ])(x) are functions only of coordinates (Subscript[x, μ]), and are independent of u, the equation is separable in u and we can write a special solution in the form φ = exp((1/2) 
              i(m^2) u) ψ(x) where ψ(x), a function of the coordinates (Subscript[x, μ]) only, satisfies () and the eigenvalue (1/2) (m^2) conjugate to u is related to the mass m of the particle. Equation () is therefore equivalent to the Klein-Gordon Eq. () provided we ask in the end only for the solution of () corresponding to the eigenvalue (1/2) (m^2) for the quantity conjugate to u.

We may now proceed to represent Eq. () in Lagrangian form in general and without regard to this eigenvalue condition. Only in the final solutions need we apply the eigenvalue condition. That is, if we have some special solution φ(x, u) of () we can select that part corresponding to the eigenvalue (1/2) (m^2) by calculating

Since () is so closely analogous to the Schrödinger equation, it is easily written in the Lagrangian form described in C, simply by working by analogy. For example if φ(x, u) is known at one value of u its value at a slightly larger u + ϵ is given by

In the first place in the special case of a free particle we can define a kernel (k^((0)))(x, (Subscript[u, 0]); (x^′), 0) for arrival from ((Subscript[x, μ])^′), 0 to (Subscript[x, μ]) at (Subscript[u, 0]) as the sum over all trajectories between these points of exp - 
        i (∫_0^((Subscript[u, 0])))(1/2) (((d (Subscript[x, μ])/d u))^2) 
              d u. Then for this case we have

∫(((4 (π^2) (Subscript[u, 0]^2) 
                      i))^(-1)) exp(
                ip·x) exp-(1/2)i((m^2) (Subscript[u, 0]) + ((Subscript[x, μ])^2)/(Subscript[u, 0]))(d^4)(Subscript[τ, x])= exp - (1/2) i (Subscript[u, 0])((m^2) - ((Subscript[p, μ])^2))

If \(TraditionalForm\`\(Φ\^\((0)\)\)[x(μ)] = 
        exp - i \(∫\_0\%\(u\_0\)\(1\/2\) \(\((dx\_μ/du)\)\^2\) du\)\) is the amplitude for a given trajectory (Subscript[x, ν])(u) for a free particle, then the amplitude in a potential is

For interpretation, the integral in () must be written as a Riemann sum, and if a perturbation expansion is made, care must be taken with the terms quadratic in the velocity, for the effect of ((Subscript[x, μ, i + 1]) - 
            (Subscript[x, μ, i])) ((Subscript[x, ν, i + 1]) - 
            (Subscript[x, ν, i])) is not of order (ϵ^2) but is \(TraditionalForm\`i\(δ\_\(μ\ ν\)\) ϵ\). The “velocity” d(Subscript[x, μ])/du becomes the momentum operator \(TraditionalForm\`p\_μ = \(+i\) ∂\(\(/\)\(∂x\_μ\)\)\) operating half before and half after (Subscript[A, μ]), just as in the non-relativistic Schrödinger equation discussed in Sec. 5. Furthermore, in exactly the same manner as in that case, but here in four dimensions, a term quadratic in (Subscript[A, μ]) arises in the second-order perturbation terms from the coincidence of two velocities for the same value of u.

As an example, the kernel (k^((A)))(x, (Subscript[u, 0]); (x^′), 0) for proceeding from (Subscript[x, μ]^′), 0 to (Subscript[x, μ]), (Subscript[u, 0]) in a potential (Subscript[A, μ]) differs from (k^((0))) to first order in (Subscript[A, μ]) by a term

The transition to quantum electrodynamics is simple for in () we already have a transition amplitude represented as a sum (over trajectories, and eventually (Subscript[u, 0])) of terms, in each of which the potential appears in exponential form. We may make use of the general relation (). Hence, for example, one finds for the case of no photons in the initial and final states, in the presence of an external potential (Subscript[B, μ]), the amplitude that a particle proceeds from ((Subscript[x, μ]^′), 0) to ((Subscript[x, μ]), (Subscript[u, 0])) is the sum over all trajectories of the quantity

