are analyzed nevertheless in anticipation of application to quantum electrodynamics.
The results are also expressed in momentum-energy variables. Equivalence to the second quantization theory of holes is proved in an appendix.
1. INTRODUCTION
THIS is the first of a set of papers dealing with the solution of problems in quantum electrodynamics. The main principle is to deal directly with the solutions to the Hamiltonian differential equations rather than with these equations themselves. Here we treat simply the motion of electrons and positrons in given external potentials. In a second paper we consider the interactions of these particles, that is, quantum electrodynamics.
The problem of charges in a fixed potential is usually treated by the method of second quantization of the eletron field, using the ideas of the theory of holes. Instead we show that by a suitable choice and interpretation of the solutions of Dirac’s equation the problem may be equally well treated in a manner which is fundamentally no more complicated than Schrödinger’s method of dealing with one or more particles. The various creation and annihilation operators in the conventional electron field view are required because the number of particles is not conserved, i.e., pairs may be created or destroyed. On the other hand charge is conserved which suggests that if we follow the charge, not the particle, the results can be simplified.
In the approximation of classical relativistic theory the creation of an electron pair (electron A, positron B) might be represented by the start of two world lines from the point of creation, 1. The world lines of the positron will then continue until it annihilates another electron, C, at a world point 2. Between the times t1 and t2 there are then three world lines, before and after only one. However, the world lines of C, B, and A together form one continuous line albeit the “positron part” B of this continuous line is directed backwards in time. Following the charge rather than the particles corresponds to considering this continuous world line as a whole rather than breaking it up into its pieces. It is as though a bombardier flying low over a road suddenly sees three roads and it is only when two of them come together and disappear again that he realizes that he has simply passed over a long switchback in a single road.
This over-all space-time point of view leads to considerable simplification in many problems. One can take into account at the same time processes which ordinarily would have to be considered separately. For example, when considering the scattering of an electron by a potential one automatically takes into account the effects of virtual pair productions. The same equation, Dirac’s, which describes the deflection of the world line of an electron in a field, can also describe the deflection (and in just as simple a manner) when it is large enough to reverse the time-sense of the world line, and thereby correspond to pair annihilation. Quantum mechanically the direction of the world lines is replaced by the direction of propagation of waves.
This view is quite different from that of the Hamiltonian method which considers the future as developing continuously from out of the past. Here we imagine the entire space-time history laid out, and that we just become aware of increasing portions of it successively. In a scattering problem this over-all view of the complete scattering process is similar to the S -matrix viewpoint of Heisenberg. The temporal order of events during the scattering, which is analyzed in such detail by the Hamiltonian differential equation, is irrelevant. The relation of these viewpoints will be discussed much more fully in the introduction to the second paper, in which the more complicated interactions are analyzed.
The development stemmed from the idea that in non-relativistic quantum mechanics the amplitude for a given process can be considered as the sum of an amplitude for each space-time path available.nt In view of the fact that in classical physics positrons could be viewed as electrons proceeding along world lines toward the past (reference 7) the attempt was made to remove, in the relativistic case, the restriction that the paths must proceed always in one direction in time. It was discovered that the results could be even more easily understood from a more familiar physical viewpoint, that of scattered waves. This viewpoint is the one used in this paper. After the equations were worked out physically the proof of the equivalence to the second quantization theory was found.nu
First we discuss the relation of the Hamiltonian differential equation to its solution, using for an example the Schrödinger equation. Next we deal in an analogous way with the Dirac equation and show how the solutions may be interpreted to apply to positrons. The interpretation seems not to be consistent unless the electrons obey the exclusion principle. (Charges obeying the Klein-Gordon equations can be described in an analogous manner, but here consistency apparently requires Bose statistics.)nv A representation in momentum and energy variables which is useful for the calculation of matrix elements is described. A proof of the equivalence of the method to the theory of holes in second quantization is given in the Appendix.
