mj We call a three-dimensional manifold in the four-dimensional space-time world simply a “surface.”
mk Suppose that a surface in the kx ky kz k-space is defined by means of the equation k Then this surface has the invariant meaning in this space, since is invariant against Lorentz transformations. The area of the surface element of this surface is given by
Now, since dS has the invariant meaning, we can thus conclude that is invariant, and this implies that the function defined by (11) is invariant.
ml Here we suppose the representation which makes the coordinates q1,q2, . . . ,qN diagonal. Thus the vector Φ is represented by a function of these coordinates.
mm The notion of local time of this kind has been occasionally introduced by Stueckelberg.(6)
mn The word state is here used in the relativistic space-time meaning. Cf Dirac’s book (second edition), §6
mo As the matrix elements are functional of v(P), we use here the square brackets.
mp R. P. Feynman, Phys. Rev. 76, 749 (1949), hereafter called I.
mq For a discussion of this modification in classical physics see R. P. Feynman, Phys. Rev. 74 939 (1948), hereafter referred to as A.
mr A brief summary of the methods and results will be found in R. P. Feynman, Phys. Rev. 74, 1430 (1948), hereafter referred to as B.
ms J. Schwinger, Phys. Rev. 74, 1439 (1948), Phys. Rev. 75, 651 (1949). A proof of this equivalence is given by F. J. Dyson, Phys. Rev. 75, 486 (1949).
mt R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948). The application to electrodynamics is described in detail by H. J. Groenewold, Koninklijke Nederlandsche Akademia van Weteschappen. Proceedings Vol. LII, 3 (226) 1949.
mu For a discussion of this modification in classical physics see R. P. Feynman, Phys. Rev. 74 939 (1948), hereafter referred to as A.
mv This is the viewpoint of the theory of the S matrix of Heisenberg.
mw It and a like term for the effect of a on b, leads to a theory which, in the classical limit, exhibits interaction through half-advanced and half-retarded potentials. Classically, this is equivalent to purely retarded effects within a closed box from which no light escapes (e.g., see A, or J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945)). Analogous theorems exist in quantum mechanics but it would lead us too far astray to discuss them now.
mx Although 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+(s256)γµ × K+(6, 2)dτ6 arising from Maxwell’s equations −□2 Aµ = 4π jµ from a “current” jµ (6) = e2K + (4, 6)γµ K+(6, 2) produced by particle b in going from 2 to 4. This is virtue of the fact that δ+ satisfies
(5)
my These considerations make it appear unlikely that the contention of J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945), that electrons do not act on themselves, will be a successful concept in quantum electrodynamics.
mz This is discussed in reference 5 in which it is pointed out that the concept of a wave function loses accuracy if there are delayed self-actions.
na H. A. Bethe, Phys. Rev. 72, 339 (1947).
nb First, 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.
nc This relation is given incorrectly in A, equation just preceding 16.
nd That 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 calculation, 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 In 2kmax − 1 = In λmin used by the author should have been In 2kmax − 5/6 = In 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.
ne The expression is not exact because the substitution of Δm 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)−1 b which is a((1 + ε) p + m)(p − m) × ((1 + ε)p + m)p(2ε + ∈2 )−2 m−4 . But since2 = 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/ε but smaller, so that its effect drops out in the limit.
nf We have used, to first order, the general expansion (valid for any operators A, B)(A + B0−1 = A−1 − A−1 BA−1 + A−1 BA−1 BA−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.
ng The 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 p1 m−1 factor replaced by mγ4 /E1 where E 1 = 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 p1 u1 = mu1 .
nh H.W. Lewis, Phys. Rev. 73, 173 (1948).
ni It 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.
nj This 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.
nk Added to the terms given below (33) there is a term for C(k2) = −λ2(k2 - λ2)-1, which is not gauge invariant. (In addition the charge renormalization has −7/6 added to the logarithm.)
nl E. A. Uehling, Phys. Rev. 48, 55 (1935), R. Serber, Phys. Rev. 48, 49 (1935).
nm There are loops completely without external interactions. For example, a pair is created virtually along with a photon. Next they annihilate, absorbing this photon. Such 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.
nn A 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) = we can still conclude (γK . . . γK) = (γt ... γt).
no The 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 relationβµβvβσ + β�
�βvβµ = δµvβσδσvβµ.
