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

TL;DR

This paper introduces a new derivation of Feynman's propagator using the groupoidal approach in Schwinger's quantum mechanics, connecting it to GNS representations of DFS states derived from a q-Lagrangian.

Contribution

It presents a novel derivation of the quantum propagator via groupoid and GNS framework, linking Feynman's sum-over-histories to Schwinger's picture.

Findings

Derivation of Feynman's propagator using groupoid methods.

Connection between DFS states and GNS representations.

Application to a Riemannian manifold example.

Abstract

A novel derivation of Feynman's sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac-Feynman-Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function $\ell$ on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian $\ell$ allow to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman's original derivation of the propagator for a point particle described by a classical Lagrangian $L$.