RADIATIVE CORRECTIONS TO SCATTERING

We can now complete the discussion of the radiative corrections to scattering. In the integrals we include the convergence factor C(k2), so that they converge for large k. Integral (12) is also not convergent because of the well-known infrared catastrophy. For this reason we calculate (as discussed in B) the value of the integral assuming the photons to have a small mass λmin  m  λ. The integral (12) becomes (e2/πi) ∫ γμ(p2 − k − m)−1a(p1 − k − m)−1 ×γμ(k2 − γ2 min)−1d4kC(k2 − λ2 min), which when integrated (see Appendix B) gives (e2/2π) times [ 2 ( ln m λmin − 1 ) ( 1 − 2θ tan2θ )

• θtanθ
• 4 tan2θ θ∫ 0 α tanαdα ] a + 1 4m (qa − aq) 2θ sin 2θ + ra, (22) where (q2)1/2 = 2m and we have assumed the matrix to operate between states of momentum p1 and p2 = p1 + q and have neglected terms of order λmin/m, m/λ, and q2/λ2. Here the only dependence on the convergence factor is in the term ra, where r = ln(λ/m) + 9/4 − 2 ln(m/λmin). (23) 20 As we shall see in a moment, the other terms (13), (14) give contributions which just cancel the ra term. The remaining terms give for small q, (e2/4π) ( 1 2m (qa − aq) + 4q2 3m2 a ( ln m λmin − 3 8 )) , (24) which shows the change in magnetic moment and the Lamb shift as inter- preted in more detail in B.15 We must now study the remaining terms (13) and (14). The integral on k in (13) can be performed (after multiplication by C(k2)) since it involves nothing but the integral (19) for the self-energy and the result is allowed to operate on the initial state u1, (so that p1u1 = mu1). Hence the factor following a(p1 − m)−1 will be just ∆m. But, if one now tries to expand 1/(p1 − m) = (p1 + m)/(p2 1 − m2) one obtains an infinite result, since p2 1 = m2. This is, however, just what is expected physically. For the quantum can be emitted and absorbed at any time previous to the scattering. Such a process has the effect of a change in mass of the electron in the state

1. It therefore changes the energy by ∆E and the amplitude to first order in ∆E by −i∆E · t where t is the time it is acting, which is infinite. That is, the major effect of this term would be canceled by the effect of change of mass ∆m. The situation can be analyzed in the following manner. We suppose that the electron approaching the scattering potential a has not been free for an infinite time, but at some time far past suffered a scattering by a potential b. If we limit our discussion to the effects of ∆m and of the virtual radiation of one quantum between two such scatterings each of the effects will be finite, though large, and their difference is determinate. The propagation from b to a is represented by a matrix a(p′ − m)−1b, (25) 15That the result given in B in Eq. (19) was in error was repeatedly pointed out to the author, in private communication, by V. F. Weisskopf and J. B. French, as their calcu- lation, completed simultaneously with the author’s early in 1948, gave a different result. French has finally shown that although the expression for the radiationless scattering B, Eq. (18) or (24) above is correct, it was incorrectly joined into Bethe’s non-relativistic result. He shows that the relation ln 2kmax − 1 = ln λmin used by the author should have been ln 2kmax − 5/6 = ln λmin. This results in adding a term −(1/6) to the logarithm in B, Eq. (19) so that the result now agrees with that of J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949) and N. H. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949). The author feels unhappily responsible for the very considerable delay in the publication of French’s result occasioned by this error. This footnote is appropriately numbered. 