Higher Spin Fields and Symplectic Geometry

TL;DR

This paper explores how higher spin fields can be understood as gauge fields arising from Hamiltonian mechanics, with symplectic geometry providing a framework for their symmetries and invariants.

Contribution

It introduces a novel perspective linking higher spin fields to Hamiltonian mechanics and symplectic geometry, defining gauge-invariant geometric quantities under symplectic transformations.

Findings

Higher spin fields act as gauge fields ensuring covariance under canonical transformations.

Symplectic invariants for volume and curvature are proposed as counterparts to traditional geometric notions.

Reparametrization invariant action describes particles in higher spin backgrounds with symplectic symmetry.

Abstract

We argue that higher spin fields originate from Hamiltonian mechanics and play a role of gauge fields ensuring covariance of geometric observables such as length and volume with respect to canonical transformations in the same way as a metric tensor in Riemannian geometry ensures covariance with respect to diffeomorphisms. We consider a reparametrization invariant action of a point particle in Hamiltonian form. Reparametrization invariance is achieved in the standard way by coupling to the auxiliary world-line metric. Identifying Hamiltonian function with a generating function for higher spin fields this action can be viewed as an action for the point particle in a higher spin background, while canonical transformations act as higher spin symmetries. We define the gauge invariant length as a proper time of a particle moving along the geodesic. Following the usual geometrical interpretation we introduce the volume form and the scalar curvature for a combined lower spin sector. As for the general case, we show that notions of local volume and scalar curvature are not compatible with symplectic transformations. We propose symplectically invariant counterparts for the total volume of the space and Einstein-Hilbert action.