Free fall in KvN mechanics and Einstein's principle of equivalence

TL;DR

This paper explores how Einstein's principle of equivalence applies in Koopman-von Neumann (KvN) classical mechanics, showing similarities with quantum mechanics and demonstrating that gravitational effects manifest as phase shifts in wave functions.

Contribution

It demonstrates that KvN mechanics, a Hilbert space formulation of classical mechanics, exhibits Einstein's equivalence principle similarly to quantum mechanics, highlighting fundamental conceptual parallels.

Findings

Wave functions in KvN mechanics acquire phase shifts in gravitational fields.

Fisher information remains unaffected by gravitational field strength.

Equivalence principle holds in KvN mechanics, paralleling quantum mechanics.

Abstract

The implementation of Einstein's principle of equivalence in Koopman-von Neumann (KvN) mechanics is discussed. The implementation is very similar to the implementation of this principle in quantum mechanics. This is not surprising, because KvN mechanics provides a Hilbert space formulation of classical mechanics that is very similar to the quantum mechanical formalism. Both in KvN mechanics and quantum mechanics, a propagator in a homogeneous gravitational field is simply related with a free propagator. As a result, the wave function in a homogeneous gravitational field in a freely falling reference frame differs from the free wave function only in phase. Fisher information, which quantifies our ability to estimate mass from coordinate measurements, does not depend on the magnitude of the homogeneous gravitational field, and this fact constitutes the formulation of Einstein's principle of equivalence, which is valid both in quantum mechanics and KvN mechanics.