Relational bundle geometric formulation of non-relativistic quantum mechanics

TL;DR

This paper develops a geometric bundle formulation of non-relativistic quantum mechanics, viewing wave functions as cocyclic forms on configuration space-time and introducing a relational perspective via the dressing field method.

Contribution

It introduces a novel geometric and relational formulation of quantum mechanics using bundle theory and the dressing field method, connecting to path integrals and reference frames.

Findings

Wave functions are represented as cocyclic tensorial 0-forms on configuration space-time.

The Schrödinger equation emerges from covariant derivatives in this geometric setting.

The relational reformulation allows any particle position to serve as a physical reference frame.

Abstract

We present a bundle geometric formulation of non-relativistic many-particles Quantum Mechanics. A wave function is seen to be a $\mathbb{C}$-valued cocyclic tensorial 0-form on configuration space-time seen as a principal bundle, while the Schr\"odinger equation flows from its covariant derivative, with the action functional supplying a (flat) cocyclic connection 1-form on the configuration bundle. In line with the historical motivations of Dirac and Feynman, ours is thus a Lagrangian geometric formulation of QM, in which the Dirac-Feynman path integral arises in a geometrically natural way. Applying the dressing field method, we obtain a relational reformulation of this geometric non-relativistic QM: a relational wave function is realised as a basic cocyclic 0-form on the configuration bundle. In this relational QM, any particle position can be used as a dressing field, i.e. as a "physical reference frame". The dressing field method naturally accounts for the freedom in choosing the dressing field, which is readily understood as a covariance of the relational formulation under changes of physical reference frame.