21 in which one is to integrate possibly over p′ (depending on details of the situation). (If the time is long between b and a, the energy is very nearly determined so that p′2 is very nearly m2.) We shall compare the effect on the matrix (25) of the virtual quanta and of the change of mass ∆m. The effect of a virtual quantum is (e2/πi) ∫ a(p′ − m)−1γμ(p′ − k − m)−1 × γμ(p′ − m)−1bk−2d4kC(k2), (26) while that of a change of mass can be written a(p′ − m)−1∆m(p′ − m)−1b, (27) and we are interested in the difference (26)-(27). A simple and direct method of making this comparison is just to evaluate the integral on k in (26) and subtract from the result the expression (27) where ∆m is given in (21). The remainder can be expressed as a multiple −r(p′2) of the unperturbed amplitude (25); −r(p′2)a(p′ − m)−1b. (28) This has the same result (to this order) as replacing the potentials a and b in (25) by (1− 1 2 r(p′2))a and (1− 1 2 r(p′2))b. In the limit, then, as p′2 → m2 the net effect on the scattering is − 1 2 ra where r, the limit of r(p′2) as p′2 → m2 (assuming the integrals have an infrared cut-off), turns out to be just equal to that given in (23). An equal term − 1 2 ra arises from virtual transitions after the scattering (14) so that the entire ra term in (22) is canceled. The reason that r is just the value of (12) when q2 = 0 can also be seen without a direct calculation as follows: Let us call p the vector of length m in the direction of p′ so that if p′2 = m(1 + )2 we have p′ = (1 + 0p and we take  as very small, being of order T −1 where T is the time between the scatterings b and a. Since (p′ −m)−1 = (p′ +m)/(p′2 −m2) ≈ (p+m)/2m2, the quantity (25) is of order −1 or T . We shall compute corrections to it only to its own order (−1) in the limit  → 0. The term (27) can be written approximately16 as (e2/πi) ∫ a(p′ − m)−1γμ(p − k − m)−1 × γμ(p′ − m)−1bk−2d4kC(k2), 16The expression is not exact because the substitution of ∆m by the integral in (19) is valid only if p operates on a state such that p can be replaced by m. The error, however, is of order a(p′ − m)−1(p − m)(p′ − m)−1b which is a((1 + )p + m)(p − m) × ((1 + )p + m)p(2 + 2)−2m−4. But since 2 = m2 we have p(p − m) = −m(p − m) = (p − m)p so the net result is approximately a(p − m)b/4m2 and is not of order 1/ but smaller, so that its effect drops out in the limit. 22 using the expression (19) for ∆m. The net of the two effects is therefore approximately17 −(e2/πi) ∫ a(p′ − m)−1γμ(p − k − m)−1p(p − k − m)−1 ×γμ(p′ − m)−1bk−2d4kC(k2), a term now of order 1/ (since (p′ − m)−1 ≈ (p + m) × (2m2)−1) and therefore the one desired in the limit. Comparison to (28) gives for r the expression (p1 + m/2m) ∫ γμ(p1 − k − m)−1(p1m−1)(p1 − k − m)−1 ×γμk−2d4kC(k2). (29) The integral can be immediately evaluated, since it is the same as the integral (12), but with q = 0, for a replaced by p1/m. The result is therefore r · (p1/m) which when acting on the state u1 is just r, as p1u1 = mu1. For the same reason the term (p1 + m)/2m in (29) is effectively 1 and we are left with −r of (23).18 In more complex problems starting with a free electron the same type of term arises from the effects of a virtual emission and absorption both previ- ous to the other processes. They, therefore, simply lead to the same factor r so that the expression (23) may be used directly and these renormalization integrals need not be computed afresh for each problem. In this problem of the radiative corrections to scattering the net result is insensitive to the cut-off. This means, of course, that by a simple rear- rangement of terms previous to the integration we could have avoided the use of the convergence factors completely (see for example Lewis19). The problem was solved in the manner here in order to illustrate how the use of 17We have used, to first order, the general expansion (valid for any operators A, B) (A + B0−1 = A−1 − A−1BA−1 + A−1BA−1BA−1 − . . . with A = p − k − m and B = p′ − p = p to expand the difference of (p′ − k − m)−1 and (p − k − m)−1.

