Comparison With The Hamiltonian Method

Electrodynamics can be looked upon in two equivalent and complemen- tary ways. One is as the description of the behavior of a field (Maxwell’s equations). The other is as a description of a direct interaction at a dis- tance (albeit delayed in time) between charges (the solutions of Lienard and Wiechert). From the latter point of view light is considered as an interaction of the charges in the source with those in the absorber. This is an imprac- tical point of view because many kinds of sources produce the same kind of effects. The field point of view separates these aspects into two simpler problems, production of light, and absorption of light. On the other hand, the field point of view is less practical when dealing with close collisions of particles (or their action on themselves). For here the source and absorber are not readily distinguishable, there is an intimate exchange of quanta. The fields are so closely determined by the motions of the particles that it is just as well not to separate the question into two problems but to consider the process as a direct interaction. Roughly, the field point of view is most practical for problems involving real quanta, while the interaction view is best for the discussion of the virtual quanta involved. We shall emphasize the interaction viewpoint in this paper, first because it is less familiar and therefore requires more discussion, and second because the important aspect in the problems with which we shall deal is the effect of virtual quanta.

The Hamiltonian method is not well adapted to represent the direct action at a distance between charges because that action is delayed. The Hamiltonian method represents the future as developing out of the present.

If the values of a complete set of quantities are known now, their values can be computed at the next instant in time. If particles interact through a de- layed interaction, however, one cannot predict the future by simply knowing the present motion of the particles. One would also have to know what the motions of the particles were in the past in view of the interaction this may have on the future motions. This is done in the Hamiltonian electrodynam- ics, of course, by requiring that one specify besides the present motion of the particles, the values of a host of new variables (the coordinates of the field oscillators) to keep track of that aspect of the past motions of the particles which determines their future behavior. The use of the Hamiltonian forces one to choose the field viewpoint rather than the interaction viewpoint.

In many problems, for example, the close collisions of particles, we are not interested in the precise temporal sequence of events. It is not of interest to be able to say how the situation would look at each instant of time during a collision and how it progresses from instant to instant. Such ideas are only useful for events taking a long time and for which we can readily obtain information during the intervening period. For collisions it is much easier to treat the process as a whole.7 The M∅ller interaction matrix for the the collision of two electrons is not essentially more complicated than the nonrelativistic Rutherford formula, yet the mathematical machinery used to obtain the former from quantum electrodynamics is vastly more complicated than Schr¨odinger’s equation with the e2/r12 interaction needed to obtain the latter. The difference is only that in the latter the action is instantaneous so that the Hamiltonian method requires no extra variables, while in the former relativistic case it is delayed and the Hamiltonian method is very cumbersome.

We shall be discussing the solutions of equations rather than the time differential equations from which they come. We shall discover that the so- lutions, because of the over-all space-time view that they permit, are as easy to understand when interactions are delayed as when they are instantaneous. As a further point, relativistic invariance will be self-evident. The Hamil- tonian form of the equations develops the future from the instantaneous present. But for different observers in relative motion the instantaneous present is different, and corresponds to a different 3-dimensional cut of space-time.

Thus the temporal analyses of different observers is different and their Hamiltonian equations are developing the process in different ways. These differences are irrelevant, however, for the solution is the same in any space time frame. By forsaking the Hamiltonian method, the wedding of relativity and quantum mechanics can be accomplished most naturally. We illustrate these points in. the next section by studying the solution of Schr¨odinger’s equation for non-relativistic particles interacting by an in- stantaneous Coulomb potential (Eq. 2). When the solution is modified to include the effects of delay in the interaction and the relativistic properties of the electrons we obtain an expression of the laws of quantum electrody- namics (Eq. 4).

2 THE INTERACTION BETWEEN CHARGES

We study by the same methods as in I, the interaction of two particles using the same notation as I. We start by considering the non-relativistic case described by the Schr¨dinger equation (I, Eq. 1). The wave function at a given time is a function ψ(xa, xb, t) of the coordinates xa and xb of each particle. Thus call K(xa, xb, t; x′a, x′b, t′) the amplitude that particle a at x′a at time t′ will get to xa at t while particle b at x′b at t′ gets to xb at t.

