Stress and Strain: T^{μν} of Higher Spin Gauge Fields

TL;DR

Stress and Strain shows that for free higher-spin gauge fields the local $T^{\\mu\\nu}$ is gauge-variant yet its integrated moments can be expressed in terms of gauge-invariant, transverse-traceless variables. The authors construct the Poincaré generators from $T^{0\\mu}$ and their moments, obtaining $P_0 = \\int d^3x\, H$ and $\\mathbf P = -\\int d^3x\, p\\nabla q$, with a spin contribution ensuring the full Poincaré algebra on shell. They illustrate the construction for spin 1 and spin 2, showing that the spin piece arises from the nontrivial transverse equal-time commutators in TT space and that a general $p\\times q$ structure extends to arbitrary spin. The work highlights that while this TT formulation separates scalar and spin content and preserves the Poincaré algebra for the integrated charges, it does not cure the gauge variance of local $T^{\\mu\\nu}$ nor produce a covariant stress tensor suitable for coupling to gravity in generic curved backgrounds, underscoring the challenges of dynamical higher-spin sources.

Abstract

We present some results concerning local currents, particularly the stress tensors T^{μν}, of free higher (>1) spin gauge fields. While the T^{μν} are known to be gauge variant, we can express them, at the cost of manifest Lorentz invariance, solely in terms of (spatially nonlocal) gauge-invariant field components, where the "scalar" and "spin" aspects of the systems can be clearly separated. Using the fundamental commutators of these transverse-traceless variables we verify the Poincare algebra among its generators, constructed from the T^0_μand their moments. The relevance to the interaction difficulties of higher spin systems is mentioned.