18The renormalization terms appearing B, Eqs. (14), (15) when translated directly into the present notation do not give twice (29) but give this expression with the central p1m−1 factor replaced by mγ4/E1 where E1 = p1μ, for μ = 4. When integrated it therefore gives ra((p1 + m)/2m)(mγ4/E1) or ra − ra(mγ4/E1)(p1 − m)/2m. (Since p1γ4 + γ4p1 = 2E1) which gives just ra, since p1u1 = mu1. 19H.W. Lewis, Phys. Rev. 73, 173 (1948).

such convergence factors, even when they are actually unnecessary, may fa- cilitate analysis somewhat by removing the effort and ambiguities that may be involved in trying to rearrange the otherwise divergent terms. The replacement of δ+ by f+ given in (16), (17) is not determined by the analogy with the classical problem. In the classical limit only the real part of δ+ (i.e., just δ) is easy to interpret. But by what should the imaginary part, 1/(πs2), of δ+ be replaced? The choice we have made here (in den- ning, as we have, the location of the poles of (17)) is arbitrary and almost certainly incorrect. If the radiation resistance is calculated for an atom, as the imaginary part of (8), the result depends slightly on the function f+. On the other hand the light radiated at very large distances from a source is independent of f+. The total energy absorbed by distant absorbers will not check with the energy loss of the source. We are in a situation analogous to that in the classical theory if the entire f function is made to contain only retarded contributions (see A, Appendix). One desires instead the analogue of 〈F 〉ret of A. This problem is being studied. One can say therefore, that this attempt to find a consistent modification of quantum electrodynamics is incomplete (see also the question of closed loops, below). For it could turn out that any correct form of f+ which will guarantee energy conservation may at the same time not be able to make the self-energy integral finite. The desire to make the methods of simplifying the calculation of quantum electrodynamic processes more widely available has prompted this publication before an analysis of the correct form for f+ is complete. One might try to take the position that, since the energy dis- crepancies discussed vanish in the limit λ → ∞, the correct physics might be considered to be that obtained by letting λ → ∞ after mass renormal- ization. I have no proof of the mathematical consistency of this procedure, but the presumption is very strong that it is satisfactory. (It is also strong that a satisfactory form for f+ can be found.) 7 THE PROBLEM OF VACUUM POLARIZATION In the analysis of the radiative corrections to scattering one type of term was not considered. The potential which we can assume to vary as aμexp(−iq·x) creates a pair of electrons (see Fig. 6), momenta pa, −pb. This pair then reannihilates, emitting a quantum q = qb − qa, which quantum scatters the original electron from state 1 to state 2. The matrix element for this process (and the others which can be obtained by rearranging the 24 order in time of the various events) is −(e2/πi)(˜u2γμu1) ∫ Sp[(pa + q − m)−1 ×γμ(pa − m)−1γμ]d4paq−2C(q2)aν . (30) This is because the potential produces the pair with amplitude proportional to aν γν the electrons of momenta pa, and −(pa + q) proceed from there to annihilate, producing a quantum (factor γμ) which propagates (factor q−2C(q2)) over to the other electron, by which it is absorbed (matrix ele- ment of γμ, between states 1 and 2 of the original electron (˜u2γμu1)). All momenta pa and spin states of the virtual electron are admitted, which means the spur and the integral on d4pa are calculated. One can imagine that the closed loop path of the positron-electron pro- duces a current 4πJμaν , (31) which is the source of the quanta which act on the second electron. The quantity Jμν = −(e2/πi) ∫ Sp[(p + q − m)−1 ×γμ(p − m)−1γμ]d4p, (32) is then characteristic for this problem of polarization of the vacuum. One sees at once that Jμν diverges badly. The modification of δ to f alters the amplitude with