Actually, knowledge of the motion of a single charge implies a great deal about the behavior of several charges. For a pair which eventually may turn out to be a virtual pair may appear in the short run as two “other particles.” As a virtual pair, that is, as the reverse section of a very long and complicated single track we know its behavior by (). We can assume that such a section can be looked at equally well, for a limited duration at least, as being due to other unconnected particles. This then implies a definite law of interaction of particles if the self-action () of a single particle is known. (This is similar to the relation of real and virtual photon processes discussed in detail in Appendix B.) It is possible that a detailed analysis of this could show that () implied that () was correct for many particles. There is even reason to believe that the law of Bose-Einstein statistics and the expression for contributions from closed loops could be deduced by following this argument. This has not yet been analyzed completely, however, so we must leave this formulation in an incomplete form. The expression for closed loops should come out to be (Subscript[C, ν]) = exp + L where L, the contribution from a single loop, is

That this possibility exists can be seen from the consideration that what looks like a real process from one point of view may appear as a virtual process occurring over a more extended time.

For example, if we wish to study a given real process, such as the scattering of light, we can, if we wish, include in principle the source, scatterer, and eventual absorber of the scattered light in our analysis. We may imagine that no photon is present initially, and that the source then emits light (the energy coming say from kinetic energy in the source). The light is then scattered and eventually absorbed (becoming kinetic energy in the absorber). From this point of view the process is virtual; that is, we start with no photons and end with none. Thus we can analyze the process by means of our formula for virtual processes, and obtain the formulas for real processes by attempting to break the analysis into parts corresponding to emission, scattering, and absorption.

To put the problem in a more general way, consider the amplitude for some transition from a state empty of photons far in the past (time (t^′)) to a similar one far in the future (t = (t^″)). Suppose the time interval to be split into three regions a, b, c in some convenient manner, so that region b is an interval (Subscript[t, 2]) > t > (Subscript[t, 1]) around the present time that we wish to study. Region a, ((Subscript[t, 1]) > t > (t^′)), precedes b, and c, ((t^″) > t > (Subscript[t, 2])), follows b. We want to see how it comes about that the phenomena during b can be analyzed by a study of transitions (Subscript[g, j i])(b) between some initial state i at time (Subscript[t, 1]), (which no longer need be photon-free), to some other final state j at time (Subscript[t, 2]). The states i and j are members of a large class which we will have to find out how to specify. (The single index i is used to represent a large number of quantum numbers, so that different values of i will correspond to having various numbers of various kinds of photons in the field, etc.) Our problem is to represent the over-all transition amplitude, g(a, b, c), as a sum over various values of i, j of a product of three amplitudes,

To become acquainted with what is involved, suppose first that we do not have a problem involving virtual photons, but just the transition of a one-dimensional Schrödinger particle going in a long time interval from, say, the origin o to the origin o, and ask what states i we shall need for intermediary time intervals. We must solve the problem () where g(a, b, c) is the sum over all trajectories going from o at (t^′) to o at (t^″) of exp i S where S = ∫L dt. The integral may be split into three parts S = (Subscript[S, a]) + (Subscript[S, b]) + (Subscript[S, c]) corresponding to the three ranges of time. Then exp(
          i S) = exp(i (Subscript[S, a])) exp(i (Subscript[S, b])) exp(i (Subscript[S, c])) and the separation () is accomplished by taking for (Subscript[g, i 0])(a) the sum over all trajectories lying in a from o to some end point (Subscript[x, (Subscript[t, 1])]) of exp(i (Subscript[S, a])), for (Subscript[g, j i])(b) the sum over trajectories in b of exp(i (Subscript[S, b])) between end points (Subscript[x, (Subscript[t, 1])]) and (Subscript[x, (Subscript[t, 2])]) and for (Subscript[g, 0 j])(c) the sum of exp(i (Subscript[S, c])) over the section of the trajectory lying in c and going from (Subscript[x, (Subscript[t, 2])]) to o. Then the sum on i and j can be taken to be the integrals on (Subscript[x, (Subscript[t, 1])]), (Subscript[x, (Subscript[t, 2])]), respectively. Hence the various states i can be taken to correspond to particles being at various coordinates x. (Of course any other representation of the states in the sense of Dirac’s transformation theory could be used equally well. Which one, whether coordinate, momentum, or energy level representation, ;is of course just a matter of convenience and wee cannot determine that simply from ().)

