A product picture for quantum electrodynamics

TL;DR

This paper introduces a 'product picture' formulation of quantum electrodynamics (QED) that simplifies the Hilbert space structure and may overcome previous quantization flaws, with implications for quantum gravity and related theories.

Contribution

It presents a new product picture of QED, providing an alternative to Coulomb gauge formulation, and discusses its potential to extend to Yang-Mills fields and quantum gravity.

Findings

Product picture of QED is equivalent to standard Coulomb gauge QED.

Identifies flaws in previous temporal gauge quantization attempts.

Suggests potential for a product picture in quantum gravity.

Abstract

We present a short account of our work to provide quantum electrodynamics with a 'product picture'. It aims to complement the longer exposition in a recent paper in 'Foundations of Physics' and to help to make that work more accessible. The product picture is a formulation of QED, equivalent to standard Coulomb gauge QED, but in which the Hilbert space arises as (a certain physical subspace of) a product of a Hilbert space for the electromagnetic field and a Hilbert space for charged matter (i.e. the Dirac field) and in which the Hamiltonian arises as the sum of an electromagnetic Hamiltonian and a charged matter Hamiltonian and an interaction term. (The Coulomb gauge formulation of QED is not a product picture because, in it, the longitudinal part of the electromagnetic field is made out of charged matter operators.) We also recall a 'Contradictory Commutator Theorem' for QED which exposes flaws in previous attempts at temporal gauge quantization of QED and we explain how our product picture appears to offer a way to overcome those flaws. Additionally, we discuss the extent to which that theorem generalizes to Yang-Mills fields. We also develop a product picture for nonrelativistic charged particles in interaction with the electromagnetic field and point out how this leads to a novel way of thinking about the theory of many nonrelativistic electrically charged particles with Coulomb interactions. We explain how the provision of a product picture for QED gives hope that one will be able likewise to have a product picture for (Yang Mills and) quantum gravity -- the latter being needed to make sense of the author's 'matter-gravity entanglement hypothesis'. In an afterword, we briefly discuss similarities and differences between that hypothesis and its predictions and ideas of Roger Penrose related to a possible role of gravity in quantum state reduction and to cosmological entropy.