Higher Spin Fields in Siegel Space, Currents and Theta Functions

TL;DR

This paper formulates the dynamics of all spins of massless fields in four dimensions within Siegel space, revealing connections to theta functions, particle-antiparticle distinctions, and higher-spin currents, with potential extensions to higher dimensions.

Contribution

It introduces a novel framework for describing massless fields in Siegel space, linking multidimensional theta functions to field solutions and currents, and explores symmetry and evolution in this setting.

Findings

Multidimensional Riemann theta functions solve massless field equations.

Unfolded equations distinguish particles and antiparticles.

Global symmetry parameters must be singular to produce nonzero currents.

Abstract

Dynamics of four-dimensional massless fields of all spins is formulated in the Siegel space of complex $4\times 4$ symmetric matrices. It is shown that the unfolded equations of free massless fields, that have a form of multidimensional Schrodinger equations, naturally distinguish between positive- and negative-frequency solutions of relativistic field equations, i.e. particles and antiparticles. Multidimensional Riemann theta functions are shown to solve massless field equations in the Siegel space. We establish the correspondence between conserved higher-spin currents in four-dimensional Minkowski space and those in the ten-dimensional matrix space. It is shown that global symmetry parameters of the current in the matrix space should be singular to reproduce a nonzero current in Minkowski space. The $\D$-function integral evolution formulae for 4d massless fields in the Fock-Siegel space are obtained. The generalization of the proposed scheme to higher dimensions and systems of higher ranks is considered.