The Dreams That Stuff is Made of
Page 67
In the non-relativistic region the expression for ΔE can be worked out as has been done by Bethe.na In the relativistic region (points 4 and 3 as close together as a Compton wave-length) the which should appear in (8) can be replaced to first order in V by K+ plus (2,1) given in I, Eq. (13). The problem is then very similar to the radiationless scattering problem discussed below.
4 EXPRESSION IN MOMENTUM AND ENERGY SPACE
The evaluation of (9), as well as all the other more complicated expressions 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 which is
(10)
which can be obtained from (3) and (5) or from I, Eq. (32) noting that from I, Eq. (34). The k-2 means (k · k)-1 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
(11)
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 integrates 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 potential,
FIG. 3 Interaction of an electron with itself. Momentum space, Eq. (11).
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:
(12)
FIG. 4 Radiative correction to scattering, momentum space.
For in this case, firstnb 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) receiving additional momentum q, propagating on then (factor (p2 − k − m)−1) with the new momentum until the quantum is reabsorbed (γµ). The quantum 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))
(13)
or if both emission and absorption occur after the scattering, (Fig. 4(c))
(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 normalization). For example, consider the Compton effect (Fig. 5 (a)) in which an electron in state p1 absorbs a quantum of momentum q1, polarization vector e 1µ 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,
(15)
FIG. 5 Compton scattering, Eq. (15).
the second term arising because the emission of e2, may also precede the absorption 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.
5 THE CONVERGENCE OF PROCESSES WITH VIRTUAL QUANTA
These expressions are, as has been indicated, no more than a reexpression 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 . 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 article, A. There the appearing in the action of interaction was replaced by 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 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 writtennc as
where g(k · k) is times the density of oscillators and may be expressed for positive k4 as (A, Eq. (16))
where and G involves values of λ large compared to m. This simply means that the amplitude for propagation of quanta of momentum k is
rather than k−2. That is, writing F+(k2) = −π−1k−2C(k2),
(16)
Every integral over an intermediate quantum which previously involved a factor d4k/k2 is now supplied with a convergence factor C(k2) where
(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 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
(18)
This condition will also be used in discussing the convergence of vacuum polarization integrals.
The expression for the self-energy matrix is now
(19)
which, since C(k2) falls off at least as rapidly as 1/k2, converges. For practical 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 representing propagation of an electron while that quantum is in the field, we can expect all su
ch integrals with their convergence factors to converge and that the result of ail such processes will now be finite and definite (excepting the 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)
(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))
(21)
6 RADIATIVE CORRECTIONS TO SCATTERING
We can now complete the discussion of the radiative corrections to scattering. In the integrals we include the convergence factor C(k2), so that they converge for large k. Integral (12) is also not convergent because of the well-known infrared catastrophy. For this reason we calculate (as discussed in B) the value of the integral assuming the photons to have a small mass λmin << m << λ. The integral (12) becomes
which when integrated (see Appendix B) gives (e2/2π) times
(22)
where (q2)1/2 = 2m and we have assumed the matrix to operate between states of momentum p1 and p2 = p1 + q and have neglected terms of order λmin/m, m/λ, and q2/λ2. Here the only dependence on the convergence factor is in the term ra, where
(23)
As we shall see in a moment, the other terms (13), (14) give contributions which just cancel the ra term. The remaining terms give for small q,
(24)
which shows the change in magnetic moment and the Lamb shift as interpreted in more detail in B.nd
We must now study the remaining terms (13) and (14). The integral on k in (13) can be performed (after multiplication by C(k2)) since it involves nothing but the integral (19) for the self-energy and the result is allowed to operate on the initial state u1, (so that p1u1 = mu1). Hence the factor following a(p1 − m)−1 will be just Δm. But, if one now tries to expand one obtains an infinite result, since . This is, however, just what is expected physically. For the quantum can be emitted and absorbed at any time previous to the scattering. Such a process has the effect of a change in mass of the electron in the state 1. It therefore changes the energy by ΔE and the amplitude to first order in ΔE by −iΔE· t where t is the time it is acting, which is infinite. That is, the major effect of this term would be canceled by the effect of change of mass Δm.
The situation can be analyzed in the following manner. We suppose that the electron approaching the scattering potential a has not been free for an infinite time, but at some time far past suffered a scattering by a potential b. If we limit our discussion to the effects of Δm and of the virtual radiation of one quantum between two such scatterings each of the effects will be finite, though large, and their difference is determinate. The propagation from b to a is represented by a matrix
(25)
in which one is to integrate possibly over p′ (depending on details of the situation). (If the time is long between b and a, the energy is very nearly determined so that p′2 is very nearly m2.)
