A quantum route to the classical Lagrangian formalism
arXiv:2106.00998 · doi:10.1142/S0217732321500917
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 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 on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold , the quadratic expansion of will reproduce the standard Lagrangians on used to describe the classical dynamics of particles.
12 pages