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arXiv · 2501.02046

Relational bundle geometric formulation of non-relativistic quantum mechanics

J. François·L. Ravera
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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.

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J. François, L. Ravera. 2025-01-03. Relational bundle geometric formulation of non-relativistic quantum mechanics. https://arxiv.org/abs/2501.02046

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