We can consider next the problem including virtual photons. That is, g(a, b, c) now contains an additional factor exp(i R) where R involves a double integral ∫∫ over all time. Those parts of the index i which correspond to the particle states can be taken in the same way as though R were absent. We study now the extra complexities in the states produced by splitting the R. Let us first (solely for simplicity of the argument) take the case that there are only two regions a, c separated by time (Subscript[t, 0]), and try to expand

The expression for (Subscript[R, a c]), is just twice () but with the integral on s extending over the range a and that for t extending over c. Thus exp(i(Subscript[R, a c])) contains the variables for times in a and in c in a quite complicated mixture. Our problem is to write exp(i(Subscript[R, a c])) as a sum over possibly a vast class of states i of the product of two parts, like ((Subscript[h, i])^′)(c) (Subscript[h, i])(
          a), each of which involves the coordinates in one interval alone.

This separation may be made in many different ways, corresponding to various possible representations of the state of the electromagnetic field. We choose a particular one. First we can expand the exponential, exp(i(Subscript[R, a c])), in a power series, as (∑_n)(( 
                    (i^n)(n!))^(-1)) ((((Subscript[R, a c])))^n). The states i can therefore be subdivided into subclasses corresponding to an integer n which we can interpret as the number of quanta in the field at time (Subscript[t, 0]). The amplitude for the case n = 0 clearly just involves exp(i(Subscript[R, a a])) and exp(i(Subscript[R, c c])) in the way that it should if we interpret these as the amplitudes for regions a and c, respectively, of making a transition between a state of zero photons and another state of zero photons.

Next consider the case n = 1. This implies an additional factor in the transitional element; the factor (Subscript[R, a c]). The variables are still mixed up. But an easy way to perform the separation suggests itself. Namely, expand the \(δ\_+\)((((t - s))^2)-((((Subscript[x, n])(t)-(Subscript[x, m])(s)))^2)) in (Subscript[R, a c]), as a Fourier integral as

The term for a given polarization and value of K (for n = 1) is clearly just \(TraditionalForm\`\(-β\_c\) \(β\_a\^*\)\) where the (Subscript[β, a]) is defined in () but with the time integral extending just over region a, while (Subscript[β, c]) is the same expression with the integration over region c. Hence the amplitude for transition during interval a from a state with no quanta to a state with one in a given state of polarization and momentum is calculated by inclusion of an extra factor \(TraditionalForm\`i\(β\_a\^*\)\) in the transition element. Absorption in region c corresponds to a factor i(Subscript[β, c]).

We next turn to the case n = 2. This requires analysis of (Subscript[R, a (c^2)]). The \(TraditionalForm\`\(δ\_+\)\) can be expanded again as a Fourier integral, but for each of the two \(TraditionalForm\`\(δ\_+\)\) in (1/2) (Subscript[R, a (c^2)]) we have a value of K which may be different. Thus we say we have two photons, one of momentum K and one momentum K' and we sum over all values of K and K'. (Similarly each photon is characterized by its own independent polarization index.) The factor (1/2) can be taken into account neatly by asserting that we count each possible pair of photons as constituting just one state at time (Subscript[t, 0]). Then the (1/2) arises for the sum over all K, K' (and polarizations) counts each pair twice. On the other hand, for the terms representing two identical photons (K=K') of like polarization, the (1/2) cannot be so interpreted. Instead we invent the rule that a state of two like photons has statistical weight (1/2) as great as that calculated as though the photons were different. This, generalized to n identical photons, is the rule of Bose statistics.

The higher values of n offer no problem. The 1/n! is interpreted combinatorially for different photons, and as a statistical factor when some are identical. For example, for all n identical one obtains a factor \(TraditionalForm\`\(\((\(n!\))\)\^\(-1\)\) \((\(-β\_c\) \(β\_a\^*\))\)\) so that \(TraditionalForm\`\(\((\(n!\))\)\^\(-\(1\/2\)\)\) \((i\(β\_a\%*\))\)\^n\) can be interpreted as the amplitude for emission (from no initial photons) of n identical photons, in complete agreement with () for (Subscript[G, m 0]).

