Quantum Electricity: Absolute Operator-Theoretic Closure of Gauge-Covariant Current, Pauli Spin Transport, Maxwell Source Completion, and Spectral-Scattering Structure

This manuscript establishes the absolute operator-theoretic closure of quantum electricity. It formally defines quantum electricity strictly as the gauge-covariant conserved charge transport generated by a self-adjoint matter Hamiltonian under electromagnetic minimal coupling, rather than treating it as a supplementary physical ontology. The paper rigorously derives the exact one-particle current, its operator-valued local density, its Heisenberg continuity law, and its variational identity as the functional derivative of the Hamiltonian with respect to the vector potential. The framework is then systematically extended to spin-1/2 Pauli matter—proving the exact decomposition of the conserved current into distinct transport and divergence-free magnetization sectors—and to second quantization, establishing the many-body continuity law. Furthermore, the manuscript demonstrates that classical electrical source fields emerge exactly as the macroscopic expectation values of the quantum current operators. This provides the rigorous source-side completion for Maxwell field evolution. The theory is also generalized to relativistic quantum electrodynamics (identifying the current as the Noether current of local U(1) gauge invariance), expressed through spectral representation, and applied to open-system transport via scattering theory and wave operators. Ultimately, the manuscript proves a unified closure: all admissible electrical behavior is completely exhausted by the combination of gauge-covariant matter current, Maxwell field evolution, constitutive response, and boundary data.