Understanding Spin in Trace Dynamics Using Division Algebras

Trace dynamics is a matrix-valued Lagrangian/Hamiltonian dynamics whose equilibrium statistical mechanics yields quantum theory via the Adler--Millard conserved charge. A persistent conceptual gap is a canonical (Noether--Hamiltonian) definition of spin: since trace dynamics is fundamentally pre-spacetime (in the sense that classical spacetime geometry is emergent), the conventional interpretation of spin as an ``internal'' angular momentum requires a precise identification of the relevant configuration ``angle'' variable and the space in which it lives. Building on earlier phase-amplitude constructions of complexified trace-dynamical variables, we propose a mathematically sharpened definition: \emph{spin is the canonical momentum conjugate to an orientation variable valued in an appropriate spin group} (e.g.\ $\Spin(3,1)$ on an emergent Lorentzian leaf, or $\Spin(3,3)$ at the 6D pre-localisation stage). This reformulation upgrades the abelian phase $\theta$ to a nonabelian group element $U(\tau)$, with angular velocity $\Omega=U^{-1}\dot U$ and intrinsic spin tensor $S=\delta\,\Tr L/\delta\Omega\in\mathfrak{spin}(p,q)$. We show (i) how the earlier $\theta$-momentum definition arises as a restriction to a one-parameter subgroup; (ii) how quantisation of spin follows from the emergent canonical (anti)commutators induced by equipartition of the Adler--Millard charge together with the topology of the true orientation manifold ($\SU(2)$ double cover); (iii) how the Pauli--Lubanski invariant is recovered and how a 6D $(3,3)$-signature generalisation naturally appears as a 3-form; and (iv) how division-algebra geometry (octonions, split bioctonions) provides a concrete scaffolding for the relevant spin groups, including the $\SO(3,3)\to \SO(3,1)\times \SO(2)$ leaf selection and the $\SU(3)_{\mathrm{geom}}$-induced internal $\mathrm{Spin}^c$ structure on $\mathbb{CP}^2$-type fibres. We also clarify the relation between Poincar\'e-mass as the translation Casimir and ``square-root mass'' charges that can arise from geometric $U(1)$ factors.