2. GREEN’S FUNCTION TREATMENT OF SCHRÖDINGER’S EQUATION
We begin by a brief discussion of the relation of the non-relativistic wave equation to its solution. The ideas will then be extended to relativistic particles, satisfying Dirac’s equation, and finally in the succeeding paper to interacting relativistic particles, that is, quantum electrodynamics.
The Schrödinger equation
(1)
describes the change in the wave function ψ in an infinitesimal time Δt as due to the operation of an operator exp(−iHΔt). One can ask also, if ψ (x1, t1) is the wave function at x1 at time t1, what is the wave function at time t2 > t1? It can always be written as
(2)
where K is a Green’s function for the linear Eq. (1). (We have limited ourselves to a single particle of coordinate x, but the equations are obviously of greater generality.) If H is a constant operator having eigenvalues En, eigenfunctions φn so that ψ (x, t1) can be expanded as Σn Cn φn (x), then ψ (x, t2)=exp(−iEn (t2 − t1)) × Cnφn(x). Since Cn= φ*n(x1 )ψ(x1, t1)d3 x1, one finds (where we write 1 for x1, t1 and 2 for x2, t2) in this case
(3)
n for t2 > t1 . We shall find it convenient for t2 < t1 to define K (2, 1) = 0 (Eq. (2) is then not valid for t2 < t1). It is then readily shown that in general K can be defined by that solution of
(4)
which is zero for t2 2 < t1, where δ(2, 1) = δ(t2 − t1 )δ(x 2 − x1) × δ(y2 − y1)δ(z2 − z1) and the subscript 2 on H2 means that the operator acts on the variables of 2 of K (2, 1). When H is not constant, (2) and (4) are valid but K is less easy to evaluate than (3).nw
We can call K (2, 1) the total amplitude for arrival at x2, t2 starting from x1, t1. (It results from adding an amplitude, expiS, for each space time path between these points, where S is the action along the path.1) The transition amplitude for finding a particle in state χ(x2, t2) at time t2, if at t1 it was in ψ (x1, t1), is
(5)
A quantum mechanical system is described equally well by specifying the function K, or by specifying the Hamiltonian H from which it results. For some purposes the specification in terms of K is easier to use and visualize. We desire eventually to discuss quantum electrodynamics from this point of view.
To gain a greater familiarity with the K function and the point of view it suggests, we consider a simple perturbation problem. Imagine we have a particle in a weak potential U(x, t), a function of position and time. We wish to calculate K (2, 1) if U differs from zero only for t between t1 and t2. We shall expand K in increasing powers of U:
(6)
To zero order in U, K is that for a free particle, K0(2, 1).4 To study the first order correction K (1) (2, 1), first consider the case that U differs from zero only for the infinitesimal time interval Δt3 between some time t3 and t3 + Δt3(t1 < t3 < t2). Then if ψ (1) is the wave function at x1, t1, the wave function at x3, t3 is
(7)
since from t1 to t3 the particle is free. For the short interval Δt3 we solve (1) as
where we put H = H0 + U, H0 being the Hamil
tonian of a free particle. Thus ψ (x, t3 + Δt3) differs from what it would be if the potential were zero (namely (1 − i H0Δt3 )ψ (x, t3 )) by the extra piece
(8)
which we shall call the amplitude scattered by the potential. The wave function at 2 is given byψ (x2 , t2) = ∫ K 0 (x2 , t2 ; x3 , t3 + Δt3 ) ψ (x3 , t3 + Δt3 d3x3.
since after t3 + Δt3 the particle is again free. Therefore the change in the wave function at 2 brought about by the potential is (substitute (7) into (8) and (8) into the equation for ψ (x2, t2)):Δψ(2) = −i ∫ K0(2, 3)U(3) K 0 (3, 1) ψ (1)d 3x1d 3x3 Δt3.
