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Bohmian Mechanics fails to compute multi-time correlations

Robert C. Helling

TL;DR

The paper shows that a spatial GHZ construction using positional observables reveals a fundamental mismatch between Bohmian trajectories and standard quantum multi-time correlations. While Bohmian mechanics can reproduce single-time (or equal-time) Born-rule statistics, it fails to reproduce the correct multi-time correlations unless a microscopic wave-function collapse mechanism is invoked after measurements, demanding explicit equations of motion for the collapsed dynamics. This challenges claims of equivalence between Bohmian realism and orthodox QM for time-ordered observables and highlights the need for collapse-like dynamics or measurement-context dependence to preserve predictive power. The results imply that experimental tests of multi-time correlations could distinguish Bohmian predictions from standard quantum mechanics and motivate further foundational work on measurement in Bohmian theories.

Abstract

The violation of Bell type inequalities in quantum systems manifests that quantum states cannot be described by classical probability distributions. Yet, Bohmian mechanics is a realistic, non-local theory of classical particle trajectories that is claimed to be indistinguishable by observations from more traditional approaches to quantum mechanics. We set up a spatial version of the GHZ system with qubits realised as positional observables that demonstrates that the Bohmian theory fails to match predictions of textbook quantum mechanics (and most likely experients) unless enlarged by a microscopic theory of collapse of the wave function after observation. For this discrepancy to occur it is essential that positions at different times do not commute.

Bohmian Mechanics fails to compute multi-time correlations

TL;DR

The paper shows that a spatial GHZ construction using positional observables reveals a fundamental mismatch between Bohmian trajectories and standard quantum multi-time correlations. While Bohmian mechanics can reproduce single-time (or equal-time) Born-rule statistics, it fails to reproduce the correct multi-time correlations unless a microscopic wave-function collapse mechanism is invoked after measurements, demanding explicit equations of motion for the collapsed dynamics. This challenges claims of equivalence between Bohmian realism and orthodox QM for time-ordered observables and highlights the need for collapse-like dynamics or measurement-context dependence to preserve predictive power. The results imply that experimental tests of multi-time correlations could distinguish Bohmian predictions from standard quantum mechanics and motivate further foundational work on measurement in Bohmian theories.

Abstract

The violation of Bell type inequalities in quantum systems manifests that quantum states cannot be described by classical probability distributions. Yet, Bohmian mechanics is a realistic, non-local theory of classical particle trajectories that is claimed to be indistinguishable by observations from more traditional approaches to quantum mechanics. We set up a spatial version of the GHZ system with qubits realised as positional observables that demonstrates that the Bohmian theory fails to match predictions of textbook quantum mechanics (and most likely experients) unless enlarged by a microscopic theory of collapse of the wave function after observation. For this discrepancy to occur it is essential that positions at different times do not commute.

Paper Structure

This paper contains 5 sections, 28 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setting the stage
  3. The Bohmian GHZ
  4. Ways out
  5. Appendix: Frauchinger-Renner states

Figures (5)

  • Figure 1: Wave functions of ground state and first excited state as well as $\sigma$
  • Figure 2: A comic of the periodic time evolution of the probability density $|\Psi_{GHZ}|^2$ as a particle in a three-dimensional box
  • Figure 3: Propability distribution at $t=0$ and $t=\pi/(2\Delta E)$
  • Figure 4: (\ref{['eq:stucorrelation']} (blue) for $t=s=u$ and the Bohmian expression $\mathbb{E}\langle\mathop{\mathrm{sgn}}\nolimits(q_1(t))\mathop{\mathrm{sgn}}\nolimits(q_2(t))\mathop{\mathrm{sgn}}\nolimits(q_3(t))\rangle$ (orange) agree well
  • Figure 5: (\ref{['eq:stucorrelation']}) for $t=0$ and $s=u$ (blue) and the Bohmian expression for the particles $\mathbb{E}\langle\mathop{\mathrm{sgn}}\nolimits(q_1(0))\mathop{\mathrm{sgn}}\nolimits(q_2(s))\mathop{\mathrm{sgn}}\nolimits(q_3(s))\rangle$ (orange) no longer agree