The quantum-mechanical paradox proposed by Einstein, Podolsky, and Rosen¹ (EPR) in 1935 is essentially a demonstration that the results of quantum mechanics are logically inconsistent with the premise that a measurement made with one instrument cannot influence the measurement made by another instrument if the measurement events are separated by a spacelike interval.² This is sometimes called the locality premise.

    In 1964 it was demonstrated by Bell³ in analyzing a Gedankenexperiment suggested by Bohm and Aharonov⁴ that locality implied inequalities in the measured probabilities of spin orientation experiments on certain physical systems. Recently, it has been shown that these Bell inequalities lead to experimental predictions which differ markedly from those of quantum mechanics.^5,6 Thus it has become feasible to confront these two divergent views of reality, quantum mechanics and the EPR locality premise, with experimental tests.⁹ A number of such experimental tests have now been performed^7,8,10-13 and the most reasonable interpretation of the experimental results is that the quantum-mechanical predictions have been confirmed. ^9,10,14

    The implication of these experimental results is that, although the EPR locality premise seems eminently reasonable, it must be wrong. However, the locality premise is not easily relinquished, for if one measurement can alter the result of another measurement across a spacelike interval, then a suitable choice of inertial reference frames can make the "effect," i.e., the altered measurement, precede in time sequence the "cause," i.e., the altering measurement, in violation of the principle of causality. Clearly then, these experimental tests, while confirming the validity of quantum mechanics, have not clarified the EPR paradox, nor do they provide us with any new insights as to how the premise of locality (or causality) could be violated in quantum-mechanical systems. It is the purpose of this paper to attempt to clarify this situation.

    The analysis of the EPR paradox which will be presented here will involve the interaction of advanced and retarded wave functions. Therefore, we must start by examining these wave functions in the context of classical electrodynamics. The electromagnetic wave equation15 for source-free space can be written in the form:

where Ѳ is the Laplacian operator providing the second space derivative in three dimensions and F represents either the electric field vector E or the magnetic field vector B of the wave. Since this differential equation is second order in both time and space, it has two independent time solutions and two independent space solutions.

    Let us restrict our consideration to one dimension by requiring that the wave motion described by Eq. (1) moves along the x axis and that the E vector of the wave is along the y axis. Then two independent time solutions of Eq. (1) might have the form:

and

where l and f are the wavelength and frequency of the wave and the alternating signs in Eqs. (2) and (3) represent the two independent time solutions mentioned above. If the source of this radiation is considered to be at the origin and emitting in the +x direction, then these waves will exist only for x>0. We can investigate the path of these waves by requiring that the argument of the sinusoidal function in Eqs. (2) and (3) be a constant phase angle and examining the (x, t) locus which this implies. The wave corresponding to E₊ and B₊ will exist only when t <0, while the wave corresponding to E_- and B_- will exist only for t>0. Thus, the E_- wave arrives at a point x in a time t after emission, while the E₊ wave arrives at x in a time t before emission (i.e., goes backward in time).

    We can also examine the energy and momentum flow produced by these waves. From Maxwell's equations,

and

so

    Therefore, the Poynting vector, which indicates the direction of energy and momentum flow of the wave, is

where x , y, and z with carets are unit vectors along the Cartesian axes.

    Therefore, the upper sign in Eqs. (2) and (3) corresponds to a wave which is emitted from the origin in the +x direction but which corresponds to energy and momentum flow in the -x direction. Thus, wave E₊(x, t) is a negative-energy (and negative-frequency) solution of Eq. (1). As mentioned above, it will arrive at a point a distance x from the source at a time t =x/c before the instant of emission. For this reason, it is called an advanced wave. Solution E_-(x, t), on the other hand, is the more familiar positive-energy solution of Eq. (1). It arrives at x a time t = x/c after the instant of emission and is called the retarded solution.

    This advanced/retarded dichotomy emerges even more clearly when one examines the Lienard-Wiechert solutions of Eq. (1) when the latter is interpreted as a differential equation involving the electromagnetic four-potential.15 The advanced and retarded potential solutions then explicitly involve the evaluation of the potential at an advanced or retarded time depending of the distance from the source and the corresponding negative or positive transit time. These potentials correspond to the negative-energy advanced solution and the positive-energy retarded solution of the wave equation which are discussed above.

    Negative-energy solutions also appear in quantum-mechanical treatments of electromagnetism. We can, for example, consider Eq. (1) to be a quantum-mechanical wave equation, and can investigate the properties of its solutions by examining their eigenvalues. We can, for example, choose plane-wave solutions to Eq. (1) which have the form

where k is the wave number (k =2p/l) of the wave and points in the direction of propagation, and f is an arbitrary phase. The alternating signs in Eq. (6) indicate the pairs of space and time solutions mentioned above.

