Non-perturbative quantization of Dirac's 1950 gauge-foliation-coupled electrodynamics: the $e^{-2}$ Hamiltonian, spinning electrons, and the self-energy question

In 1950 Dirac reformulated the classical electrodynamics of a charged spin-zero (Pauli-Weisskopf) field on arbitrary space-like surfaces, coupling the gauge function to the choice of surface by $g-f=eS$ and eliminating the matter phase so that charge is carried by longitudinal field variables. The resulting Hamiltonian contains an interaction term proportional to the inverse square of the electronic charge, $e^{-2}(B_s^s)^2R^{-2}$, so the usual perturbative expansion in powers of $e$ is structurally unavailable. The paper left open: a non-perturbative quantization of this Hamiltonian; the analogous canonical formulation for spinning electrons with Fermi statistics; the consequences of restricting admissible gauge transformations to those producing space-like surfaces; and whether the resulting quantum theory controls or removes the infinite self-energies of quantum electrodynamics. These four questions remain unresolved in the literature: Dirac's own continuation and the modern Fock-Stueckelberg constraint-relaxation or 'matter without matter' revival both work on a different, later 'new classical theory of electrons' (1951-1954) rather than on the 1950 construction, and the only quantization attempts in that later line are lowest-order perturbative.

Background

In the late 1940s the quantum theory of charged particles was computed by perturbation theory: the coupling between the electromagnetic four-potential $A_\mu$ and charged matter was treated as a power series in the electronic charge $e$. This framework gave successful predictions but also infinite self-energies of charged particles, which the newly developed renormalization procedure absorbed into redefined parameters. Dirac regarded this as physically unsatisfactory and sought a reformulation of electrodynamics whose quantization would be intrinsically better behaved.

Two structural facts motivate the 1950 paper. First, in the covariant Tomonaga-Schwinger formulation the quantum state is assigned to an arbitrary space-like surface (a three-dimensional surface in space-time whose every tangent direction is 'slower than light') and the Hamiltonian formalism describes how the state changes when that surface is deformed; the choice of the family of surfaces is a large freedom of the description. Second, electrodynamics has its own gauge freedom: the potential may be shifted $A_\mu \to A_\mu + \partial S/\partial x_\mu$ with an arbitrary scalar function $S$ without changing any field strength. These two large independent freedoms leave the dynamical description with a redundancy of variables.

The charged matter in Dirac's construction is the Pauli-Weisskopf field: the relativistic field of spin-zero charged particles obeying Bose statistics (the charged complex Klein-Gordon field), which Dirac treated as a classical field pending quantization. Writing the field as $\psi = R\,e^{-i\theta}$ with real amplitude $R$ and phase $\theta$, and using surface-adapted components of the potential (tangential components $A^r$ and the normal component $A_l$, with conjugate momenta $B_r$ and the constraint $B_l=0$), Dirac introduced the hypothesis that couples the two freedoms: if the surfaces are $\tau=f(x)$ and $\tau=g(x)$, a change of foliation is accompanied by a gauge transformation with $g-f=eS$, so a gauge transformation acquires the geometric meaning of shifting the foliation $f \to f+eS$. Admissible gauge functions are thereby restricted to those for which $f+eS$ still defines space-like surfaces. Choosing the foliation so that each surface carries constant phase ($\theta=\tau$), the phase pair $(\theta,\rho)$ decouples and can be eliminated, the charge density being carried by the longitudinal field momentum divergence $B_s^s$ through the Maxwell constraint $B_s^s-e\rho=0$. For flat surfaces $x^0=\tau$ the reduced Hamiltonian reads

$H = \int \left\{ \frac{1}{4}F_{rs}F^{rs} - \frac{1}{2}B_sB^s + \frac{1}{2}P^2 - \frac{1}{2}R_rR^r + \frac{1}{2}m^2R^2 + \frac{1}{2}e^{-2}(B_s^s)^2R^{-2} - \frac{1}{2}e^2R^2A_rA^r \right\}\,d^3u$

where $P$ is the momentum conjugate to $R$, $F_{rs}$ the surface field strength, $m$ the particle mass, and the remaining first-class constraints (combinations of canonical variables that vanish on the constraint surface and generate the residual gauge and surface-deformation freedoms) are appended as generators. The interaction now consists of the two algebraically unusual terms $\frac{1}{2}e^{-2}(B_s^s)^2R^{-2}$ and $-\frac{1}{2}e^2R^2A_rA^r$. Because the first carries the inverse square of the charge, the interaction does not vanish as $e \to 0$: the standard perturbative expansion in powers of $e$, on which the calculation methods of quantum electrodynamics rest, is unavailable for this Hamiltonian. The 1950 paper leaves the quantization of this Hamiltonian, the analogous construction for spinning electrons (a Dirac spinor cannot be written as one real amplitude times one phase, so the elimination step has no immediate analogue), the analysis of the gauge-foliation restriction, and the effect on the self-energy infinities all open.

Problem Statement

The problem is the completion and assessment of Dirac's 1950 reformulation of electrodynamics, as left open by that paper. Its single scientific objective is to decide whether the gauge-foliation-coupled classical theory of a charged spin-zero field can be turned into a quantum theory of charged particles, and whether that quantum theory improves on the ultraviolet behaviour of standard quantum electrodynamics. Concretely, the open questions are:

1. Quantization without an expansion in $e$. Construct — or prove impossible — the quantization and the solution of the Schrödinger equation for the reduced Hamiltonian

$H = \int \left\{ \frac{1}{4}F_{rs}F^{rs} - \frac{1}{2}B_sB^s + \frac{1}{2}P^2 - \frac{1}{2}R_rR^r + \frac{1}{2}m^2R^2 + \frac{1}{2}e^{-2}(B_s^s)^2R^{-2} - \frac{1}{2}e^2R^2A_rA^r \right\}\,d^3u$

(together with its remaining first-class constraints, in particular the Maxwell constraint $B_s^s-e\rho=0$; here $F_{rs}$ is the electromagnetic field strength on the surface, $A_r$ and $B_r$ are the surface-adapted tangential components of the potential and their conjugate momenta with $B_s^s$ their surface divergence, $R$ and $P$ are the amplitude of the charged field and its conjugate momentum, $m$ is the particle mass, and $e$ is the electronic charge), which because of the $e^{-2}$ interaction term cannot be treated by the usual perturbative expansion in the electronic charge; this includes defining the operator $R^{-2}$ and an ordering for $(B_s^s)^2R^{-2}$.

2. Spinning matter. Construct the corresponding relativistic canonical formulation on space-like surfaces for electrons — spinning particles with Fermi statistics — identifying what replaces the amplitude-phase elimination of the spin-zero case.

3. The gauge-foliation restriction. Determine the consequences and possible limitations of the hypothesis $g-f=eS$ — which couples a change of the foliation function $f \to g$ to a gauge transformation with gauge function $S$, where $e$ is the electronic charge — that restricts admissible gauge transformations to those producing space-like surfaces: its impact on global properties, on admissible boundary conditions at infinity, and on the full class of allowed gauge functions in the quantum theory.

4. Self-energy. Determine whether the resulting non-perturbative quantum theory — if Question 1 yields one — controls or removes the usual infinite self-energies of quantum electrodynamics; if Question 1 instead establishes that quantization is impossible, state what that impossibility implies for the self-energy problem.

References

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