Space-Time Approach to Quantum Electrodynamics on Superphysics

Comparison With The Hamiltonian Method

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Electrodynamics can be looked upon in two equivalent and complemen- tary ways. One is as the description of the behavior of a field (Maxwell’s equations). The other is as a description of a direct interaction at a dis- tance (albeit delayed in time) between charges (the solutions of Lienard and Wiechert). From the latter point of view light is considered as an interaction of the charges in the source with those in the absorber. This is an imprac- tical point of view because many kinds of sources produce the same kind of effects. The field point of view separates these aspects into two simpler problems, production of light, and absorption of light. On the other hand, the field point of view is less practical when dealing with close collisions of particles (or their action on themselves). For here the source and absorber are not readily distinguishable, there is an intimate exchange of quanta. The fields are so closely determined by the motions of the particles that it is just as well not to separate the question into two problems but to consider the process as a direct interaction. Roughly, the field point of view is most practical for problems involving real quanta, while the interaction view is best for the discussion of the virtual quanta involved. We shall emphasize the interaction viewpoint in this paper, first because it is less familiar and therefore requires more discussion, and second because the important aspect in the problems with which we shall deal is the effect of virtual quanta.

EXPRESSION IN MOMENTUM AND ENERGY SPACE

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The evaluation of (9), as well as all the other more complicated expres- sions arising in these problems, is very much simplified by working in the momentum and energy variables, rather than space and time. For this we shall need the Fourier Transform of δ+(s2 21) which is −δ+(s2 21) = π−1 ∫ exp(−ik · x21)k−2d4k, (10) which can be obtained from (3) and (5) or from I, Eq. (32) noting that I+(2, 1) for m2 = 0 is δ+(s2 21) from I, Eq. (34). The k−2 means (k · k)−1 Figure 4: Radiative correction to scattering, momentum space. or more precisely the limit as δ → 0 of (k · k + iδ)−1. Further d4k means (2π)−2dk1dk2dk3dk4. If we imagine that quanta are particles of zero mass, then we can make the general rule that all poles are to be resolved by considering the masses of the particles and quanta to have infinitesimal negative imaginary parts. Using these results we see that the self-energy (9) is the matrix element between u and u of the matrix (e2/πi) ∫ γμ(p − k − m)−1γμk−2d4k, (11) 15 where we have used the expression (I, Eq. (31)) for the Fourier transform of K+. This form for the self-energy is easier to work with than is (9). The equation can be understood by imagining (Fig. 3) that the electron of momentum p emits (γμ) a quantum of momentum k, and makes its way now with momentum p − k to the next event (factor (p − k − m)−1) which is to absorb the quantum (another γμ). The amplitude of propagation of quanta is k−2. (There is a factor e2/πi for each virtual quantum). One inte- grates over all quanta. The reason an electron of momentum p propagates as 1/(p − m) is that this operator is the reciprocal of the Dirac equation operator, and we are simply solving this equation. Likewise light goes as 1/k2, for this is the reciprocal D’Alembertian operator of the wave equation of light. The first γμ, represents the current which generates the vector po- tential, while the second is the velocity operator by which this potential is multiplied in the Dirac equation when an external field acts on an electron. Using the same line of reasoning, other problems may be set up directly in momentum space. For example, consider the scattering in a potential A = Aμγμ varying in space and time as a exp(−iq · x). An electron initially in state of momentum p1 = p1μγμ will be deflected to state p2 where p2 = p1 + q. The zero-order answer is simply the matrix element of a between states 1 and 2. We next ask for the first order (in e2) radiative correction due to virtual radiation of one quantum. There are several ways this can happen. First for the case illustrated in Fig. 4(a), find the matrix: Figure 5: Compton scattering, Eq. (15). (e2/πi) ∫ γμ(p2 − k − m)−1a(p1 − k − m)−1γμk−2d4k. (12) 16 For in this case, first13 a quantum of momentum k is emitted (γμ), the electron then having momentum p1 − k and hence propagating with factor (p1 − k − m)−1. Next it is scattered by the potential (matrix a) receiv- ing additional momentum q, propagating on then (factor (p2 − k − m)−1) with the new momentum until the quantum is reabsorbed (γμ). The quan- tum propagates from emission to absorption (k−2) and we integrate over all quanta (d4k), and sum on polarization μ. When this is integrated on k4, the result can be shown to be exactly equal to the expressions (16) and (17) given in B for the same process, the various terms coming from residues of the poles of the integrand (12). Or again if the quantum is both emitted and reabsorbed before the scattering takes place one finds (Fig. 4(b)) (e2/πi) ∫ a(p1 − m)−1γμ(p1 − k − m)−1γμk−2d4k, (13) or if both emission and absorption occur after the scattering, (Fig. 4(c)) (e2/πi) ∫ γμ(p2 − k − m)−1γμ(p2 − m)−1ak−2d4k. (14) These terms are discussed in detail below. We