    The waves F₂ and F3 still have an undefined phase factor f. Assume that there is an electron at the origin of Fig. 1 which is oscillating with a position y(t)= yo Cos(2pf t) such as to produce these waves. It is well known15 that the retarded wave will lag the oscillation in phase by 90°, so Re(F2) is proportional to +Sin(2pf t) and Exp(if2) = -i. The advanced wave F3, which is the time reverse and therefore the complex conjugate of F₂, will have a corresponding phase factor Exp(if3) = +i. Therefore, we may define the retarded- and advanced-wave solution in terms of these waves:

and

    If we follow the space-time trajectory of these waves from negative x and t to positive x and t, we will see a continuous wave, expect that it has a 180º phase change at the origin in Fig. 1, i.e., the location of the source which produces the advanced and retarded waves. Thus, a superimposed wave which cancels F_adv will tend to reinforce F_ret and vice versa.

    We can investigate the energies and momenta of these two waves by operating on them with the total energy operator H º -i(h/2p)d/dt and the momentum operator P º i(h/2p)Ñ.  Doing this, we find that

    We note that the energy of a photon is E =hf and its momentum is p = (h/2p)k . Thus, the retarded solution is characteristic of a light photon having a momentum vector p and positive energy, while the advanced solution is characteristic of a light photon having negative energy and a momentum vector -p, i.e., in the opposite direction from that of the retarded wave. Although the energy and momentum were obtained for a specific example, the result is quite general.

    In Fig. 1 waves F₁ and F₂ both have positive energy, while waves F₃ and F₄ both have negative energy. This bears on the "zig-zag" problem posed by Gold^16,27; there is no solution of the electromagnetic wave equation which has the characteristic of negative energy and also moves in the "future" light cone. This is because the time direction and the characteristic energy are intimately connected and share the same sign. In the example given in Eq. (6) the sign of the second exponential determines both the time direction and the sign of the characteristic energy of the wave.

    The conventional interpretation of the above solutions is that the retarded solution corresponds to the process of emission of electromagnetic radiation, e.g., from an accelerated charge, and the advanced solution describes the absorption of electromagnetic radiation, so that the characteristic negative energy of that solution has the effect of increasing the energy of the absorber. Thus, we would have

    In one sense the conventional approach does apparently have time symmetry, because an observer viewing a movie made of a microscopic emission of radiation followed by its absorption would not be able to say whether the movie was running forward or backward in time sequence. This apparent symmetry and the association of advanced radiation with absorption has led to some confusion in the literature¹⁷ as to the time symmetry of the conventional approach to electrodynamics. However, in a deeper sense it should be clear that the conventional boundary conditions on electromagnetic radiation are not time symmetric, since they predict that if we pass an alternating current through an antenna we will observe retarded waves diverging from the antenna to infinity and toward the infinite future, not advanced waves converging on the antenna from infinity and from the infinite past. The choice of the conventional boundary condition imposes an ad hoc electromagnetic direction of time.

    In 1945 a paper was published by Wheeler and Feynman¹⁸ describing what has come to be known as Wheeler-Feynman absorber theory, or simply absorber theory.¹⁹ This approach to electrodynamics, which was anticipated to some extent by the work of Dirac,²⁰ Fokker, ²¹ and of Tetrode,²² proposes a time-symmetric boundary condition which asserts that a proper electromagnetic wave is composed of a half-amplitude retarded wave and a half-amplitude advanced wave, and that such waves are characteristic of both emission and absorption processes.

    The 1945 Wheeler-Feynman paper¹⁸ was preoccupied with radiative reaction and damping and demonstrated in four separate ways that the damping arose not from the interaction of the radiating particle with its own field but from its interaction with the advanced wave(s) produced by the distant absorber(s). It asserted as a postulate that the radiating particle did not interact with its own field, and placed a great deal of importance on this postulate because of its implications for the self-energy problem of the electron. However, Feynman later pointed out²³ that this non-self-interaction postulate is probably invalid, as demonstrated in certain situations arising in quantum electrodynamics (e.g., the Lamb shift) in which the interaction of an electron with its own field is required.

    For the purpose of the present work, the validity of the non-self-interaction postulate is considered irrelevant. As will be shown below, the time symmetry of the emitted radiation requires that there be no net reaction and damping when a time-symmetric pair of waves, advanced and retarded, are emitted simultaneously, and so the non-self-interaction postulate is not needed. Further, the electron's self-energy is needed to explain the Lamb shift, and so the "out" of eliminating the self-energy problem by invoking the non-self-interaction postulate looks considerably less attractive.

    However, there is a related problem for which absorber theory offers an advantage. Dirac, in his analysis of the radiation of an accelerated electron,²⁰ pointed out that the conventional approach to electrodynamics is troubled not only by the self-energy divergence but also by analogous singularities in the radiation field near the radiating electron. He showed that by including the advanced-wave contributions to the radiation field (which is equivalent to using the Wheeler-Feynman time-symmetric boundary condition), these radiation-field singularities (but not the self-energy singularity) were eliminated. For this reason, Konopinski²⁴ in his Lorentz-covariant treatment of electron radiation has adopted this "Lorentz-Dirac" approach, and points out that this elimination of the radiation-field singularities amounts to a de facto renormalization of the theory.

