  We provide a theoretical foundation for the notion of the quantum coherent
state of the electrostatic field of a static external charge distribution
introduced in a 1998 paper and rederive formulae there for the inner products
of a pair of such states. Contrary to what one might expect, these inner
products are non-zero whenever the total charges of the two charge
distributions are equal, even if the charge distributions themselves differ. We
actually display two different frameworks for these same coherent states, in
the second of which Gauss's law only holds in expectation value. We propose an
experiment capable of ruling that out. The first framework leads to a 'product
picture' for full QED -- i.e. a reformulation of standard QED in which it has a
total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a
free charged-matter Hamiltonian and an interaction term, acting on a 'physical
subspace' of the full tensor product of charged-matter and
electromagnetic-field Hilbert spaces. (The traditional Coulomb gauge
formulation of QED isn't a product picture because, in it, the longitudinal
part of the electric field is a function of the charged matter operators.) We
do this for both Maxwell-Dirac and Maxwell-Schr\"odinger QED. For all states in
the physical subspace of each of these systems, the charged matter is entangled
with longitudinal photons and Gauss's law holds on the physical subspace as an
operator equation; albeit the electric field operator and the Hamiltonian,
while self-adjoint on the physical subspace, fail to be self-adjoint on the
full tensor-product Hilbert space. Analogues of our coherent state inner
products and of the product picture play a role in the author's matter-gravity
entanglement hypothesis. Also, the product picture amounts to a temporal gauge
quantization of QED which appears to be free from the difficulties of previous
versions.
