Schwinger's Picture of Quantum Mechanics III: The statistical interpretation

TL;DR

This paper explores Schwinger's algebra of selective measurements within a groupoid framework, revealing a natural emergence of Sorkin's quantum measure and providing new insights into the statistical interpretation of quantum mechanics.

Contribution

It introduces a novel interpretation of Schwinger's algebra using quantum measures and decoherence functionals, connecting to Sorkin's framework and analyzing specific quantum examples.

Findings

Quantum measures can be derived from states of the algebra of virtual transitions.

Factorizable states possess a Dirac-Feynman exponential of the action form.

Schwinger's transformation functions are interpreted as transition amplitudes.

Abstract

Schwinger's algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the theory, was presented in two previous papers (arXiv:1905.12274 and arXiv:1907.03883). In this paper, a closer look to the statistical interpretation of the theory is taken and it is found that an interpretation in terms of Sorkin's quantum measure emerges naturally. It is proven that a suitable class of states of the algebra of virtual transitions of the theory allows to define quantum measures by means of the corresponding decoherence functionals. Quantum measures satisfying a reproducing property are described and a class of states, called factorizable states, possessing the Dirac-Feynman `exponential of the action' form are characterized. Finally, Schwinger's transformation functions are interpreted similarly as transition amplitudes defined by suitable states. The simple examples of the qubit and the double slit experiment are described in detail, illustrating the main aspects of the theory.