A sum-over-histories account of an EPR(B) experiment

We analyze a variant of the EPRB experiment within a framework for quantum mechanics that rests on a radical interpretation of the sum over histories. Within this framework, reality is (just as classically) a single history, e.g. a definite collection of particles undergoing definite motions; and quantum dynamics appears as a kind of stochastic law of motion for that history, a law formulated in terms of non-classical probability-amplitudes. No state vectors enter this framework, and their attendant nonlocality is therefore absent as well.1. That is, before pair creations and annihilations were discovered. (The electronic and nuclear spins might also be regarded as new aspects of their kinematics. But perhaps spin is better construed, within the sum-over-histories framework, as a quality of a more dynamical character, namely as a generalized sort of probability-amplitude.)2. A possible escape would be the so-called Everett interpretation, in which the collapse never occurs, but its effects are supposed to be recovered via a more careful analysis of closed systems in which measurement-like processes take place. Among other things, this approach tends to lead either to the view that nothing really happens 1] or to the view that everything really happens 2] (which perhaps is not that different from the former view).3. For example, the rule, collapse occurs along the past light cone (in the Heisenberg picture), appears to be consistent.4. And Bell's inequality shows thatany theory formulated in terms of an instantaneous state evolving in time would encounter the same trouble. Indeed, the trouble shows up even more glaringly if one adapts Bell's argument to spin-1 systems, using the results of Kochen and Specker10]. In order to use the Kochen-Specker results in the EPR manner one needs a scheme for measuring the relevant observables, but this can be accomplished by means of suitably concatenated Stern-Gerlach analyzers with recombining beams 13]. Then, as Allen Stairs has pointed out 14], even the perfect correlations become impossible to reproduce, and no reference to probability theory is needed to establish a contradiction with locality. Recently, an analogous experiment using three spin 1/2 particles instead of two spin 1 particles has also been given 15].5. No technical problem obstructs an extension to fermionic fields (indeed the functional integral formalism for Quantum Field Theory is probably the most popular at present), but the realistic interpretation of the individual histories seems to get lost. One way out would be if all fermions were composites or collective excitations of fields quantized according to bosonic commutation relations. Another would be if the particle formulation were taken as basic, with the complementary field formulation being merely a mathematical artifice (at least for fermions).6. In the approach of Gell-Mann-Hartle and Griffiths for example, only a small subset of the possible partitions is granted meaning, in such a way that all interference terms are suppressed and quantum probabilities reduce to classical ones.7. In stating these rules we consider an idealized situation in which the spatio-temporal indeterminacy of particle-locationwithin a given one of our trajectories is ignored; or if you prefer, you can take the experiment as only a Gedanken one affording a simplified illustration of how EPR-like correlations are understood within the sum-over-histories framework. In this connection recall also that the semiclassical propagator is in fact exact for a free particle.8. This can be interpreted either as part of the specification of the initial conditions, or (as suggested by a referee) merely as an example of relativization of probabilities.9. Thus a state vector may be defined as an equivalence-class of sets of partial histories.10. One such generalization applies to open systems, for example to a particle in contact with a heat reservoir. For this example see 11], wherein the two-way path formalism of §5 above is used, and the influence of the reservoir results in an effective dynamics for the particle in which the forward and backward portions of its world-line are coupled to each other by a certain interaction term in the amplitude. In this type of situation a density-operator (though not a state vector ) can still be introduced, but it no longer summarizes all the relevant information about the past (and correspondingly its evolution lacks the Markov property that(t + dt) is determined by(t) alone). For quantum gravity, it may be that not even such a non-Markov will be exactly definable, and only the global probabilities themselves will make sense.11. Ironically it is just this property of the amplitudes which, as mentioned above, makes possible the introduction of the state vectors whose collapse then introduces such a strong appearance ofnonlocality into the theory.