To obtain the amplitude for transitions in which neither the initial nor the final state is empty of photons we must consider the more general case of the division into three time regions (). This time we see that the factor which involves the coordinates in an entangled manner is exp i(
          (Subscript[R, a b]) + (Subscript[R, b c]) + (Subscript[R, a c])). It is to be expanded in the form (∑_i)(∑_j)(Subscript[h, i]^″)(
          c) (Subscript[h, i j]^′)(b) (Subscript[h, j])(a). Again the expansion in power series and development in Fourier series with a polarization sum will solve the problem. Thus the exponential is (∑_r)(∑_((Subscript[l, 1])))(∑_((Subscript[l, 2])))\(TraditionalForm\`\(\((iR\_\(a\ c\))\)\^r\) \(\((iR\_\(a\ b\))\)\^l\_1\) \(\((iR\_\(b\ c\))\)\^l\_2\) \(\((\(l\_1!\))\)\^\(-1\)\) \(\((\(l\_2!\))\)\^\(-1\)\) \((\(r!\))\)\^\(-1\)\). Now the R are written as Fourier series, one of the terms containing (Subscript[l, 1]) + (Subscript[l, 2]) + r variables K. Since (Subscript[l, 1]) + r involve a, (Subscript[l, 2]) + r involve c and (Subscript[l, 1]) + (Subscript[l, 2]) involve b, this term will give the amplitude that (Subscript[l, 1]) + r photons are emitted during the interval a, of those (Subscript[l, 1]) are absorbed during b but the remaining r, along with (Subscript[l, 2]) new ones emitted during b go on to be absorbed during the interval c. We have therefore n = (Subscript[l, 1]) + r photons in the state at time (Subscript[t, 1]) when b begins, and m = (Subscript[l, 2]) + r at (Subscript[t, 2]) when b is over. They each are characterized by momentum vectors and polarizations. When these are different the factors \(TraditionalForm\`\(\((\(l\_1!\))\)\^\(-1\)\) \(\((\(l\_2!\))\)\^\(-1\)\) \((\(r!\))\)\^\(-1\)\) are absorbed combinatorially. When some are equal we must invoke the rule of the statistical weights. For example, suppose all (Subscript[l, 1]) + (Subscript[l, 2]) + r photons are identical. Then \(TraditionalForm\`R\_\(a\ b\) = 
        i\(β\_b\) \(β\_a\^*\)\), \(TraditionalForm\`R\_\(b\ c\) = 
        i\(β\_c\) \(β\_b\^*\)\), \(TraditionalForm\`R\_\(a\ c\) = 
        i\(β\_c\) \(β\_a\^*\)\) so that our sum is

Putting m = (Subscript[l, 2]) + r, n = (Subscript[l, 1]) + r, this is the sum on n and m of

We have therefore found rules for real photons in terms of those for virtual. The real photons are a way of representing and keeping track of those aspects of the past behavior which may influence the future.

If one starts from a theory involving an arbitrary modification of the direct interaction \(TraditionalForm\`\(δ\_+\)\) (or in more general situations) it is possible in this way to discover what kinds of states and physical entities will be involved if one tries to represent in the present all the information needed to predict the future. With the Hamiltonian method, which begins by assuming such a representation, it is difficult to suggest modifications of a general kind, for one cannot formulate the problem without having a complete representation of the characteristics of the intermediate states, the particles involved in interaction, etc. It is quite possible (in the author’s opinion, it is very likely) that we may discover that in nature the relation of past and future is so intimate for short durations that no simple representation of a present may exist. In such a case a theory could not find expression in Hamiltonian form.