In the case that the potential exists for an extended time, it may be looked upon as a sum of effects from each interval Δt3 so that the total effect is obtained by integrating over t3 as well as x3. From the definition (2) of K then, we find
(9)
where the integral can now be extended over all space and time, dτ3 = d3x3dt3. Automatically there will be no contribution if t3 is outside the range t1 1 to t2 because of our definition, K 0 (2, 1) = 0 for t2 < t1.
We can understand the result (6), (9) this way. We can imagine that a particle travels as a free particle from point to point, but is scattered by the potential U. Thus the total amplitude for arrival at 2 from 1 can be considered as the sum of the amplitudes for various alternative routes. It may go directly from 1 to 2 (amplitude K0(2, 1), giving the zero order term in (6)). Or (see Fig. 1 (a)) it may go from 1 to 3 (amplitude K0(3, 1)), get scattered there by the potential (scattering amplitude −iU(3) per unit volume and time) and then go from 3 to 2 (amplitude K0(2, 3)). This may occur for any point 3 so that summing over these alternatives gives (9).
Again, it may be scattered twice by the potential (Fig. 1 (b)). It goes from 1 to 3 (K0 (3,1)), gets scattered there (−iU(3)) then proceeds to some other point, 4, in space time (amplitude K0(4, 3)) is scattered again (−iU (4)) and then proceeds to 2 (K0(2, 4)). Summing over all possible places and times for 3, 4 find that the second order contribution to the total amplitude K(2) (2, 1) is
(10)
FIG. 1 The Schrödinger (and Dirac) equation can be visualized as describing the fact that plane waves are scattered successively by a potential. Figure 1 (a) illustrates the situation in first order. K0(2, 3) is the amplitude for a free particle starting at point 3 to arrive at 2. The shaded region indicates the presence of the potential A which scatters at 3 with amplitude −iA(3) per cm3sec. (Eq. (9)). In (b) is illustrated the second order process (Eq. (10)), the waves scattered at 3 are scattered again at 4. However, in Dirac one-electron theory K0(4, 3) would represent electrons both of positive and of negative energies proceeding from 3 to 4. This is remedied by choosing a different scattering kernel K + (4, 3), Fig. 2.
This can be readily verified directly from (1) just as (9) was. One can in this way obviously write down any of the terms of the expansion (6).nx
3. TREATMENT OF THE DIRAC EQUATION
We shall now extend the method of the last section to apply to the Dirac equation. All that would seem to be necessary in the previous equations is to consider H as the Dirac Hamiltonian, ψ as a symbol with four indices (for each particle). Then K0 can still be defined by (3) or (4) and is now a 4–4 matrix which operating on the initial wave function, gives the final wave function. In (10), U(3) can be generalized to A4 (3) − α · A(3) where A4, A are the scalar and vector potential (times e, the electron charge) and α are Dirac matrices.
To discuss this we shall define a convenient relativistic notation. We represent four-vectors like x, t by a symbol xµ, where µ = 1, 2, 3, 4 and x4 = t is real. Thus the vector and scalar potential (times e) A, A4 is Aµ. The four matrices β α, β can be considered as transforming as a four vector γµ (our γµ differs from Pauli’s by a factor i for µ = 1, 2, 3). We use the summation convention aµbµ = a4b4 − a1b1 − a2b2 − a3b3 = a · b. In particular if aµ is any four vector (but not a matrix) we write a = aµγµ so that a is a matrix associated with a vector (a will often be used in place of aµ as a symbol for the vector). The γµ satisfy γµγν + γν γµ = 2δµν where δ44 = + 1, δ11 = δ22 = δ33 = − 1, and the other δµν are zero. As a consequence of our summation convention δµν a ν = a µ and δµµ = 4. Note that ab + ba = 2a · b and that a2 = aµaµ = a · a is a pure number. The symbol ∂ /∂ xµ, will mean ∂ /∂ t for µ = 4, and −∂/∂x, − ∂/∂y, − ∂/∂z for µ = 1, 2, 3. Call ∇ = γµ∂ /∂ xµ = β ∂ /∂ t + βα · ∇. We shall imagine hereafter, purely for relativistic convenience, that φn * in (3) is replaced by its adjoint .
