Completing Dirac's quantum theory of point electric charges and magnetic poles: the consistency and calculability of the quantum field theory of electric and magnetic charge

Dirac's 1948 paper "The Theory of Magnetic Poles" formulated the classical and quantum dynamics of interacting point electric charges and point magnetic poles, with dynamical Dirac strings as unphysical gauge variables. Its published abstract acknowledged "unsolved difficulties, concerned with the interaction of a point charge or a point pole with the field it produces itself": point sources were regularized by an unspecified smearing kernel $\gamma(x)$ with the point-particle limit left formal, and the theory fixed only the product $e_0g_0$ of the elementary electric and magnetic quanta. The subsequent literature identified further unfinished aspects of the program: whether a particle can carry both electric and magnetic charge, the radiation-damping dynamics of poles, and the spin and statistics of poles. The open core is the consistency and calculability of the quantum field theory of point electric and magnetic charges: no known Lagrangian is simultaneously local and manifestly Lorentz invariant; perturbation theory fails because the magnetic fine-structure constant is $\alpha_g\approx 34$ and individual graphs depend on unphysical string choices; string independence is proven only formally; the renormalization of dual quantum electrodynamics is unsettled; and monopole scattering and production processes cannot currently be computed beyond special limits (soft-photon resummation; low-multiplicity pairwise-helicity on-shell amplitudes).

Background

A magnetic monopole is a hypothetical particle carrying net magnetic charge — a source of diverging magnetic field, the magnetic analogue of the electron. Its existence would make Maxwell's equations symmetric between electricity and magnetism, but it breaks the standard device of writing the magnetic field as the curl of a vector potential, since $\nabla\cdot\mathbf{B}\neq 0$.

Dirac showed in 1931 that a monopole of pole strength $g$ can still be described with a vector potential if one attaches to each pole an unphysical line of singular flux — the Dirac string — whose position is arbitrary. Quantum interference of a charge $e$ transported around the string is unobservable only if the Dirac quantization condition $eg = \frac{n\hbar c}{2},\qquad n\in\mathbb{Z},$ holds (Gaussian units). This makes the quantization of electric charge a consequence of the existence of even one monopole, which was Dirac's stated motivation.

In 1948 Dirac went far beyond this kinematic argument and formulated the full dynamics of a system of point charges and point poles: an action whose field-interaction term is smeared with a kernel $\gamma(x-x')$ intended to approach the four-dimensional delta function (so the point-particle limit was left formal), worldlines of both kinds of particle, and strings whose positions are dynamical but unphysical gauge variables. From this he derived the Hamiltonian equations of motion, canonical momenta and Poisson brackets, and quantum wave equations for spin-$1/2$ charges and poles, with the quantization condition appearing in product form $e_0g_0=n\hbar c/2$ for the elementary quanta. The published abstract explicitly acknowledged the point-source self-field difficulties; the unfinished program can be organized into five fronts — the point-source self-field divergences (the kernel $\gamma$ never specified, so the infinities and the renormalization scheme were open); whether an elementary particle can carry both charge and pole; the missing radiation-damping dynamics for poles; the fact that only the product $e_0g_0$ is fixed; and the provisional spin assignment. Taking $g_0^2=(137/4)\hbar c$ (so that the minimal pole matches the observed electronic charge), the magnetic fine-structure constant is $\alpha_g=g_0^2/\hbar c\approx 34$: any on-shell monopole process is strongly coupled in the magnetic sector.

A particle carrying both electric charge and magnetic charge is a dyon. For dyons, pairwise consistency replaces Dirac's condition with the Dirac-Schwinger-Zwanziger (DSZ) condition $e_1g_2-e_2g_1=n\hbar c/2$.

