P. A. M. Dirac's 'A new classical theory of electrons. III' (Proc. R. Soc. Lond. A 223, 438-445, 1954) completes a classical reformulation of electrodynamics in which electrons are several interpenetrating continuous charged streams obeying Maxwell's equations and the Lorentz force law, reduced to a closed Poisson algebra of surface variables Ps,α∗,β∗P^s,\alpha^*,\beta^* on an arbitrary spacelike hypersurface. The series was explicitly intended (Part I abstract) to be 'more suitable ... as a basis for a passage to the quantum theory', but Part III leaves that passage open: the reduced algebra is intrinsically noncanonical (the momentum-density variables PsP^s do not Poisson-commute), the stream Hamiltonian differs structurally from the point-charge Hamiltonian ('some discrepancy', in the published abstract), and the classical theory contains ee and mm only through e/me/m, with no spin and no Fermi statistics. The open problem is to construct, or prove impossible, a consistent quantization of the 1954 stream theory that yields spin-1/2 electrons obeying Fermi statistics, resolves the Hamiltonian discrepancy, and recovers point-like electron behavior and standard electron-photon phenomenology in an appropriate limit. No later work quantizing the 1954 stream variables is known, and the problem remains open.

Background

In the late 1940s quantum electrodynamics (QED, the quantum field theory of electrons and photons) had become quantitatively successful after renormalization, but P. A. M. Dirac regarded its ultraviolet divergences as a symptom that the underlying classical theory - point electrons coupled to the Maxwell field, with their infinite electromagnetic self-energy - was defective. Between 1951 and 1954 he published a three-part series, 'A new classical theory of electrons', whose stated purpose (in the published abstract of Part I) was to obtain 'a simplified classical theory of electrons, which appears to be more suitable than the usual one as a basis for a passage to the quantum theory.'

Part I (1951) begins from the observation that in charge-free electrodynamics the potentials AμA_\mu are not fixed by the field but are subject to gauge transformations, so the theory has more dynamical variables than are physically needed. Dirac destroys the gauge freedom by imposing the non-linear condition AμAμ=k2A_\mu A^\mu = k^2; the superfluous variables then acquire physical meaning as a unit timelike vector field describing a flow of electric charge (in his contemporary Nature note 'Is there an Æther?' he speculated that this velocity field describes a real medium). With k=m/ek = m/e the stream obeys Lorentz's equations of motion, so an electron is represented by the flow of a charged continuum rather than by a point particle. Part II (1952) generalized the single stream to allow vorticity (rotational flow, described by additional scalar field variables of the Clebsch type - auxiliary potentials in terms of which a velocity field with vorticity can be represented) and noted in its published abstract that the theory still involves ee and mm only in the ratio e/me/m.

Part III (1954) treats several interpenetrating streams - charge flows that can pass through one another with different local velocities. The world-lines of the aa-th stream are labeled by three scalar fields ηar(x)\eta_{ar}(x), constant along each stream line; from them a stream 4-vector qaμ=ϵμνρσ ∂νηa1 ∂ρηa2 ∂σηa3q_{a\mu}=\epsilon_{\mu\nu\rho\sigma}\,\partial^\nu\eta_{a1}\,\partial^\rho\eta_{a2}\,\partial^\sigma\eta_{a3} is built, where ϵμνρσ\epsilon_{\mu\nu\rho\sigma} is the totally antisymmetric Levi-Civita symbol. A manifestly covariant action principle for the electromagnetic potential AμA_\mu and the stream variables yields both Maxwell's equations, with all streams as sources, and the Lorentz force law for each stream. Dirac then passes to a Hamiltonian formulation on an arbitrary space-like three-dimensional surface (a hypersurface in space-time such that no light signal connects two of its points, with coordinates usu^s), following his parametrized form of field dynamics in which the state is given on such a surface and the Hamiltonian appears as a constraint generating deformation of the surface.

After eliminating the redundant stream-label parametrization, the dynamics is carried by a closed set of surface densities: the momentum density Pas=ξar ηarsP^s_a = \xi^r_a\,\eta_{ar}{}^s (where ξar\xi^r_a is the momentum conjugate to the label field ηar\eta_{ar}), the surface electric density αa∗\alpha^*_a, and the surface rest-mass density βa∗\beta^*_a. Their Poisson brackets (the antisymmetric bracket [F,G][F,G] that governs classical dynamics; square brackets are used here as in the paper for brackets of densities) close on these variables alone:

[Pr(u), Ps(u′)]=∂rδ(u−u′) Ps(u)+∂sδ(u−u′) Pr(u′),[P^r(u),\,P^s(u')] = \partial^r\delta(u-u')\,P^s(u) + \partial^s\delta(u-u')\,P^r(u'),

