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Preliminary Statement: The connection between the classical electric fields and
the photon wave function is obtained by a careful canonical quantization.
The canonical formulation of the electromagnetic fields in
multiply connected cavities (with non--trivial Betti numbers) is possible
for the transversal and for the cohomological
(=transversal+longitudinal) components of the fields, arising
from the Helmholtz-Hodge decomposition. Let us assume ideal
conductor boundary conditions at the cavity walls.
The cohomological magnetic field is then the curl of a constant cohomological
vector potential. Physically it arises from persistent surface
currents. If the magnetic field is localized, its cohomological component
and the cohomological vector potential may be non-localized.
The cohomological electric field is the time derivative of a
different time--dependent cohomological vector potential.
The transversal electric field and transversal vector potential
and the cohomological electric field and dynamical cohomological
vector potential form both canonically conjugate field variables
and can be canonically quantized.
After Weyl quantization and selection of a regular representation
the canonical fields are combined into a hermitian smeared field
operator. The longitudinal electric field stays classical.
A particle structure is an additional structure and arises only for
the transversal fields in a unique Fock representation, using
a distinguished complexification of the test function space.
The latter specifies the decomposition of the hermitian field
operator into creation and annihilation operators, and thus
also the Fock vacuum. The complexification is determined by
the diagonalization of the free Maxwell dynamics.
The 1-photon wave functions consist of the transversal A-iE.
The 1-photon
Hamiltonian is hbar times i times the generator of the free Maxwell
dynamics in its unique diagonalized form.
It is the square root of curlcurl with appropriate boundary conditions.
The classical fields are re-obtained as the coherent part of
the quantized field, emitted from mesoscopic
radiation models in non-Fock representations.