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 )

• 1 6 ln λ2 m2 ] +δστ [(λ2 + m2) ln(λ2m−2 + 1) − C′λ2] (32a) where we assume λ2  m2 and have put some terms into the arbitrary constant C′ which is independent of λ2 (but in principle could depend on q2 and which drops out in the integral on G(λ)dλ. We have set q2 = 4m2 sin2 θ. In a very similar way the integral with m2 in the numerator can be worked out. It is, of course, necessary to differentiate this m2 also when calculating I′ and I′′. There results −(8i) ∫ m2d4p((p − 1 2 q)2 − m2)−1((p + 1 2 q)2 − m2)−1 = 4m2(1 − θctnθ) − q2/3 + 2(λ2 + m2) ln(λ2m−2 + 1) − C′′λ2), (33a) with another unimportant constant C′′. The complete problem requires the further integral, −(8i) ∫ (1; pσ )d4p((p − 1 2 q)2 − m2)−1((p + 1 2 q)2 − m2)−1 = (1, 0)(4(1 − θctnθ) + 2 ln(λ2m−2)). (34a) The value of the integral (34a) times m2 differs from (33a), of course, because the results on the right are not actually the integrals on the left, hut rather equal their actual value minus their value for m2 = m2 + λ2. Combining these quantities, as required by (32), dropping the constants C′, C′′ and evaluating the spur gives (33). The spurs are evaluated in the usual way, noting that the spur of any odd number of γ matrices vanishes and Sp(AB) = Sp(BA) for arbitrary A, B. The Sp(1) = 4 and we also have 1 2 Sp[(p1 + m1)(p2 − m2)] = p1 · p2 − m1m2, (35a) 47 1 2 Sp[(p1 + m1)(p2 − m2)(p4 − m4)] = (p1 · p2 − m1m2)(p3 · p4 − m3m4) −(p1 · p3 − m1m3)(p2 · p4 − m2m4) + (P − 1 · p4 − m1m4)(p2 · p3 − m2m3), (36a) where pi, mi are arbitrary four-vectors and constants. It is interesting that the terms of order λ2 ln λ2 go out, so that the charge renormalization depends only logarithmically on λ2. This is not true for some of the meson theories. Electrodynamics is suspiciously unique in the mildness of its divergence. D. More Complex Problems Matrix elements for complex problems can be set up in a manner analogous to that used for the simpler cases. We give three illustrations; higher order corrections to the M∅ller scattering, to the Compton scattering, and the interaction of a neutron Figure 8: The interaction between two electrons to order (e2/ℏc)2. One adds the contribution of every figure involving two virtual quanta, Appendix D. with an electromagnetic field. For the M∅ller scattering, consider two electrons, one in state u1 of momentum p1 and the other in state u2 of momentum p2. Later they are found in states u3, p3 and u4, p4. This may happen (first order in e2/ℏc) because they exchange a quantum of momentum q = p1 − p3 = p4 − p2 in the manner of Eq. (4) and 48 Fig. 1. The matrix element for this process is proportional to (translating (4) to momentum space) (˜u4γμu2)(˜u3γμu1)q−2. (37a) We shall discuss corrections to (37a) to the next order in e2/ℏc. (There is also the possibility that it is the electron at 2 which finally arrives at 3, the electron at 1 going to 4 through the exchange of quantum of momentum p3 − p2. The amplitude for this process, (˜u4γμu1)(˜u3γμu2)(p3 − p2)−2 must be subtracted from (37a) in accordance with the exclusion principle. A similar situation exists to each order so that we need consider in detail only the corrections to (37a), reserving to the last the subtraction of the same terms with 3, 4 exchanged.) One reason that (37a) is modified is that two quanta may be exchanged, in the manner of Fig. 8a. The total matrix element for all exchanges of this type is (e2/πi) ∫ (˜u3γν (p1 − k − m)−1γμu1)(˜u4γν (p2 + k − m)−1γμu2) ·k−2(q − k)−2d4k, (38a) as is clear from the figure and the general rule that electrons of momentum p contribute in amplitude (p − m)−1 between interactions γμ and that quanta of momentum k contribute k−2. In integrating on d4k and summing over μ and ν, we add all alternatives of the type of Fig. 8a. If the time of absorption, γμ, of the quantum k by electron 2 is later than the absorption, γμ, of q − k, this corresponds to the virtual state p2 +k being a positron (so that (38a) contains over thirty terms of the conventional method of analysis). In integrating over all these alternatives we have considered all possible distor- tions of Fig. 8a which preserve the order of events along the trajectories. We have not included the possibilities corresponding to Fig. 8b, however. Their contribution is (e2/πi) ∫ (˜u3γν (p1 − k − m)−1γν u1) ×(˜u4γμ(p2 + q − k − m)−1γν u2)k−2(q − k)−2d4k, (39a) as is readily verified by labeling the diagram. The contributions of all possible ways that an event can occur are to be added. This means that one adds with equal weight the integrals corresponding to each topologically distinct figure. To this same order there are also the possibilities of Fig. 8d which give (e2/πi) ∫ (˜u3γν (p2 − k − m)−1γμ(p1 − k − m)−1γν u1) × (˜u4γμu2)k−2q−2d4k. This integral on k will be seen to be precisely the integral (12) for the radiative corrections to scattering, which we have worked out. The term may be combined with the renormalization terms resulting from the difference of the effects of mass change and the terms, Figs. 8f and 8g. Figures 8e, 8h, and 8i are similarly analyzed. Finally the term Fig. 8c is clearly related to our vacuum polarization problem, and when integrated gives a term proportional to (˜u4γμu2)(˜u3γν u1)Jμν q−4. If the 49 Figure 9: Radiative correction to the Compton scattering term (a) of Fig.

5. Appendix D. charge is renormalized the term ln(λ/m) in Jμν in (33) is omitted so there is no remaining dependence on the cut-off. The only new integrals we require are the convergent integrals (38a) and (39a). They can be simplified by rationalizing the denominators and combining them by (14a). For example (38a) involves the factors (k2 − 2p1 · k)−1(k2 + 2p2 · k)−1k−2(q2 + k2 − 2q · k)−2. The first two may be com- bined by (14a) with a parameter x, and the second pair by an expression obtained by differentation (l5a) with respect to b and calling the parameter y. There results a factor (k2 −2px ·k)−2(k2 +yq2 −2yq ·k)−4 so that the integrals on d4k now involve two factors and can be performed by the methods given earlier in the appendix. The subsequent integrals on the parameters x and y are complicated and have not been worked out in detail. Working with charged mesons there is often a considerable reduction of the number of terms. For example, for the interaction between protons resulting from the exchange of two mesons only the term corresponding to Fig. 8h remains. Term 8a, for example, is impossible, for if the first proton emits a positive meson the second cannot absorb it directly for only neutrons can absorb positive mesons. As a second example, consider the radiative correction to the Compton scat- tering. As seen from Eq. (15) and Fig. 5 this scattering is represented by two terms, so that we can consider the corrections to each one separately. Figure 9 shows the types of terms arising from corrections to the term of Fig. 5a. Calling k 50 the momentum of the virtual quantum, Fig. 9a gives an integral ∫ γμ(p2 − k − m)−1e2(p1 + q1 − k − m)−1e1(p1 − k − m)−1γμk−2d4k, convergent without cut-off and reducible by the methods outlined in this appendix. The other terms are relatively easy to evaluate. Terms b and c of Fig. 9 are closely related to radiative corrections (although somewhat more difficult to evaluate, for one of the states is not that of a free electron, (p1 + q)2 6 = m2). Terms e, f , are renormalization terms. From term d must be subtracted explicitly the effect of mass ∆m, as analyzed in Eqs. (26) and (27) leading to (28) with p′ = p1 + q, a = e2, b = e1. Terms g, h give zero since the vacuum polarization has zero effect on free light quanta, q2 1 = 0, q2 2 = 0. The total is insensitive to the cut-off λ. The result shows an infra-red catastrophe, the largest part of the effect. When cut-off at λmin, the effect proportional to ln(m/λmin) goes as (e2/π) ln(m/λmin)(1 − 2θ ctn2θ), (40a) times the uncorrected amplitude, where (p2 −p1)2 = 4m2 sin2 θ. This is the same as for the radiative correction to scattering for a deflection p2 − p1. This is physically clear since the long wave quanta are not effected by short-lived intermediate states. The infra-red effects arise30 from a final adjustment of the field from the asymptotic coulomb field characteristic of the electron of momentum p1 before the collision to that characteristic of an electron moving in a new direction p2 after the collision. The complete expression for the correction is a very complicated expression involving transcendental integrals. As a final example we consider the interaction of a neutron with an electromag- netic field in virtue of the fact that the