On the Geometric Origin of Spin

Abstract

We develop a geometric account of spin for localized matter. Earlier work in this framework equips the configuration space of complex metrics with a positive Euclidean kinetic form and models a matter particle as a finite-energy Hopf configuration. We pull this kinetic form back to the soliton moduli space and prove that it supplies a strictly positive contribution to the collective rotational inertia. For maps \(S^3\to S^2\), the reduced configuration space contains a noncontractible \(\mathbb Z_2\) rotation loop. The local transgression $$frac{\mathcal A_*}{32\pi^2}\int d\theta\wedge A\wedge F$$ fixes the normalization of the Euclidean–Hopf phase, while the two characters of the rotation fundamental group define the two Finkelstein–Rubinstein quantizations. We select the nontrivial character for the fundamental Hopf matter sector and realize it as the flat torsion part of the Euclidean–Hopf line bundle. The corresponding period is \(\pi Q_H\mathcal A_*\) modulo \(2\pi\mathcal A_*\), with holonomy \((-1)^{Q_H}\). Odd Hopf sectors therefore support half-integer rotational representations, whereas even sectors support integer representations. Topology and the selected character determine the spin parity; the complete positive inertia and the soliton stabilizer determine the lowest admissible level. In the minimal matter sector \(Q_H=1\), if the stabilizer admits \(j=1/2\) and bounded corrections preserve the rotor ordering, this doublet is the rotational ground state. Matching its collective generators to the little-group generators on an isolated massive Poincaré subspace gives the relativistic spin. With the independent normalization \(\mathcal A_*=\hbar\), the ground multiplet satisfies \(J^2=3\hbar^2/4\) and \(J_z=\pm\hbar/2\). The construction also distinguishes topologically protected matter cores from topologically trivial response modes. Keywords Intrinsic angular momentum; Hopf solitons; Finkelstein–Rubinstein quantization; Euclidean inertia; geometric phase; collective coordinates; Pauli–Lubanski vector.