have now achieved our simplification of the form of writing matrix elements arising from virtual processes. Processes in which a number of real quanta is given initially and finally offer no problem (assuming correct nor- malization). For example, consider the Compton effect (Fig. 5(a)) in which an electron in state p1 absorbs a quantum of momentum q1, polarization vector e1μ so that its interaction is e1μγμ = e1, and emits a second quantum of momentum −q2 polarization e2 to arrive in final state of momentum p2. The matrix for this process is e2(p1 + q1 − m)−1e1. The total matrix for the Compton effect is, then, e2(p1 + q1 − m)−1e1 + e1(p1 + q2 − m)−1e3, (15) the second term arising because the emission of e2, may also precede the ab- sorption of e1 (Fig. 5(b)). One takes matrix elements of this between initial and final electron states (p1 + q1 = p2 − q2), to obtain the Klein Nishina formula. Pair annihilation with emission of two quanta, etc., are given by the same matrix, positron states being those with negative time component of p. Whether quanta are absorbed or emitted depends on whether the time component of q is positive or negative. 13First, next, etc., here refer not to the order in true time but to the succession of events along the trajectory of the electron. That is, more precisely, to the order of appearance of the matrices in the expressions. 17 5 THE CONVERGENCE OF PROCESSES WITH VIRTUAL QUANTA These expressions are, as has been indicated, no more than a re-expression of conventional quantum electrodynamics. As a consequence, many of them are meaningless. For example, the self-energy expression (9) or (11) gives an infinite result when evaluated. The infinity arises, apparently, from the coincidence of the δ–function singularities in K+(4, 3) and δ+(s2 43). Only at this point is it necessary to make a real departure from conventional electrodynamics, a departure other than simply rewriting expressions in a simpler form. We desire to make a modification of quantum electrodynamics analogous to the modification of classical electrodynamics described in a previous arti- cle, A. There the δ(s2 12) appearing in the action of interaction was replaced by f (s2 12) where f (x) is a function of small width and great height. The obvious corresponding modification in the quantum theory is to replace the δ+(s2) appearing the quantum mechanical interaction by a new function f=(s2). We can postulate that if the Fourier transform of the classical f (s2 12) is the integral over ail k of F (k2)exp(−ik · x12)d4k, then the Fourier transform of f+(s2) is the same integral taken over only positive frequencies k4 for t2 > t1 and over only negative ones for t2 < t1 in analogy to the relation of δ+(s2) to δ(s2). The function f (s2) = f (x · x) can be written 14 as f (x · x) = (2π)−2 ∞∫ k4=0 ∫ sin(k4|x4|) × cos(K · x)dk4d3Kg(k · k), where g(k · k) is k−1 4 times the density of oscillators and may be expressed for positive k4 as (A, Eq. (16)) g(k2) = ∞∫ 0 (δ(k2) − δ(k2 − λ2))G(λ)dλ, where ∞∫ 0 G(λ)dλ = 1 and G involves values of λ large compared to m. This simply means that the amplitude for propagation of quanta of momentum 14This relation is given incorrectly in A, equation just preceding 16. 18 k is −F+(k3) = π−1 ∫ ∞ 0 (k−2 − (k2 − λ2)−1)G(λ)dλ, rather than k−2. That is, writing F+(k2) = −π−1k−2C(k2), −f+(s2 12) = π−1 ∫ exp(−ik · x12)k−2C(k2)d4k. (16) Every integral over an intermediate quantum which previously involved a factor d4k/k2 is now supplied with a convergence factor C(k2) where C(k2) = ∫ ∞ 0 −λ2(k2 − λ2)−1G(λ)dλ. (17) The poles are defined by replacing k2 by k2 + iδ in the limit δ → 0. That is λ2 may be assumed to have an infinitesimal negative imaginary part. The function f+(s2 12) may still have a discontinuity in value on the light cone. This is of no influence for the Dirac electron. For a particle satisfying the Klein Gordon equation, however, the interaction involves gradients of the potential which reinstates the δ function if f has discontinuities. The condition that f is to have no discontinuity in value on the light cone implies k2C(k2) approaches zero as k2 approaches infinity. In terms of G(λ) the condition is ∞∫ 0 λ2G(λ)dλ = 0. (18) This condition will also be used in discussing the convergence of vacuum polarization integrals. The expression for the self-energy matrix is now (e2/πi) ∫ γμ(p − k − m)−1γμk−2d4kC(k2), (19) which, since C(k2) falls off at least as rapidly as 1/k2, converges. For prac- tical purposes we shall suppose hereafter that C(k2) is simply −λ2/(k2 −λ2) implying that some average (with weight G(λ)dλ) over values of λ may be taken afterwards. Since in all processes the quantum momentum will be contained in at least one extra factor of the form (p − k − m)−1 represent- ing propagation of an electron while that quantum is in the field, we can expect all such integrals with their convergence factors to converge and that the result of ail such processes will now be finite and definite (excepting the 19 processes with closed loops, discussed below, in which the diverging integrals are over the momenta of the electrons rather than the quanta). The integral of (19) with C(k2) = −λ2(k2 − λ2)−1 noting that p2 = m2, λ m and dropping terms of order m/λ is (see Appendix A) (e2/2π)[4m(ln(λ/m) + 1 2 ) − p(ln(λ/m) + 5/4)]. (20) When applied to a state of an electron of momentum p satisfying pu = mu, it gives for the change in mass (as in B, Eq. (9)) ∆m = m(e2/2π)(3 ln(λ/m) + 3 4 ). (21)

