Higher spins and Finsler geometry

TL;DR

This paper explores the connection between Finsler geometry and higher-spin fields, showing that linear Finsler Ricci tensors reproduce the Fronsdal equations and discussing nonlinear interactions and gauge issues.

Contribution

It demonstrates how Finsler geometry can encode higher-spin equations and analyzes the nonlinear structure and gauge redundancy issues without fixing spacetime dimensions.

Findings

Finsler Ricci tensor yields Fronsdal equations for all spins.

Nonlinear terms suggest possible interacting higher-spin structure.

Gauge redundancy persists due to Stueckelberg mechanism.

Abstract

Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This creates a serious issue: non-transverse modes are not eliminated, at least for the versions of Finsler dynamics examined in this paper.