If the particles are free and do not interact this is K(xa, xb, t; x′ a, x′ b, t′) = K0a(xa, t; x′ a, t′)K0b(xb, t; x′ b, t′) where K0a is the K0 function for particle a considered as free. In this case we can obviously define a quantity like K, but for which the time t need not be the same for particles a and b (likewise for t′); e.g., K0(3, 4; 1, 2) = K0a(3, 1)K0b(4, 2) (1)

can be thought of as the amplitude that particle a goes from x1 at t1 to x3 at t3 and that particle b goes from x2 at t2 to x4 at t4.

When the particles do interact, one can only define the quantity K(3, 4; 1, 2) precisely if the interaction vanishes between t1 and t2 and also between t3 and t4. In a real physical system such is not the case. There is such an enormous advantage, however, to the concept that we shall continue to use it, imagining that we can neglect the effect of interactions between t1 and t2 and between t3 and t4. For practical problems this means choosing such long time intervals t3 − t1 and t4 − t2 that the extra interactions near the end points have small relative effects.

As an example, in a scattering problem it may well be that the particles are so well separated initially and finally that the interaction at these times is negligible. Again energy values can be defined by the average rate of change of phase over such long time intervals that errors initially and finally can be neglected. Inasmuch as any physical problem can be defined in terms of scattering processes we do not lose much in a general theoretical sense by this approximation. If it is not Figure 1: The fundamental interaction Eq. (4). Exchange of one quantum between two electrons.

made it is not easy to study interacting particles relativistically, for there is nothing significant in choosing t1 = t3 if x1 6 = x3 as absolute simultaneity of events at a distance cannot be defined invariantly. It is essentially to avoid this approximation that the complicated structure of the older quantum electrodynamics has been built up. We wish to describe electrodynamics as a delayed interaction between particles. If we can make the approximation of assuming a meaning to K(3, 4; 1, 2) the results of this interaction can be expressed very simply.

To see how this may be done, imagine first that the interaction is simply that given by a Coulomb potential e2/r where r is the distance between the particles. If this be turned on only for a very short time ∆t0 at time t0 the first order correction to K(3, 4; 1, 2) can be worked out exactly as was Eq. (9) of I by an obvious generalization to two particles: K(1)(3, 4; 1, 2) = −ie2 ∫ ∫ K0a(3, 5)K0b(4, 6)r−1 56 ×K0a(5, 1)K0b(6, 2)d3x5d3x6∆t0, 8 where t5 = t6 = t0. If now the potential were on at all times (so that strictly K is not defined unless t4 = t3 and t1 = t2), the first-order effect is obtained by integrating on t0, which we can write as an integral over both t5 and t6 if we include a delta-function δ(t5 − t6) to insure contribution only when t5 = t6 Hence, the first-order effect of interaction is (calling t5 − t6 = t56): K(1)(3, 4; 1, 2) = −ie2 ∫ ∫ K0a(3, 5)K0b(4, 6)r−1 56 ×δ(t56)K0a(5, 1)K0b(6, 2)dτ5dτ6, (2) where dτ = d3xdt.

We know, however, in classical electrodynamics, that the Coulomb po- tential does not act instantaneously, but is delayed by a time r56, taking the speed of light as unity. This suggests simply replacing r−1 56 δ(t56) in (2) by something like r−1 56 δ(t56 − r56) to represent the delay in the effect of b on a. This turns out to be not quite right,8 for when this interaction is repre- sented by photons they must be of only positive energy, while the Fourier transform of δ(t56 −r56) contains frequencies of both signs. It should instead be replaced by δ+(t56 − r56) where δ+(x) = ∞∫ 0 e−iωxdω/π = lim →0 (πi)−1 x − i = δ(x) + (πix)−1. (3) This is to be averaged with r−1 56 δ+(−t56 − r56) which arises when t5 < t6 and corresponds to a emitting the quantum which b receives. Since (2r)−1(δ+(t − r) + δ+(−t − r)) = δ+(t2 − r2), this means r−1 56 δ(t56) is replaced by δ+(s2 56) where s2 56 = t2 56 −r2 56 is the square of the relativistically invariant interval between points 5 and 6. Since in clas- sical electrodynamics there is also an interaction through the vector poten- tial, the complete interaction (see A, Eq. (I)) should be (1 − (v5 · v6)δ+(s2 56), or in the relativistic case, (1 − αa · αb)δ+(s2 56) = βaβbγaμγbμδ+(s2 56). 8It 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.