Space-Time Approach to Quantum Electrodynamics

Wed, 30 Sep 2026 00:00:00 +0000

Abstract

The Self–Energy Problem

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Having a term representing the mutual interaction of a pair of charges, we must include similar terms to represent the interaction of a charge with itself. For under some circumstances what appears to be two distinct electrons 9Although in the expressions stemming from (4) the quanta are virtual, this is not actually a theoretical limitation. One way to deduce the correct rules for real quanta from (4) is to note that in a closed system all quanta can be considered as virtual (i.e., they have a known source and are eventually absorbed) so that in such a system the present description is complete and equivalent to the conventional one. In particular, the relation of the Einstein A and B coefficients can be deduced. A more practical direct deduction of the expressions for real quanta will be given in the subsequent paper. It might be noted that (4) can be rewritten as describing the action on a, K(1)(3, 1) = i ∫ K+(3, 5) × A(5)K+(5, 1)dτ5 of the potential Aμ(5) = e2 ∫ K+(4, 6)δ+(s2 56)γμ × K+(6, 2)dτ6 arising from Maxwell’s equations −2Aμ = 4πjμ from a “current” jμ(6) = e2K+(4, 6)γμK+(6, 2) produced by particle b in going from 2 to 4. This is virtue of the fact that δ+ satisfies −2 2δ+(s2 21) = 4πδ(2, 1). (5)

