Relativized Wave Function: a First Step in a Unifying Approach to General Relativity and Bohmian Mechanics
Résumé
We propose a new way to unify General Relativity (GR) with Bohmian Mechanics (BM) that remains faithful to GR’s tensorial and deterministic philosophy. In our approach, the conventional quantum wave function emerges as the scalar contraction of a tensorial field $\psi^\mu$, namely $\Psi=\psi_\nu\psi^\nu$, thereby encoding quantum dynamics in a geometric object. Crucially, the tensor $\psi^\nu$ interacts directly with spacetime via the stress-energy tensor, with an auxiliary vector field $a^\mu$ serving as the mediator between the quantum and gravitational sectors. We analyze the equations for one single particle without accounting for spin. As an illustrative example, we show that under suitable approximations our formulation reproduces the non-relativistic Schr\"odinger equation with a Coulomb potential\textemdash as encountered in the hydrogen atom. We further establish a dynamical equation for $\psi^\nu$ and a corresponding volume-like equation for $a^\mu$.
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