A gentle introduction to Schwinger's formulation of quantum mechanics: The groupoid picture

TL;DR

This paper reviews Schwinger's formulation of quantum mechanics, highlighting that the underlying mathematical structure is a groupoid, from which both Hilbert spaces and C*-algebras naturally emerge.

Contribution

It introduces the groupoid framework as the foundational structure behind Schwinger's approach, unifying different formulations of quantum mechanics.

Findings

Hilbert space and C*-algebra are derived from the groupoid structure

Schwinger's symbolism is interpreted through groupoid theory

Provides a new perspective on quantum measurement structures

Abstract

In this short letter we review Schwinger's formulation of Quantum Mechanics and we argue that the mathematical structure behind Schwinger's "Symbolism of Atomic Measurements" is that of a groupoid. In this framework, both the Hilbert space (Schr\"{o}dinger picture) and the $C^{*}$-algebra (Heisenberg picture) of the system turn out to be derived concepts, that is, they arise from the underlying groupoid structure.