(30)
This is because the potential produces the pair with amplitude proportional to aνγν the electrons of momenta pa , and − (pa + q) proceed from there to annihilate, producing a quantum (factor γµ) which propagates (factor q−2C(q2)) over to the other electron, by which it is absorbed (matrix element of γµ, between states 1 and 2 of
FIG. 6 Vacuum polarization effect on scattering, Eq. (30).
the original electron (ũ2γµu1)). All momenta pa and spin states of the virtual electron are admitted, which means the spur and the integral on d4 pa are calculated.
One can imagine that the closed loop path of the positron-electron produces a current
(31)
which is the source of the quanta which act on the second electron.
The quantity
(32)
is then characteristic for this problem of polarization of the vacuum.
One sees at once that Jµν diverges badly. The modification of δ to f alters the amplitude with which the current jµ, will affect the scattered electron, but it can do nothing to prevent the divergence of the integral (32) and of its effects.
One way to avoid such difficulties is apparent. From one point of view we are considering all routes by which a given electron can get from one region of space-time to another, i.e., from the source of electrons to the apparatus which measures them. From this point of view the closed loop path leading to (32) is unnatural. It might be assumed that the only paths of meaning are those which start from the source and work their way in a continuous path (possibly containing many time reversals) to the detector. Closed loops would be excluded. We have already found that this may be done for electrons moving in a fixed potential.
Such a suggestion must meet several questions, however. The closed loops are a consequence of the usual hole theory in electrodynamics. Among other things, they are required to keep probability conserved. The probability that no pair is produced by a potential is not unity and its deviation from unity arises from the imaginary part of Jµν. Again, with closed loops excluded, a pair of electrons once created cannot annihilate one another again, the scattering of light by light would be zero, etc. Although we are not experimentally sure of these phenomena, this does seem to indicate that the closed loops are necessary. To be sure, it is always possible that these matters of probability conservation, etc., will work themselves out as simply in the case of interacting particles as for those in a fixed potential. Lacking such a demonstration the presumption is that the difficulties of vacuum polarization are not so easily circumvented.ni
An alternative procedure discussed in B is to assume that the function K +(2, 1) used above is incorrect and is to be replaced by a modified function having no singularity on the light cone. The effect of this is to provide a convergence factor C(p2 − m2) for every integral over electron momenta.nj This will multiply the integrand of (32) by C(p2 − m2)C((p + q)2 − m2), since the integral was originally δ(pa − pb + q)d4pad4pb and both pa and pb get convergence factors. The integral now converges but the result is unsatisfactory.nk
One expects the current (31) to be conserved, that is qµjµ = 0 or qµJµν = 0. Also one expects no current if a is a gradient, or aν = qν, times a constant. This leads to the condition Jµνqν = 0 which is equivalent to qµJµν = 0 since Jµν is symmetrical. But when the expression (32) is integrated with such convergence factors it does not satisfy this condition. By altering the kernel from K to another, K′, which does not satisfy the Dirac equation we have lost the gauge invariance, its consequent current conservation and the general consistency of the theory.
One can see this best by calculating Jµνqv directly from (32). The expression within the spur becomes (p + q − m)−1q(p − m)−1γµ which can be written as the difference of two terms: (p − m)−1γµ − (p + q − m)−1γµ Each of these terms would give the same result if the integration d4p were without a convergence factor, for the first can be converted into the second by a shift of the origin of p, namely p′ = p + q. This does not result in cancelation in (32) however, for the convergence factor is altered by the substitution.
