Homological Quantum Mechanics

TL;DR

This paper introduces a homological approach to quantum mechanics using BV algebra cohomology, enabling computation of expectation values and deriving the Unruh effect, with applications to quantum field theory.

Contribution

It develops a homotopy transfer framework from BV algebra to phase space functions, providing a new cohomological formulation of quantum mechanics without gauge symmetry.

Findings

Computed two-point functions for harmonic oscillator states.

Derived the Unruh effect using the homological framework.

Validated the approach through perturbation theory and Wick's theorem.

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.