The later frameworks define the problem's modern context. Schwinger (1966) constructed a quantum field theory of particles carrying either electric or magnetic charge; Zwanziger (1968) extended the construction to particles with both, and (1971) gave a local Lagrangian formulation using two electromagnetic potentials and a fixed four-vector that selects the string direction — local, but not manifestly Lorentz invariant, whereas Dirac's 1948 Lagrangian is Lorentz invariant but non-local. Wu and Yang (1975, 1976) gave a string-free formulation in fiber-bundle language (the magnetic field is described by overlapping patches of vector potential glued by gauge transformations) and introduced monopole harmonics (the angular wavefunctions of a charged particle in a monopole field). 't Hooft and Polyakov (1974) showed that unified non-Abelian gauge theories — grand unified theories, GUTs — contain monopoles as smooth soliton solutions (localized, particle-like field lumps), with charge quantization following from the gauge group; Julia and Zee (1975) found the corresponding classical dyon solutions. Weinberg (1965) observed that the one-photon-exchange charge-monopole scattering amplitude is not Lorentz invariant — the "Weinberg paradox" — which Terning and Verhaaren (2019) resolved by resumming the emission of soft (low-energy) photons to all orders, given charge quantization. Monopole processes radiate enormous numbers of soft photons — an infrared (IR) problem — and the resulting energy loss damps scattering so strongly that fixed-order perturbation theory is meaningless (Blagojević et al. 1982). Most recently, Csaki et al. (2021) constructed charge-monopole and monopole-monopole scattering amplitudes on shell (for physical asymptotic particle states, from Lorentz symmetry, unitarity — conservation of probability — and little-group structure, without a Lagrangian), using the pairwise helicity $q_{12}=e_1g_2-e_2g_1$, a multi-particle representation of the Poincaré group found by Zwanziger (1972); the "little group" is the subgroup of Lorentz transformations fixing a state's momentum, which determines its spin degrees of freedom, and Zwanziger's pairwise version determines the allowed helicities of charge-pole pairs.

Against this backdrop, the question is whether Dirac's 1948 program — a consistent, calculable quantum theory of point electric charges and magnetic poles — can be completed.

Problem Statement

Dirac's 1948 paper formulates, but does not complete, a quantum theory of interacting point electric charges and point magnetic poles. Its published abstract acknowledges "unsolved difficulties, concerned with the interaction of a point charge or a point pole with the field it produces itself". The unfinished program is organized here into five fronts: (1) the point-source self-field problem — the action and the quantum wave equations use an unspecified smeared kernel $\gamma(x)$ with the point-particle limit $\gamma\to\delta_4$ left formal, so the self-field infinities and the renormalization scheme are open; (2) whether an elementary particle can simultaneously carry a nonzero electric charge and a nonzero magnetic pole — the question addressed by the later dyon literature; (3) the radiation-damping dynamics for poles, needed for quantitative binding energies and scattering, developed only in special limits by the later literature; (4) the theory fixes only the product $e_0g_0$, not the separate electric and magnetic quanta; (5) the spin (and statistics) of poles — spin-$1/2$ in Dirac's wave equations — whose general status was opened by Goldhaber.

Does there exist a consistent, calculable quantum theory of point electric charges and magnetic poles — including dyons wherever the DSZ condition admits them — completing Dirac's 1948 program, specifically, a formulation in which (a) the point-source self-fields are finite or renormalized in a controlled scheme (replacing Dirac's formal $\gamma$-smearing); (b) computed physical observables — scattering amplitudes including radiation corrections, and monopole production processes — are independent of the unphysical Dirac strings and Lorentz covariant; and (c) radiation-dominated monopole processes (radiation reaction/damping) are computable within the full theory, beyond the established special limits of soft-photon/eikonal resummation and low-multiplicity pairwise-helicity on-shell amplitudes? Any such theory must of course reproduce the established consistency results — the Dirac and DSZ quantization conditions, the Goldhaber spin-statistics constraints for charge-monopole composites, and the Lorentz covariance of soft-photon-resummed amplitudes — which follow from existing theorems rather than constituting additional completion conditions. Or, alternatively, does a precise no-go theorem show that no theory in a fixed, objectively specified formulation class has these properties? The admissible class is the class of Lagrangian-based (local or non-local) and on-shell formulations of the quantum field theory of point electric and magnetic charges, and it is required to contain Dirac's non-local Lorentz-invariant 1948 action, Zwanziger's local two-potential theory, and the on-shell pairwise-helicity amplitudes of Csaki et al. (2021); a no-go theorem over this class resolves the task. Determining the actual numerical value of $g_0$ beyond the quantized product, and the experimental discovery of monopoles, are empirical questions outside this theoretical construction.

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