Spin–1/2 Particle Fields in Maxwell–Schwartz Theory: Dirac–Schwartz Dynamics, Probability Currents and Entanglement

We develop a Maxwell–Schwartz field-theoretic formulation of spin-1/2 quantum systems within the framework of tempered distribution spaces and Schwartz linear algebra. In this approach, the fundamental objects of the theory are Maxwellian fields, namely elements of S′(M₄,C³), while scalar quantum states arise as derived projections carrying the spectral structure of the theory. Relativistic dynamics is formulated through square-root Hamiltonian operators defined via spectral calculus. On a distinguished subspace of Maxwellian fields, the curl operator realizes the momentum magnitude operator, allowing a direct correspondence between scalar and vectorial relativistic dynamics. This leads to a Schrödinger-type formulation of relativistic evolution without requiring Dirac’s matrix factorization as a primitive assumption. Spin-1/2 structures emerge naturally from two-component Cartesian configurations of Maxwellian fields. The Pauli algebra is realized as an internal symmetry acting on such pairs, and spin observables correspond to explicit transformations of field components. A first-order Dirac-type dynamics is obtained through operator factorization, yielding a Maxwellian realization of relativistic two-component equations, including the massless Weyl regime. The tensorial structure of the theory provides a natural setting for entanglement. Bell-type correlations are derived intrinsically from Maxwellian field configurations, reproducing the standard quantum predictions and their violation of Bell inequalities. A central result of the present work is the energetic interpretation of quantum probability. The spectral coefficients of a state determine the amplitudes of its Maxwellian realization, whose energy density is quadratic in these amplitudes. The Born rule is thus recovered as a normalized energy distribution law. Moreover, the electromagnetic continuity equation induces a probability continuity equation, with the probability current identified as the normalized Poynting flux. This transport interpretation extends to entangled states, where Bell correlations are interpreted as correlations in the transport of energy across field channels. Finally, the framework is extended to massive Maxwellian fields, where probability currents are expressed through nonlocal spectral kernels associated with square-root Hamiltonians, yielding a unified continuity structure for both massless and massive regimes. These results suggest that spinorial quantum structures, probabilistic interpretation, and quantum correlations may be understood as emerging from the internal organization, spectral structure, and energy transport of Maxwellian fields in the Schwartz distributional setting.