If we interpret a to mean aµβµ, rather than aµγµ, for any aµ, all of the equations in momentum 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 implies that equations in coordinate space will be valid of the function K+ (2,1) is given as K+(2, 1) = [(i∇2 + m) − m–1 (∇2 + with ∇2 = 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 (i∇ − m)ψ = aψ which is formally identical to the Dirac Equation. See W. Pauli, Rev. Mod. Phys. 13, 203 (1940).
np M. Slotnick and W. Heitler, Phys. Rev. 75, 1645 (1949).
nq H. A. Bethe, Bull. Am. Phys. Soc. 24, 3, Z3 (Washington, 1949).
nr The 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−2qµqνγν 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 propagation of virtual particles of momentum q from state ν to µ is represented by multiplication by the 4-4 matrix (or tensor) Pµν = (δµν − µ2 qµqν) × (q2 − µ2)−1. The first-order interaction (from the Proca equation) with an electromagnetic potential a exp(ik·x) corresponds to multiplication by the matrix Eµν = (q2 · a + q1 · a)δµν − q2νaµ − q1νaν, where q1 and q2 = q1 + k are the momenta before and after the interaction. Finally, two potentials a, b may act simultaneously, with matrix .
ns F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).
nt R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
nu The equivalence of the entire procedure (including photon interactions) with the work of Schwinger and Tomonaga has been demonstrated by F. J. Dyson, Phys. Rev. 75, 486 (1949).
nv These are special examples of the general relation of spin and statistics deduced by W. Pauli, Phys. Rev. 58, 716 (1940).
nw For a non-reladvisdc free particle, where φn = exp(ip-x), En = p2/2m, (3) gives, as is well knownK0(2,1) = ∫ exp [ip -x1-ip·x2)ip2(t2-t1)/2m]d3p(2π)-3
for t2 > t1 , and K0 = 0 for t2 < t1 .
nx We are simply solving by successive approximations an integral equation (deducible directly from (1) with H=H0+Uand (4) with H = H0 ),ψ (2) = −i ∫ K0 (2, 3)U (3)ψ (3)d τ3 + ∫ K0 (2, 1)ψ (1)d3 x1 ,
where the first integral extends over all space and all times t3 greater than the t1 appearing in the second term, and t2 > t1 .
ny It has often been noted that the one-electron theory apparently gives the same matrix elements for this process as does hole theory. The problem is one of interpretation, especially in a way that will also give correct results for other processes, e.g., self-energy.
nz The idea that positrons can be represented as electrons with proper time reversed relative to true time has been discussed by the author and others, particularly by Stückelberg. E. C. C. Stückelberg, Helv. Phys. Acta 15, 23 (1942); R. P. Feynman, Phys. Rev. 74, 939 (1948). The fact that classically the action (proper time) increases continuously as one follows a trajectory is reflected in quantum mechanics in the fact that the phase, which is |En| |t2 − t1 |, always increases as the particle proceeds from one scattering point to the next.
oa By multiplying (12) on the right by (−i∇1 − m) and noting that ∇1 δ(2, 1) = −∇2 δ(2, 1) show that K + (2, 1) also satisfies K + (2, 1)(−i∇1 − m) = iδ(2, 1), where the ∇1 operates on variable 1 in K + (2, 1) but is written after that function to keep the correct i order of the γ matrices. Multiply this equation by ψ (1) and Eq. (11) (with A = 0, calling the variables 1) by K + (2, 1), subtract and integrate over a region of space-time. The integral on the left-hand side can be transformed to an integral over the surface of the region. The right-hand side is ψ (2) if the point 2 lies within the region, and is zero otherwise. (What happens when the 3-surface contains a light line and hence has no unique normal need not concern us as these points can be made to occur so far away from 2 that their contribution vanishes.)