which the current jμ, will affect the scattered electron, but it can do nothing to prevent the divergence of the integral (32) and of its effects. One way to avoid such difficulties is apparent. From one point of view we are considering all routes by which a given electron can get from one region of space-time to another, i.e., from the source of electrons to the apparatus which measures them. From this point of view the closed loop path leading to (32) is unnatural. It might be assumed that the only paths of meaning are those which start from the source and work their way in a continuous path (possibly containing many time reversals) to the detector. Closed loops would be excluded. We have already found that this may be done for electrons moving in a fixed potential. Such a suggestion must meet several questions, however. The closed loops are a consequence of the usual hole theory in electrodynamics. Among other things, they are required to keep probability conserved. The proba- bility that no pair is produced by a potential is not unity and its deviation from unity arises from the imaginary part of Jμν . Again, with closed loops 25 Figure 6: Vacuum polarization effect on scattering, Eq. (30). excluded, a pair of electrons once created cannot annihilate one another again, the scattering of light by light would be zero, etc. Although we are not experimentally sure of these phenomena, this does seem to indicate that the closed loops are necessary. To be sure, it is always possible that these matters of probability conservation, etc., will work themselves out as simply in the case of interacting particles as for those in a fixed potential. Lack- ing such a demonstration the presumption is that the difficulties of vacuum polarization are not so easily circumvented.20 An alternative procedure discussed in B is to assume that the function K+(2, 1) used above is incorrect and is to be replaced by a modified function K′

• having no singularity on the light cone. The effect of this is to provide a convergence factor C(p2 − m2) for every integral over electron momenta.21 This will multiply the integrand of (32) by C(p2 − m2)C((p + q)2 − m2), since the integral was originally δ(pa − pb + q)d4pad4pb and both pa and pb get convergence factors. The integral now converges but the result is unsatisfactory.22 One expects the current (31) to be conserved, that is qμjμ = 0 or 20It would be very interesting to calculate the Lamb shift accurately enough to be sure that the 20 megacycles expected from vacuum polarization are actually present. 21This technique also makes self-energy and radiationless scattering integrals finite even without the modification of δ+ to f+ for the radiation (and the consequent convergence factor C(k2) for the quanta). See B. 22Added to the terms given below (33) there is a term 1 4 (λ3 −2μ2 + 1 3 q2)δμν for C(k2) = −λ2(k2 − λ2)−1 , which is not gauge invariant. (In addition the charge renormalization has −7/6 added to the logarithm.) 26 qμJμν = 0. Also one expects no current if a is a gradient, or aν = qν , times a constant. This leads to the condition Jμν qν = 0 which is equivalent to qμJμν = 0 since Jμν is symmetrical. But when the expression (32) is in- tegrated with such convergence factors it does not satisfy this condition. By altering the kernel from K to another, K′, which does not satisfy the Dirac equation we have lost the gauge invariance, its consequent current conservation and the general consistency of the theory. One can see this best by calculating Jμν qν directly from (32). The ex- pression within the spur becomes (p + q − m)−1q(p − m)−1γμ which can be written as the difference of two terms: (p − m)−1γμ − (p + q − m)−1γμ Each of these terms would give the same result if the integration d4p were without a convergence factor, for the first can be converted into the second by a shift of the origin of p, namely p′ = p + q. This does not result in cancelation in (32) however, for the convergence factor is altered by the substitution. A method of making (32) convergent