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

TL;DR

This work develops a first-quantized, bosonic spinning-particle model that realizes Lichnerowicz's algebra for symmetric tensors via geodesic motion with a complex tangent vector. Quantization maps wavefunctions to sections of the symmetric-tensor bundle $SM$, turning Noether charges into geometric operators that deform the Fourier--Jacobi algebra $sp(2,\mathbb{R})^J$ in curved spaces. A key advancement is the introduction of a trace-free reformulation using the depth operator $\kappa$ and the square root of the Casimir ${\cal C}$, which leads to the associative algebra $\widetilde{\mathcal U}(sp(2,\mathbb{R})^J)$ and simplifies operator ordering for high powers of the generators. These results offer a practical framework for higher-spin constructions and suggest symmetric-tensor analogues of de Rham cohomology, along with avenues to generalize to broader orthosymplectic structures.

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.

Figures (3)

• Figure 1: An ant laden with a complex tangent vector.
• Figure 2: The root diagram for the Fourier--Jacobi Lie algebra.
• Figure 3: Defining relations for the $\widetilde{\, \cal U\, }\!(sp(2,{\mathbb R})^J)$ algebra.