Quantum Electrodynamics for Dyons

• Quantum Electrodynamics for Dyons (dQED) is a framework describing theories where particles carry both electric and magnetic charge and exhibit duality symmetry.
• Multiple formulations, including external-field treatments, doubled U(1)×U(1) theories, and higher-dimensional constructions, elucidate dQED’s effective actions and quantization conditions.
• dQED analysis reveals novel phenomena like topological phases, anomaly inflow, vacuum birefringence, and refined charge lattice structures essential for quantum scattering and boundary physics.

Searching arXiv for recent and foundational papers on quantum electrodynamics with dyons, duality, and related effective formulations. Quantum electrodynamics for dyons, commonly abbreviated dQED, denotes quantum theories of electromagnetic interactions in which the spectrum includes particles carrying both electric and magnetic charge. In the broadest sense, the term covers several closely related frameworks: compact $U(1)$ gauge theories with electric charges, monopoles, and dyons; duality-symmetric one-particle or many-body formulations with electric and magnetic potentials; non-Abelian ultraviolet completions in which dyons arise as solitons; and anomalous or boundary realizations in which the dyonic spectrum is fixed by higher-dimensional topology. Across these realizations, recurring structural themes are the coexistence of electric and magnetic sources, Dirac or Schwinger–Zwanziger charge quantization, nontrivial electric–magnetic duality, and, in some constructions, the necessity of doubled gauge potentials or higher-dimensional anomaly inflow (Heras, 2019, Bi et al., 2015, Kravec et al., 2014, Azevedo et al., 12 Jul 2026).

1. Historical and conceptual scope

The modern dQED literature does not define a single unique theory, but rather a family of constructions that realize dyonic electromagnetism under different microscopic assumptions. A controlled one-particle external-field treatment is developed through a duality-symmetric quantum phase for a point dyon moving around a “dual solenoid,” where the central observable is a topological phase depending on $q\Phi_{\rm m}-g\Phi_{\rm e}$ (Heras, 2019). A string-free field-theoretic realization with two Abelian gauge sectors, $U(1)\times U(1)$, is used to derive a one-loop Euler–Heisenberg effective action for a Dirac dyon coupled simultaneously to an “electric” photon $A_\mu^{(1)}$ and a “magnetic” photon $A_\mu^{(2)}$ (Azevedo et al., 12 Jul 2026). Other works treat dyons as solitonic excitations of non-Abelian Yang–Mills–Higgs systems, where the long-distance limit is an Abelian dyonic gauge theory whose charge lattice is inherited from the root and weight data of the ultraviolet gauge group (Olszewski, 2010, Olszewski, 2015, Adachi et al., 27 May 2025).

A distinct line of work studies dyonic QED as an anomalous boundary state. In this setting, 3+1-dimensional compact $U(1)$ gauge theory with electric charges, monopoles, and dyons is realized on the boundary of a 4+1-dimensional bosonic short-range-entangled phase. The boundary theory may exhibit exact electric–magnetic duality and projective symmetry assignments for both electric and magnetic sectors, with anomaly inflow from the bulk enforcing a dyonic spectrum that cannot be realized in a strictly 3+1-dimensional bosonic microscopic model (Bi et al., 2015, Kravec et al., 2014).

These constructions differ substantially in formalism, but they share the same defining content: a $U(1)$ gauge sector with both electric and magnetic sources, a nontrivial dyon lattice, and observables organized by duality-invariant combinations of charges and fluxes. This suggests that “dQED” is best understood as a class of quantum electromagnetic theories with dyons rather than a single canonical Lagrangian.

