Quantizing Dirac's 1951 constrained-potential theory of electrons: spin, charge quantization, and self-energy

Dirac's 1951 'A new classical theory of electrons' destroys the gauge invariance of the electromagnetic four-potential by imposing the algebraic constraint $A_\mu A^\mu=k^2$, so that the former gauge freedom itself carries electric charge ($j^\mu=-\lambda A^\mu$), and it yields a canonical Hamiltonian whose distinctive square-root factor $(A_sA^s+k^2)^{1/2}$ Dirac explicitly flagged for replacement by spin variables in the passage to quantum theory. The problem is that unfinished passage: quantize the 1951 constrained theory so that the square-root Hamiltonian becomes an operator structure incorporating electron spin, determine whether charge quantization and the value of $e$ (for example, in terms of Planck's constant $h$) emerge — the classical theory fixes only the ratio $e/m$ — and decide whether the infinities of electron self-interaction are removed or ameliorated.

Background

By 1951 Paul Dirac regarded quantum electrodynamics (the quantum theory of electrons and the electromagnetic field, QED) as unsatisfactory even after renormalization — the successful bookkeeping rules that absorb divergent quantities into finite physical parameters. In the opening of 'A new classical theory of electrons' he wrote that although renormalization 'has been very successful in setting up rules for handling the infinities and subtracting them away', the 'resulting theory is an ugly and incomplete one' which 'cannot be considered as a satisfactory solution of the problem of the electron' — the problem being the infinite electromagnetic self-energy of the point charge. He therefore rebuilt the classical theory of electrons, intending it as a better starting point for quantization.

In standard classical electrodynamics the field is described either by the field tensor $F_{\mu\nu}$ or by the four-potential $A_\mu$ (from which the fields follow by differentiation); the potential is defined only up to a gauge transformation $A_\mu\to A_\mu+\partial_\mu\chi$, a redundancy with no physical content, while charges are added separately as point particles. Dirac's 1951 move was to destroy the gauge invariance by imposing the algebraic condition

$A_{\mu}A^{\mu}=k^{2}$

on the potential (with $k$ a universal constant), enforced in the action by a Lagrange-multiplier field $\lambda(x)$ (a field whose variation imposes the constraint). The field equation becomes $\partial_{\nu}F^{\mu\nu}=\lambda A^{\mu}$, so the combination $j^{\mu}=-\lambda A^{\mu}$ is a conserved charge-current density: electric charge now lives entirely in the former gauge freedom of the potential, with no independent particle variables. For an infinitesimal test charge moving in a prescribed external potential $A^*_{\mu}$ the theory reproduces Lorentz-force motion through a Hamilton–Jacobi equation (the equation whose solution, a scalar function $S$, generates the motion), $(\partial_{\mu}S+A^*_{\mu})(\partial^{\mu}S+A^{*\mu})=k^2$, which identifies $k=m/e$ with the electron's mass-to-charge ratio; consequently the classical theory involves $e$ and $m$ only through the ratio $e/m$. Electrons are described as a continuous stream whose velocity field is fixed by one Hamilton–Jacobi solution, not as individually tracked point particles.

Dirac then gave the theory a canonical Hamiltonian formulation, designed as the springboard for quantization. The spatial components $A^r$ ($r=1,2,3$) serve as coordinates, their conjugate momentum is the electric field $B_r=F_{r0}$, and the time component $A_0$ is eliminated through the constraint — which is where a square root enters, $A_0=\pm\left(A_{s}A^{s}+k^{2}\right)^{1/2}$. The resulting Hamiltonian functional (equation (18) of the paper) is, up to sign conventions,

