A quantum route to the classical Lagrangian formalism

TL;DR

This paper introduces a novel approach linking quantum mechanics and classical Lagrangian formalism through groupoid and Lie algebroid structures, providing a new perspective on the role of the Lagrangian in quantum systems.

Contribution

It develops a groupoidal framework that connects quantum histories with classical Lagrangians via a new q-Lagrangian function on the Lie algebroid.

Findings

The q-Lagrangian determines a state on the von Neumann algebra of histories.

Quadratic expansion of the Lie algebroid function reproduces classical Lagrangians.

Provides a geometric bridge between quantum and classical descriptions.

Abstract

Using the recently developed groupoidal description of Schwinger's picture of Quantum Mechanics, a new approach to Dirac's fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function $\ell$ on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call {\itshape q-Lagrangian}, can be described in terms of a new function $\mathcal{L}$ on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold $M$, the quadratic expansion of $\mathcal{L}$ will reproduce the standard Lagrangians on $TM$ used to describe the classical dynamics of particles.