SMuK 2021 – scientific programme

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 18: Constructive Tools for Quantum Field Theory

MP 18.1: Talk

Friday, September 3, 2021, 11:00–11:25, H6

A product picture for quantum electrodynamics — •Bernard Kay — Department of Mathematics, University of York, York, England

We exhibit a product picture for QED – i.e. a reformulation in which it has a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free Dirac Hamiltonian and an interaction term, acting on a Hilbert space which is a subspace (the “physical subspace”) of the full tensor product of an electron/positron Hilbert space and an electromagnetic-field Hilbert space. The traditional Coulomb-gauge formulation of QED isn’t a product picture in this sense because, in it, the longitudinal part of the electric field is a function of the Dirac ψ-field. We prove (at a formal level) our product picture is equivalent to Coulomb gauge QED. Also, in the product picture: (i) In all states in the physical subspace, the ψ field is entangled with longitudinal photons; (ii) Gauss’s law holds as an operator equation; (iii) The electric field operator and the full Hamiltonian aren’t self-adjoint on the full tensor-product Hilbert space, but they are self-adjoint on the physical subspace and so that’s OK. Also (iv) The product picture resembles temporal gauge quantization but seems free from the well-known difficulties of that; (v) In the nonrelativistic limit, one obtains a reformulation of Schroedinger many body theory in which the usual Coulomb potential is absent and, instead, a term representing the kinetic energy of longitudinal photons is present so that, say, the binding energy of the Hydrogen atom is seen to arise as a byproduct of the entanglement of the proton and electron with longitudinal photons.