Hence we have for electrons obeying the Dirac equation, K(1)(3, 4; 1, 2) = ie2 ∫ ∫ K+a(3, 5)K+b(4, 6)γaμγbμ ×δ+(s2 56)K+a(5, 1)K+b(6, 2)dτ5dτ6, (4) where γaμ and γbμ, are the Dirac matrices applying to the spinor correspond- ing to particles a and b, respectively (the factor βaβb being absorbed in the definition, I Eq. (17), of K+). This is our fundamental equation for electrodynamics. It describes the effect of exchange of one quantum (therefore first order in e2) between two electrons. It will serve as a prototype enabling us to write down the corre- sponding quantities involving the exchange of two or more quanta between two electrons or the interaction of an electron with itself. It is a consequence of conventional electrodynamics. Relativistic invariance is clear. Since one sums over μ it contains the effects of both longitudinal and transverse waves in a relativistically symmetrical way. We shall now interpret Eq. (4) in a manner which will permit us to write down the higher order terms. It can be understood (see Fig. 1) as saying that the amplitude for “a” to go from 1 to 3 and “b” to go from 2 to 4 is altered to first order because they can exchange a quantum. Thus, “a” can go to 5 (amplitude (K+(5, 1)) emit a quantum (longitudinal, transverse, or scalar γaμ) and then proceed to 3 (K+(3, 5)). Meantime “b” goes to 6 (K+(6, 2)), absorbs the quantum (γbμ) and proceeds to 4 (K+(4, 6)). The quantum meanwhile proceeds from 5 to 6, which it does with amplitude δ+(s256). We must sum over all the possible quantum polarizations it and positions and times of emission 5, and of absorption 6. Actually if t5 > t6 it would be better to say that “a” absorbs and “b” emits but no attention need be paid to these matters, as all such alternatives are automatically contained in (4). The correct terms of higher order in e2 or involving larger numbers of electrons (interacting with themselves or in pairs) can be written down by the same kind of reasoning. They will be illustrated by examples as we proceed. In a succeeding paper they will all be deduced from conventional quantum electrodynamics.

Calculation, from (4), of the transition element between positive energy free electron states gives the M∅ller scattering of two electrons, when ac- count is taken of the Pauli principle. The exclusion principle for interacting charges is handled in exactly the same way as for noninteracting charges (I). For example, for two charges it requires only that one calculate K(3, 4; 1, 2) − K(4, 3; 1, 2) to get the net 10 amplitude for arrival of charges at 3 and 4. It is disregarded in intermedi- ate states. The interference effects for scattering of electrons by positrons discussed by Bhabha will be seen to result directly in this formulation. The formulas are interpreted to apply to positrons in the manner discussed in I. As our primary concern will be for processes in which the quanta are virtual we shall not include here the detailed analysis of processes involving real quanta in initial or final state, and shall content ourselves by only stating the rules applying to them.9 The result of the analysis is, as expected, that they can be included by the same line of reasoning as is used in discussing the virtual processes, provided the quantities are normalized in the usual manner to represent single quanta. For example, the amplitude that an electron in going from 1 to 2 absorbs a quantum whose vector potential, suitably normalized, is cμexp(−ik · x) = Cμ(x) is just the expression (I, Eq. (13)) for scattering in a potential with A (3) replaced by C (3). Each quantum interacts only once (either in emission or in absorption), terms like (I, Eq. (14)) occur only when there is more than one quantum involved. The Bose statistics of the quanta can, in all cases, be disregarded in intermediate states. The only effect of the statistics is to change the weight of initial or final states. If there are among quanta, in the initial state, some a which are identical then the weight of the state is (1/n!) of what it would be if these quanta were considered as different (similarly for the final state).