$H=\int\left\{\tfrac{1}{4}F_{rs}F^{rs}-\tfrac{1}{2}B_{r}B^{r}-\left(A_{s}A^{s}+k^{2}\right)^{1/2}\,\partial_{r}B^{r}\right\}\,dx_{1}dx_{2}dx_{3},$

whose distinctive feature is the square-root factor multiplying the momentum divergence. Dirac presents the theory as 'more suitable than the usual one as a basis for a passage to the quantum theory' and states in his own words: 'To make this passage one will presumably have to replace the square root in the Hamiltonian with something involving spin variables.' The historical template for replacing a square root by a linear operator is Dirac's own 1928 route from the Klein–Gordon equation (the relativistic wave equation for spin-0 particles) to the Dirac equation (the wave equation of the electron carrying spin $1/2$); but that was a single-particle square root, whereas here the object to be made operatorial is a functional of the potential and the electric field. Schrödinger's 1952 comment deepened the difficulty: the modified Maxwell equations of Dirac's theory coincide with those of Klein–Gordon scalar electrodynamics (the field theory of a charged spin-0 field coupled to the electromagnetic field) in the unitary gauge (the gauge in which the charged scalar field is real), suggesting that the natural quanta of the field equations are spin-0 — so the emergence of spin-$1/2$ (and with it the Fermi–Dirac statistics, antisymmetric many-electron states, that electrons in fact obey), of charge quantization, and of finite self-energy in the quantum theory are all open questions. Dirac's own sequels of 1952 (vorticity in the electron stream, a more general action) and 1954 (interpenetrating streams) remained classical and did not touch them. The one line of work that did quantize a theory carrying Dirac's condition — Nambu's 1968 quantum electrodynamics in the nonlinear gauge and its descendants — supplied the charge-current externally rather than letting it emerge from the constraint, and therefore does not answer the 1951 questions either.

Problem Statement

The problem is Dirac's unfinished passage from his 1951 constrained-potential classical theory of electrons to quantum theory. The classical theory to be quantized is fixed by the source: the free Maxwell Lagrangian $\mathscr{L}=-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}$ with the constraint $A_{\mu}A^{\mu}=k^{2}$ (with $k=m/e$) enforced by a Lagrange-multiplier field $\lambda$, giving the field equation $\partial_{\nu}F^{\mu\nu}=\lambda A^{\mu}$ with charge-current $j^{\mu}=-\lambda A^{\mu}$ carried entirely by the formerly redundant gauge degrees of freedom (no independent charged matter), and the canonical Hamiltonian (equation (18) of the paper), which in the paper's notation is, up to sign conventions,

$H=\int\left\{\tfrac{1}{4}F_{rs}F^{rs}-\tfrac{1}{2}B_{r}B^{r}-\left(A_{s}A^{s}+k^{2}\right)^{1/2}\,\partial_{r}B^{r}\right\}\,dx_{1}dx_{2}dx_{3},$

where $A^r$ ($r=1,2,3$) are the spatial components of the potential, $B_r$ is the electric field conjugate to them, and the square-root factor $\left(A_{s}A^{s}+k^{2}\right)^{1/2}$ arises from eliminating the time component $A_0$.

Construct the quantum theory descending from this classical theory and settle the questions Dirac explicitly left open:

(i) how to quantize the constrained Hamiltonian — replacing the square-root structure by an operatorial form, and determining what spin content and statistics the resulting quantum theory in fact carries — 'To make this passage one will presumably have to replace the square root in the Hamiltonian with something involving spin variables';

(ii) whether the quantized theory yields electric charge quantization and determines the value of $e$ — for example, by expressing $e$ in terms of Planck's constant $h$ — given that the classical theory fixes only the ratio $e/m$;

(iii) whether the new formulation, upon quantization, ameliorates or removes the infinities associated with electron self-interaction that motivated it.

The scientific target is a fully specified quantum theory whose classical limit is the 1951 theory, together with its consequences for spin content, charge spectrum, and electron self-energy. A rigorous negative answer settles the corresponding part of the problem equally: a proof that no such quantization with the stated properties exists, or a demonstration — by explicit construction or by a no-go proof — that the quantized theory's spin content and statistics are not the electron's (i), that the theory does not fix $e$ or cannot express it in terms of $h$ (ii), or that finiteness fails (iii). The question concerns this specific 1951 theory, in which charge is emergent from the destroyed gauge freedom; quantizing other reformulations of electrodynamics does not answer it — in particular, quantizations that impose $A_\mu A^\mu=k^2$ merely as a gauge condition while keeping the charge-current external or charged matter independent (Nambu's nonlinear-gauge QED and its descendants) address a neighboring theory, not this one.

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