APPLICATION TO MESON THEORIES

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The theories which have been developed to describe mesons and the in- teraction of nucleons can be easily expressed in the language used here. Cal- culations, to lowest order in the interactions can be made very easily for the 34 various theories, but agreement with experimental results is not obtained. Most likely all of our present formulations are quantitatively unsatisfactory. We shall content ourselves therefore with a brief summary of the methods which can be used. The nucleons are usually assumed to satisfy Dirac’s equation so that the factor for propagation of a nucleon of momentum p is (p − M )−1 where M is the mass of the nucleon (which implies that nucleons can be created in pairs). The nucleon is then assumed to interact with mesons, the various theories differing in the form assumed for this interaction. First, we consider the case of neutral mesons. The theory closest to electrodynamics is the theory of vector mesons with vector coupling. Here the factor for emission or absorption of a meson is gγμ, when this meson is “polarized” in the μ direction. The factor g the “mesonic charge,” replaces the electric charge e. The amplitude for propagation of a meson of momen- tum q in intermediate states is (q2 − μ2)−1 (rather than q−2 as it is for light) where μ is the mass of the meson. The necessary integrals are made finite by convergence factors C(q2 − μ2) as in electrodynamics. For scalar mesons with scalar coupling the only change is that one replaces the γμ by 1 in emission and absorption. There is no longer a direction of polarization, μ, to sum upon. For pseudo-scalar mesons, pseudoscalar coupling replace γμ by γ5 = iγxγyγz γt. For example, the self-energy matrix of a nucleon of momentum p in this theory is (g2/πi) ∫ γ5(p − k − M )−1γ5d4k(k2 − μ2)−1C(k2 − μ2). Other types of meson theory result from the replacement of γμ, by other expressions (for example by 1 2 (γμγν − γν γμ) with a subsequent sum over all μ and ν for virtual mesons). Scalar mesons with vector coupling result from the replacement of γμ by μ−1q where q is the final momentum of the nucleon minus its initial momentum, that is, it is the momentum of the meson if absorbed, or the negative of the momentum of a meson emitted. As is well known, this theory with neutral mesons gives zero for all processes, as is proved by our discussion on longitudinal waves in electrodynamics. Pseudoscalar mesons with pseudo-vector coupling corresponds to γμ being replaced by μ−1γ5q white vector mesons with tensor coupling correspond to using (2μ)−1(γμq − qγμ). These extra gradients involve the danger of producing higher divergencies for real processes. For example, γ5q gives a logarithmically divergent interaction of neutron and electron.27 Although 27M. Slotnick and W. Heitler, Phys. Rev. 75, 1645 (1949). 35 these divergencies can be held by strong enough convergence factors, the results then are sensitive to the method used for convergence and the size of the cut-off values of λ. For low order processes μ−1γ5q is equivalent to the pseudoscalar interaction 2M μ−1γ5 because if taken between free particle wave functions of the nucleon of momenta p1 and p2 = p1 + q, we have (˜u2γ5qu1) = (˜u2γ5(p2 − p1)u1) = −(˜u2p2γ5u1) −(˜u2γ5p1u1) = −2M (˜u2γ5u1) since γ5 anticommutes with p2 and p2 operating on the state 2 equivalent to M as is 1 on the state 1. This shows that the γ5 interaction is unusually weak in the non-relativistic limit (for example the expected value of γ5 for a free nucleon is zero), but since γ2 5 = 1 is not small, pseudoscalar theory gives a more important interaction in second order than it does in first. Thus the pseudoscalar coupling constant should be chosen to fit nuclear forces including these important second order processes.28 The equivalence of pseudoscalar and pseudo-vector coupling which holds for low order pro- cesses therefore does not hold when the pseudoscalar theory is giving its most important effects. These theories will therefore give quite different results in the majority of practical problems. In calculating the corrections to scattering of a nucleon by a neutral vector meson field (γμ) due to the effects of virtual mesons, the situation is just as in electrodynamics, in that the result converges without need for a cut-off and depends only on gradients of the meson potential. With scalar (1) or pseudoscalar (γμ) neutral mesons the result diverges logarithmically and so must be cut off. The part sensitive to the cut-off, however, is di- rectly proportional to the meson potential. It may thereby be removed by a renormalization of mesonic charge g. After this renormalization the results depend only on gradients of the meson potential and are essentially inde- pendent of cut-off. This is in addition to the mesonic charge renormalization coming from the production of virtual nucleon pairs by a meson, analogous to the vacuum polarization in electrodynamics. But here there is a further difference from electrodynamics for scalar or pseudoscalar mesons in that the polarization also gives a term in the induced current proportional to the meson potential representing therefore an additional renormalization of the mass of the meson which usually depends quadratically on the cut-off. Next consider charged mesons in the absence of an electromagnetic field. One can introduce isotopic spin operators in an obvious way. (Specifically 28H. A. Bethe, Bull. Am. Phys. Soc. 24, 3, Z3 (Washington, 1949). 