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We provide a theoretical foundation for the notion of the quantum coherentstate of the electrostatic field of a static external charge distributionintroduced in a 1998 paper and rederive formulae there for the inner productsof a pair of such states. Contrary to what one might expect, these innerproducts are non-zero whenever the total charges of the two chargedistributions are equal, even if the charge distributions themselves differ. Weactually display two different frameworks for these same coherent states, inthe second of which Gauss's law only holds in expectation value. We propose anexperiment capable of ruling that out. The first framework leads to a 'productpicture' for full QED -- i.e. a reformulation of standard QED in which it has atotal Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, afree charged-matter Hamiltonian and an interaction term, acting on a 'physicalsubspace' of the full tensor product of charged-matter andelectromagnetic-field Hilbert spaces. (The traditional Coulomb gaugeformulation of QED isn't a product picture because, in it, the longitudinalpart of the electric field is a function of the charged matter operators.) Wedo this for both Maxwell-Dirac and Maxwell-Schr\"odinger QED. For all states inthe physical subspace of each of these systems, the charged matter is entangledwith longitudinal photons and Gauss's law holds on the physical subspace as anoperator equation; albeit the electric field operator and the Hamiltonian,while self-adjoint on the physical subspace, fail to be self-adjoint on thefull tensor-product Hilbert space. Analogues of our coherent state innerproducts and of the product picture play a role in the author's matter-gravityentanglement hypothesis. Also, the product picture amounts to a temporal gaugequantization of QED which appears to be free from the difficulties of previousversions.

Quantum electrostatics, Gauss's law, and a product picture for quantum electrodynamics; or, the temporal gauge revised