## Quantum electrostatics and a product picture for quantum electrodynamics: or, the temporal gauge revised

Research output: Working paper

Date | Published - 16 Mar 2020 |
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Publisher | arXiv |
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Number of pages | 24 |
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Original language | English |
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We introduce a suitable notion of quantum coherent state to describe the electrostatic field of a static classical charge distribution, thereby underpinning the author's 1998 formulae for the inner product of a pair of such states. (We also correct an incorrect factor of 4π.) Contrary to what one might expect, this is non-zero whenever the two total charges are equal, even if the charge distributions themselves are different. We then address the problem of furnishing QED with a "product structure", i.e. a formulation in which there is a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a total Hilbert space which is the tensor product of an electromagnetic Hilbert space and a charged-matter Hilbert space. (The traditional Coulomb-gauge formulation of QED doesn't have a product structure in this sense because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) Motivated by all this, and both for a charged Dirac field and for a system of non-relativistic charged balls, we transform Coulomb-gauge QED into an equivalent formulation which we call the "product picture". This involves a full Hilbert space which is the tensor product of a Hilbert space of transverse and longitudinal photons with a Hilbert space for charged matter and in this sits a physical subspace (in all states of which the charged matter is entangled with longitudinal photons) on which Gauss's law holds strongly, together with a total Hamiltonian which has a product structure, albeit the electric field operator while self-adjoint on the physical subspace, isn't self-adjoint on the full Hilbert space. The product-picture Hamiltonian resembles the temporal gauge Hamiltonian, but the product picture is free from the difficulties in pre-existing temporal-gauge quantizations.

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