In 1955 P. A. M. Dirac reformulated quantum electrodynamics (QED) entirely in terms of manifestly gauge-invariant dynamical variables, constructing a dressed electron operator ψ∗(x)=ψ(x)eiC(x)\psi^*(x)=\psi(x)e^{iC(x)} whose kernel condition forces the simultaneous creation of the electron and its Coulomb field, and he identified four connected open issues: (i) the vacuum problem — no precise definition of the reference ("standard ket") state satisfying both the equations of motion and the quantum subsidiary conditions; (ii) infinite c-number constants produced by operator reordering that spoil the subsidiary conditions and the conserved generators; (iii) an incomplete characterization of the space of physical one-electron states and their accompanying photon distributions; and (iv) the absence of a consistent, finite-energy dynamics for singular line-like (Faraday-line) dressings, together with the question whether such a discrete-line representation of the electromagnetic field can account for the quantization of electric charge. The precise surviving core — a complete, mathematically well-defined, manifestly gauge-invariant theory of QED's physical charged states, including the status of singular line dressings — remains open.

Background

Quantum electrodynamics (QED) is the quantum field theory of the electron (a spin-12\tfrac{1}{2} Dirac field ψ\psi) coupled to the electromagnetic field. The electromagnetic field can be described either by the field strengths or by potentials AμA^{\mu}, which are redundant variables: a gauge transformation Aμ→Aμ+∂μSA^{\mu}\to A^{\mu}+\partial^{\mu}S with an arbitrary function S(x)S(x) leaves the physics unchanged while rotating the phase of the electron field. Standard canonical quantization removes this redundancy by fixing a gauge (e.g. the Coulomb gauge, in which the divergence of the vector potential vanishes and the electrostatic interaction appears as an instantaneous Coulomb potential); the price is that the basic operators are no longer gauge invariant and Gauss's law, ∇⋅E=e ψ†ψ\nabla\cdot\mathbf{E}=e\,\psi^{\dagger}\psi, survives only as a constraint (a "subsidiary condition") that physical states must satisfy.

In 1955 Dirac proposed instead to build the canonical theory entirely from gauge-invariant variables. His central construction is a gauge-invariant electron-creation operator

ψ∗(x)=ψ(x) eiC(x),C(x)=∫cr(x,x′) Ar(x′) d3x′,\psi^{*}(x)=\psi(x)\,e^{iC(x)},\qquad C(x)=\int c_{r}(x,x')\,A^{r}(x')\,d^{3}x',

where the kernel must satisfy the distributional divergence condition ∂r′cr(x,x′)=eℏ−1δ3(x−x′)\partial_{r'}c_{r}(x,x')=e\hbar^{-1}\delta^{3}(x-x'); the spherically symmetric solution is the Coulomb kernel cr=e4πℏxr′−xr∣x′−x∣3c_{r}=\frac{e}{4\pi\hbar}\frac{x'_{r}-x_{r}}{|x'-x|^{3}}. Acting on a suitable reference state (Dirac's "standard ket", the analogue of the vacuum in his formalism), the operator ψ∗(x)\psi^{*}(x) creates an eigenstate of the longitudinal electric field with the classical Coulomb field of one electron as eigenvalue: gauge-invariant electron creation automatically creates the electron together with its Coulomb field. Dirac also noted singular solutions of the same kernel condition concentrated on a line, C(x)=−eℏ−1∫−∞xAr dxr′C(x)=-e\hbar^{-1}\int_{-\infty}^{x}A^{r}\,dx'_{r} along a path, which concentrate the electric flux into a single Faraday line of force — a line running from the charge out to infinity along which the electric field is carried.

