arXiv:2404.08815

Star exponentials from propagators and path integrals

Jasel Berra–Montiel, Hugo García‐Compeán, Alberto Molgado

Published Apr 12, 2024• 0 citations

Abstract

In this paper we address the relation between the star exponentials emerging within the Deformation Quantization formalism and Feynman's path integrals associated with propagators in quantum dynamics. In order to obtain such a relation, we start by visualizing the quantum propagator as an integral transform of the star exponential by means of the symbol corresponding to the time evolution operator and, thus, we introduce Feynman's path integral representation of the propagator as a sum over all the classical histories. The star exponential thus constructed has the advantage that it does not depend on the convergence of formal series, as commonly understood within the context of Deformation Quantization. We include some basic examples to illustrate our findings, recovering standard results reported in the literature. Further, for an arbitrary finite dimensional system, we use the star exponential introduced here in order to find a particular representation of the star product which resembles the one encountered in the context of the quantum field theory for a Poisson sigma model.

DOI: 10.48550/arxiv.2404.08815

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arXiv:2404.08815

Star exponentials from propagators and path integrals

Jasel Berra–Montiel, Hugo García‐Compeán, Alberto Molgado

Published Apr 12, 2024• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2404.08815

Related papers

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arXiv:1003.1156Letters in Mathematical PhysicsSep 29, 20108 citations

In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of

SU(2)

and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on

S^2

and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on

{\mathbb{R}^n}

, with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in

{\hbar}

, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the

{t \to 0}

limit approaches the δ distribution. As such, our construction gives explicitly the full

{\hbar\to 0}

asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.