Thus the Dirac equation for a particle, mass m, in an external field A = Aµγµ is
(11)
and Eq. (4) determining the propagation of a free particle becomes
(12) the index 2 on ∇2 indicating differentiation with respect to the coordinates x2µ which are represented as 2 in K +(2, 1) and δ(2, 1).
The function K+(2, 1) is defined in the absence of a field. If a potential A is acting a similar function, say K + (A) (2, 1) can be defined. It differs from K +(2, 1) by a first order correction given by the analogue of (9) namely
(13)
representing the amplitude to go from 1 to 3 as a free particle, get scattered there by the potential (now the matrix A(3) instead of U (3)) and continue to 2 as free. The second order correction, analogous to (10) is
(14)
and so on. In general K + (A) satisfies
(15) and the successive terms (13), (14) are the power series expansion of the integral equation
(16)
which it also satisfies.
We would now expect to choose, for the special solution of (12), K+ = K 0 where K0(2, 1) vanishes for t2 < t1 and for t2 > t1 is given by (3) where φn and En are the eigenfunctions and energy values of a particle satisfying Dirac’s equation, and φn* is replaced by .
The formulas arising from this choice, however, suffer from the drawback that they apply to the one electron theory of Dirac rather than to the hole theory of the positron. For example, consider as in Fig. 1 (a) an electron after being scattered by a potential in a small region 3 of space time. The one electron theory says (as does (3) with K+ = K0) that the scattered amplitude at another point 2 will proceed toward positive times with both positive and negative energies, that is with both positive and negative rates of change of phase. No wave is scattered to times previous to the time of scattering. These are just the properties of K0 (2, 3).
On the other hand, according to the positron theory negative energy states are not available to the electron after the scattering. Therefore the choice K+ = K0 is unsatisfactory. But there are other solutions of (12). We shall choose the solution defining K+ (2, 1) so that K+ (2, 1) for t2 > t1 is the sum of (3) over positive energy states only. Now this new solution must satisfy (12) for all times in order that the representation be complete. It must therefore differ from the old solution K0 by a solution of the homogeneous Dirac equation. It is clear from the definition that the difference K0 − K + is the sum of (3) over all negative energy states, as long as t2 > t1. But this difference must be a solution of the homogeneous Dirac equation for all times and must therefore be represented by the same sum over negative energy states also for t2 < t1. Since K0 = 0 in this case, it follows that our new kernel, K+(2, 1), for t2< t1 is the negative of the sum (3) over negative energy states. That is,
(17)
With this choice of K+ our equations such as (13) and (14) will now give results equivalent to those of the positron hole theory.
That (14), for example, is the correct second order expression for finding at 2 an electron originally at 1 according to the positron theory may be seen as follows (Fig. 2). Assume as a special example that t2 > t1 and that the potential vanishes except in interval t2 − t1 so that t4 and t3 both lie between t1 and t2.
First suppose t4 > t3 (Fig. 2(b)). Then (since t3 3 > t1) the electron assumed originally in a positive energy state propagates in that state (by K+(3, 1)) to position 3 where it gets scattered (A(3)). It then proceeds to
4, which it must do as a positive energy electron. This is correctly described by (14) for K+(4, 3) contains only positive energy components in its expansion, as t4 > t3. After being scattered at 4 it then proceeds on to 2, again necessarily in a positive energy state, as t2 > t4.
In positron theory there is an additional contribution due the possibility of virtual pair production (Fig. 2(c)). A pair could be created by the potential A (4) at 4, the electron of which is that found later at 2. The positron (or rather, the hole) proceeds to 3 where it annihilates the electron which has arrived there from 1.
The Dreams That Stuff is Made of Page 71