The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra

TL;DR

This paper explores the algebraic structures arising from differential operators on symmetric tensors, introducing a new associative algebra formulation that enhances understanding of geometric and quantum mechanical operator relations.

Contribution

It presents a novel associative algebra reformulation of the rank 2 deformation of the Fourier-Jacobi algebra, improving upon traditional operator ordering methods.

Findings

Deformation of Fourier-Jacobi algebra in curved spaces

Introduction of a new associative algebra for operator ordering

Application to higher spin particle theories

Abstract

Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.