Homological Quantum Mechanics

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.

Cosmological Perturbations in Double Field Theory

“…We believe that a most promising framework is that of the homotopy algebra formulation of gauge theories, see e.g. [37], because here there are natural 'homological' approaches to the definition of gauge invariant variables [33] and to the computation of quantum mechanical expectation values [38].…”

Braided quantum electrodynamics

“…To properly implement Fermi statistics, we need to carefully define both the action of the extended contracting homotopy H and the braided BV Laplacian ∆ BV when acting on spinor fields. 12 There are two important extra signs which arise in this case: an extra weight (−1) a−1 in each term of the sum (2.37) defining the extended contracting homotopy H, and an extra weight (−1) (a−1)+(b−1−a) = (−1) b in each term of the sum in (2.39) defining the braided BV Laplacian ∆ BV . The only non-trivial pairings (3.17) are between ψ, ψ and their corresponding antifields ψ+ , ψ + .…”

“…Homotopical methods based on L ∞ -algebras and A ∞ -algebras have been playing an increasingly significant role in our understanding of the algebraic and kinematic structures inherent in scattering amplitudes and correlation functions of quantum field theory; for an incomplete sample of recent works see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. At a given order of perturbation theory, these can be calculated in a purely algebraic fashion without resorting to canonical quantization or path integral techniques: the quantum Batalin-Vilkovisky (BV) formalism gives an explicit homological construction of correlators which algebraically generates Feynman diagrams.…”

Correlation functions of scalar field theories from homotopy algebras

“…On the other hand, the description of on-shell scattering amplitudes in terms of homotopy algebras is based on the fact that Feynman diagrams are algebraically generated in this approach [9,23,24], and we expect that there is a way to generate Feynman diagrams for correlation functions using homotopy algebras as well. Furthermore, in the framework of the Batalin-Vilkovisky formalism [25,26,27] correlation functions have been discussed [28,29], and we again expect that there is a way to describe correlation functions in the framework of homotopy algebras which can be thought of as being dual to the Batalin-Vilkovisky formalism. In this paper we demonstrate that it is indeed the case that we can describe correlation functions in terms of homotopy algebras, and we present highly explicit expressions for correlation functions of scalar field theories in perturbation theory using quantum A ∞ algebras.…”