2. Charge lattice, duality, and dyonic observables

A dyon is specified by electric and magnetic charges, commonly denoted $(q,g)$ in external-field treatments or $(q_e,q_m)$ in field-theoretic treatments. In compact $U(1)$ theories, the spectrum forms a charge lattice subject to Dirac quantization; in one formulation this is written as

$q\Phi_{\rm m}-g\Phi_{\rm e}$0

In the self-dual boundary construction of compact QED, the basic lattice generators are an electric particle $q\Phi_{\rm m}-g\Phi_{\rm e}$1 with $q\Phi_{\rm m}-g\Phi_{\rm e}$2, a monopole $q\Phi_{\rm m}-g\Phi_{\rm e}$3 with $q\Phi_{\rm m}-g\Phi_{\rm e}$4, and a dyon $q\Phi_{\rm m}-g\Phi_{\rm e}$5 with $q\Phi_{\rm m}-g\Phi_{\rm e}$6, and the lattice is symmetric under electric–magnetic interchange or quarter-rotation depending on the symmetry realization (Bi et al., 2015). In anomalous all-fermion electrodynamics, the same $q\Phi_{\rm m}-g\Phi_{\rm e}$7 lattice appears, but the minimal electric charge, monopole, and dyon are all fermions; that charge–statistics pattern is self-consistent as an effective theory yet obstructed as a purely 3+1-dimensional bosonic UV completion (Kravec et al., 2014).

A central quantum-mechanical dyonic observable is the duality-invariant topological phase acquired when a dyon encircles a dual solenoid carrying electric and magnetic flux. For a dyon of charges $q\Phi_{\rm m}-g\Phi_{\rm e}$8 winding $q\Phi_{\rm m}-g\Phi_{\rm e}$9 times around a configuration enclosing $U(1)\times U(1)$0, the wavefunction acquires

$U(1)\times U(1)$1

This phase is topological because it depends only on winding number and nonlocal because it is nonzero even though $U(1)\times U(1)$2 along the dyon trajectory (Heras, 2019). In this sense, dQED generalizes the Aharonov–Bohm effect: the ordinary AB phase and its magnetic dual arise as special duality frames of the same dyonic phase.

Duality acts naturally on both charges and fluxes. In the external-field treatment, continuous $U(1)\times U(1)$3 electric–magnetic duality rotates $U(1)\times U(1)$4 and $U(1)\times U(1)$5 such that the bilinear $U(1)\times U(1)$6 is invariant (Heras, 2019). In self-dual boundary QED, the duality may instead be an exact microscopic symmetry: in one example time reversal exchanges electric and magnetic charges, and in another an internal $U(1)\times U(1)$7 acts as

$U(1)\times U(1)$8

with corresponding rotations of $U(1)\times U(1)$9 (Bi et al., 2015). In axion-coupled dyonic theories, the duality group is more naturally $A_\mu^{(1)}$0, acting on the complexified coupling $A_\mu^{(1)}$1 and on the dyon charge lattice (Heidenreich et al., 2023).

3. Field-theoretic realizations

One major distinction among dQED realizations concerns the gauge-field description. In external-field quantum mechanics, a dyon moving in prescribed fields is described using both the ordinary magnetic vector potential $A_\mu^{(1)}$2 and an electric dual vector potential $A_\mu^{(1)}$3, with

$A_\mu^{(1)}$4

A local canonical Hamiltonian does not exist in arbitrary backgrounds with both electric and magnetic charges, so a nonlocal Lagrangian containing a line integral is introduced. In field-free regions outside the dual solenoid, however, the Hamiltonian reduces to the local form

$A_\mu^{(1)}$5

which suffices to derive spectra, interference shifts, and scattering amplitudes controlled by $A_\mu^{(1)}$6 (Heras, 2019).

A local doubled-potential field theory is realized in the Cabibbo–Ferrari–Salam–Govaerts framework, where electromagnetism is promoted to a $A_\mu^{(1)}$7 gauge theory with gauge fields $A_\mu^{(1)}$8 and $A_\mu^{(1)}$9. A single Dirac field $A_\mu^{(2)}$0 carries charge vector $A_\mu^{(2)}$1, and the covariant derivative is

$A_\mu^{(2)}$2

The physical electric and magnetic fields are not identified with those of a single sector, but as the combinations

$A_\mu^{(2)}$3

This eliminates Dirac strings at the level of the classical description and supports a perturbative one-loop computation of the effective action (Azevedo et al., 12 Jul 2026).