A method of making (32) convergent without spoiling the gauge invariance has been found by Bethe and by Pauli. The convergence factor for light can be looked upon as the result of superposition of the effects of quanta of various masses (some contributing negatively). Likewise if we take the factor C(p2 − m2) = −λ2(p2−m2−λ2)−1 so that (p2 − m2)−1C(p2 − m2) = (p2 − m2)−1 − (p2 − m2 − λ2)−1 we are taking the difference of the result for electrons of mass m and mass (λ2 + m2)1/2. But we have taken this difference for each propagation between interactions with photons. They suggest instead that once created with a certain mass the electron should continue to propagate with this mass through all the potential interactions until it closes its loop. That is if the quantity (32), integrated over some finite range of p, is called Jµν(m2) and the corresponding quantity over the same range of p, but with m replaced by (m2 + λ2)1/2 is Jµν(m2 + λ2) we should calculate
(32′)
the function G(λ) satisfying and .
Then in the expression for the range of p integration can be extended to infinity as the integral now converges. The result of the integration using this method is the integral on dλ over G(λ) of (see Appendix C)
(33)
with q2 = 4m2 sin2 θ.
The gauge invariance is clear, since qµ(qµqν − q2δµν) = 0. Operating (as it always will) on a potential of zero divergence the (qµqν − δµνq2)aν , is simply −q2aµ, the D’Alembertian of the potential, that is, the current producing the potential. The term therefore gives a current proportional to the current producing the potential. This would have the same effect as a change in charge, so that we would have a difference Δ(e2) between e2 and the experimentally observed charge, e2 + Δ(e2) , analogous to the difference between m and the observed mass. This charge depends logarithmically on the cut-off, Δ(e2)/e2 = −(2e⅔π) ln(λ/m). After this renormalization of charge is made, no effects will be sensitive to the cut-off.
After this is done the final term remaining in (33), contains the usual effectsnl of polarization of the vacuum. It is zero for a free light quantum (q2 = 0). For small q2 it behaves as (2/15)q2 (adding to the logarithm in the Lamb effect). For q2 > (2m)2 it is complex, the imaginary part representing the loss in amplitude required by the fact that the probability that no quanta are produced by a potential able to produce pairs ((q 2)1/2 > 2m) decreases with time. (To make the necessary analytic continuation, imagine m to have a small negative imaginary part, so that (1 − q2/4m2 − 1)1/2 becomes −i(q2/4m2 − 1)1/2 as q2 goes from below to above 4m2. Then θ = π/2 + iu where sin hu = +(q2/4m2 − 1)1/2 , and −1/tanθ = itanhu = +i(q2 − 4m2)1/2(q2)−1/2).
Closed loops containing a number of quanta or potential interactions larger than two produce no trouble. Any loop with an odd number of interactions gives zero (I, reference 9). Four or more potential interactions give integrals which are convergent even without a convergence factor as is well known. The situation is analogous to that for self-energy. Once the simple problem of a single closed loop is solved there are no further divergence difficulties for more complex processes.nm
8 LONGITUDINAL WAVES
In the usual form of quantum electrodynamics the longitudinal and transverse waves are given separate treatment. Alternately the condition (∂Aµ/∂xµ)ψ = 0 is carried along as a supplementary condition. In the present form no such special considerations are necessary for we are dealing with the solutions of the equation −□2 Aµ = 4πjµ, with a current jµ, which is conserved ∂jµ/∂xµ = 0. That means at least □2(∂Aµ/∂xµ) = 0 and in fact our solution also satisfies ∂Aµ/∂xµ = 0.
To show that this is the case we consider the amplitude for emission (real or virtual) of a photon and show that the divergence of this amplitude vanishes. The amplitude for emission for photons polarized in the µ direction involves matrix elements of γµ. Therefore what we have to show is that
the corresponding matrix elements of qµγµ = q vanish. For example, for a first order effect we would require the matrix element of q between two states p1 and p2 = p1 + q. But since q = p2 − p1 and (ũ2p1u1) = m(ũ2u1) = (ũ2p2u1) the matrix element vanishes, which proves the contention in this case. It also vanishes in more complex situations (essentially because of relation (34), below) (for example, try putting e2 = q2 in the matrix (15) for the Compton Effect).