Dirac's paper explicitly left four connected issues open. (i) The vacuum problem: no precise definition of the standard ket was given that satisfies both the equations of motion and the quantum subsidiary conditions, so the representation of physical states as gauge-invariant operators acting on it remained formal. (ii) Infinite constants: reordering operator products (normal ordering — moving creation operators to the left of annihilation operators, required because operator products at the same point are divergent) produces infinite cc-number terms in the subsidiary condition and in the Hamiltonian, which spoil the vanishing of classically constrained terms (for example in the total momentum) and obstruct conserved generators in the quantum theory. (iii) The dressing space: ψ∗(x)\psi^{*}(x) with a fixed kernel describes only a restricted class of photon configurations accompanying the electron; a general physical one-electron state must be built by multiplying by further functionals of the electromagnetic field operators, and the structure of this space was not characterized. (iv) The line solutions: the standard electric-field energy 12∫E2 d3x\tfrac{1}{2}\int\mathbf{E}^{2}\,d^{3}x diverges for flux concentrated on a line, so a modified Hamiltonian would be needed to make discrete Faraday lines dynamically admissible; Dirac speculated that such a discrete-line picture might explain why electric charge is quantized (observed only in integer multiples of the elementary charge ee).

Later work gave this bundle a precise vocabulary. An infraparticle is a charged particle that, by Gauss's law, cannot be an eigenstate of the mass operator — its mass spectrum is continuous because it cannot be separated from photons of arbitrarily low energy (soft photons). A photon cloud is the asymptotic configuration of the electromagnetic field accompanying a charge, quantified by the electric flux per solid angle at spacelike infinity. A superselection sector is a class of states that cannot be coherently superposed with states of other classes by any physical operation; in QED the shape of the photon cloud labels superselection sectors, and there are uncountably many of them. A string-localized field is a quantum field that depends on a spacetime point and a spatial direction and is localized along a half-line (the "string") rather than at a point; modern work uses such fields to give Dirac's formal line-integral dressings a rigorous non-perturbative meaning. The adiabatic limit is the removal of the space-time cutoff with which the interaction is switched on, a delicate step in theories with massless particles. Standard scattering frameworks — LSZ reduction (which requires a sharp mass shell) and Haag-Ruelle theory (which requires fields satisfying a free equation of motion) — both fail for charged infraparticles, which is why the scattering theory of the dressed electron is itself part of the problem.

Problem Statement

Within Dirac's manifestly gauge-invariant canonical formulation of quantum electrodynamics, construct the quantum theory of the physical charged states and characterize it completely, where completeness is fixed by the four connected requirements below and all four must be resolved jointly. Concretely, the four connected requirements left open by the source are: (i) give a precise definition of the reference (vacuum/standard ket) state and of the physical state space on which the gauge-invariant operators act, so that every physical state has an unambiguous representation; (ii) control the infinite c-number constants generated by operator reordering in the subsidiary condition and the Hamiltonian, so that the quantum constraints and the conserved generators (for example the total momentum) behave as their classical counterparts do; (iii) characterize the full space of physical one-electron states — the complete class of photon distributions/dressings that may accompany the gauge-invariant dressed electron ψ∗(x)\psi^{*}(x), their superselection structure and spectral (infraparticle) properties — and define the Hamiltonian dynamics on this space, including its scattering theory; and (iv) determine whether singular, line-like dressings (discrete Faraday lines of force) can be made dynamically admissible by a consistent modification of the electromagnetic field energy that renders their energy finite, and whether such a discrete-line representation of the field can account for the quantization of electric charge. The goal is the completion of Dirac's program: resolving requirements (i)-(iv) jointly, within his manifestly gauge-invariant canonical formulation or in a precisely specified continuation of it that retains his gauge-invariant dressed charged field as its starting point and states explicitly how it departs from the rest of Dirac's construction. "Resolved jointly" is fixed per requirement, so that progress on a part is distinguishable from closing it: (i) is resolved by an explicit definition of the reference state together with the physical state space on which the gauge-invariant operators act; (ii) by a control of the reordering constants such that the subsidiary condition and the conserved generators are well defined and carry their classical counterparts' content; (iii) by an explicit characterization of the full one-electron dressing space and its superselection and infraparticle structure together with a Hamiltonian dynamics on it whose scattering theory is required, not illustrative; and (iv) by a definite verdict on each of its two parts, finite-energy admissibility of singular line-like dressings and the account of charge quantization, a negative verdict resolving a part only when supported by a proof of impossibility rather than by the absence of a construction. Delivering fewer than all four requirements, or a partial result on any one of them, does not resolve the task.

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