the one hand, both the foundations and the applications of the theory have been simplified by being presented in a completely relativistic way; on the other, the divergence difficulties have been at least partially overcome. In the reports so far published, emphasis has naturally been placed on the second of these advances; the magnitude of the first has been somewhat obscured by the fact that the new methods have been applied to problems which were beyond the range of the older theories, so that the simplexity of the methods was hidden by the complexity of the problems. Furthermore, the theory of Feynman differs so profoundly in its formulation from that of Tomonaga and Schwinger, and so little of it has been published, that its particular advantages have not hitherto been available to users of the other formulations. The advantages of the Feynman theory are simplicity and ease of application, while those of Tomonaga-Schwinger are generality and theoretical completeness.
The present paper aims to show how the Schwinger theory can be applied to specific problems in such a way as to incorporate the ideas of Feynman. To make the paper reasonably self-contained it is necessary to outline the foundations of the theory, following the method of Tomonaga; but this paper is not intended as a substitute for the complete account of the theory shortly to be published by Schwinger. Here the emphasis will be on the application of the theory, and the major theoretical problems of gauge invariance and of the divergencies will not be considered in detail. The main results of the paper will be general formulas from which the radiative reactions on the motions of electrons can be calculated, treating the radiation interaction as a small perturbation, to any desired order of approximation. These formulas will be expressed in Schwinger’s notation, but are in substance identical with results given previously by Feynman. The contribution of the present paper is thus intended to be twofold: first, to simplify the Schwinger theory for the benefit of those using it for calculations, and second, to demonstrate the equivalence of the various theories within their common domain of applicability.oj
II. OUTLINE OF THEORETICAL FOUNDATIONS
Relativistic quantum mechanics is a special case of non-relativistic quantum mechanics, and it is convenient to use the usual non-relativistic terminology in order to make clear the relation between the mathematical theory and the results of physical measurements. In quantum electrodynamics the dynamical variables are the electromagnetic potentials Aµ(r) and the spinor electron-positron field ψα(r); each component of each field at each point r of space is a separate variable. Each dynamical variable is, in the Schrödinger representation of quantum mechanics, a time-independent operator operating on the state vector Φ of the system. The nature of Φ (wave function or abstract vector) need not be specified; its essential property is that, given the Φ of a system at a particular time, the results of all measurements made on the system at that time are statistically determined. The variation of Φ with time is given by the Schrödinger equation
(1)
where H(r) is the operator representing the total energy-density of the system at the point r. The general solution of(1) is
(2)
with Φ0 any constant state vector.
Now in a relativistic system, the most general kind of measurement is not the simultaneous measurement of field quantities at different points of space. It is also possible to measure independently field quantities at different points of space at different times, provided that the points of space-time at which the measurements are made lie outside each other’s light cones, so that the measurements do not interfere with each other. Thus the most comprehensive general type of measurement is a measurement of field quantities at each point r of space at a time t(r), the locus of the points (r, t(r)) in space-time forming a 3-dimensional surface σ which is space-like (i.e., every pair of points on it is separated by a space-like interval). Such a measurement will be called “an observation of the system on σ.” It is easy to see what the result of the measurement will be. At each point rʹ the field quantities will be measured for a state of the system with state vector Φ(t(r’)) given by (2). But all observable quantities at rʹ are operators which commute with the energy-density operator H(r) at every point r different from r’, and it is a general principle of quantum mechanics that if B is a unitary operator commuting with A, then for any state Φ the results of measurements of A are the same in the state Φ as in the state BΦ. Therefore, the results of measurement of the field quantities at rʹ in the state Φ(t(rʹ)) are the same as if the state of the system were
(3)
which differs from Φ(t(rʹ)) only by a unitary factor commuting with these field quantities. The important fact is that the state vector Φ(σ) depends only on σ and not on r’. The conclusion reached is that observations of a system on σ give results which are completely determined by attributing to the system the state vector Φ(σ) given by (3).
The Tomonaga-Schwinger form of the Schrödinger equation is a differential form of (3). Suppose the surface σ to be deformed slightly near the point r into the surface σʹ, the volume of space-time separating the two surfaces being V. Then the quotient[Φ (σʹ) − Φ(σ)/V
tends to a limit as V → 0, which we denote by ∂Φ/∂σ(r) and call the functional derivative of Φ with respect to σ at the point r. From (3) it follows that
(4)
and (3) is, in fact, the general solution of (4).
The whole meaning of an equation such as (4) depends on the physical meaning which is attached to the statement “a system has a constant state vector Φ0.” In the present context, this statement means “results of measurements of field quantities at any given point of space are independent of time.” This statement is plainly non-relativistic, and so (4) is, in spite of appearances, a non-relativistic equation.