without spoiling the gauge in- variance has been found by Bethe and by Pauli. The convergence fac- tor for light can be looked upon as the result of superposition of the ef- fects of quanta of various masses (some contributing negatively). Like- wise if we take the factor C(p2 − m2) = −λ2(p2 − m2 − λ2)−1 so that (p2 − m2)−1C(p2 − m2) = (p2 − m2)−1 − (p2 − m2 − λ2)−1 we are taking the difference of the result for electrons of mass m and mass (λ2 + m2)1/2. But we have taken this difference for each propagation between interactions with photons. They suggest instead that once created with a certain mass the electron should continue to propagate with this mass through all the potential interactions until it closes its loop. That is if the quantity (32), integrated over some finite range of p, is called Jμν (m2) and the correspond- ing quantity over the same range of p, but with m replaced by (m2 + λ2)1/2 is Jμν (m2 + λ2) we should calculate JP μν = ∞∫ 0 [Jμν (m2) − Jμν (m2 + λ2)]G(λ)dλ, (32′) the function G(λ) satisfying ∫ ∞ 0 G(λ)dλ = 1 and ∫ ∞ 0 G(λ)λ2dλ = 0. Then in the expression for JP μν the range of p integration can be extended to infinity as the integral now converges. The result of the integration using 27 this method is the integral on dλ over G(λ) of (see Appendix C) JP μν = − e2 π (qμqν − δμν q2) ( − 1 3 ln λ2 m2 − [ 4m2 + 2q2 3q2 ( 1 − θ tanθ ) − 1 9 ]) , (33) with q2 = 4m2 sin2 θ. The gauge invariance is clear, since qμ(qμqν −q2δμν ) = 0. Operating (as it always will) on a potential of zero divergence the (qμqν − δμν q2)aν , is simply −q2aμ, the D’Alembertian of the potential, that is, the current producing the potential. The term − 1 3 (ln(λ2/m2))(qμqν − q2δμν ) therefore gives a current proportional to the current producing the potential. This would have the same effect as a change in charge, so that we would have a difference ∆(e2) between e2 and the experimentally observed charge, e2 + ∆(e2), analogous to the difference between m and the observed mass. This charge depends logarithmically on the cut-off, ∆(e2)/e2 = −(2e2/3π) ln(λ/m). After this renormalization of charge is made, no effects will be sensitive to the cut-off. After this is done the final term remaining in (33), contains the usual effects23 of polarization of the vacuum. It is zero for a free light quan- tum (q2 = 0). For small q2 it behaves as (2/15)q2 (adding − 1 5 to the logarithm in the Lamb effect). For q2 > (2m)2 it is complex, the imagi- nary part representing the loss in amplitude required by the fact that the probability that no quanta are produced by a potential able to produce pairs ((q2)1/2 > 2m) decreases with time. (To make the necessary analytic continuation, imagine m to have a small negative imaginary part, so that (1 − q2/4m2 − 1)1/2 becomes −i(q2/4m2 − 1)1/2 as q2 goes from below to above 4m2. Then θ = π/2 + iu where sin hu = +(q2/4m2 − 1)1/2, and −1/tanθ = itanhu = +i(q2 − 4m2)1/2(q2)−1/2). Closed loops containing a number of quanta or potential interactions larger than two produce no trouble. Any loop with an odd number of inter- actions gives zero (I, reference 9). Four or more potential interactions give integrals which are convergent even without a convergence factor as is well known. The situation is analogous to that for self-energy. Once the sim- ple problem of a single closed loop is solved there are no further divergence difficulties for more complex processes.24 23E. A. Uehling, Phys. Rev. 48, 55 (1935), R. Serber, Phys. Rev. 48, 49 (1935). 