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arXiv:quant-ph/9801070 · Physical Review A

Hypersurface Bohm-Dirac models

Detlef Dürr, Sheldon Goldstein, Karin Berndl, Nino Zanghı̀

Published Oct 1, 1999• 130 citations
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Abstract

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.

DOI
10.1103/physreva.60.2729

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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.
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.
A candidate for a realistic relativistic quantum theory is the hypersurface Bohm-Dirac model. Its formulation uses a foliation of space-time into space-like hypersurfaces. In order to apply the theory and to make contact with the usual quantum formalism, one needs a framework for the description of subsystems. The presence of spin together with the foliation renders the subsystem description more complicated than in the non-relativistic case with spin. In this paper, we provide such a framework in terms of an appropriate conditional density matrix and an effective wave function as well as clarify their relation, thereby generalizing previous subsystem descriptions in the non-relativistic case.
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.
In this paper we intend to extend some ideas of a recently proposed Lorentz-invariant Bohmian model, obeying Klein-Gordon equations, but considering particles with a spin different than zero. First we build a Bohmian model for a single particle, closely related to the Bohm-Dirac model, but using Nikolic's ideas on a space-time probability density and the introduction of a synchronization parameter σfor the particle trajectory. We prove that, in the case of a spinor whose component phases become equal when \plank \to 0, it is possible to use the usual Bohmian Mechanics procedures to obtain the classical limit. We extend these ideas for a two-particle system, based on two non-local but covariant Dirac-like equations.