36 replace the neutral γ5, say, by τiγ − 5 and sum over i = 1, 2, where τ1 = τ++τ−, τ2 = i(τ+−τ−) and τ+ changes neutron to proton (τ+ on proton = 0) and τ− changes proton to neutron.) It is just as easy for practical problems simply to keep track of whether the particle is a proton or a neutron on a diagram drawn to help write down the matrix element. This excludes certain processes. For example in the scattering of a negative meson from q1 to q2 by a neutron, the meson q2 must be emitted first (in order of operators, not time) for the neutron cannot absorb the negative meson q1 until it becomes a proton. That is, in comparison to the Klein Nishina formula (15), only the analogue of second term (see Fig. 5(b)) would appear in the scattering of negative mesons by neutrons, and only the first term (Fig. 5 (a)) in the neutron scattering of positive mesons. The source of mesons of a given charge is not conserved, for a neutron capable of emitting negative mesons may (on emitting one, say) become a proton no longer able to do so. The proof that a perturbation q gives zero, discussed for longitudinal electromagnetic waves, fails. This has the consequence that vector mesons, if represented by the interaction γμ, would not satisfy the condition that the divergence of the potential is zero. The interaction is to be taken29 as γμ − μ−2qμq in emission and as γμ in absorp- tion if the real emission of mesons with a non-zero divergence of potential is to be avoided. (The correction term μ−2qμq gives zero in the neutral case.) The asymmetry in emission and absorption is only apparent, as this is clearly the same thing as subtracting from the original γμ . . . γμ, a term μ−2q . . . q. That is, if the term −μ−2qμq is omitted the resulting theory 29The vector meson field potentials ϕμ satisfy −∂/∂xν (∂ϕμ/∂xν − ∂ϕν /∂xμ) − μ2ϕμ = −4πsμ, where sμ, the source for such mesons, is the matrix element of γμ between states of neutron and proton. By taking the divergence ∂/∂xμ of both sides, conclude that ∂ϕν /∂xν = 4πμ−2∂sν /∂xν , so that the original equation can lie rewritten as 2ϕμ − μ2ϕμ = −4π(sμ + μ−2∂/∂xμ(∂sν /∂xν )). The right hand side gives in momentum representation γμ − μ−2qμqν γν the left yields the (q2 − μ2)−1 and finally the interaction sμϕμ in the Lagrangian gives the γμ on absorption. Proceeding in this way find generally that particles of spin one can be represented by a four-vector uμ (which, for a free particle of momentum q satisfies q · u = 0). The propaga- tion of virtual particles of momentum q from state ν to μ is represented by multiplication by the 4-4 matrix (or tensor) Pμν = (δμν − μ2qμqν ) × (q2 − μ2)−1. The first-order interac- tion (from the Proca equation) with an electromagnetic potential a exp(ik · x) corresponds to multiplication by the matrix Eμν = (q2 · a + q1 · a)δμν − q2ν aμ − q1ν aν , where q1 and q2 = q1 + k are the momenta before and after the interaction. Finally, two potentials a, b may act simultaneously, with matrix E′ μν = −(a · b)δμν + bμaν 37 describes a combination of mesons of spin one and spin zero. The spin zero mesons, coupled by vector coupling q, are removed by subtracting the term μ−2q . . . q. The two extra gradients q . . . q make the problem of diverging integrals still more serious (for example the interaction between two protons corre- sponding to the exchange of two charged vector mesons depends quadrati- cally on the cut-off if calculated in a straightforward way). One is tempted in this formulation to choose simply γμ . . . γμ and accept the admixture of spin zero mesons. But it appears that this leads in the conventional formalism to negative energies for the spin zero component. This shows one of the advan- tages of the method of second quantization of meson fields over the present formulation. There such errors of sign are obvious while here we seem to be able to write seemingly innocent expressions which can give absurd re- sults. Pseudovector mesons with pseudovector coupling correspond to using γ5(γμ − μ−2qμq) for absorption and γ5γμ for emission for both charged and neutral mesons. In the presence of an electromagnetic field, whenever the nucleon is a proton it interacts with the field in the way described for electrons. The meson interacts in the scalar or pseudoscalar case as a particle obeying the Klein-Gordon equation. It is important here to use the method of calculation of Bethe and Pauli, that is, a virtual meson is assumed to have the same “mass” during all its interactions with the electromagnetic field. The result for mass μ and for (μ2 + λ2)1/2 are subtracted and the difference integrated over the function G(λ)dλ. A separate convergence factor is not provided for each meson propagation between electromagnetic interactions, otherwise gauge invariance is not insured. When the coupling involves a gradient, such as γ − 5q where q is the final minus the initial momentum of the nucleon, the vector potential A must be subtracted from the momentum of the proton. That is, there is an additional coupling ±γ5A (plus when going from proton to neutron, minus for the reverse) representing the new possibility of a simultaneous emission (or absorption) of meson and photon. Emission of positive or absorption of negative virtual mesons are repre- sented in the same term, the sign of the charge being determined by temporal relations as for electrons and positrons. Calculations are very easily carried out in this way to lowest order in g2 for the various theories for nucleon interaction, scattering of mesons by nucleons, meson production by nuclear collisions and by gamma-rays, nu- clear magnetic moments, neutron electron scattering, etc., However, no good agreement with experiment results, when these are available, is obtained. Probably all of the