neutron may emit a virtual negative meson. We choose the example of pseudoscalar mesons with pseudovector coupling. The change in amplitude due to an electromagnetic field A = a exp(−iq · x) determines the scattering of a neutron by such a field. In the limit of small q it wilt vary as qa − aq which represents the interaction of a particle possessing a magnetic mo- ment. The first-order interaction between an electron and a neutron is given by the same calculation by considering the exchange of a quantum between the electron and the nucleon. In this case aμ is q−2 times the matrix element of γμ between the initial and final states of the electron, the states differing in momentum by q. The interaction may occur because the neutron of momentum p1 emits a neg- ative meson becoming a proton which proton interacts with the field and then reabsorbs the meson (Fig. 10a). The matrix for this process is (p2 = p1 + q), ∫ (γ5k(p2 − k − M )−1a(p1 − k − M )−1(γ5k)(k2 − μ2)−1d4k. (41a) Alternatively it may be the meson which interacts with the field. We assume that it does this in the manner of a scalar potential satisfying the Klein Gordon Eq. 30F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937). 51 (35), (Fig. l0b) − ∫ (γ5k2)(p1 − k1 − M )−1(γ5k1)(k2 2 − μ2)−1 ×(k2 · a + k1 · a)(k2 1 − μ2)−1d4k1, (42a) where we have put k2 = k1 + q. The change in sign arises because the virtual meson is negative. Finally there are two terms arising from the γ5a part of the pseudovector coupling (Figs. 10c, 10d) ∫ (γ5k)(p2 − k − M )−1(γ5a)(k2 − μ2)−1d4k, (43a) and ∫ (γ5a)(p1 − k − M )−1(γ5k)(k2 − μ2)−1d4k, (44a) Using convergence factors in the manner discussed in the section on meson theories each integral can be evaluated and the results combined. Expanded in powers of q the first term gives the magnetic moment of the neutron and is insensitive to the cut-off, the next gives the scattering amplitude of stow electrons on neutrons, and depends logarithmically on the cut-off. The expressions may be simplified and combined somewhat before integration. This makes the integrals a little easier and also shows the relation to the case of pseudoscalar coupling. For example in (41a) the final γ5k can be written as γ5(k − p1 + M ) since p1 = M when operating on the initial neutron state. This is (p1 − k − M )γ5 + 2mγ5 since γ5 anticommutes with p1 and k. The first term cancels the (p1 − k − M )−1 and gives a term which just cancels (43a). In a like manner the leading factor γ5k in (41a) is written as −2M γ − 5 − γ5(p2 − k − M ), the second term leading to a simpler term containing no (p2 − k − M )−1 factor and combining with a similar one from (44a). One simplifies the γ5k1 and γ5k2 in (42a) in an analogous way. There finally results terms like (41a), (42a) but with pseudoscalar coupling 2M γ5 instead of γ5k, no terms like (43a) or (44a) and a remainder, representing the difference in effects of pseudovector and pseudoscalar coupling. The pseudoscalar terms do not depend sensitively on the cut-off, but the difference term depends on it logarithmically. The difference term affects the electron-neutron interaction but not the magnetic moment of the neutron. Interaction of a proton with an electromagnetic potential can be similarly an- alyzed. There is an effect of virtual mesons on the electromagnetic properties of the proton even in the case that the mesons are neutral. It is analogous to the radiative corrections to the scattering of electrons due to virtual photons. The sum of the magnetic moments of neutron and proton for charged mesons is the same as the proton moment calculated for the corresponding neutral mesons. In fact it is readily seen by comparing diagrams. that for arbitrary q, the scattering matrix to first order in the electromagnetic potential for a proton according to neutral meson theory is equal, if the mesons were charged, to the sum of the matrix for a neutron 52 Figure 10: According to the meson theory a neutron interacts with an elec- tromagnetic potential a by first emitting a virtual charged meson. The figure illustrates the case for a pseudoscalar meson with pseudovector cou- pling. Appendix D. and the matrix for a proton. This is true, for any type or mixtures of meson cou- pling, to all orders in the coupling (neglecting the mass difference of neutron and proton). 53