24There are loops completely without external interactions. For example, a pair is created virtually along with a photon. Next they annihilate, absorbing this photon. Such 28 8 LONGITUDINAL WAVES In the usual form of quantum electrodynamics the longitudinal and transverse waves are given separate treatment. Alternately the condition (∂Aμ/∂xμ)Ψ = 0 is carried along as a supplementary condition. In the present form no such special considerations are necessary for we are dealing with the solutions of the equation −2Aμ = 4πjμ, with a current jμ, which is conserved ∂jμ/∂xμ = 0. That means at least 2(∂Aμ/∂xμ) = 0 and in fact our solution also satisfies ∂Aμ/∂xμ = 0. To show that this is the case we consider the amplitude for emission (real or virtual) of a photon and show that the divergence of this amplitude vanishes. The amplitude for emission for photons polarized in the μ direction involves matrix elements of γμ. Therefore what we have to show is that the corresponding matrix elements of qμγμ = q vanish. For example, for a first order effect we would require the matrix element of q between two states p1 and p2 = p1 +q. But since q = p2 −p1 and (˜u2p1u1) = m(˜u2u1) = (˜u2p2u1) the matrix element vanishes, which proves the contention in this case. It also vanishes in more complex situations (essentially because of relation (34), below) (for example, try putting e2 = q2 in the matrix (15) for the Compton Effect). To prove this in general, suppose ai, i = 1 to N are a set of plane wave disturbing potentials carrying momenta qi, (e.g., some may be emissions or absorptions of the same or different quanta) and consider a matrix for the transition from a state of momentum p0 to pN such as aN ΠN −1 t=1 (pi −m)−1ai, where pi = pi−1 + qi (and in the product, terms with larger i are written to the left). The most general matrix element is simply a linear combination of these. Next consider the matrix between states p0 and pN + q in a situation in which not only are the ai, acting but also another potential a exp(−iq · x) where a = q. This may act previous to all ai in which case it gives aN Π(pi + q − m)−1ai(p0 + q − m)−1q which is equivalent to +aN Π(pi + q − m)−1ai since +(p0 + q − m)−1q is equivalent to (p0 + q − m)−1 × (p0 + q − m) as p0 is equivalent to m acting on the initial state. Likewise if it acts after all the potentials it gives q(pN − m)−1aN Π(pi − m)−1ai which is equivalent to −aN Π(pi − m)−1ai since pN + q − m gives zero on the final state. Or again loops are disregarded on the grounds that they do not interact with anything and are thereby completely unobservable. Any indirect effects they may have via the exclusion principle have already been included. 29 it may act between the potential ak and ak+1 for each k. This gives N −1∑ k=1 aN ΠN −1 i=k+1(pi + q − m)−1ai(pk + q − m)−1 ×q(pk − m)−1ak Πk−1 j=1 (pj − m)−1aj . However, (pk + q − m)−1q(pk − m)−1 = (pk − m)−1 − (pk + q − m)−1, (34) so that the sum breaks into the difference of two sums, the first of which may be converted to the other by the replacement of k by k − 1. There remain only the terms from the ends of the range of summation, +aN N −1 Π i=1 (pi − m)−1ai − aN N −1 Π i=1 (pi + q − m)−1ai. These cancel the two terms originally discussed so that the entire effect is zero. Hence any wave emitted will satisfy ∂Aμ/∂xμ = 0. Likewise lon- gitudinal waves (that is, waves for which Aμ = ∂φ/∂xμ or a = q cannot be absorbed and will have no effect, for the matrix elements for emission and absorption are similar. (We have said little more than that a poten- tial Aμ = ∂ϕ/∂xμ has no effect on a Dirac electron since a transformation ψ′ = exp(−iφ)ψ removes it. It is also easy to see in coordinate representa- tion using integrations by parts.) This has a useful practical consequence in that in computing probabil- ities for transition for unpolarized light one can sum the squared matrix over all four directions rather than just the two special polarization vectors. Thus suppose the matrix element for some process for light polarized in direction eμ, is eμMμ. If the light has wave vector qμ, we know from the argument above that qμMμ = 0. For unpolarized light progressing in the z direction we would ordinarily calculate M 2 x + M 2 y . But we can as well sum M 2 x + M 2 y + M 2 z − M 2 t for qμMμ implies Mt = Mz since qt = qz for free quanta. This shows that unpolarized light is a relativistically invariant con- cept, and permits some simplification in computing cross sections for such light. Incidentally, the virtual quanta interact through terms like γμ . . . γμk−2d4k. Real processes correspond to poles in the formulae for virtual processes. The pole occurs when k2 = 0, but it looks at first as though in the sum on all four values of μ, of γμ . . . γμ we would have four kinds of polarization instead of two. Now it is clear that only two perpendicular to k are effective. 