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arXiv · 1505.08102

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

John LaChapelle
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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.

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John LaChapelle. 2015-05-27. Functional Integral Approach to $C^*$-algebraic Quantum Mechanics I: Heisenberg and Poincar\'{e}. https://arxiv.org/abs/1505.08102

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