formulations are incorrect. An uncertainty arises since 38 the calculations are only to first order in g2, and are not valid if g2/ℏc is large. The author is particularly indebted to Professor H. A. Bethe for his explanation of a method of obtaining finite and gauge invariant results for the problem of vacuum polarization. He is also grateful for Professor Bethe’s criticisms of the manuscript, and for innumerable discussions during the development of this work. He wishes to thank Professor J. Ashkin for his careful reading of the manuscript. APPENDIX In this appendix a method will be illustrated by which the simpler integrals appearing in problems in electrodynamics can be directly evaluated. The integrals arising in more complex processes lead to rather complicated functions, but the study of the relations of one integral to another and their expression in terms of simpler integrals may be facilitated by the methods given here. As a typical problem consider the integral (12) appearing in the first order radiationless scattering problem: ∫ γμ(p2 − k − m)−1a(p1 − k − m)−1γμk−2d4kC(k2), (1a) where we shall take C(k2) to be typically −λ2(k2 − λ2)−1 and d4k means (2π)−2dk1dk2dk3dk4. We first rationalize the factors (p − k − m)−1 = (p − k + m) ((p − k)2 − m2)−1 obtaining, ∫ γμ(p2 − k + m)a(p1 − k + m)γμk−2d4kC(k2) ×((p1 − k)2 − m2)−1((p2 − k)2 − m2)−1. (2a) The matrix expression may be simplified. It appears to be best to do so after the integrations are performed. Since AB = 2A · B − BA where A · B = AμBμ is a number commuting with all matrices, find, if R is any expression, and A a vector, since γμA = −Aγμ + 2Aμ, γμARγμ = −AγμRγμ + 2RA. (3a) Expressions between two γμ’s can be thereby reduced by induction. Particularly useful are γμγμ = 4 γμAγμ = −2A γμABγμ = 2(AB + BA) = 4A · B γμABCγμ = −2CBA (4a) 39 where A, B, C are any three vector-matrices (i.e., linear combinations of the four γμs), In order to calculate the integral in (2a) the integral may be written as the sum of three terms (since k = kσ γσ ), γμ(p2 + m)a(p1 + m)γμJ1 − [γμγσ a(p1 + m)γμ +γμ(p2 + m)aγσ γμ]J2 + γμγσ aγτ γμJ3, (5a) where J(1; 2; 3) = ∫ (1; kσ ; kσ kτ )k−2d4kC(k2) ×((p2 − k)2 − m2)−1((p1 − k)2 − m2)−1. (6a) That is for J1 the (1; kσ ; kσ kτ ) is replaced by 1, for J2 by kσ and for J3 by kσ kτ . More complex processes of the first order involve more factors like ((p3 − k)2 − m2)−1 and a corresponding increase in the number of k’s which may appear in the numerator, as kσ kτ kν . . .. Higher order processes involving two or more virtual quanta involve similar integrals but with factors possibly involving k + k′ instead of just k, and the integral extending on k−2d4kC(k2)k′−2d4kC(′ 2). They can be simplified by methods analogous to those used on the first order integrals. The factors (p − k)2 − m2 may be written (p − k)2 − m2 = k2 − 2p · k − ∆, (7a) where ∆ = m2 − p2, ∆1 = m2 1 − p2 1, etc., and we can consider dealing with cases of greater generality in that the different denominators need not have the same value of the mass m. In our specific problem (6a) p2 1 = m2 that ∆1 = 0, but we desire to work with greater generality. Now for the factor C(k2)/k2 we shall use −λ2(k2 − λ2)−1k−2. This can be written as −λ2/(k2 − λ2)k2 = k−2C(k2) = − λ2 ∫ 0 dL(k2 − L)−2. (8a) Thus we can replace k−2C(k2) by (k2 − L)−2 and at the end integrate the result with respect to L from zero to λ2. We can for many practical purposes consider λ2 very large relative to m2 or p2. When the original integrai converges even without the convergence factor, it will be obvious since the L integration will then be convergent to infinity. If an infra-red catastrophe exists in the integral one can simply assume quanta have a small mass λmin and extend the integral on L from λ2 min to λ2, rather than from zero to λ2. We then have to do integrals of the form ∫ (1; kσ ; kσ kτ d4k(k2 − L)−2(k2 − 2p1 · k − ∆1)−1 × (k2 − 2p2 · k − ∆2)−1, (9a) where by (1; kσ ; kσ kτ ) we mean that in the place of this symbol either 1, or kσ or kσ kτ may stand in different cases. In more complicated problems there may be 40 more factors (k2 − 2pi · k − ∆i)−1 or other powers of these factors (the (k2 − L)−2 may be considered as a special case of such a factor with pi = 0, ∆i = L) and further factors like kσ kτ kρ . . . in the numerator. The poles in all the factors are made definite by the assumption that L, and the ∆’s have infinitesimal negative imaginary parts. We shall do the integrals of successive complexity by induction. We start with the simplest convergent one, and show ∫ d4k(k2 − L)−3 = (8iL)−1. (10a) For this integral is ∫ (2π)−2dk4d3K(k2 4 − K · K − L)−3 where the vector K, of magnitude K = (K · K)1/2 is k1, k2, k3. The integral on k4 shows third order poles at k4 = +(K2 + L)1/2 and k4 = −(K2 + L)1/2. Imagining, in accordance with our definitions, that L has a small negative imaginary part only the first is below the real axis. The contour can be closed by an infinite semi-circle below this axis, without change of the value of the integral since the contribution from the semi-circle vanishes in the limit. Thus the contour can be shrunk about the pole k4 = +(K2 + L)1/2 and the resulting k4, integral is −2πi times the residue at this pole. Writing k4 = (k2 + L)1/2 + and expanding (k2 4 − K2 − L)−3 = −3( + 2(K2 + L)1/2)−3 in powers of , the residue, being the coefficient of the term −1, is seen to be 6(2(K2 + L)1/2)−5 so our integral is −(3i/32π) ∞∫ 0 4πK2dK(K2 + L)−5/2 = (3/8i)(1/3L) establishing (10a). We also have ∫ kσ d4k(k2 − L)−3 = 0 from the symmetry in the k space. We write these results as (8i) ∫ (1; kσ )d4k(k2 − L)−3 = (1; 0)L−1, (11a) where in the brackets (1; kσ ) and (1; 0) corresponding entries are to he used. Substituting k = k′ − p in (11a) and calling L − p2 = ∆ shows that (8i) ∫ (1; kσ )d4k(k2 − 2p · k − ∆)−3 = (1; pσ )(p2 + ∆)−1. (12a) By differentiating both sides of (12a) with respect to ∆ or with respect to pτ there follows directly (24i) ∫ (1; kσ ; kσ kτ )d4k(k2 − 2p · k − ∆)−4 = −(1; pσ ; pσ pτ − 1 2 δστ (p2 + ∆))(p2 + ∆)−2. (13a) Further differentiations give directly successive integrals including more k factors in the numerator and higher powers of (k2 − 2p · k − ∆) in the denominator. 