    Another difference in approach between the present work and previous treatments of absorber theory^18,19,25-27 is that the latter papers employ very general (but rather non-transparent) formalism and are concerned with the interaction between the emitter and a large number of absorbing sites. Here, on the other hand, we will use the simplest and most transparent formalism that is consistent with the points to be made and will concentrate on a "minimum" emitter-absorber "transaction." More elaborate emitter-absorber events such as those discussed in previous works on absorber theory^18,19,25,27 (cf. the two examples given in Ref. 18) are linear superpositions of these minimum transactions.

    The time-symmetric boundary conditions postulated by Wheeler and Feynman do not impose an ad hoc time direction. They may be restated as follows: (1) The process of emission produces an electromagnetic wave consisting of a half-amplitude retarded wave and a half-amplitude advanced wave which lie along the same four-vector but with opposite time directions. (2) The process of absorption is identical to that of emission and occurs in such a way that the wave produced by the absorber is 180° out of phase with the wave incident on it from the emitter. (3) An advanced wave may be reinterpreted as a retarded wave by reversing the signs of the energy and momentum (and therefore the time direction) of the wave, and likewise a retarded wave may be reinterpreted as an advanced wave. Thus in the Wheeler-Feynman scheme, emission and absorption will correspond to the time-symmetric combinations

    These describe both emission and absorption with the same time-symmetric combination of advanced and retarded radiation. In interacting with this time-symmetric field which it has produced, the emitter (or absorber) cannot change its energy or momentum, for such changes are intrinsically unsymmetric in time and therefore cannot result from interactions with a time-symmetric field. Thus, this simultaneous emission of a pair of waves, advanced and retarded, can produce no energy or momentum change in the emitter.

    The emission of these time-symmetric electromagnetic waves produces some immediate problems in its correspondence with observation, for the emitter experiences neither recoil (i.e., momentum transfer) nor energy loss in the act of emission. Thus an emitter, e.g., an oscillating electron, could emit such radiation indefinitely without "noticing," since neither its energy nor its momentum would be affected by such emissions. Clearly, this does not fit with observations.

    However, if absorption of the emitted retarded wave occurs sometime later, the correspondence with observation is restored. Let us refer to Fig. 2, in which an emitter + absorber event is illustrated. The absorber, according to rule (2) above, can be considered to perform the absorption by producing a canceling retarded wave which is exactly 180° out of phase with an incident radiation, so that the incident wave "stops" at the absorber. But the Wheeler-Feynman time-symmetric boundary condition tells us that the production of this canceling wave will be accompanied by the production of an advanced wave, which will carry negative energy in the reverse time direction and travel back, both in space and in time, to the point and the instant of emission. This advanced wave, according to rule (3) above, may be reinterpreted as a retarded wave traveling in the opposite direction and will reinforce the initial retarded wave, raising it from half to full amplitude. When the new advanced wave "passes" the point (and instant) of emission, it will be superimposed on the initial half-amplitude advanced wave and, because of the 180° phase difference imposed by the absorber, it will cancel this wave completely.

    Thus, an observer viewing this process will perceive no advanced radiation, but will describe the event as the emission of a full-amplitude retarded wave by the emitter, with appropriate energy loss and recoil, followed by the absorption of this retarded wave by the absorber at some later time, with accompanying energy gain and recoil. The recoils during emission and absorption occur because the respective emitter and absorber, presumably charged particles such as electrons, move in the electromagnetic fields of the waves, advanced and retarded, respectively, sent to them by the other charged particle, as demonstrated by Wheeler and Feynman.¹⁸ The energy loss during emission and gain during absorption occur because the un-canceled full-amplitude wave carries energy from the emitter to the absorber. From one point of view, the emission-absorption process can be thought of as a standing wave in space-time, with the boundaries of the wave being the "terminating" emitter and absorber which bounce the wave back (as advanced radiation) and forward (as retarded radiation) between them.

    It is sometimes stated that Wheeler-Feynman absorber theory requires that there be an absorber for each emitted wave. This is not strictly true, as can be seen by considering another kind of transaction which can be deduced from absorber theory, and which is illustrated in Fig. 3. Here we have the same emission and absorption events as those described in Fig. 2, except that the sign of the waves produced by the absorber is reversed. Because of this, it is the waves connecting the emitter and absorber which are canceled, while the advanced wave from the emitter toward negative time and retarded wave from the absorber in the positive time direction are brought up to full amplitude. This will be called a type II transaction, as contrasted with the previously described transaction which we will henceforth designate as type I. It has the problem that neither of the emitted waves may be terminated by later absorbers (or by earlier emitters) which is unlikely in most physically realistic cases. For the purposes of the present discussion, therefore, we will give no further consideration to type II transactions. We note that Wheeler and Feynman¹⁸ and a number of subsequent authors have discussed rather complicated emitter-absorber situations, but that these can always be reduced to a linear superposition of the type I and type II transactions described above.

