Lagrangian in quantum mechanics is a connection one-form

TL;DR

This paper reformulates Dirac's quantum mechanical Lagrangian using the language of vector bundles, revealing it as an operator-valued connection one-form, and explores phase transformations and relativistic limits.

Contribution

It introduces a geometric perspective by expressing the quantum Lagrangian as a connection one-form in vector bundle language, linking phases to differential forms.

Findings

Action is an operator-valued connection one-form.

Phases from frame changes are total differentials.

Correct non-relativistic phase for uniform acceleration.

Abstract

We recast Dirac's Lagrangian in quantum mechanics in the language of vector bundles and show that the action is an operator-valued connection one-form. Phases associated with change of frames of reference are seen to be total differentials in the transformation of the action. The relativistic case is discussed and we show that it gives the correct phase in the non-relativistic limit for uniform acceleration.