A more algebraic route uses quaternionic or split-octonionic unification of electric and magnetic variables. In a split-octonion formulation, the generalized dyon potential is encoded as a Zorn matrix

$A_\mu^{(2)}$4

with $A_\mu^{(2)}$5 and $A_\mu^{(2)}$6 electric and magnetic four-potentials, and the field equations reduce to compact matrix equations such as $A_\mu^{(2)}$7 and $A_\mu^{(2)}$8, equivalent to generalized Maxwell equations with both electric and magnetic sources (Chanyal et al., 2010). Quaternionic approaches similarly combine electric and magnetic potentials as $A_\mu^{(2)}$9 and derive massive, duality-symmetric wave equations for the potentials, currents, and fields (Chanyal, 2017, Chanyal, 2017). These constructions are not full second-quantized QFTs in the modern sense, but they supply alternative first-quantized or algebraic dyonic electrodynamics.

4. Non-Abelian and higher-dimensional origins

Many controlled dyon spectra arise not from fundamental Abelian dyons, but from solitons of non-Abelian gauge theories broken to $U(1)$0. In Yang–Mills–Higgs theory with adjoint Higgs field $U(1)$1, the generalized ’t Hooft tensor defines an Abelian field strength $U(1)$2 associated with a chosen Cartan direction $U(1)$3. For suitable root-theoretic choices of $U(1)$4, $U(1)$5 satisfies Maxwell equations and supports topological monopole and dyon charges (Olszewski, 2010). In $U(1)$6, the magnetic charge is

$U(1)$7

and for roots of unit length this reduces to $U(1)$8. Electric charge quantization is set by Cartan eigenvalues in the chosen representation, and in the BPS limit the dyon mass takes the standard form $U(1)$9, where $U(1)$0 (Olszewski, 2010).

String-theoretic realizations provide a closely related non-Abelian dyon sector. Exact dyon solutions on coincident D4-branes arise from the non-Abelian Dirac–Born–Infeld plus Wess–Zumino action, and after compactifying one worldvolume direction the low-energy theory reduces to a 3+1-dimensional Yang–Mills–Higgs+$U(1)$1 theory. The electric and magnetic charges satisfy a Witten-effect-shifted quantization,

$U(1)$2

and the complex coupling

$U(1)$3

transforms under $U(1)$4, exhibiting the familiar Montonen–Olive structure expected of a symmetric dyonic theory (Olszewski, 2015).

Higher-dimensional gauge–Higgs unification gives a different route to dyon quantization. In 5D pure $U(1)$5 Yang–Mills on $U(1)$6, the Higgs field is identified with $U(1)$7, and the BPS monopole becomes a self-dual configuration in the 4D Euclidean space $U(1)$8. This implies topological quantization of the monopole mass and of the Higgs vacuum expectation value: $U(1)$9 For dyons, the electric-to-magnetic ratio $(q,g)$0, defined by $(q,g)$1, is usually continuous; the 5D analysis shows that an induced Chern–Simons term and a periodic potential for an auxiliary $(q,g)$2 Wilson line can discretize $(q,g)$3, and hence, via the Witten effect, discretize the dyon electric charge: $(q,g)$4 This yields a discrete BPS dyon mass tower,

$(q,g)$5

with small non-BPS corrections in the numerically studied examples (Adachi et al., 27 May 2025).

These constructions show that dQED often appears as an infrared limit of a non-Abelian or higher-dimensional theory rather than as a standalone microscopic Abelian model. A plausible implication is that the physically most robust dyonic charge lattices are those inherited from topology, root systems, and higher-dimensional couplings rather than imposed by hand.