    It might be argued that the Wheeler-Feynman formulation of absorber theory is strictly a classical one, and is therefore inappropriate to discussions of quantum-mechanical paradoxes. However, the approach lends itself quite naturally to a quantum-mechanical formulation since it is basically just an alteration of the choice of boundary conditions applied to the solutions of classical or quantum-mechanical wave equations. In particular, Hoyle and Narlikar^29,30 and Davies^31-33 have presented quantum-mechanical formulations of absorber theory. Hoyle and Narlikar have demonstrated, using the Feynman path-integral technique, that absorber theory can be applied to the description of spontaneous transitions in atoms. They point out that second quantization of the field is absent in their formulation, but demonstrate that they are able to successfully describe a process usually thought to require a description involving second quantization. They have also demonstrated in their second paper³⁰ that all of the rules of quantum electrodynamics derived by Feynman can also be obtained from this quantum-mechanical formulation of absorber theory. Davies has presented a quantum-mechanical formulation using the S-matrix approach³¹ and has extended this formulation to the relativistic domain.³² He has also been able to derive from his formulation of absorber theory³³ the usual expression for the real photon processes of quantum electrodynamics. This body of work provides fairly convincing evidence that there are no barriers to treating absorber theory in a full quantum-mechanical framework.

    In the discussion that follows, therefore, we will take as given that a complete quantum-mechanical formulation of absorber theory can be accomplished, and will concentrate on the insights into quantum-mechanical paradoxes that such a formulation provides.

    Absorber theory, because it involves advanced radiation, is not without its causality problems. At this point, however, we would like to make a distinction between two forms of the principle of causality which are often used interchangeably. We will call these the principles of strong and weak causality.

    There is an analogy here to the situation with the group and phase velocities of electromagnetic waves in a wave guide¹⁵: The phase velocity can exceed c but cannot carry macroscopic information; the group velocity represents the speed of travel of macroscopic information but is never greater than c. If the phase of the wave can be considered to carry microscopic information (e.g., phase information which will affect interference phenomena) then its velocity represents a violation of strong causality. In any case, weak causality is not violated. We wish to emphasize that while there is abundant experimental evidence in support of the principle of weak causality, there is at present no experimental evidence for strong causality. Thus, strong-causality violations are not a compelling reason for rejecting any particular approach.

    In absorber theory there are always violations of strong causality since advanced radiation transfers microscopic information as well as energy in the negative time direction, but there can be no violations of weak causality as long as the absorption is complete in the "future" time direction, since absorber theory in this limit gives predictions identical with those of conventional electromagnetic theory.^18,19 (Causality problems arising from incomplete future absorption are discussed more fully in Ref. 19.) Thus, assuming complete future absorption, absorber theory only implies violations of the strong-causality principle. Weak causality remains intact since there is no possibility of using the advanced waves to transmit macroscopic information. This is the reason that absorber theory is of interest in the context of the Einstein-Podolsky-Rosen paradox, for it is just such a violation of strong but not weak causality which is needed to explain the results of the tests of the Bell inequality.

    While the type I transaction described in Sec. III above is a simple one, the same "handshake" procedure can apply to a much more complicated process such as the simultaneous emission of two or more photons. This is relevant because in all but one case the experimental tests of the Bell inequality mentioned in Sec. I above involved the emission of pairs of polarization-correlated photons. Such a transaction would require a double "confirmation" from the two absorbers, or it would not take place. Note specifically that the two absorptions need not occur simultaneously in order to produce simultaneous confirmations at the point of emission because advanced radiation is the carrier of the confirmation and travels backward in time to the instant of emission, no matter how long after emission the absorption event occurred.

    From the point of view of absorber theory it is not difficult to answer this question using the conceptual framework provided by the preceding discussion. The excited calcium atom will emit a number of probe waves corresponding to the possible emission of a pair of photons in various directions with various allowed polarization correlations. If "verifying" advanced waves are sent back by the pair of absorbers, then the transaction is complete and the double detection event has occurred. If the verifying waves do not match an allowed polarization correlation then they are not verifying the same transaction and will not, except accidentally, be correlated in time. This is illustrated in Fig. 5. In this situation the "two-bounce" standing wave described in Sec. II above becomes a "three-bounce" standing wave with one emission and two absorption boundaries.

    The two lightlike four-0vectors in this sum are:

and

and thus their sum is

    The squares of the space and time parts of this sum are, respectively,

and

    But since r₁r₂ ³ (r₁.r₂), this four-vector will be spacelike unless r₁ and r₂ are exactly in the same direction, i.e., parallel, (which is not the case for any of the experimental tests of the Bell inequality). The communication path from detector D2 to D1, i.e., A₁₀ + R₀₂ will also be spacelike.