To prove this in general, suppose ai, i = 1 to N are a set of plane wave disturbing potentials carrying momenta qi, (e.g., some may be emissions or absorptions of the same or different quanta) and consider a matrix for the transition from a state of momentum p0 to pN such as , where pi = pi −1 + qi (and in the product, terms with larger i are written to the left). The most general matrix element is simply a linear combination of these. Next consider the matrix between states p0 and pN + q in a situation in which not only are the ai, acting but also another potential a exp(−iq·x) where a = q. This may act previous to all ai in which case it gives aNΠ(pi + q − m)−1ai(p0 + q − m)−1q which is equivalent to +aNΠ(pi + q − m)−1ai since + (p0 + q − m)−1q is equivalent to (p0 + q − m)−1 × (p0 + q − m) as p0 is equivalent to m acting on the initial state. Likewise if it acts after all the potentials it gives q(pN − m)−1aNΠ(pi − m)−1ai which is equivalent to −aNΠ(pi − m)−1ai since pN + q − m gives zero on the final state. Or again it may act between the potential ak and ak+1 for each k. This gives
However,
(34)
so that the sum breaks into the difference of two sums, the first of which may be converted to the other by the replacement of k by k − 1. There remain only the terms from the ends of the range of summation,
These cancel the two terms originally discussed so that the entire effect is zero. Hence any wave emitted will satisfy ∂Aµ/∂xµ = 0. Likewise longitudinal waves (that is, waves for which Aµ = ∂φ/∂xµ or a = q cannot be absorbed and will have no effect, for the matrix elements for emission and absorption are similar. (We have said little more than that a potential Aµ = ∂ϕ/∂xµ has no effect on a Dirac electron since a transformation ψ′ = exp(−iφ)ψ removes it. It is also easy to see in coordinate representation using integrations by parts.)
This has a useful practical consequence in that in computing probabilities for transition for unpolarized light one can sum the squared matrix over all four directions rather than just the two special polarization vectors. Thus suppose the matrix element for some process for light polarized in direction eµ, is eµMµ. If the light has wave vector qµ, we know from the argument above that qµ Mµ = 0. For unpolarized light progressing in the z direction we would ordinarily calculate . But we can as well sum for qµMµ implies Mt = Mz since qt = qz for free quanta. This shows that unpolarized light is a relativistically invariant concept, and permits some simplification in computing cross sections for such light.
Incidentally, the virtual quanta interact through terms like γµ . . . γµk−2d4k. Real processes correspond to poles in the formulae for virtual processes. The pole occurs when k2 = 0, but it looks at first as though in the sum on all four values of µ, of γµ . . . γµ we would have four kinds of polarization instead of two. Now it is clear that only two perpendicular to k are effective.
The usual elimination of longitudinal and scalar virtual photons (leading to an instantaneous Coulomb potential) can of course be performed here too (although it is not particularly useful). A typical term in a virtual transition is γµ . . . γµk−2d4k where the . . . represent some intervening matrices. Let us choose for the values of µ, the time t, the direction of vector part K, of k, and two perpendicular directions 1, 2. We shall not change the expression for these two 1, 2 for these are represented by transverse quanta. But we must find (γt . . . γt) − (γK . . . γK). Now k = k4 γt − KγK, where K = (K · K)1/2, and we have shown above that k replacing the γµ. gives zero.nn Hence KγK is equivalent to k4γt and
so that on multiplying by k−2d4k = d4k(k42 − K2)−1 the net effect is −(γt . . . γt)d4k/K2. The γt means just scalar waves, that is, potentials produced by charge density. The fact that 1/K2 does not contain k4 means that k4 can be integrated first, resulting in an instantaneous interaction, and the d3K/K2 is just the momentum representation of the Coulomb potential, 1/r.
