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arXiv:1307.1714 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Can Bohmian mechanics be made relativistic?

Detlef Dürr, Sheldon Goldstein, Travis Norsen, Ward Struyve, Nino Zanghì

Published Dec 11, 2013• 166 citations
arXivHTMLPDF

Abstract

In relativistic space-time, Bohmian theories can be formulated by introducing a privileged foliation of space-time. The introduction of such a foliation-as extra absolute space-time structure-would seem to imply a clear violation of Lorentz invariance, and thus a conflict with fundamental relativity. Here, we consider the possibility that, instead of positing it as extra structure, the required foliation could be covariantly determined by the wave function. We argue that this allows for the formulation of Bohmian theories that seem to qualify as fundamentally Lorentz invariant. We conclude with some discussion of whether or not they might also qualify as fundamentally relativistic.

DOI
10.1098/rspa.2013.0699

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In relativistic space-time, Bohmian theories can be formulated by introducing a privileged foliation of space-time. The introduction of such a foliation-as extra absolute space-time structure-would seem to imply a clear violation of Lorentz invariance, and thus a conflict with fundamental relativity. Here, we consider the possibility that, instead of positing it as extra structure, the required foliation could be covariantly determined by the wave function. We argue that this allows for the formulation of Bohmian theories that seem to qualify as fundamentally Lorentz invariant. We conclude with some discussion of whether or not they might also qualify as fundamentally relativistic.
Lorentz invariance is considered a fundamental property of relativistic quantum theory. However, in standard quantum theory this invariance is only partially realized: while relativistic wave equations like the Dirac equation are Lorentz invariant, the collapse postulate is not. In the non-relativistic domain, alternative theories have been formulated, like Bohmian mechanics and spontaneous collapse theories, which dispense with the problematic collapse postulate. Extending these theories to the relativistic domain appears challenging, particularly with respect to implementing Lorentz invariance, mainly due to the unavoidable non-locality implied by Bell’s theorem. However, space-time theories can also trivially be formulated in a Lorentz-invariant way. If Lorentz invariance is to impose a genuine constraint on the content of a physical theory, one should aim for what can be called substantive Lorentz invariance. However, articulating a precise definition of this notion is notoriously difficult. This paper investigates two candidate criteria: Anderson’s criterion based on the identification of absolute objects, and a relativity principle for isolated subsystems. These criteria are applied to evaluate several Lorentz-invariant Bohmian models, a spontaneous collapse model, and a version of the Many-Worlds theory. With the exception of two Bohmian approaches, these models satisfy both criteria. Nevertheless, some Bohmian models that meet these conditions still do not appear to be substantively Lorentz-invariant, suggesting that the proposed criteria may not fully capture the intended concept. Nevertheless, they help clarify what aspects of relativity theory may need to be given up in passing from classical to quantum theory.
Bohmian mechanics can be generalized to a relativistic theory without preferred foliation, with a price of introducing a puzzling concept of spacetime probability conserved in a scalar time. We explain how analogous concept appears naturally in classical statistical mechanics of relativistic particles, with scalar time being identified with the proper time along particle trajectories. The conceptual understanding of relativistic Bohmian mechanics is significantly enriched by this classical insight. In particular, the analogy between classical and Bohmian mechanics suggests the interpretation of Bohmian scalar time as a quantum proper time different from the classical one, the two being related by a nonlocal scale factor calculated from the wave function. In many cases of practical interest, including the macroscopic measuring apparatus, the fundamental spacetime probability explains the more familiar space probability as an emergent approximate description. Requiring that the quantum proper time in the classical limit should reduce to the classical proper time, we propose that only massive particles have Bohmian trajectories. An analysis of the macroscopic measuring apparatus made up of massive particles restores agreement with the predictions of standard quantum theory.
The extension of Bohmian mechanics to the relativistic domain, and to quantum field theory presents specific theoretical difficulties. The non-local nature of the pilot-wave guidance equations inherently requires a temporal ordering of spacelike separated events. Historically, this requirement has been satisfied by introducing an arbitrary preferred foliation of spacetime, which leads to covariance concerns. Furthermore, the standard Bohmian particle model struggles to accommodate particle creation and annihilation events, which are characteristic of quantum field theory (QFT). This paper constructs a theoretical framework that is rigorously Lorentz-invariant, and compatible with QFT. The proposed synthesis couples the Tomonaga-Schwinger equation for the relativistic wave functional with a hypersurface Bohm-Dirac model (HBDM). To ensure Lorentz invariance without adding absolute spatiotemporal structure, the necessary foliation is covariantly determined by the universal wave function. To address particle non-conservation, the deterministic trajectories are supplemented by a Bell-type stochastic Markov jump process defined on Fock space, utilizing minimal jump rates derived directly from the interaction Hamiltonian. A hybrid model is adopted, assigning localized particle beables to fermions, and continuous field beables to bosons. The exact equivariance of the probability distributions across the covariantly defined leaves of the foliation is proven, demonstrating that the model reproduces the empirical predictions of standard regularized QFT. In an appendix, we discuss the modeling of neutrinos. Finally, this framework is interpreted with the multi-fold theory. In this context, the covariantly determined foliation emerges physically from a discrete, non-commutative spacetime concretized by 2D random walks of massless Higgs bosons. The multi-fold mechanism and the E/G conjecture (entanglement is gravity) provide a local, physical model for the non-local Bohmian guidance equations, while space-time matter induction and scattering explain the origin of the proposed hybrid field-particle beables as patterns and condensates of these random walks. Sizes or scales also align with multi-fold modeling of QFT with 2D random walks, and spacetime non-commutativity to support Quantum Physics and Fermions. These add to many examples where the multi-fold preons as massless Higgs bosons can microscopically interpretate Quantum Physics behaviors.
We define a class of Lorentz-invariant Bohmian quantum models for N entangled but noninteracting Dirac particles. Lorentz invariance is achieved for these models through the incorporation of an additional dynamical space-time structure provided by a foliation of space time. These models can be regarded as the extension of Bohm's model for N Dirac particles, corresponding to the foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to more general foliations. As with Bohm's model, there exists for these models an equivariant measure on the leaves of the foliation. This makes possible a simple statistical analysis of position correlations analogous to the equilibrium analysis for (the nonrelativistic) Bohmian mechanics.