30 The usual elimination of longitudinal and scalar virtual photons (leading to an instantaneous Coulomb potential) can of course be performed here too (although it is not particularly useful). A typical term in a virtual transition is γμ . . . γμk−2d4k where the . . . represent some intervening matrices. Let us choose for the values of μ, the time t, the direction of vector part K, of k, and two perpendicular directions 1, 2. We shall not change the expression for these two 1, 2 for these are represented by transverse quanta. But we must find (γt . . . γt) − (γK . . . γK). Now k = k4γt − KγK, where K = (K · K)1/2, and we have shown above that k replacing the γμ. gives zero.25 Hence KγK is equivalent to k4γt and (γt . . . γt) − (γK . . . γK) = ((K2 − k2 4 )/K2)(γt . . . γy), so that on multiplying by k−2d4k = d4k(k2 4 − K2)−1 the net effect is −(γt . . . γt)d4k/K2. The γt means just scalar waves, that is, potentials pro- duced by charge density. The fact that 1/K2 does not contain k4 means that k4 can be integrated first, resulting in an instantaneous interaction, and the d3K/K2 is just the momentum representation of the Coulomb potential, 1/r. 25A little more care is required when both γμ’s act on the same particle. Define x = k4γt + KγK, and consider (k . . . x) + x . . . k). Exactly this term would arise if a system, acted on by potential x carrying momentum −k, is disturbed by an added potential k of momentum +k (the reversed sign of the momenta in the intermediate factors in the second term x . . . k has no effect since we will later integrate over all k). Hence as shown above the result is zero, hut since (k . . . x) + (x . . . k) = k2 4 (γt . . . γt) − K2(γK . . . γK) we can still conclude (γK . . . γK) = k2 4 K−2(γt . . . γt). 31 9 KLEIN GORDON EQUATION The methods may be readily extended to particles of spin zero satisfying the Klein Gordon equation,26 2ψ − m2ψ = i∂(Aμψ)/∂xμ + iAμ∂ψ/∂xμ − AμAμψ. (35) The important kernel is now I+(2, 1) denned in (I, Eq. (32)). For a free particle, the wave function ψ(20 satisfies +2ψ − m2ψ = 0. At a point, 2, inside a space time region it is given by ψ(2) = ∫ [ψ(1)∂I+(2, 1)/∂x1μ − (∂ψ/∂x1μ)I+(2, 1)]Nμ(1)d3V1, (as is readily shown by the usual method of demonstrating Green’s the- orem) the integral being over an entire 3-surface boundary of the region (with normal vector Nμ). Only the positive frequency components of ψ con- tribute from the surface preceding the time corresponding to 2, and only negative frequencies from the surface future to 2. These can be interpreted as electrons and positrons in direct analogy to the Dirac case. The right-hand side of (35) can be considered as a source of new waves and a series of terms written down to represent matrix elements for processes of increasing order. There is only one new point here, the term in AμAμ by which two quanta can act at the same time. As an example, suppose three quanta or potentials, aμexp(−iqa · x), bμexp(−iqb · x), and cμexp(iqe · x) are to act in that order on a particle of original momentum p0μ, so that pa = p0 + qa, and pb = pa + qb; the final momentum being pc = pb + qc. 26The equations discussed in this section were deduced from the formulation of the Klein Gordon equation given in reference 5, Section 14. The function ψ in this section has only one component and is not a spinor. An alternative formal method of making the equations valid for spin zero and also for spin 1 is (presumably) by use of the Kemmer-Duffin matrices βμ satisfying the commutation relation βμβν βσ + βσ βν βμ = δμν βσ + δσν βμ. If we interpret a to mean aμβμ, rather than aμγμ, for any aμ, all of the equations in mo- mentum space will remain formally identical to those for the spin 1/2; with the