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Relational bundle geometric formulation of non-relativistic quantum mechanics

J. François, L. Ravera

TL;DR

This work presents a geometric, bundle-based reformulation of non-relativistic quantum mechanics using cocyclic bundle geometry, where wavefunctions are cocyclic 0-forms on configuration-space-time and Schrödinger dynamics arise from a flat cocyclic covariant derivative tied to a cocyclic connection. The Dirac–Feynman path integral naturally emerges from this framework, linking classical action to a cocyclic structure and enabling a Lagrangian view of QM. The Dressing Field Method is employed to derive a relational formulation, yielding basic cocyclic fields that encode relational degrees of freedom and provide covariance under changes of physical reference frames. The approach yields a relational quantum mechanics with a clear notion of frame covariance and a relational path integral, offering a quantum-democratic viewpoint in which any subsystem can serve as a reference without privileging an external observer. Potential extensions include relatiionalizing spins, extended bodies, and a relativistic version, with a broader program of relational quantization across physics.

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 Schrödinger 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.

Relational bundle geometric formulation of non-relativistic quantum mechanics

TL;DR

This work presents a geometric, bundle-based reformulation of non-relativistic quantum mechanics using cocyclic bundle geometry, where wavefunctions are cocyclic 0-forms on configuration-space-time and Schrödinger dynamics arise from a flat cocyclic covariant derivative tied to a cocyclic connection. The Dirac–Feynman path integral naturally emerges from this framework, linking classical action to a cocyclic structure and enabling a Lagrangian view of QM. The Dressing Field Method is employed to derive a relational formulation, yielding basic cocyclic fields that encode relational degrees of freedom and provide covariance under changes of physical reference frames. The approach yields a relational quantum mechanics with a clear notion of frame covariance and a relational path integral, offering a quantum-democratic viewpoint in which any subsystem can serve as a reference without privileging an external observer. Potential extensions include relatiionalizing spins, extended bodies, and a relativistic version, with a broader program of relational quantization across physics.

Abstract

We present a bundle geometric formulation of non-relativistic many-particles Quantum Mechanics. A wave function is seen to be a -valued cocyclic tensorial 0-form on configuration space-time seen as a principal bundle, while the Schrödinger 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.
Paper Structure (27 sections, 179 equations, 2 figures)

This paper contains 27 sections, 179 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bundle geometry and the dressing field method
  3. Cocyclic bundle geometry in a nutshell
  4. Spaces of remarkable differential forms
  5. Connections and covariant derivatives
  6. Gauge transformations
  7. Local structure
  8. The dressing field method: basics
  9. Dressing field and basic dressed fields
  10. Residual transformations of the 1st kind
  11. Residual transformations of the 2nd kind
  12. Bundle geometric non-relativistic Mechanics and Quantum Mechanics
  13. Bundle geometry of configuration space-time $\mathscr{C}$
  14. Classical mechanics on $\mathscr{C}$: The action as a cocyclic connection
  15. Quantum mechanics on $\mathscr{C}$: Schrödinger equation as a cocyclic covariant derivative
  16. ...and 12 more sections

Figures (2)

  • Figure 1: The $2$-fibration structure of the configuration space-time bundle $\mathscr{C}$ of $N$ classical particles.
  • Figure 2: Realisations of the relational space-time configuration bundle via dressing and frame covariance.