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

Bernard S. Kay|arXiv (Cornell University)|Mar 16, 2020

Quantum Mechanics and Applications24 references9 citations

TL;DR

This paper establishes a rigorous quantum foundation for electrostatic coherent states describing static charge distributions, deriving their inner product formula—correcting a prior 4π error—and proposing a novel product picture for quantum electrodynamics (QED) in which the Hamiltonian acts on a tensor product Hilbert space, with Gauss’s law holding as an operator equation in the physical subspace. The framework resolves longstanding issues in temporal gauge quantization and shows that entanglement between charged matter and longitudinal photons underlies non-zero inner products even for distinct charge distributions with equal total charge.

ABSTRACT

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.

Motivation & Objective

• To provide a rigorous theoretical foundation for electrostatic coherent states in quantum electrodynamics (QED), correcting a prior 4π error in inner product formulae.
• To resolve the paradox that coherent states for distinct charge distributions with equal total charge can have non-zero inner products, contrary to naive orthogonality expectations.
• To construct a product picture for QED—where the full Hilbert space is a tensor product of matter and electromagnetic field sectors—such that the Hamiltonian is self-adjoint on the physical subspace and Gauss’s law holds as an operator equation.
• To distinguish two frameworks: one where Gauss’s law holds in expectation only (non-self-adjoint electric field), and another where it holds as an operator equation (non-self-adjoint Hamiltonian), and propose an experiment to rule out the former.
• To demonstrate equivalence of the new product picture with Coulomb gauge QED for both Dirac and non-relativistic charged balls, showing entanglement between matter and longitudinal photons.

Proposed method

• Construct electrostatic coherent states as quantum states in a Fock space of non-dynamical longitudinal photons, using a modified free-field quantization that includes a longitudinal mode structure.
• Derive the inner product between two such coherent states using two distinct frameworks: one with Gauss’s law in expectation value (non-self-adjoint electric field), and one with Gauss’s law as an operator equation (non-self-adjoint Hamiltonian).
• Use canonical quantization in the temporal gauge to define a total Hamiltonian as a sum of free electromagnetic, free matter, and interaction terms, acting on a physical subspace of the tensor product Hilbert space.
• Show that the electric field operator −˜π is self-adjoint on the physical subspace but not on the full tensor product space, while Gauss’s law holds as an operator equation in the physical subspace.
• Demonstrate equivalence between the product picture and Coulomb gauge QED for both a Dirac field and a system of non-relativistic charged balls, via explicit mapping of states and operators.
• Propose an experimental test based on measuring the expectation value of the electric field operator in superpositions of coherent states to rule out the framework where Gauss’s law holds only in expectation.

Experimental results

Research questions

• RQ1 Why do electrostatic coherent states with equal total charge but different charge distributions have non-zero inner products, contrary to naive orthogonality expectations?
• RQ2 Can a product picture for QED be constructed such that the full Hamiltonian is self-adjoint on the physical subspace, and Gauss’s law holds as an operator equation?
• RQ3 What is the physical significance of the two frameworks—one with Gauss’s law in expectation and one with it as an operator equation—leading to the same inner product formulae?
• RQ4 How does the entanglement between charged matter and longitudinal photons manifest in the product picture, and how does it differ from the Coulomb gauge formulation?
• RQ5 Can the framework where Gauss’s law holds only in expectation be experimentally ruled out, and what observable differences would distinguish it from the operator-form Gauss’s law framework?

Key findings

• The inner product between electrostatic coherent states with equal total charge is generally non-zero, even for distinct charge distributions, and the formula corrects a prior 4π error in earlier work.
• Two distinct theoretical frameworks yield identical inner product formulae: one where Gauss’s law holds only in expectation (with non-self-adjoint electric field), and one where it holds as an operator equation (with non-self-adjoint Hamiltonian).
• The product picture for QED is constructed such that the physical subspace is a subspace of the tensor product of matter and electromagnetic Hilbert spaces, with a total Hamiltonian that is self-adjoint on the physical subspace.
• In the product picture, the longitudinal electric field is not an epiphenomenon but a quantum degree of freedom described by coherent states, which are eigenstates of annihilation operators.
• For both the Dirac field and non-relativistic charged balls, the product picture is unitarily equivalent to Coulomb gauge QED, and the entanglement between matter and longitudinal photons is explicitly shown.
• The framework where Gauss’s law holds only in expectation can be experimentally ruled out by measuring the expectation value of the electric field operator in superpositions of coherent states, as it would predict different outcomes than the operator-form framework.