ob This term actually vanishes as can be seen as follows. In any spur the sign of all γ matrices may be reversed. Reversing the sign of γ in K + (2, 1) changes it to the transpose of K + (1, 2) so that the order of all factors and variables is reversed. Since the integral is taken over all τ1 , τ2 , and τ3 this has no effect and we are left with (−1)3 from changing the sign of A. Thus the spur equals its negative. Loops with an odd number of potential interactors give zero. Physically this is because for each loop the electron can go around one way or in the opposite direction and we must add these amplitudes. But reversing the motion of an electron makes it behave like a positive charge thus changing the sign of each potential interaction, so that the sum is zero if the number of interactions is odd. This theorem is due to W. H. Furry, Phys. Rev. 51, 125 (1937).
oc A closed expression for L in terms of is hard to obtain because of the factor (1/n) in the nth term. However, the perturbation in L, ΔL due to a small change in potential ΔA, is easy to express. The (1/n) is canceled by the fact that ΔA can appear in any of the n potentials. The result after summing over n by (13), (14) and using (16) is
(29)
. The term K + (1, 1) actually integrates to zero.
od I+ (x, t) is (2i)−1 (D1 (x, t) − iD(x, t)) where D1 and D are the functions defined by W. Pauli, Rev. Mod. Phys. 13, 203 (1941).
oe If the − iδ is kept with m here too the function I+ approaches zero for infinite positive and negative times. This may be useful in general analyses in avoiding complications from infinitely remote surfaces.
of See, for example, G. Wentzel, Einfuhrung in die Quantentheorie der Wellenfelder (Franz Deuticke, Leipzig, 1943), Chapter V.
og Sin-itiro Tomonaga, Prog. Theoret. Phys. 1, 27 (1946); Koba, Tati, and Tomonaga, Prog. Theoret. Phys. 2, 101 198 (1947); S. Kanesawa and S. Tomonaga, Prog. Theoret. Phys. 3, 1, 101 (1948); S. Tomonaga, Phys. Rev. 74, 224 (1948).
oh Julian Schwinger, Phys. Rev. 73, 416 (1948); Phys. Rev. 74, 1439 (1948). Several papers, giving a complete exposition of the theory, are in course of publication.
oi R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 74, 939, 1430 (1948); J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945). These articles describe early stages in the development of Feynman’s theory, little of which is yet published.
oj After this paper was written, the author was shown a letter, published in Progress of Theoretical Physics 3, 205 (1948) by Z. Koba and G. Takeda. The letter is dated May 22, 1948, and briefly describes a method of treatment of radiative problems, similar to the method of this paper.
Results of the application of the method to a calculation of the second-order radiative correction to the Klein-Nishina formula are stated. All the papers of Professor Tomonaga and his associates which have yet been published were completed before the end of 1946. The isolation of these Japanese workers has undoubtedly constituted a serious loss to theoretical physics.
ok See, for example, Gregor Wentzel, Einführung in die Quantentheorie der Wellenfelder (Franz Deuticke, Wien, 1943), pp. 18–26.
ol Werner Heisenberg, Zeits. f. Physik 120, 513 (1943), 120, 673 (1943), and Zeits. f. Naturforschung 1, 608 (1946).
om Here Schwinger’s notation Ψ = �
�* β is used.
on See Wolfgang Pauli, Rev. Mod. Phys. 13, 203 (1941), Eq. (96), p. 224.
oo Schwinger’s results agree with those of the earlier, theoretically unsatisfactory treatment of vacuum polarization. The best account of the earlier work is V. F. Weisskopf, Kgl. Danske Sels. Math.-Fys. Medd. 14, No. 6 (1936).
op Robert Serber, Phys. Rev. 48, 49 (1935); E. A. Uehling, Phys. Rev. 48, 55 (1935).
oq W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241 (1947).
or P. Kusch and H. M. Foley, Phys. Rev. 74, 250 (1948).
os J. E. Nafe and E. B. Nelson, Phys. Rev. 73, 718 (1948); Aage Bohr, Phys. Rev. 73, 1109 (1948).
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