The Dreams That Stuff is Made of
Page 70
from (14a) where
(21a)
We therefore need the integrals
(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, Δ = 0:
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−y2 which leads to a diverging integral on x as x → 0. This infra-red catastrophe is analyzed by using λ2min for the lower limit of the L integral. For the last term the upper limit of L must be kept as λ2. Assuming λ2min « py2 « λ2 the x integrals which remain are trivial, as in the self-energy case. One finds
(23a )
(24a )
(25a )
The integrals on y give,
(26a )
(27a )
(28a )
(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 that 2p1 · q = −q2 so that since py = p1 + q(1 −y ), p2y = m 2 − q2y (1–y ). The substitution 2y − 1 = tanθ where θ is defined by 4m2 sin2 θ = q2 is useful for it means p2y = m2 sec2 α/ sec2 θ and 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:
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.
C. Vacuum Polarization
The expressions (32) and (32’) for Jµν in the vacuum polarization problem require the calculation of the integral
(32)
where we have replaced p by p −1/2q to simplify the calculation somewhat. We shall indicate the method of calculation by studying the integral,
The factors in the denominator, p2− p · q − m2+ 1/4q2 and p2 + p · q − m2+ 1/4q2 are combined as usual by (8a) but for symmetry we substitute x = ½ (1 + η), (1 − x ) = ½ (1 − η) and integrate η from −1 to +1:
(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)d λ. 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 λ 2. By differentiating (30a), with respect to m2 find,
This still diverges, but we can differentiate again to get
(31a )
(where D = ¼(η2 − 1)q2 + m2 + L ), which now converges and has been evaluated by (13a) with p = ½ ηq and Δ = m2 + L − ¼ q2. Now to get I’ we may integrate 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 −C λ2 in I(m2) − I(m 2 + λ2) which vanishes since we will integrate the results times G(λ)d λ and ∫0∞ λ2 G (λ)d λ = 0. This means that the logarithm appearing on integrating L in (31a) presents no problem. We may take
a subsequent integral on L and finally on η presents no new problems. There results
(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)d λ. We have set q2 = 4m2 sin2θ .
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
(33a )
with another unimportant constant C". The complete problem requires the further integral,
(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 + λ 2.
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 S p (AB) = S p (BA) for arbitrary A, B. The Sp (1) = 4 and we also have
(35a )
(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 Møller scattering, to the Compton scattering, and the interaction of a neutron with an electromagnetic field.
For the Møller 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 e 2 /ħ c ) because they exchange a quantum of momentum q = p1 − p3 = p4 − p2 in the manner of Eq. (4) and Fig. 1. The matrix element for this process is proportional to (translating (4) to momentum space)
(37a )
We shall discuss corrections to (37a) to the next order in e 2 /ħ c . (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, (ũ4γµu1)(ũ3γµu2)(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
(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
FIG. 8 The interaction between two electrons to order (e2 /ħc)2. One adds the contribution of every figure involving two virtual quanta, Appendix D.
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 distortions 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
(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
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 (ũ4γµu2) (ũ3γνu1) Jµνq−4. If the charge is renormalized the term ln(λ/m) in Jµν i
n (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)−1 k−2 (q2 + k2 − 2q · k)−2. The first two may be combined by (14a) with a parameter x, and the second pair by an expression obtained by differentation (15a) 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 d 4k 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 scattering. 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 the momentum of the
FIG. 9 Radiative correction to the Compton scattering term (a) of Fig. 5. Appendix D.
virtual quantum, Fig. 9a gives an integral
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 ≠ m2). Terms e, f, are renormalization terms. From term d must be subtracted explicitly the effect of mass Δm, as analyzed in Eqs. (26) and (27) leading to (28) with pʹ = p1 + q, a = e2, b = e1. Terms g, h give zero since the vacuum polarization has zero effect on free light quanta,. 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
(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 arisens 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 electromagnetic 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 moment. 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 negative 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),
(41a )
FIG. 10 According to the meson theory a neutron interacts with an electromagnetic potential a by first emitting a virtual charged meson. The figure illustrates the case for a pseudoscalar meson with pseudovector coupling. Appendix D.
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. (35), (Fig. 10b)
(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)
(43a )
and
(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 γ5 k 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 −2Mγ − 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 2Mγ5 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 analyzed. 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 and the matrix for a proton. This is true, for any type or mixtures of meson coupling, to all orders in the coupling (neglecting the mass difference of neutron and proton).
THE THEORY OF POSITRONS
BY
RICHARD FEYNMAN
The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms.
In this solution, the “negative energy states” appear in a form which may be pictured (as by Stückelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the electron. A particle moving forward in time (electron) in a potential may be scattered forward in time (ordinary scattering) or backward (pair annihilation). When moving backward (positron) it may be scattered ba
ckward in time (positron scattering) or forward (pair production). For such a particle the amplitude for transition from an initial to a final state is analyzed to any order in the potential by considering it to undergo a sequence of such scatterings.
The amplitude for a process involving many such particles is the product of the tránsition amplitudes for each particle. The exclusion principle requires that antisymmetric combinations of amplitudes be chosen for those complete processes which differ only by exchange of particles. It seems that a consistent interpretation is only possible if the exclusion principle is adopted. The exclusion principle need not be taken into account in intermediate states. Vacuum problems do not arise for charges which do not interact with one another, but theseReprinted with permission from the American Physical Society: Feynman, Physical Review, Volume 76, p. 749, 1949. ©1949, by the American Physical Society.