The simplest way to introduce a new state vector ψ which shall be a relativistic invariant is to require that the statement “a system has a constant state vector ψ” shall mean “a system consists of photons, electrons, and positrons, traveling freely through space without interaction or external disturbance.” For this purpose, let
(5)
where H0 is the energy-density of the free electromagnetic and electron fields, and H1 is that of their interaction with each other and with any external disturbing forces that may be present. A system with constant ψ is, then, one whose H1 is identically zero; by (3) such a system corresponds to a Φ of the form
(6)
It is therefore consistent to write generally
(7)
thus defining the new state vector ψ of any system in terms of the old Φ. The differential equation satisfied by ψ is obtained from (4), (5), (6), and (7) in the form
(8)
Now if q(r) is any time-independent field operator, the operatorq (x0) = (T (σ))−1 q (r)T (σ )
is just the corresponding time-dependent operator as usually defined in quantum electrodynamics.ok It is a function of the point x0 of space-time whose coordinates are (r, ct(r)), but is the same for all surfaces σ passing through this point, by virtue of the commutation of H1(r) with H0(rʹ) for rʹ ≠ r. Thus (8) may be written
(9)
where H1(x0) is the time-dependent form of the energy-density of interaction of the two fields with each other and with external forces. The left side of (9) represents the degree of departure of the system from a system of freely traveling particles and is a relativistic invariant; H1(x0) is also an invariant, and thus is avoided one of the most unsatisfactory features of the old theories, in which the invariant H1 was added to the non-invariant H0. Equation (9) is the starting point of the Tomonaga-Schwinger theory.
III. INTRODUCTION OF PERTURBATION THEORY
Equation (9) can be solved explicitly. For this purpose it is convenient to introduce a one-parameter family of space-like surfaces filling the whole of space-time, so that one and only one member σ (x) of the family passes through any given point x. Let σ0, σ1, σ2, ... be a sequence of surfaces of the family, starting with σ0 and proceeding in small steps steadily into the
past. By
is denoted the integral of H1 (x) over the 4-dimensional volume between the surfaces σ1 and σ0; similarly, by
are denoted integrals over the whole volume to the past of σ0 and to the future of σ0, respectively. Consider the operator
(10)
the product continuing to infinity and the surfaces σ0, σ1, . . . being taken in the limit infinitely close together. U satisfies the differential equation
(11)
and the general solution of (9) is
(12)
with ψ0 any constant vector.
Expanding the product (10) in ascending powers of H1 gives a series
(13)
Further, U is by (10) obviously unitary, and
(14)
It is not difficult to verify that U is a function of σ0 alone and is independent of the family of surfaces of which σ0 is one member. The use of a finite number of terms of the series (13) and (14), neglecting the higher terms, is the equivalent in the new theory of the use of perturbation theory in the older electrodynamics.
The operator U(∞), obtained from (10) by taking σ0 in the infinite future, is a transformation operator transforming a state of the system in the infinite past (representing, say, converging streams of particles) into the same state in the infinite future (after the particles have interacted or been scattered into their final outgoing distribution). This operator has matrix elements corresponding only to real transitions of the system, i.e., transitions which conserve energy and momentum. It is identical with the Heisenberg S matrix.ol
IV. ELIMINATION OF THE RADIATION INTERACTION
In most of the problem of electrodynamics, the energy-density H1(x0) divides into two parts—
(15)
(16)
the first part being the energy of interaction of the two fields with each other, and the second part the energy produced by external forces. It is usually not permissible to treat He as a small perturbation as was done in the last section. Instead, H i alone is treated as a perturbation, the aim being to eliminate H i but to leave H e in its original place in the equation of motion of the system.
Operators S(σ) and S(∞) are defined by replacing H1 by Hi in the definitions of U(σ) and U(∞). Thus S(σ) satisfies the equation
(17)
Suppose now a new type of state vector Ω(σ) to be introduced by the substitution
(18)
By (9), (15), (17), and (18) the equation of motion for Ω(σ) is
(19)
The elimination of the radiation interaction is hereby achieved; only the question, “How is the new state vector Ω(σ) to be interpreted?,” remains.