exception of those in which a denominator (p − m)−1 has been rationalized to (p + m)(p2 − m2)−1 since p2 is no longer equal to a number, p · p. But p3 does equal (p · p)p that (p − m)−1 may now be interpreted as (mp + m2 + p2 − p · p)(p · p − m2)−1. This im- plies that equations in coordinate space will be valid of the function K+(2, 1) is given as K+(2, 1) = [(iO2 + m) − m−1(O2 + 2 2)]iI+(2, 1) with O2 = βμ∂/∂x2μ. This is all in virtue of the fact that the many component wave function ψ (5 components for spin 0, 10 for spin 1) satisfies (iO − m)ψ = aψ which is formally identical to the Dirac Equation. See W. Pauli, Rev. Mod. Phys. 13, 203 (1940). 32 The matrix element is the sum of three terms (p2 = pμpμ) (illustrated in Fig. 7) (pc · c + pb · c)(p2 b − m2)−1(pb · b + pa · b) × (p2 a − m2)−1(pa · a + p0 · a) −(pc · c + pb · c)(p2 b − m2)−1(b · a) − (c · b)(p2 a − m2)−1(pa · a + p0 · a). (36) The first comes when each potential acts through the perturbation i∂(Aμψ)/∂xμ + iAμ∂ψ/∂xμ. These gradient operators in momentum space mean respectively the momentum after and before the potential Aμ oper- ates. The second term comes from bμ and aμ acting at the same instant and arises from the AμAμ term in (a). Together bμ and aμ carry momentum qbμ + qaμ so that after b · a operates the momentum is p0 + qa + qb or pb. The final term comes from cμ and bμ operating together in a similar manner. The term AμAμ thus permits a new type of process in which two quanta can be emitted (or absorbed, or one absorbed, one emitted) at the same time.

There is no a · c term for the order a, b, c we have assumed. In an actual problem there would be other terms like (36) but with alterations in the order in which the quanta a, b, c act. In these terms a · c would appear. As a further example the self-energy of a particle of momentum pμ is (e2/2πim) ∫ [(2p − k)μ((p − k)2 − m2)−1 × (2p − k)μ − δμμ]d4kk−2C(k2), where the δμμ comes from the AμAμ term and represents the possibility of the simultaneous emission and absorption of the same virtual quantum. This integral without the C(k2) diverges quadratically and would not converge if C(k2) = −λ2/(k2 − λ2). Since the interaction occurs through the gradients of the potential, we must use a stronger convergence factor, for example C(k2) = λ4(k2 − λ2)−2, or in general (17) with ∫ ∞ 0 λ2G(λ)dλ = 0. In this

case the self-energy converges but depends quadratically on the cut-off λ and is not necessarily small compared to m. The radiative corrections to scattering after mass renormalization are insensitive to the cut-off just as for the Dirac equation.

When there are several particles one can obtain Bose statistics by the rule that if two processes lead to the same state but with two electrons ex- changed, their amplitudes are to be added (rather than subtracted as for Fermi statistics). In this case equivalence to the second quantization treat- ment of Pauli and Weisskopf should be demonstrable in a way very much like that given in I (appendix) for Dirac electrons. The Bose statistics mean that the sign of contribution of a closed loop to the vacuum polarization is 33 the opposite of what it is for the Fermi case (see I). It is (pb = pa + q) Jμν = e2 2πim ∫ [(pbμ + paμ)(pbν + paν )(p2 a − m2)−1 ×(p2 b − m2)−1 − δμν (p2 a − m2)−1 − δμν (p2 b − m2)−1]d4pa giving, JP μν = e2 π (qμqν − δμν q2) [ 1 6 ln λ2 m2 + 1 9 − 4m2 − q2 3q2 ( 1 − θ tanθ )] , the notation as in (33). The imaginary part for (q2)1/2 > 2m is again positive representing the loss in the probability of finding the final state to be a vacuum, associated with the possibilities of pair production. Fermi statistics would give a gain in probability (and also a charge renormalization of opposite sign to that expected).

Figure 7: Klein-Gordon particle in three potentials, Eq, (36). The coupling to the electromagnetic field is now, for example, p0 · a + pa · a, and a new possibility arises, (b), of simultaneous interaction with two quanta a · b. The propagation factor is now (p · p − m2)−1 for a particle of momentum pμ.