41 The integrals so far only contain one factor in the denominator. To obtain results for two factors we make use of the identity a−1b−1 = ∫ 1 0 dx(ax + b(1 − x))−2, (14a) (suggested by some work of Schwinger’s involving Gaussian integrals). This rep- resents the product of two reciprocals as a parametric integral over one and will therefore permit integrals with two factors to be expressed in terms of one. For other powers of a, b we make use of all of the identities, such as a−2b−1 = ∫ 1 0 2xdx(ax + b(1 − x))−3, (15a) deducible from (14a) by successive differentiations with respect to a or b. To perform an integral, such as (8i) ∫ (1; kσ )d4k(k2 − 2p1 · k − ∆1)−2(k2 − 2p2 · k − ∆2)−1, (16a) write, using (15a), (k2 − 2p1 · k − ∆1)−2(k2 − 2p2 · k − ∆2)−1 = ∫ 1 0 2xdx(k2 − 2px · k − ∆x)−3, where px = xp1 + (1 − x)p2 and ∆x = x∆1 + (1 − x)∆2, (17a) (note that ∆x is not equal to m2 − p2 x) so that the expression (l6a) is (8i) ∫ 1 0 2xdx(1; kσ )d4k(k2 − 2px · k∆x)−3 which may now be evaluated by (12a) and is (16a) = ∫ 1 0 (1; pxσ )2xdx(p2 x + ∆x)−1, (18a) where, px, ∆x are given in (17a). The integral in (18a) is elementary, being the integral of ratio of polynomials, the denominator of second degree in x. The general expression although readily obtained is a rather complicated combination of roots and logarithms. Other integrals can be obtained again by parametric differentiation. For exam- ple differentiation of (16a), (18a) with respect to ∆2 or p2τ gives (8i) ∫ (1; kσ ; kσ kτ )d4k(k2 − 2p1 · k − ∆1)−2(k2 − 2p2 · k − ∆2)−2 = − ∫ 1 0 (1; pxσ ; pxσ pxτ − 1 2 δστ (x2p2 + ∆x)) × 2x(1 − x)dx(p2 x + ∆x)−2, (19a) again leading to elementary integrals. 42 As an example, consider the case that the second factor is just (k2 − L)−2 and in the first put p1 = p, ∆1 = ∆. Then px = xp, ∆x − x∆ + (1 − x)L. There results (8i) ∫ (1; kσ ; kσ kτ )d4k(k2 − L)−2(k2 − 2p · k − ∆)−2 = − ∫ 1 0 (1; xpσ ; x2pσ pτ − 1 2 δστ (x2p2 + ∆x)) × 2x(1 − x)dx(x2p2 + ∆x)−2. (20a) Integrals with three factors can be reduced to those involving two by using (14a) again. They, therefore, lead to integrals with two parameters (e.g., see application to radiative correction to scattering below). The methods of calculation given in this paper are deceptively simple when applied to the lower order processes. For processes of increasingly higher orders the complexity and difficulty increases rapidly, and these methods soon become impractical in their present form. A. Self-Energy The self-energy integral (19) is (e2/πi) ∫ γμ(p − k − m)−1γμk−2d4kC(k2), (19) so that it requires that we find (using the principle of (8a)) the integral on L from 0 to λ2 of ∫ γμ(p − k + m)γμd4k(k2 − L)−2(k2 − 2p · k)−1, since (p − k)2 − m2 = k2 − 2p · k, as p2 = m2. This is of the form (16a) with ∆1 = L, p1 = 0, ∆2 = 0, p2 = p so that (18a) gives, since px = (1 − x)p, ∆x = xL, (8i) ∫ (1; kσ )d4k(k2 − L)−2(k2 − 2p · k)−1 = ∫ 1 0 (1; (1 − x)pσ 02xdx((1 − x)2m2 xL)−1, or performing the integral on L, as in (8), (8i) ∫ (1; kσ )d4kk−2C(k2)(k2−2p·k)−1 = ∫ 1 0 (1; (1−x)pσ )2dx ln xλ2 + (1 − x)2m2 (1 − x)2m2 . Assuming now that λ2 m2 we neglect (1 − x)2m2 relative to xλ2 in the ar- gument of the logarithm, which then becomes (λ2/m2)(x/(1 − x)2). Then since ∫ 1 0 dx ln(x(1 − x)−2) = 1 and ∫ 1 0 (1 − x)dx ln(x(1 − x)−2) = −(1/4) find (8i) ∫ (1; kσ )k−2C(k2)d4k(k2 − 2p · k)−1 = ( 2 ln λ2 m2 + 2; pσ ( ln λ2 m2 − 1 2 )) , 43 so that substitution into (19) (after the (p − k−m)−1 in (19) is replaced by (p − k+ m)(k2 − 2p · k)−1) gives (19) = (e2/8π)γμ[(p + m)(2 ln(λ2/m2) + 2) − p(ln(λ2/m2) − 1 2 )]γμ = (e2/8π)[8m(ln(λ2/m2) + 1) − p(2 ln(λ2m2) + 5)], (20) using (4a) to remove the γμ’s. This agrees with Eq. (20) of the text, and gives the self-energy (21) when p is replaced by m. B. Corrections to Scattering The term (12) in the radiationless scattering, after rationalizing the matrix denominators and using p2 1 = p2 2 = m2 requires the integrals (9a), as we have discussed. This is an integral with three denominators which we do in two stages. First the factors (k2 − 2p1 · k) and (k2 − 2p2 · k) are combined by a parameter y; (k2 − 2p1 · k)−1(k2 − 2p2 · k)−1 = ∫ 1 0 dy(k2 − 2py · k)−2, from (14a) where py = yp1 + (1 − y)p2. (21a) We therefore need the integrals (8i) ∫ (1; kσ ; kσ kτ )d4k(k2 − L)−2(k2 − 2py · k)−2, (22a) which we will then integrate with respect to y from 0 to 1. Next we do the integrals (22a) immediately from (20a) with p = py , ∆ = 0: (22a) = − ∫ 1 0 ∫ 1 0 (1; xpyσ ; x2pyσ pyτ − 1 2 δστ (x2p2 y + (1 − x)L))2x(1 − x)dx(x2p2 y + L(1 − x))−2dy. We now turn to the integrals on L as required in (8a). The first term, (1), in (1; kσ ; kσ kτ ) gives no trouble for large L, but if L is put equal to zero there results x−2p−2 y which leads to a diverging integral on x as x → 0. This infra-red catastrophe is analyzed by using λ2 min for the lower limit of the L integral. For the last term the upper limit of L must be kept as λ2. Assuming λ2 min p2 y λ2 the x integrals which remain are trivial, as in the self-energy case. One finds −(8i) ∫ (k2 − λ2 min)−1d4kC(k2 − λ2 min)(k2 − 2p1 · k)−1(k2 − 2p2 · k)−1

