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arXiv · 2106.00998

A quantum route to the classical Lagrangian formalism

Florio M. Ciaglia·Fabio Di Cosmo·Alberto Ibort·Giuseppe Marmo·Luca Schiavone·Alessandro Zampini
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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.

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Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone, Alessandro Zampini. 2021-06-02. A quantum route to the classical Lagrangian formalism. https://doi.org/10.1142/s0217732321500917

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