# Field Equations for the Simplest Multi-Particle Higher-Spin Systems

## TL;DR

This paper derives product laws for specific multi-particle higher-spin algebras and analyzes their equations of motion, revealing their roles in describing off-shell higher-spin systems and conserved currents.

## Contribution

It introduces the product law for the algebras $M_2(A)$ and $M^2(A)$ and analyzes the resulting equations of motion for these multi-particle higher-spin systems.

## Key findings

- $M_2(A)$ equations describe off-shell higher-spin systems.
- $M^2(A)$ equations describe conserved currents.
- Off-shell degrees of freedom are represented by rank-two fields.

## Abstract

We derive the product law for the simplest multi-particle higher-spin algebra $M_2(A)$ and its factor-algebra $M^2(A)$ with rank-one fields factored out. Equations of motion for the systems resulting from these algebras are analysed. We conclude that the equations for $M_2(A)$ describe the conformal off-shell higher-spin system with the rank-two fields representing the off-shell degrees of freedom of the originally massless system. On the other hand the equations resulting from $M^2(A)$ describe the infinite system of conserved currents unrelated to massless fields.