    More generally, the sum of a lightlike four-vector with a negative time component and a light-like four-vector with a positive time component will be a spacelike four-vector, unless the two four-vectors are exactly antiparallel in four-dimensional space. Thus, absorber theory provides a mechanism whereby detector D1 can communicate microscopically with detector D2, and vice versa, over a spacelike interval in just the way needed to explain the EPR paradox.

    The conceptual framework provided by absorber theory, as outlined above, has provided a means of understanding the EPR paradox. Its application, however, is not limited to that particular conceptual problem of quantum mechanics but can also be applied to the understanding of other quantum-mechanical paradoxes.

    Let us consider, for example, the famous "Schrödinger's cat" paradox³⁴ and Wigner's variant of this basic conceptual problem, which is sometimes called the "Wigner's friend" paradox³⁵. The latter involves a variation of the former by introducing a second observer who observes what happens to the first observer (who replaces the cat) and by having both observers report their observations to a third observer. Both of these problems involve the role of the observer(s) in the experiment and particularly their role in the collapse of the state vector (or "wave packet") by the act of observation. From the point of view of absorber theory, neither of these Gedankenexperimente is particularly troublesome because the collapse of the state vector is implemented in just the right way by the confirmation with advanced waves of the quantum-mechanical transaction. It is the absorber, not the observer, which collapses the state vector, but absorption is an essential part of observation.

    The role of the absorber in the collapse of the state vector is perhaps even better illustrated by considering a new quantum-mechanical paradox which has recently been proposed by Wheeler, which he calls a "delayed choice" experiment.³⁶ Since this paradox is not yet widely known, it would perhaps be appropriate to briefly describe it here: A standard Young two-slit interference experiment is modified (a) by arranging the light source to emit only one photon at a time, (b) by modifying the photographic emulsion which records the interference pattern so that it is mounted in a pivoting lattice of strips like a "Venetian blind," and (c) by placing behind this lattice two photomultiplier tubes with collimators and lenses arranged so that each may receive light from only one of the two slits. Thus, with the lattice closed to experiment records the interference pattern resulting from the wave function of the emitted photons passing through both slits, while when the lattice is opened the experiment determines the slit through which each photon passes (either slit 1 or slit 2).

    So far there is no problem, since the experiments cannot be performed on the same photon, the lattice being either opened or closed. Further, while the experiments are complementary and mutually exclusive, both are feasible measurements. The paradox arises in the following way: The observer (who has a very fast reaction time) waits until after the photon has passed through the slits to decide which of the two complementary experiments is to be performed. This means that the photon must "commit itself" to passage through a single slit or both slits before it is decided which experiment is to be performed.

    Wheeler's conclusion from considering this and several other delayed choice Gedankenexperimente is that the objective reality of the wave function during the passage of the photon through the slit system is brought into question; that the wave function is essentially made real by the "irreversible act of amplification" which tells us which slit was transited or by the "indelible record" made by the interference pattern on the photographic emulsion.

    It is informative to analyze the delayed-choice experiment described above within the conceptual framework provided by absorber theory. The retarded "probe wave" from the light source spreads out in various directions, and in particular passes through both of the slits on its way to the detection apparatus. If the detector lattice is closed, then the wave impinges on the photographic emulsion and is absorbed. In the process of absorption, the emulsion generates the advanced waves that confirm the transaction and these pass back through both slits to the light source, so that the event involves passage through both slits.

    If the lattice is open, then the wave travels to the photomultiplier detectors, and one of these absorbs the wave, generating the confirming advanced wave. However, because of the collimator system associated with the photomultiplier, this advanced wave is able to travel back to the source through only one of the slits (the one at which the detector system is aimed). Thus, the transaction in this case involves the passage of the photon through only one slit.

    Notice that in this analysis the choice of which detector system is used can be made either before or after the retarded wave has passed through the slit system without affecting the analysis. This, of course, is because of the role of the advanced wave in traveling backward in time to the instant of emission to confirm the transaction across a negative lightlike interval.

    The above example may be taken as illustrative of the power of the conceptual framework provided by absorber theory for dealing with quantum-mechanical paradoxes. We will limit our discussion to the above paradoxes. However, we have not as yet been able to discover any such paradox which cannot be satisfactorily dealt with in this conceptual framework. This gives us some assurance that quantum-mechanical wave functions can be viewed as having some objective reality beyond their role as a mathematical tool for calculating experimental results, at least within the context of absorber theory.

