A quantum route to the classical Lagrangian formalism

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