[α∗(u), Ps(u′)]=∂s(δ(u−u′) α∗(u)),[β∗(u), Ps(u′)]=∂s(δ(u−u′) β∗(u)),[\alpha^*(u),\,P^s(u')] = \partial^s\big(\delta(u-u')\,\alpha^*(u)\big),\qquad [\beta^*(u),\,P^s(u')] = \partial^s\big(\delta(u-u')\,\beta^*(u)\big),

[α∗(u), α∗(u′)]=[β∗(u), β∗(u′)]=[α∗(u), β∗(u′)]=0,[\alpha^*(u),\,\alpha^*(u')] = [\beta^*(u),\,\beta^*(u')] = [\alpha^*(u),\,\beta^*(u')] = 0,

with δ\delta the Dirac delta function on the surface and ∂s\partial^s the surface derivative. The Hamiltonian constraint that generates normal deformation of the surface contains the characteristic square-root term ∑aβa∗2−(Pas+αa∗As)(Pas+αa∗As)\sum_a\sqrt{\beta_a^{*2}-(P_a^s+\alpha_a^*A^s)(P_{as}+\alpha_a^*A_s)} (with AsA^s the tangential component of the potential), and the momentum-velocity relation Pas=βa∗vas−αa∗AsP^s_a=\beta^*_a v^s_a-\alpha^*_a A^s is the continuum analogue of the point-particle relation ps=mvs−eAsp^s = mv^s - eA^s.

The brackets above are not of canonical form: the momentum densities PsP^s do not Poisson-commute among themselves, and their algebra is that of generators of relabeling transformations of a continuum (the same structural type as the momentum constraints of parametrized field theories, whose brackets involve density weights (coefficients that transform like densities on the surface under reparametrization) and structure functions (bracket coefficients that depend on the dynamical variables themselves) rather than structure constants), so no change of variables restores the elementary canonical form [qi,pj]=δij[q_i,p_j]=\delta_{ij} familiar from point-particle mechanics. The published abstract of Part III records the consequence: 'The Hamiltonian has considerable analogy to that for point-charge electrons, but there is some discrepancy.'

Quantization of a classical theory normally proceeds by promoting canonical variables to operators whose commutators reproduce 1/(iℏ)1/(i\hbar) times the Poisson brackets. For the 1954 stream theory this route is blocked in three linked ways: (i) the reduced algebra is noncanonical and closes with derivative-of-delta structure functions, so there is no standard correspondence rule; (ii) the theory is formulated with first-class constraints (relations among the variables that generate gauge and reparametrization freedom) on arbitrary space-like surfaces, whose quantum implementation must be internally consistent (anomaly-free); and (iii) the theory contains no spin and no exchange antisymmetry - electron spin (the intrinsic angular momentum ℏ/2\hbar/2 encoded in the Dirac equation) and Fermi statistics (antisymmetry of multi-electron states under exchange, implying the Pauli exclusion principle) would have to emerge from quantizing the collective stream variables - while classically ee and mm enter only through the ratio e/me/m (Part II abstract), so the classical theory does not fix the charge ee separately from ℏ\hbar. These are exactly the respects in which the series was meant to serve as 'a basis for a passage to the quantum theory', and they are what the 1954 paper leaves unfinished.

Problem Statement

Dirac's 1954 paper supplies a complete classical theory of interpenetrating electron streams but no way to pass from it to the quantum theory of electrons that the series was designed to seed. The problem is: construct, or prove impossible, a consistent quantization of the 1954 stream theory - a quantum theory in which the operators (or functional-integral variables) corresponding to the reduced stream variables Pas,αa∗,βa∗P^s_a,\alpha^*_a,\beta^*_a and the electromagnetic field reproduce, in the classical limit ℏ→0\hbar\to 0, the Poisson brackets and the square-root Hamiltonian constraint quoted in the Background as transcriptions of the Hamiltonian formulation derived in Part III, on arbitrary space-like hypersurfaces, with an internally consistent (anomaly-free) implementation of the gauge and reparametrization constraints.

For such a quantization to count as a quantization of Dirac's electron theory, rather than merely of an associated field system, it must in addition satisfy two linked physical conditions:

  1. The charged excitations it produces must be spin-12\tfrac12 electrons obeying Fermi statistics (antisymmetric exchange, Pauli exclusion), not bosonic or spinless stream modes - the electron of the theory's title.

  2. It must resolve the 'some discrepancy' between the stream Hamiltonian and the point-charge electron Hamiltonian recorded in the paper's abstract, by exhibiting an appropriate limit in which electrons behave as point particles and the theory reproduces the established low-energy phenomenology of electron-photon physics. Reproducing, at leading order, the Coulomb interaction between electrons and the tree-level electron-photon scattering amplitudes of QED suffices to discharge this phenomenological condition.

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