2015/05/27 by J. LaChapelle, LaChapelle, John
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical and Theoretical Analysis #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.1505.08102
openalex publication_date 2015/05/27 · openalex created_date 2022/09/13 · openalex updated_date 2026/07/28
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^∗-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^∗-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 Poincaré groups.