Quantum mechanics without spacetime IV : a noncommutative Hamilton-Jacobi equation

TL;DR

This paper proposes a noncommutative generalization of the Hamilton-Jacobi equation for relativistic particles, offering a background-independent formulation of quantum mechanics that aligns with standard quantum theory under specific assumptions.

Contribution

It introduces a novel noncommutative Hamilton-Jacobi equation as a background-independent formulation of quantum mechanics for relativistic particles.

Findings

Noncommutative Hamilton-Jacobi equation is equivalent to standard quantum mechanics under certain metric and commutation assumptions.

Provides a background-independent framework for quantum mechanics.

Bridges noncommutative geometry with quantum theory.

Abstract

It has earlier been argued that there should exist a formulation of quantum mechanics which does not refer to a background spacetime. In this paper we propose that, for a relativistic particle, such a formulation is provided by a noncommutative generalisation of the Hamilton-Jacobi equation. If a certain form for the metric in the noncommuting coordinate system is assumed, along with a correspondence rule for the commutation relations, it can be argued that this noncommutative Hamilton-Jacobi equation is equivalent to standard quantum mechanics.