Functional Integral Approach to $C^*$-algebraic Quantum Mechanics I: Heisenberg and Poincar\'{e}

TL;DR

This paper develops a hybrid formalism combining algebraic and path integral approaches to quantum mechanics using topological groups, applied to Heisenberg and Poincaré groups.

Contribution

It introduces a novel functional integral framework based on topological groups that unifies $C^*$-algebraic and path integral formulations of quantum mechanics.

Findings

Constructs functional integral representations of $C^*$-algebras.

Applies the formalism to non-relativistic quantum mechanics.

Extends the approach to relativistic quantum mechanics with Poincaré group.

Abstract

The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous $C^\ast$-algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral representations of $C^\ast$-algebras. The merger achieves a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role. To illustrate the formalism, we apply the framework to non-relativistic and relativistic quantum mechanics via the Heisenberg and Poincar\'{e} groups.