# Schwinger's Picture of Quantum Mechanics II: Algebras and Observables

## TL;DR

This paper develops a groupoid-based algebraic framework for quantum mechanics, analyzing its dynamical aspects, states, and transition functions, and illustrating the approach with qubit and harmonic oscillator examples.

## Contribution

It introduces a dynamical formalism within Schwinger's algebraic approach using groupoids, connecting it with traditional quantum pictures and exploring quantum-classical transition.

## Key findings

- Hamiltonian evolution emerges naturally in the formalism.
- The approach provides a simple way to discuss quantum-to-classical transition.
- Examples include qubit and harmonic oscillator analyzed within the new framework.

## Abstract

The kinematical foundations of Schwinger's algebra of selective measurements were discussed in a previous paper (arXiv:1905.12274) and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analysed. For that, the algebra generated by the observables, as well as the notion of state, are dicussed, and the structure of the transition functions, that plays an instrumental role in Schwinger's picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac-Schr\"odinger and Born-Jordan-Heisenberg pictures is discussed.