We shall compare the effect on the matrix (25) of the virtual quanta and of the change of mass Δm. The effect of a virtual quantum is
(26)
while that of a change of mass can be written
(27)
and we are interested in the difference (26)–(27). A simple and direct method of making this comparison is just to evaluate the integral on k in (26) and subtract from the result the expression (27) where Δm is given in (21). The remainder can be expressed as a multiple −r(p′2) of the unperturbed amplitude (25);
(28)
This has the same result (to this order) as replacing the potentials a and b in (25) by and In the limit, then, as p′2 → m2 the net effect on the scattering is where r, the limit of r(p′2) as p′2 → m2 (assuming the integrals have an infrared cut-off), turns out to be just equal to that given in (23). An equal term arises from virtual transitions after the scattering (14) so that the entire ra term in (22) is canceled.
The reason that r is just the value of (12) when q2 = 0 can also be seen without a direct calculation as follows: Let us call p the vector of length m in the direction of p′ so that if p′2 = m(1 + ∈)2 we have p′ = (1 + ε0p and we take ε as very small, being of order T−1 where T is the time between the scatterings b and a. Since (p′ − m)−1 = (p′ + m)/(p′2 − m2) ≈ (p + m)/2m2∈, the quantity (25) is of order ∈−1 or T. We shall compute corrections to it only to its own order (∈−1) in the limit ∈ → 0. The term (27) can be written approximatelyne as
using the expression (19) for Δm. The net of the two effects is therefore approximatelynf
a term now of order 1/ε (since (p′ − m)−1 ≈ (p + m) × (2m2∈)−1) and therefore the one desired in the limit. Comparison to (28) gives for r the expression
(29)
The integral can be immediately evaluated, since it is the same as the integral (12), but with q = 0, for a replaced by p1/m. The result is therefore r · (p1/m) which when acting on the state u1 is just r, as p1u1 = mu1. For the same reason the term (p1 + m)/2m in (29) is effectively 1 and we are left with −r of (23).ng
In more complex problems starting with a free electron the same type of term arises from the effects of a virtual emission and absorption both previous to the other processes. They, therefore, simply lead to the same factor r so that the expression (23) may be used directly and these renormalization integrals need not be computed afresh for each problem.
In this problem of the radiative corrections to scattering the net result is insensitive to the cut-off. This means, of course, that by a simple rearrangement of terms previous to the integration we could have avoided the use of the convergence factors completely (see for example Lewisnh). The problem was solved in the manner here in order to illustrate how the use of such convergence factors, even when they are actually unnecessary, may facilitate analysis somewhat by removing the effort and ambiguities that may be involved in trying to rearrange the otherwise divergent terms.
The replacement of δ+ by f+ given in (16), (17) is not determined by the analogy with the classical problem. In the classical limit only the real part of δ+ (i.e., just δ) is easy to interpret. But by what should the imaginary part, 1/(πs2), of δ+ be replaced? The choice we have made here (in denning, as we have, the location of the poles of (17)) is arbitrary and almost certainly incorrect. If the radiation resistance is calculated for an atom, as the imaginary part of (8), the result depends slightly on the function f+. On the other hand the light radiated at very large distances from a source is independent of f+. The total energy absorbed by distant absorbers will not check with the energy loss of the source. We are in a situation analogous to that in the classical theory if the entire f function is made to contain only retarded contributions (see A, Appendix). One desires instead the analogue of 〈F〉ret of A. This problem is being studied.
One can say therefore, that this attempt to find a consistent modification of quantum electrodynamics is incomplete (see also the question of closed loops, below). For it could turn out that any correct form of f+ which will guarantee energy conservation may at the same time not be able to make the self-energy integral finite. The desire to make the methods of simplifying the calculation of quantum electrodynamic processes more widely available has prompted this publication before an analysis of the correct form for ƒ+ is complete. One might try to take the position that, since the energy discrepancies discussed vanish in the limit λ → ∞, the correct physics might be considered to be that obtained by letting λ → ∞ after mass renormalization. I have no proof of the mathematical consistency of this procedure, but the presumption is very strong that it is satisfactory. (It is also strong that a satisfactory form for f+ can be found.)
7 THE PROBLEM OF VACUUM POLARIZATION
In the analysis of the radiative corrections to scattering one type of term was not considered. The potential which we can assume to vary as aµexp(−iq · x)
creates a pair of electrons (see Fig. 6), momenta pa , − pb. This pair then reannihilates, emitting a quantum q = qb − qa , which quantum scatters the original electron from state 1 to state 2. The matrix element for this process (and the others which can be obtained by rearranging the order in time of the various events) is