∫ 1 0 p−2 y dy ln(p2 y /λ2 min) (23a) 44 −(8i) ∫ kσ k−2d4kC(k2)(k2 − 2p1 · k)−1(k2 − 2p2 · k)−1 = 2 ∫ 1 0 pyσ p−2 y dy, (24a) −(8i) ∫ kσ kτ k−2d4kC(k2)(k2 − 2p1 · k)−1(k2 − 2p2 · k)−1

∫ 1 0 pyσ pyτ p−2 y dy − 1 2 δστ ∫ 1 0 dy ln(λ2p−2 y ) + 1 4 δστ . (25a) The integrals on y give, ∫ 1 0 p−2 y dy ln(p2 y λ−2 min) = 4(m2 sin 2θ)−1 [ θ ln(mλ−1 min) − ∫ θ 0 αtanαdα ] , (26a) ∫ 1 0 pyσ p−2 y dy = θ(m2 sin 2θ)−1(p1σ + P2σ ), (27a) ∫ 1 0 pyσ pyτ p−2 y dy = θ(2m2 sin 2θ)−1(p1σ + p1τ (p2σ + p2τ ) +q−2qσ qτ (1 − θctnθ), (28a) ∫ 1 0 dy ln(λ2p−2 y ) = ln(λ2/m2) + 2(1 − θctnθ). (29a) These integrals on y were performed as follows. Since p2 = p1 + q where q is the momentum carried by the potential, it follows from p2 2 = p2 1 = m2 that 2p1 ·q = −q2 so that since py = p1+q(1−y), p2 y = m2−q2y(1−y). The substitution 2y−1 = tanθ where θ is defined by 4m2 sin2 θ = q2 is useful for it means p2 y = m2sec2α/sec2θ and p−2 y dy = (m2 sin 2θ))−1dα where α goes from −θ to +θ. These results are substituted into the original scattering formula (2a), giving (22). It has been simplified by frequent use of the fact that p1 operating on the initial state is m and likewise p2 when it appears at the left is replacable by m. (Thus, to simplify: γμp2ap1γμ = −2p1ap2 by (4a), = −2(p2 − q)a(p1 + q) = −2(m − q)a(m + q). A term like qaq = −q2a + 2(a · q)q is equivalent to just −q2a since q = p2 − p1 = m − m has zero matrix element.) The renormalization term requires the corresponding integrals for the special case q = 0. 45 C. Vacuum Polarization The expressions (32) and (32′) for Jμν in the vacuum polarization problem require the calculation of the integral Jμν (m2) = e2 πi ∫ Sp[γμ(p − 1 2 q + m)γν p + 1 2 q + m)]d4p ×((p − 1 2 q)2 − m2)−1((p + 1 2 q)2 − m2)−1, (32) where we have replaced p by p − 1 2 q to simplify the calculation somewhat. We shall indicate the method of calculation by studying the integral, I(m2) = ∫ pσ pτ d4p((p − 1 2 q)2 − m2)−1((p + 1 2 q)2 − m2)−1. The factors in the denominator, p2 − p · q − m2 + 1 4 q2 and p2 + p · q − m2 + 1 4 q2 are combined as usual by (8a) but for symmetry we substitute x = 1 2 (1 + η), (1 − x) = 1 2 (1 − η) and integrate η from −1 to +1: I(m2) = +1∫ −1 pσ pτ d4p(p2 − ηp · q − m2 + 1 4 q2)−2dη/2. (30a) But the integral on p will not be found in our list for it is badly divergent. However, as discussed in Section 7, Eq. (34′) we do not wish I(m2) but rather ∞∫ 0 [I(m2) − I(m2 + λ2)]G(λ)dλ. We can calculate the difference I(m2) − I(m2 + λ2) by first calculating the derivative I′(m2 + L) of I with respect to m2 at m2 + L and later integrating L from zero to λ2. By differentiating (30a), with respect to m2 find, I′(m2 + L) = +1∫ −1 pσ pτ d4p(p2 − ηp · q − m2 − L + 1 4 q2)−3dη. This still diverges, but we can differentiate again to get I′′(m2 + L) = 3 +1∫ −1 pσ pτ d4p(p2 − ηp · q − m2 − L + 1 4 q2)−4dη = −(8i)−1 +1∫ −1 ( 1 4 η2qσ qτ D−2 − 1 2 δστ D−1)dη (31a) (where D = 1 4 (η2 − 1)q2 + m2 + L), which now converges and has been evaluated by (13a) with p = 1 2 ηq and ∆ = m2 + L − 1 4 q2. Now to get I′ we may integrate 46 I′′ with respect to L as an indefinite integral and we may choose any convenient arbitrary constant. This is because a constant C in I′ will mean a term −Cλ2 in I(m2) − I(m2 + λ2) which vanishes since we will integrate the results times G(λ)dλ and ∞∫ 0 λ2G(λ)dλ = 0. This means that the logarithm appearing on integrating L in (31a) presents no problem. We may take I′(m2 + L) = (8i)−1 +1∫ −1 [ 1 4 η2qσ qτ D−1 + 1 2 δστ ln D]dη + Cδστ , a subsequent integral on L and finally on η presents no new problems. There results −(8i) ∫ pσ pτ d4p((p − 1 2 q)2 − m2)−1((p + 1 2 q)2 − m2)−1 = (qσ qτ − δστ q2) [ 1 9 − 4m2 − q2 3q2 ( 1 − θ tanθ )

RADIATIVE CORRECTIONS TO SCATTERING

Wed, 30 Sep 2026 00:00:00 +0000

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 )