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Forward-Time Equivalent of a "Retrocausal" Diffusion Hidden Variable Model for Quantum Mechanics

William S. DeWitt, Benjamin H. Feintzeig

TL;DR

The authors show that the retrocausal stochastic hidden-variable model proposed for quantum mechanics has an exact forward-time counterpart that reproduces the same trajectories using only initial-time data, by applying time-reversal results for diffusion processes. The forward dynamics features a guiding term tied to the phase-space distribution that can be interpreted as a mean-field interaction in an ensemble, linking to McKean-Vlasov and propagation-of-chaos frameworks. Through a detailed amplifier example, they demonstrate numerical and analytical connections between the backward-time model, the forward-guided diffusion, and the mean-field N-system, highlighting deep ties to pilot-wave and many-worlds interpretations. The work provides practical computation methods and broad interpretive insight into stochastic hidden-variable models in quantum foundations.

Abstract

A recently proposed stochastic hidden variable model for quantum mechanics has been claimed to involve "retrocausality" due to the appearance of equations of motion with future-time boundary conditions. We formulate an equivalent system of forward-time equations of motion that gives rise to the same trajectories as solutions, but involves only initial-time boundary conditions. The forward-time dynamics involves a guidance term for the dynamical variables, determined by the phase-space distribution corresponding to a quantum wavefunction. We show, however, that this particular guidance term can be recovered as the mean-field limit of averaged pairwise interactions among an ensemble of finitely many particles.

Forward-Time Equivalent of a "Retrocausal" Diffusion Hidden Variable Model for Quantum Mechanics

TL;DR

The authors show that the retrocausal stochastic hidden-variable model proposed for quantum mechanics has an exact forward-time counterpart that reproduces the same trajectories using only initial-time data, by applying time-reversal results for diffusion processes. The forward dynamics features a guiding term tied to the phase-space distribution that can be interpreted as a mean-field interaction in an ensemble, linking to McKean-Vlasov and propagation-of-chaos frameworks. Through a detailed amplifier example, they demonstrate numerical and analytical connections between the backward-time model, the forward-guided diffusion, and the mean-field N-system, highlighting deep ties to pilot-wave and many-worlds interpretations. The work provides practical computation methods and broad interpretive insight into stochastic hidden-variable models in quantum foundations.

Abstract

A recently proposed stochastic hidden variable model for quantum mechanics has been claimed to involve "retrocausality" due to the appearance of equations of motion with future-time boundary conditions. We formulate an equivalent system of forward-time equations of motion that gives rise to the same trajectories as solutions, but involves only initial-time boundary conditions. The forward-time dynamics involves a guidance term for the dynamical variables, determined by the phase-space distribution corresponding to a quantum wavefunction. We show, however, that this particular guidance term can be recovered as the mean-field limit of averaged pairwise interactions among an ensemble of finitely many particles.

Paper Structure

This paper contains 6 sections, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. "Retrocausal" Diffusion Model
  3. Equivalent Forward-Time Process
  4. Mean-Field Interpretation
  5. Example: Amplifier Dynamics
  6. Conclusion

Figures (1)

  • Figure 1: Comparisons of numerical solutions for the various formulations of microdynamics for the amplifier model of § \ref{['sec:example']}. From left to right, we include the "retrocausal" model from Drummond and Reid DrRe20 in Eq. \ref{['eq:amplifierSDEq']} (solved backward in time via the Euler–Maruyama method), the forward guidance model in Eq. \ref{['eq:amplifier_micro']} (solved forward in time via Euler–Maruyama with guidance), and the mean-field model in Eq. \ref{['eq:NsystemSDE']} for $N=100$ and $N=10,000$ (solved forward in time via $N$-dimensional coupled Euler–Maruyama). For all cases, 10 trajectories are shown. As is evident from the visualization, these formulations produce the same stochastic trajectories, with the mean-field model producing better agreement as $N$ increases. Simulation code is available at https://github.com/dewitt-lab/Q-functionology .