An exactly similar analysis can be made just as easily starting with the general forms (), (). Also a coordinate representation of the photons could have been used instead of the familiar momentum one. One can deduce the rules (), (). Nothing essentially different is involved physically, however, so we shall not pursue the subject further here. Since they imply [23] all the rules for real photons, Eqs. (), (), () constitute a compact statement of all the laws of quantum electrodynamics. But they give divergent results. Can the result after charge and mass renormalization also be expressed to all orders in (e^2)/ℏc in a simple way?

An electron acting upon itself is, from one point of view, a complex system of a particle and a field of an indefinite number of photons. To find a differential law of propagation of such a system we must ask first what quantities known at one instant will permit the calculation of these same quantities an instant later. Clearly, a knowledge of the position of the particle is not enough. We should need to specify: (1) the amplitude that the electron is at x and there are no photons in the field, (2) the amplitude the electron is at x and there is one photon of such and such a kind in the field, (3) the amplitude there are two photons, etc. That is, a series of functions of ever increasing numbers of variables. Following this view, we shall be led to the wave equation of the theory of second quantization.

We may also take a different view. Suppose we know a quantity (Subscript[Φ, (e^2)])[B, x], a spinor function of (Subscript[x, μ]), and functional of (Subscript[B, μ])(1), defined as the amplitude that an electron arrives at (Subscript[x, μ]), with no photon in the field when it moves in an arbitrary external unquantized potential (Subscript[B, μ])(1). We allow the electron also to interact with itself, but (Subscript[Φ, (e^2)]) is the amplitude at a given instant that there happens to be no photons present. As we have seen, a complete knowledge of this functional will also tell us the amplitude that the electron arrives at x and there is just one photon, of form ((Subscript[A, μ])^(P H))(1) present. It is, from (), ∫(δ(Subscript[Φ, (e^2)])[B, 
                  x]/δ(Subscript[B, μ])(
                1)) ((Subscript[A, μ])^(P H))(1) d (Subscript[τ, 1]).

Higher numbers of photons correspond to higher functional derivatives of (Subscript[Φ, (e^2)]). Therefore, (Subscript[Φ, (e^2)])[B, x] contains all the information requisite for describing the state of the electron-photon system, and we may expect to find a differential equation for it. Actually it satisfies (\(TraditionalForm\`∇\(\(=\)\(\(γ\_μ\) ∂\(\(/\)\(∂x\_μ\)\)\)\)\), B = (Subscript[γ, μ]) (Subscript[B, μ])),

Expansion of the solution of () in a power series in B and (e^2) starting from a free particle solution for a single electron, produces a series of terms which agree with the rules of II for action of potentials and virtual photons to various orders. It is another matter to use such an equation for the practical solution of a problem to all orders in (e^2). It might be possible to represent the self-energy problem as the variational problem for m, stemming from (). The \(TraditionalForm\`\(δ\_+\)\) will first have to be modified to obtain a convergent result.

We are not in need of the general solution of (). (In fact, we have it in (), () in terms of the solution (Subscript[T, 0])[B] = (Subscript[Φ, 0])[B, x] of the ordinary Dirac equation (i ∇-m) (Subscript[Φ, 0])[B, x] = 
        B (Subscript[Φ, 0])[B, x]. The general solution is too complicated, for complete knowledge of the motion of a self-acting electron in an arbitrary potential is essentially all of electrodynamics (because of the kind of relation of real and virtual processes discussed for photons in Appendix , extended also to real and virtual pairs). Furthermore, it is easy to see that other quantities also satisfy (). Consider a system of many electrons, and single out some one for consideration, supposing all the others go from some definite initial state i to some definite final state f. Let (Subscript[Φ, (e^2)])[B, x] be the amplitude that the special electron arrives at x, there are no photons present, and the other electrons go from i to f when there is an external potential (Subscript[B, μ]) present (which (Subscript[B, μ]) also acts on the other electrons). Then (Subscript[Φ, (e^2)]) also satisfies (). Likewise the amplitude with closed loops (all other electrons go vacuum to vacuum) also satisfies () including all vacuum polarization effects. The various problems correspond to different assumptions as to the dependence of (Subscript[Φ, (e^2)])[B, x] on (Subscript[B, μ]) in the limit of zero (e^2). The Eq. () without further boundary conditions is probably too general to be useful.