A quantum route to the classical Lagrangian formalism

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 [Formula: see text] 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 q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.

Feynman's Propagator in Schwinger's picture of Quantum Mechanics

“…(1) t 1 ,t 0 , as the concrete history with domain [t 0 , t 2 ], reference time t 0 , determined by the map (w (2) • w (1) ) t 0 (s), given by: (w (2) • w (1) ) t 0 (s) = w…”

“…The previous formula can be interpreted naturally in terms of K-paths, that is, the composition of two K-paths is just the concatenation of the corresponding paths and the composition of their liftings. Note that the restriction of the composition of the histories w (2) • w (1) to the interval [t 0 , t 1 ] is just w (1) . Moreover if we write the K-path of w (2) • w (1) with respect to the reference time t 1 , the change of reference time formula (4.9) will give, t 1 ≤ s ≤ t 2 :…”

“…Recently it has been shown how the introduction of a q-Lagrangian function [2] on the groupoid of configurations K of a quantum system allows us to recover a c-Lagragian function L on the Lie algebroid of the given system that leads to a classical Lagrangian function L providing the standard mechanical description of the system on the tangent bundle of "velocity" configurations of the system.…”

Causality in Schwinger’s Picture of Quantum Mechanics

“…This paper addresses the problem of the role of causality in quantum mechanics, taking advantage of the recent developments on the categorical/groupoidal description of quantum systems inspired by Schwinger’s picture of quantum mechanics (see, for instance, [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 ] and references therein).…”

“…In the groupoidal picture of quantum mechanical systems, a groupoid is associated with each family of experimental settings used to describe the system (see later, Section 3.2 , and [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ] for details). The objects x of the groupoid are the outcomes of the measurements performed on the system, and its arrows, or morphisms, are the physical transitions that the system experiences.…”