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arXiv · 2112.11495

Homological Quantum Mechanics

Christoph Chiaffrino·Olaf Hohm·Allison F. Pinto
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Abstract

We provide a formulation of quantum mechanics based on the cohomology of the Batalin-Vilkovisky (BV) algebra. Focusing on quantum-mechanical systems without gauge symmetry we introduce a homotopy retract from the chain complex of the harmonic oscillator to finite-dimensional phase space. This induces a homotopy transfer from the BV algebra to the algebra of functions on phase space. Quantum expectation values for a given operator or functional are computed by the function whose pullback gives a functional in the same cohomology class. This statement is proved in perturbation theory by relating the perturbation lemma to Wick's theorem. We test this method by computing two-point functions for the harmonic oscillator for position eigenstates and coherent states. Finally, we derive the Unruh effect, illustrating that these methods are applicable to quantum field theory.

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Christoph Chiaffrino, Olaf Hohm, Allison F. Pinto. 2021-12-21. Homological Quantum Mechanics. https://arxiv.org/abs/2112.11495

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