# Non-Markovian wave-function collapse models are Bohmian-like theories in disguise

Antoine Tilloy1,2 and Howard M. Wiseman3

1Max-Planck-Institut für Quantenoptik, Garching, Germany
2Munich Center for Quantum Science and Technology (MCQST), Munich, Germany
3Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia

### Abstract

Spontaneous collapse models and Bohmian mechanics are two different solutions to the measurement problem plaguing orthodox quantum mechanics. They have, a priori nothing in common. At a formal level, collapse models add a non-linear noise term to the Schrödinger equation, and extract definite measurement outcomes either from the wave function ($e.g.$ mass density ontology) or the noise itself (flash ontology). Bohmian mechanics keeps the Schrödinger equation intact but uses the wave function to guide particles (or fields), which comprise the primitive ontology. Collapse models modify the predictions of orthodox quantum mechanics, whilst Bohmian mechanics can be argued to reproduce them. However, it turns out that collapse models and their primitive ontology can be exactly recast as Bohmian theories. More precisely, considering (i) a system described by a non-Markovian collapse model, and (ii) an extended system where a carefully tailored bath is added and described by Bohmian mechanics, the stochastic wave-function of the collapse model is exactly the wave-function of the original system conditioned on the Bohmian hidden variables of the bath. Further, the noise driving the collapse model is a linear functional of the Bohmian variables. The randomness that seems progressively revealed in the collapse models lies entirely in the initial conditions in the Bohmian-like theory. Our construction of the appropriate bath is not trivial and exploits an old result from the theory of open quantum systems. This reformulation of collapse models as Bohmian theories brings to the fore the question of whether there exists unromantic' realist interpretations of quantum theory that cannot ultimately be rewritten this way, with some guiding law. It also points to important foundational differences between true' (Markovian) collapse models and non-Markovian models.

Spontaneous collapse models and Bohmian mechanics are two very different, and at first sight unrelated, physical theories that aim to solve the quantum measurement problem. Collapse models add a noisy non-linear term to the Schrödinger equation to cause the wave function of macroscopic objects to evolve into classical-like states. Bohmian mechanics retains the Schrödinger equation but adds real particles in 3D space, grounding macroscopic classicality in microscopic reality. Each particle is guided by its own wavefunction in 3D space, but each of these is a “conditional wavefunction” –– the global wavefunction evaluated by substituting in the Bohmian value for the position of every other particle.
Unlike Bohmian mechanics, collapse models modify the predictions of orthodox quantum mechanics. To avoid predictions that have already been empirically ruled out, non-Markovian collapse models (NMCMs), with smooth-in-time noise, have been proposed. However, such NMCMs are mathematically challenging to derive and simulate.
In this paper we show that NMCMs can be easily derived and understood within a Bohmian framework, as follows. In standard NMCMs, one adds to the system an auxiliary quantum bath in order to construct the system collapse dynamics, so that averaging over the noise is the same as tracing over that ‘non-physical’ bath. Here we introduce Bohmian variables for this auxiliary bath only, with standard Bohmian dynamics. Then the “conditional wavefunction” for the system, conditioned on those auxiliary Bohmian variables, reproduces exactly the NMCM.
Our work explains complicated stochastic non-Markovian dynamics in terms of simple deterministic Markovian dynamics of a larger system with hidden variables. This suggests that to take NMCMs seriously one should take seriously the auxiliary bath as a physical system, as well as its Bohmian variables.

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[2] Guillermo Albareda, Kevin Lively, Shunsuke A. Sato, Aaron Kelly, and Angel Rubio, "Conditional Wave Function Theory: A Unified Treatment of Molecular Structure and Nonadiabatic Dynamics", Journal of Chemical Theory and Computation 17 12, 7321 (2021).

[3] Guillermo Albareda, Kevin Lively, Shunsuke A. Sato, Aaron Kelly, and Angel Rubio, "Conditional wavefunction theory: a unified treatment of molecular structure and nonadiabatic dynamics", arXiv:2107.01094, (2021).

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