Emergent Electromagnetism from an $S^{3}$ Sigma Model: Hopf Geometry, Gauge Structure, and Casimir Physics

This paper develops a geometric field-theoretic framework in which classical electromagnetism emerges dynamically from a constrained (S^{3}) nonlinear sigma model, without postulating a fundamental gauge field. The central construction exploits the Hopf fibration (S^{3}!\to!S^{2}), from which a natural (U(1)) gauge structure arises: local rotations along the Hopf fiber act as gauge transformations, the electromagnetic potential is obtained as the pullback of the Hopf connection, and the field strength appears as its curvature. Linearization about a constant background reveals a canonical decomposition of internal fluctuations into vertical (pure gauge) and horizontal (physical) components. This geometric splitting yields exactly two propagating degrees of freedom, providing an intrinsic derivation of photon polarizations without gauge fixing, constraints, or auxiliary assumptions. The framework is then applied to Casimir physics in a parallel-plate geometry. Perfect-conductor boundary conditions are implemented directly at the level of admissible sigma-model variations, which eliminates boundary gauge freedom and restricts fluctuations to the physical horizontal sector. The resulting mode spectrum reproduces the standard transverse-electric (TE) and transverse-magnetic (TM) families with the correct multiplicity and dispersion, leading to the conventional Casimir energy and force upon renormalized vacuum subtraction. In this way, the work establishes a closed conceptual chain—from internal geometry and topology, through emergent gauge symmetry and polarization structure, to observable vacuum forces—demonstrating that key features of electromagnetism can arise entirely from constrained internal geometry.