It is clear from (19) that a system with a constant Ω is a system of electrons, positrons, and photons, moving under the influence of their mutual interactions, but in the absence of external fields. In a system where two or more particles are actually present, their interactions alone will, in general, cause real transitions and scattering processes to occur. For such a system it is rather “unphysical” to represent a state of motion including the effects of the interactions by a constant state vector; hence, for such a system the new representation has no simple interpretation. However, the most important systems are those in which only one particle is actually present, and its interaction with the vacuum fields gives rise only to virtual processes. In this case the particle, including the effects of all its interactions with the vacuum, appears to move as a free particle in the absence of external fields, and it is eminently reasonable to represent such a state of motion by a constant state vector. Therefore, it may be said that the operator,
(20)
on the right of (19) represents the interaction of a physical particle with an external field, including radiative corrections. Equation (19) describes the extent to which the motion of a single physical particle deviates, in the external field, from the motion represented by a constant state-vector, i.e., from the motion of an observed “free” particle.
If the system whose state vector is constantly Ω undergoes no real transitions with the passage of time, then the state vector Ω is called “steady.” More precisely, Ω is steady if, and only if, it satisfies the equation
(21)
As a general rule, one-particle states are steady and many-particle states unsteady. There are, however, two important qualifications to this rule.
First, the interaction (20) itself will almost always cause transitions from steady to unsteady states. For example, if the initial state consists of one electron in the field of a proton, HT will have matrix elements for transitions of the electron to a new state with emission of a photon, and such transitions are important in practice. Therefore, although the interpretation of the theory is simpler for steady states, it is not possible to exclude unsteady states from consideration.
Second, if a one-particle state as hitherto defined is to be steady, the definition of S(σ) must be modified. This is because S(∞) includes the effects of the electromagnetic self-energy of the electron, and this self-energy gives an expectation value to S(∞) which is different from unity (and indeed infinite) in a one-electron state, so that Eq. (21) cannot be satisfied. The mistake that has been made occurred in trying to represent the observed electron with its electromagnetic self-energy by a wave field with the same characteristic rest-mass as that of the “bare” electron. To correct the mistake, let δm denote the electromagnetic mass of the electron, i.e., the difference in rest-mass between an observed and a “bare” electron. Instead of (5), the division of the energy-density H(r) should have taken the form H(r) = (H0(r) + δmc2ψ*(r))βψ(r)) + (H1(r) − δmc2ψ*(r)βψ(r)). The first bracket on the right here represents the energy-density of the free electromagnetic and electron fields with the observed electron restmass, and should have been used instead of H0(r) in the definition (6) of T(σ). Consequently, the second bracket should have been used instead of H1(r) in Eq. (8).
The definition of S(σ) has therefore to be altered by replacing Hi(x0) byom
(22)
The value of δm can be adjusted so as to cancel out the self-energy effects in S(∞) (this is only a formal adjustment since the value is actually infinite), and then Eq. (21) will be valid for one-electron states. For the photon self-energy no such adjustment is needed since, as proved by Schwinger, the photon self-energy turns out to be identically zero.
The foregoing discussion of the self-energy problem is intentionally only a sketch, but it will be found to be sufficient for practical applications of the theory. A fuller discussion of the theoretical assumptions underlying this treatment of the problem will be given by Schwing in his forthcoming papers. Moreover, it must realized that the theory as a whole cannot be put into a finally satisfactory form so long as divergencies occur in it, however skilfully these divergencies are circumvented; therefore, the present treatment should be regarded as justified by its success in applications rather than by its theoretical derivation.
The important results of the present paper up to this point are Eq. (19) and the interpretation of the state vector Ω. The state vector ψ of a system can be interpreted as a wave function giving the probability amplitude of finding any particular set of occupation numbers for the various possible states of free electrons, positrons, and photons. The state vector Ω of a system with a given ψ on a given surface σ is, crudely speaking, the ψ which the system would have had in the infinite past if it had arrived at the given ψ on σ under the influence of the interaction HI (x0) alone.
The definition of Ω being unsymmetrical between past and future, a new type of state vector Ωʹ can be defined by reversing the direction of time in the definition of Ω. Thus the Ωʹ of a system with a given ψ on a given σ is the ψ which the system would reach in the infinite future if it continued to move under the influence of H1(x0) alone. More simply, Ωʹ can be defined by the equation
(23)
Since S(∞) is a unitary operator independent of σ,the state vectors Ω and Ωʹ are really only the same ve
ctor in two different representations or coordinate systems. Moreover, for any steady state the two are identical by (21).
V. FUNDAMENTAL FORMULAS OF THE SCHWINGER AND FEYNMAN THEORIES
The Schwinger theory works directly from Eqs. (19) and (20), the aim being to calculate the matrix elements of the “effective external potential energy” HT between states specified by their state vectors Ω. The states considered in practice always have Ω of some very simple kind, for example, Ω representing systems in which one or two free-particle states have occupation number one and the remaining free-particle states have occupation number zero. By analogy with (13), S(σ0) is given by
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