An Abstract Interface to Higher Spin Gauge Field Theory

TL;DR

This paper develops an abstract, formal framework for higher spin gauge field theories, separating principles from implementation, and maps it to homotopy Lie algebras, highlighting the mathematical structure while noting the lack of a concrete physical realization.

Contribution

It introduces an abstract formalism for higher spin gauge fields and connects it to homotopy Lie algebras, providing a new mathematical perspective.

Findings

Formalism describes a wide class of classical field theories

Mapping to homotopy Lie algebras demonstrates the theory's mathematical existence

An initial iterative approach to interactions is proposed

Abstract

A comprehensive approach to the theory of higher spin gauge fields is proposed. By explicitly separating out details of implementation from general principles, it becomes possible to focus on the bare minimum of requirements that such a theory must satisfy. The abstraction is based on a survey of the progress that has been achieved since relativistic wave equations for higher spin fields were first considered in the nineteen thirties. As a byproduct, a formalism is obtained that is abstract enough to describe a wide class of classical field theories. The formalism, viewed as syntax, can then be semantically mapped to a category of homotopy Lie algebras, thus showing that the theory in some sense exists, at least as an abstract mathematical structure. Still, a concrete physics-like, implementation remains to be constructed. Lacking deep physical insight into the problem, an implementation in terms of generalized vertex operators is set up within which a brute force iterative determination of the first few orders in the interaction can be attempted.