9 KLEIN GORDON EQUATION
The methods may be readily extended to particles of spin zero satisfying the Klein Gordon equation,no
(35)
The important kernel is now I+(2, 1) denned in (I, Eq. (32)). For a free particle, the wave function ψ(20 satisfies +2ψ − m2ψ = 0. At a point, 2, inside a space time region it is given by
(as is readily shown by the usual method of demonstrating Green’s theorem) the integral being over an entire 3-surface boundary of the region (with normal vector Nµ). Only the positive frequency components of ψ contribute from the surface preceding the time corresponding to 2, and only negative frequencies from the surface future to 2. These can be interpreted as electrons and positrons in direct analogy to the Dirac case.
The right-hand side of (35) can be considered as a source of new waves and a series of terms written down to represent matrix elements for processes of increasing order. There is only one new point here, the term in AµAµ by which two quanta can act at the same time. As an example, suppose three quanta or potentials, aµexp(−iqa · x), bµexp(−iqb · x), and cµexp(iqe · x) are to act in that order on a particle of original momentum p0µ , so that pa = p0 + qa , and pb = pa + qb; the final momentum being pc = pb + qc. The matrix element is the
FIG. 7 Klein-Gordon particle in three potentials, Eq, (36). The coupling to the electromagnetic field is now, for example, p0 · a + pa · a, and a new possibility arises, (b), of simultaneous interaction with two quanta a · b. The propagation factor is now (p · p − m2)−1 for a particle of momentum pµ.
sum of three terms (p2 = pµpµ) (illustrated in Fig. 7)
(36)
The first comes when each potential acts through the perturbation i∂(Aµψ)/∂xµ + iAµ∂ψ/∂xµ. These gradient operators in momentum space mean respectively the momentum after and before the potential Aµ operates. The second term comes from bµ and aµ acting at the same instant and arises from the AµAµ term in (a). Together bµ and aµ carry momentum qbµ + qaµ so that after b · a operates the momentum is p0 + qa + qb or pb. The final term comes from cµ and bµ operating together in a similar manner. The term AµAµ thus permits a new type of process in which two quanta can be emitted (or absorbed, or one absorbed, one emitted) at the same time. There is no a · c term for the order a, b, c we have assumed. In an actual problem there would be other terms like (36) but with alterations in the order in which the quanta a, b, c act. In these terms a · c would appear.
As a further example the self-energy of a particle of momentum pµ is
where the δµµ comes from the AµAµ term and represents the possibility of the simultaneous emission and absorption of the same virtual quantum. This integral without the C(k2) diverges quadratically and would not converge if C(k2) = −λ2/(k2 − λ2). Since the interaction occurs through the gradients of the potential, we must use a stronger convergence factor, for example C(k2) = λ4(k2 − λ2)−2, or in general (17) with . In this case the self-energy converges but depends quadratically on the cut-off λ and is not necessarily small compared to m. The radiative corrections to scattering after mass renormalization are insensitive to the cut-off just as for the Dirac equation.
When there are several particles one can obtain Bose statistics by the rule that if two processes lead to the same state but with two electrons exchanged, their amplitudes are to be added (rather than subtracted as for Fermi statistics). In this case equivalence to the second quantization treatment of Pauli and Weisskopf should be demonstrable in a way very much like that given in I (appendix) for Dirac electrons. The Bose statistics mean that the sign of contribution of a closed loop to the vacuum polarization is the opposite of what it is for the Fermi case (see I
). It is (pb = pa + q)
giving,
the notation as in (33). The imaginary part for (q2)1/2 > 2m is again positive representing the loss in the probability of finding the final state to be a vacuum, associated with the possibilities of pair production. Fermi statistics would give a gain in probability (and also a charge renormalization of opposite sign to that expected).
10 APPLICATION TO MESON THEORIES
The theories which have been developed to describe mesons and the interaction of nucleons can be easily expressed in the language used here. Calculations, to lowest order in the interactions can be made very easily for the 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 momentum 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 = iγxγyγzγt. For example, the self-energy matrix of a nucleon of momentum p in this theory is(g2/πi) ∫ γ5 (p − k − M)−1 γ5d4 k (k2 − µ2)−1 C (k2 − µ2).
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