Bohm's pilot wave theory extended to back from the future effects

Jack Sarfatti

February 24 at 9:37pm · San Francisco, CA ·

http://philsci-archive.pitt.edu/…/Causally_Symmetric_Bohm_M…

The aim of this paper is to construct a version of Bohm’s model that also includes the existence of backwards-in-time influences in addition to the usual forwards causation. The motivation for this extension is to remove the need in the existing model for a preferred reference frame. As is well known, Bohm’s explanation for the nonlocality of Bell’s theorem necessarily involves instantaneous changes being produced at space-like separations, in conflict with the “spirit” of special relativity even though these changes are not directly observable. While this mechanism is quite adequate from a purely empirical perspective, the overwhelming experimental success of special relativity (together with the theory’s natural attractiveness), makes one reluctant to abandon it even at a “hidden” level. There are, of course, trade-offs to be made in formulating an alternative model and it is ultimately a matter of taste as to which is preferred. However, constructing an explicit example of a causally symmetric formalism allows the pros and cons of each version to be compared and highlights the consequences of imposing such symmetry1. In particular, in addition to providing a natural explanation for Bell nonlocality, the new model allows us to define and work with a mathematical description in 3-dimensional space, rather than configuration space, even in the correlated many- particle case.

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Jack Sarfatti The structure of the paper is as follows. In Sec. 2, the basic causally symmetric scheme is introduced in terms of initial and final boundary conditions. Sec. 3 then highlights the ways in which the corresponding initial and final wavefunctions will propagate. The basic equations of the alternative model are deduced in Sec. 4 in close analogy to theformalism of the standard Bohm model. Sec. 5 then points out how the notion of retrocausality has been given an explicit mathematical form and Sec. 6 checks some elementary matters of consistency. The discussion in Sec. 7 indicates how backwards-in-time effects provide a meaning for the notion of negative probability. Sec. 8 then explains the way in which an objection to an earlier and related model of de Broglie is now overcome. After dealing with some technical details in Sec. 9, the analysis in Sec. 10 shows how the model explains Bell’s nonlocality in a way that is Lorentz invariant, as well as being local from a 4-dimensional point of view. The generalization of the formalism to n particles is given in Sec. 11, followed by an outline in Sec. 12 of ways in which the model has inherently weaker predictive power. A relativistic version is formulated in Sec. 13 for the single-particle Dirac case. Finally, conclusions are presented in Sec. 14

February 25 at 9:57pm · Edited · Like
Jack Sarfatti Bohm’s model (Ref. 1) makes the assumption that a particle always has a definite, but hidden, trajectory. It then specifies the particle’s velocity in terms of the wavefunction ψ. Our aim here is to provide a consistent generalization of this formalism that incorporates backwards-in-time effects, or retrocausality, into the model. The state of the particle at any time will then be partly determined by the particle's future experiences as well as by its past. The way in which this helps with Bell’s nonlocality will then be outlined in Sec. 10.

February 25 at 10:08pm · Edited · Like
Jack Sarfatti As a first step towards developing such a formalism, we must deal with the question: what aspects of a particle's future are relevant?2 Possible factors could be the type of measurement to be performed next, the nature of the particle's interaction with the next particle it encounters and perhaps the nature of all future measurements and interactions. This seems a daunting prospect at first. However, an indication of the best way to proceed is obtained by looking at the usual way we take account of a particle's past experiences: we work with an initial wavefunction ψi which summarizes the particle's relevant past. More formally speaking, ψi specifies the initial boundary conditions. Therefore, by symmetry, it seems natural to supplement ψi with a "final" wavefunction ψf specifying the final boundary conditions. To keep the arrangement time-symmetric, the final wavefunction ψf will be restricted, like ψi, to being a solution of the time-dependent Schrödinger equation. The procedure to be followed here then is to construct a version of Bohm’s model containing both ψi and ψf.

February 25 at 10:22pm · Edited · Like
Jack Sarfatti Note that the new wavefunction ψf being introduced here is independent of the usual wavefunction ψi and should not be confused with the result of evolving ψi deterministically to a later time. Thus, at any single time t, there are two distinct wavefunctions: (i) the initial wavefunction ψi(x,t), which summarizes the initial boundary conditions existing at some earlier time t1 and which has been evolved forwards from t1 to t and (ii) the final wavefunction ψf(x,t), which summarizes the final boundary conditions at some later time t2 and which has been evolved back from t2 to t. The model to be developed here will be deterministic once both wavefunctions are specified, together with the particle’s position at one instant of time. In particular, specifying ψi at time t1 and ψf at time t2 will then determine the particle’s velocity at any intermediate time.

February 25 at 10:35pm · Edited · Like
Jack Sarfatti Like the standard Bohm model, the causally symmetric version will be a “no collapse” model, with empty branches of wavefunctions after measurements being ignored as irrelevant. The model does not give any special status to measurement interactions, observers or the macroscopic world3. Indeed, it is intended to be as similar as possible to the standard Bohm formulation, apart from the obvious fact that such a retrocausal model cannot be deterministic when only the initial conditions are given.

February 25 at 10:43pm · Like
Jack Sarfatti §5 Retrocausal Influence on Particle VelocityTo demonstrate that the particle velocity defined by (15) really is retrocausally affected by future circumstances, consider two separate particles each having an identical initial wavefunction ψi from time t1 onwards. If we choose to perform measurements of different non-commuting observables on the particles at a later time t2, they will have different final wavefunctions ψf extending back from t2 to t1 (these being eigenfunctions of the respective observables measured). Since the velocity expression (15) is obviously dependent on ψf, it then follows that the velocity values at any intermediate time between t1 and t2 will be different for the two particles. Hence the type of measurement chosen at t2 has a bearing on the physical reality existing at an earlier time, which constitutes retrocausality. This example also indicates the way in which our initial notion of retrocausality has been given a specific mathematical form. Note further in this example that it is not possible to interpret the two ψf's as instead originating at the earlier time t1, independent of the future measurements at t2, and then propagating forwards in time. This is because these ψf’s are eigenfunctions of two different observables that will subsequently be chosen freely by the experimenter at t2. It would be inexplicable why, for each particle, the ψf that arises randomly at t1 always happens to be an eigenfunction of the correct observable to be nominated and measured later at t2. The only explanation is that each ψf must be retrocausally determined by the choice at t2.§6 Consistency with ObservationIt will be demonstrated briefly here that the probability expression (14) is quite consistent with what is observed when a measurement is actually performed. It should be kept in mind that the position probability distributions of Bohm-type models describe the position of a particle at all times, so that most of the times are between measurements. In terms of experimental agreement, it doesn’t matter what is predicted there, since the distribution is hidden.