Spin-Induced Coordinate Noncommutativity of the Covariant Centroid

Abstract

The Tulczyjew-Dixon centroid of a massive spinning particle, built as a composite of canonical Poincaré variables, carries the exact coordinate algebra $\{X^μ,X^ν\}=-S^{μν}/p^2$, whose noncommutativity bivector is the transverse spin tensor of the particle itself rather than an external background. That bracket and its Compton-scale magnitude are already known, and we recover them here as the starting point of the analysis, not as a result. What the composite realization settles is the domain on which the algebra may be used. Closure of the Jacobi identity requires $p^{-2}$ to remain operator valued, and the exactness domain is the convex timelike interior $Ω_μ$, on which states supported on a single mass shell are distributional. We show that freezing $p^2$ to its shell value inside the centroid leaves every two-operator commutator unchanged on shell, so that the projected position dictionary, the Berry cancellation and the endpoint transversality identity are untouched, while the same substitution fails from three insertions onward, with residue the truncated Jacobiator $2ε^{μνρσ}W_σ/(p^2)^2$. This two-versus-three criterion replaces the qualitative restriction to shell-preserving matrix elements and states which constructions built on a fixed-shell bivector $Θ_m$ remain valid. We then follow the algebra through positive-energy projection, packet-conditioned Moyal fibers, Compton routing holonomy and projected nonassociative products, recording for each layer whether it falls on the safe side of the criterion. No prediction beyond QED follows. In the geometric branch the physical amplitude reduces to the QED amplitude, and the transverse matching coefficients remain uncomputed.