    In the discussion above we have shown for the case of electromagnetic radiation that the problem posed by the EPR paradox can be solved by the application of Wheeler-Feynman absorber theory. We here assert that the Wheeler-Feynman protocol for an emission-absorption transaction is not a peculiarity of electromagnetism. Rather, the Wheeler-Feynman emission-absorption protocol is a general feature of the emission and absorption of all particles and waves, whether fermions or bosons, whether charged or uncharged, whether massive or massless. The justification for this assertion is that the Wheeler-Feynman description of emission and absorption accounts for the violations of locality in the Bell inequality experiments involving light waves (i.e., massless uncharged bosons), but the experimental results of Lamehi-Rachti and Mittig¹³ demonstrate that locality is violated also in a Bell inequality test involving protons (i.e., massive charged fermions). The Wheeler-Feynman description therefore must be generalized to make it applicable to the latter experimental result.

    However, there are problems with such a generalization. The electromagnetic wave equation has advanced as well as retarded solutions because it is a second-order differential equation in the time variable. The corresponding wave equation for particles with nonzero rest mass is the Schrödinger equation. In its field-free time-dependent form, the Schrödinger equation can be written

were -(h/2p)²Ñ² = P² is the momentum-squared operator, i(h/2p) d/dt = H is the total energy operator, M is the particle mass, and y is the quantum-mechanical wave function of the particle of interest. Clearly, this equation is only first order in time and would have only a single solution corresponding to the positive energy or retarded solution of the electromagnetic wave equation. Thus it would seem that the absorber theory arguments could not be applied to the case of massive particles.

    However, we know that the Schrödinger equation is not correct, since it is not a proper relativistically invariant wave equation. For spin-½ particles, the appropriate relativistically invariant equation is the Dirac equation,³⁷ which can be written in the form

where c is the velocity of light, P is the momentum operator (= -i(h/2p)Ñ), M is the rest mass of the particle of interest, and a and b are dimensionless spin-dependent 4´4 matrices which are independent of momentum, energy, position, and time. This leads to a set of four coupled differential equations involving the initial and final spin states of the particle of interest, and these equations, like the electromagnetic wave equation, have both positive- and negative-energy solutions. The negative-energy solutions of the Dirac equation are conventionally interpreted as corresponding to antimatter waves (positrons, antiprotons, etc.).³⁹ We note that Pauli and Weisskopf⁴⁰ have shown that the quantized field energy is always positive, even when the eigenvalue of the energy operator H has a negative sign. Thus when we call a particular solution of the wave equation a positive- or negative-energy solution, we refer to the eigenvalue of the H operator.

    For particles having spins other than ½, e.g., bosons, the situation is more confused because there are a number of alternative relativistically invariant wave equations found in the literature.

    A full catalog of such wave equations is beyond the scope of this paper, but several which are appropriate for massive spinless bosons are of particular interest. A wave equation sometimes used in quantum mechanics at relativistic energies is the relativistic Schrödinger equation⁴¹ (sometimes also called the Thomas equation). In field-free space it has the time-dependent form

where y is the quantum-mechanical wave function of the particle of interest, -(h/2p)²Ñ^{2 =} P² is the momentum-squared operator, M is the rest mass of the particle, i(h/2p) d/dt = H is the total-energy operator, and the positive square root is assumed. This equation, like the nonrelativistic Schrödinger equation, is first order in time and therefore has only positive-energy solutions (because we assumed the positive square root). It would therefore be inappropriate for a Wheeler-Feynman type of emitter-absorber transaction.

    A more satisfactory alternative is the time-dependent Klein-Gordon equation,³⁷ which is essentially the operator-square of the relativistic Schrödinger equation and in field-free space has the form

where the symbols are as defined above and -(h/2p)² d²/dt² is the total-energy-squared operator. This equation, like the Dirac equation and the electromagnetic wave equation, has both positive-and negative-energy solutions and would therefore be appropriate for a generalization of the Wheeler-Feynman approach. Notice that when M=0 the Klein-Gordon equation becomes effectively the same as the electromagnetic wave equation (1). We note that there are other more general wave equations, such as the Bethe-Salpeter equation,⁴² which also have the desired property of giving both positive- and negative-energy solutions.

FIG. 7. Minkowski diagram of a type I transaction involving the emission and absorption of an electron. The emitter produces half-amplitude wave functions A_e and Re . The absorber, responding to R_e produces wave functions R_a and A_a such that R_a cancels Re in region 3. Wave A_a reinforces R_e in region 2 and cancels A_e in region 1. An observer sees an electron traveling from emitter to absorber through region 2. (Dashed lines indicate 180° phase shift.)

 

    Here again we see that we can form a transaction between emitter and absorber by a superposition of advanced and retarded waves. We can apply this view to the Bell inequality experiment of Lamehi-Rachti and Mittig,¹³ in which a pair of spin-correlated protons in a relative S state are observed in detection events separated by a space-like interval, paralleling our analysis of the Freedman-Clauser experiment discussed in Sec. V above. Here, however, the waves connecting the detectors with the source span timelike intervals, since the protons move with velocities less than c.

