Abstract
In non-relativistic Bohmian mechanics the universe is represented by a probability space whose sample space is composed of the Bohmian trajectories. In relativistic Bohmian mechanics an entire class of empirically equivalent probability spaces can be defined, one for every foliation of spacetime. In the literature the hypothesis has been advanced that a single preferred foliation is allowed, and that this foliation derives from the universal wave function by means of a covariant law. In the present paper the opposite hypothesis is advanced, i.e., no law exists for the foliations and therefore all the foliations are allowed. The resulting model of the universe is basically the “union” of all the probability spaces associated with the foliations. This hypothesis is mainly motivated by the fact that any law defining a preferred foliation is empirically irrelevant. It is also argued that the absence of a preferred foliation may reduce the well known conflict between Bohmian mechanics and relativity.
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Notes
In the relativistic formulation the symbols \(\mathbb {K}\) and \(\mathbb {B}\) are redefined.
Here a world line is considered to be a subset of M, and its intersection with any hypersurface is exactly one point.
This point can be better understood by means of the following analogy: in non-relativistic Bohmian mechanics the set \(\mathbb {B}\) defined in Sect. 2 is the set of the trajectories which are dynamically allowed by the guiding equation. If the guiding equation is removed as a law, the set of the allowed trajectories becomes the entire kinematic space \(\mathbb {K}\).
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Galvan, B. Relativistic Bohmian Mechanics Without a Preferred Foliation. J Stat Phys 161, 1268–1275 (2015). https://doi.org/10.1007/s10955-015-1369-8
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DOI: https://doi.org/10.1007/s10955-015-1369-8