Classical and Quantized Maxwell Fields Deduced from Algebraic Many-Photon Theory

The deduction starts with the (non-relativistic) one-photon Hilbert space $\mathcal{H}$, equipped with the one-photon Hamiltonian and basic symmetry generators, as the only input information. We recall in the functorially associated boson Fock space the multi-photon dynamics and symmetry transformations, as well as the field operator (as the scaled self-adjoint part of the creation operator) and the q(uasi)-classical states. There is no reference to a presupposed classical Maxwell theory. By abstraction, we go over to the algebraic formulation of the multi-photon theory in terms of a C\*-Weyl algebra. Its test function space $E\subset \mathcal{H}$ is constructed as a nuclear Fréchet space, in which – via infrared damping – the dynamics and symmetries are nuclear continuous and their generators bounded. Each w\*-closed, singular subspace of the continuous dual $E'$ determines non-Fock coherent states and their mixtures lead to a representation von Neumann algebra with non-trivial center. The symmetry generators restricted to the center can be transformed into the Maxwell form by means of a symplectic transformation and involve the well-known conservation quantities of electrodynamics. This identifies the central part of the represented photon field operator as composed of the two classical canonical electrodynamic field components. We have obtained, therefore, in free space a kind of fusion of the multi-photon theory and the Maxwell theory of transverse electrodynamic fields, where the latter arise as derived quantities. By means of a Bogoliubov transformation one also gets a fusion of the quantized with the classical Maxwell theory, deduced from the photon concept. A sketch of non-relativistic gauging is added in the appendix to gain longitudinal, cohomological, and scalar potentials.