    As was done for the Freedman-Clauser experiment, let us examine the communication path from detector D1 to detector D2 via advanced and retarded waves represented by the timelike four-vectors A₁₀ and R₀₂. These four-vectors can be written as

and

where b₁ and b₂ are the velocities (b_i< 1) of the protons traveling to detectors D1 and D2. The sum of these two timelike four-vectors is not necessarily spacelike, as was the case for light waves, but will be spacelike if the detectors are in opposite directions with equal path lengths and the two protons travel to the detectors with the same velocity. In that case, r₁= - r₂ and b=b₁=b₂, so the sum of the two four-vectors will be

which is clearly a spacelike interval. This is illustrated by the Minkowski diagram shown in Fig. 6(b).

    We wish to acknowledge that the above description of detector-detector "communication" is very close to a more restricted and specific one given by Costa de Beauregard.^43,44 He has pointed out that the timelike symmetry of electron and positron wave functions in the Feynman picture can, in principle, account for violations of EPR locality and has certain implications about the CP invariance of such events. However, his conclusion is based on the consideration of the electron-positron waves in a creation-annihilation event. It therefore involves "true" positron wave functions having a time direction and energy that is opposite the advanced positron waves in the present description.

    Thus, we see that the generalized Wheeler-Feynman approach has provided an explanation of the results of all of the Bell inequality tests, whether involving light waves or protons. The conclusion then is that the concept of locality is invalid in quantum mechanics because there is communication of microscopic information between detectors over spacelike intervals arising from the verification of a quantum-mechanical "transaction" provided by advanced-wave functions.
 

    The generalization of Wheeler-Feynman absorber theory presented here is not really a revision of the conventional theory, but only the application of slightly different boundary conditions to the solution of the wave equations and a reinterpretation of the results. Therefore, it would be very surprising if there were any substantiative changes in the quantum-mechanical predictions of experimental re-suits. Thus, a definitive experimental test of the approach may be difficult to arrange.

    The requirement stated above that all wave equations must have negative-energy solutions is, at least in principle, experimentally testable. For instance, the relativistic Schrödinger equation predicts relativistic corrections to Rutherford scattering which are different and may be experimentally distinguishable from those predicted by the Klein-Gordon equation or the Bethe-Salpeter equation.⁴⁵ However, the existence of boson antiparticles, e.g., pions, kaons, h's, etc., require this type of equation for adequate treatment in any case, so an experimental proof that the wave equations have negative-energy solutions could hardly be taken as verification of generalized absorber theory.

    There is one other effect which is of possible experimental importance in this context. Absorber theory, unlike conventional quantum mechanics, predicts that in a situation where there is a deficiency of future absorption in a particular spatial direction, there will be a corresponding decrease in emission in that direction.

    As a simple (classical) case of the above, an oscillating electron which was alone in an other wise "empty" (i.e., completely non-absorbing) universe would not radiate at all.¹⁸ If a second electron were introduced, the first electron would be able to radiate only in the direction of this second "absorber" electron. In an "open" universe in which the absorbing matter is not distributed isotropically so that there was no absorption in a particular direction in space, we should find that an emitter will "refuse" to emit in that direction, because there must be (ignoring type II transactions for the moment) an absorber on the other end of every emission event to complete the transaction.

    An attempt to observe this effect experimentally was made by Partridge using 9.7-GHz microwaves transmitted from a large conical horn antenna, so that the microwaves were beamed in various spatial directions and the power output of the transmitter accurately monitored.⁴⁶ Partridge found that there was no evidence for decreased emission in any direction, to any accuracy of a few parts in 109. However, this experiment has been criticized by Davies⁴⁷ as necessarily yielding a null result because of absorption by the Earth of any advanced radiation approaching the back side of the experiment. In essence, Davies argues that the most general test of absorber theory would include the possibility of type II transactions, as discussed in Sec. III above. This would require an emitter system that was symmetric in opposite spatial directions, which, for microwaves, would probably require that the experiment be performed in deep space.

    The author and collaborators⁴⁸ are currently performing an experiment similar to that of Partridge, which satisfies the Davies criteria by employing neutrinos rather than microwaves as the "broadcast" medium. Since the Earth is quite transparent to low-energy neutrinos it is relatively straightforward to mount a bi-directionally symmetric experiment on the surface of the Earth. The "transmitter" is a radioactive source involving a pure Gamow-Teller b-decay transition which simultaneously emits neutrinos and direction-correlated b particles. Thus deficiencies in neutrinos emission will be reflected as asymmetries in the angular distribution of emitted b particles. An experiment of this type has several advantages over that of Partridge. (1) The Davies symmetry is easy to obtain in the experimental design. (2) The cosmological red-shift decreases the absorption probability because the low-energy cross section for the absorption of red-shifted photons goes up as 1/E because of the inverse bremsstrahlung process, while weak-neutral-current arguments imply that the low-energy neutrino scattering cross section at low energies should go down as E². (3) Since the neutrino is a fermion (and weakly interacting) it is intrinsically very difficult to absorb because the "left-over" half unit of spin is difficult for the absorber to dispose of without reemission. This experiment, of course, is not strictly speaking a test of the Wheeler-Feynman theory, which applied only to light waves, but rather a test of the generalized absorber theory as it applies to the emission and absorption of neutrinos.

