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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Acknowledgments
We would like to thank Roberto Bonezzi, Tomas Codina and Felipe Diaz-Jaramillo for useful discussions and Owen Gwilliam and Kasia Rejzner for comments on the first version of this paper.
This work is funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 771862) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), “Rethinking Quantum Field Theory”, Projektnummer 417533893/GRK2575.
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Chiaffrino, C., Hohm, O. & Pinto, A.F. Homological quantum mechanics. J. High Energ. Phys. 2024, 137 (2024). https://doi.org/10.1007/JHEP02(2024)137
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DOI: https://doi.org/10.1007/JHEP02(2024)137