    It should also be pointed out that both the Partridge experiment⁴⁶ and this neutrino version of it⁴⁸ are "long-shots," since most cosmological models of the universe^28,47,49,50 would predict negative results for both experiments. Thus, the prospects for a definitive experimental test of generalized absorber theory appear at present to be rather unpromising, and the validity of this alternative approach to quantum mechanics may have to rest, at least for the moment, on its value in providing a framework for the resolution and understanding of quantum-mechanical paradoxes.

    In the preceding discussion we have demonstrated that generalizing Wheeler-Feynman absorber theory to make it a quantum-mechanical theory applying to all particles and waves has provided a conceptual framework within which a number of quantum-mechanical paradoxes can be resolved. In particular the Einstein-Podolsky-Rosen paradox,1 the "Schrödinger's Cat" paradox,³⁴ and indeed all other quantum-mechanical paradoxes examined including Wheeler's delayed- choice experiments,³⁶ can be understood by interpreting the lack of locality and the decomposition of the wave packet as arising from the action of advanced waves which verify the quantum-mechanical transactions. We have shown that the communication path between detectors in the Bell inequality experiments can span a spacelike interval and produce the quantum-mechanical result through the addition of two lightlike or timelike four-vectors having time components of opposite sign, thus accounting for the locality violations implied by the experimental results.

    Accepting quantum-mechanical absorber theory as a favored alternative to the usual field-theory approach to quantum-mechanical phenomena has some implications of interpretation that should be seriously considered. As has been pointed out by other authors^{18,19,27,29,31} absorber theory is basically an "action-at-a-distance" formulation. It demotes the concept of a field from the status of a real entity with its own degrees of freedom to that of a mathematical convenience, a conceptual prop for thinking about transactions between emitters and absorbers. Whether this is acceptable must ultimately rest on the relative predictive-ness of the two alternative approaches.

    However, the absorber theory approach raises questions as well as settles them. In closing, therefore, we would like to enumerate three of the more troublesome questions raised by the generalized absorber theory presented here.

(1)  If only a single particle is emitted by a system and future absorbers provide more than one "verification," how is the conflict of multiple verifications resolved so that only a single "transaction" is verified?

(2)  If absorber theory is applied to very weakly absorbed particles such as neutrinos, how can the observed emission of such particles be reconciled with their low probability of future absorption, particularly in the open-universe models which are supported by some experimental evidence?

(3)  How can the observed dominance of retarded radiation be accounted for in terms of absorber theory, when the big-bang model would imply at least as much as absorption in the past as in the future?

    Although problems (2) and (3) mentioned above do not currently have answers, we do not consider them to be without solution. In fact, their answers may be connected. In a subsequent publication51 we will seek to deal with them using the conceptual framework provided by the present work.

    This work was supported in part by the Division of Nuclear Sciences of the U.S. Department of Energy. The author would like to acknowledge critical readings and useful comments and suggestions from a number of colleagues. I am indebted to Dr. Eric B. Norman for the initial discussions that prompted the writing of this paper, as well as for his later suggestions. I would like to thank Dr. D. W. Storm, Dr. A. J. Lazzarini, Professor W. J. Braithwaite, Professor Lawrence Wilets, and Professor David Bodansky for their comments and suggestions. I am particularly grateful to Professor John A. Wheeler for a lively discussion on delayed-choice experiments, for his comments, and for his continuing help and encouragement. I am indebted to Professor E. J. Konopinski for sending me in advance the manuscript of the chapter on radiative motion from his forthcoming book.²⁴ I am grateful to Sir Rudolph Peierls, that valiant defender of the Copenhagen Interpretation, for several wide ranging discussion of the ideas in the present paper and their implications.

    Finally, I would like to thank Dr. Paul Davies for a critical evaluation of this paper and also for bringing to my attention the following remark by Professor O. R. Frisch, made at the opening of a 1968 Cambridge colloquium on the philosophical problems of quantum mechanics⁵²:

    "Or should we say ... that particles are in collusion, that they, as it were, communicate in some way or another over great distances? (Incidentally, if you try to work out a signaling system with more than the speed of light on this basis it does not work. I have tried it.)  This does not look like a good description.  At the same time I am aware that Feynman and Wheeler in their famous paper make use of a kind of signaling with the reverse speed of light ('into the past'). Whether that can be used in the present context, I do not know.  I am not here to give answers, I am here to throw out questions."