5. Boundary dQED, anomalies, and projective quantum numbers

A particularly consequential realization of dQED arises at boundaries of 4+1-dimensional bosonic short-range-entangled phases. In one construction, a 4+1D $(q,g)$6 nonlinear sigma model at $(q,g)$7 induces a 3+1D boundary Wess–Zumino–Witten term. Writing $(q,g)$8 and representing $(q,g)$9 by a $(q_e,q_m)$0 field $(q_e,q_m)$1, the boundary supports a compact $(q_e,q_m)$2 gauge field $(q_e,q_m)$3 whose electric charge is $(q_e,q_m)$4. The Dirac monopole of $(q_e,q_m)$5 coincides with a hedgehog of $(q_e,q_m)$6, described by another $(q_e,q_m)$7 field $(q_e,q_m)$8, so the boundary contains both electric and magnetic matter fields on equal footing (Bi et al., 2015).

The central result is that both the electric charge and the monopole transform projectively under global symmetry, with the projective structure fixed by the bulk. In the $(q_e,q_m)$9 example, time reversal acts as exact electric–magnetic duality,

$U(1)$0

and on the field strengths

$U(1)$1

The canonical commutator and Maxwell equations are invariant under this action, so the self-duality is not merely an infrared equivalence but a microscopic symmetry of the boundary state (Bi et al., 2015).

The dyon $U(1)$2 inherits a nontrivial internal Hilbert space described by a $U(1)$3D $U(1)$4 WZW model. The resulting bound state is a spin-$U(1)$5 doublet, and time reversal acts projectively as

$U(1)$6

Thus the dyon is a Kramers doublet fermion. The same bulk construction also supports a $U(1)$7 example in which the internal $U(1)$8 symmetry itself generates electric–magnetic duality. In both cases, the projective symmetry assignment of $U(1)$9, $q\Phi_{\rm m}-g\Phi_{\rm e}$00, and $q\Phi_{\rm m}-g\Phi_{\rm e}$01 is anomalous in strictly 3+1D bosonic systems and therefore diagnoses higher-dimensional inflow (Bi et al., 2015).

A closely related but symmetry-independent anomaly appears in all-fermion electrodynamics. There, the minimal electric charge, monopole, and dyon are all fermions. The charge lattice is consistent as a 3+1D effective theory, but the theory cannot be regulated by purely bosonic microscopic degrees of freedom. The obstruction can be exposed by the behavior of the partition function on $q\Phi_{\rm m}-g\Phi_{\rm e}$02, where the required $q\Phi_{\rm m}-g\Phi_{\rm e}$03 periodicity in $q\Phi_{\rm m}-g\Phi_{\rm e}$04 is incompatible with the geometry of a non-spin manifold for a bosonic regulator, and by a 4+1D anomaly inflow in which monopole-string processes transport fermion number between opposite boundaries (Kravec et al., 2014).

These boundary realizations establish that dQED is not always a standalone 3+1D gauge theory. In some of its most symmetric and tightly constrained forms, it is intrinsically anomalous and must be understood as boundary electrodynamics of a higher-dimensional topological phase.

6. Effective actions, nonlinear response, and axion couplings

The one-loop effective action provides a direct probe of dQED beyond kinematics. In $q\Phi_{\rm m}-g\Phi_{\rm e}$05 dQED, integrating out the Dirac dyon yields a generalized Euler–Heisenberg effective Lagrangian. In terms of sector invariants

$q\Phi_{\rm m}-g\Phi_{\rm e}$06

the weak-field expansion contains two copies of the usual Euler–Heisenberg quartic terms plus mixing operators between the two gauge sectors (Azevedo et al., 12 Jul 2026). The theory reduces smoothly to ordinary spinor QED when $q\Phi_{\rm m}-g\Phi_{\rm e}$07, but when both charges are present the vacuum response depends on hybrid combinations of the two gauge sectors.

Strong-field instability is also modified. For parallel electric and magnetic backgrounds in the doubled theory, the proper-time poles occur at

$q\Phi_{\rm m}-g\Phi_{\rm e}$08

and the imaginary part of the effective Lagrangian, signaling dyon–antidyon pair production, is

$q\Phi_{\rm m}-g\Phi_{\rm e}$09

The analog of the Schwinger limit is therefore controlled by the weighted electric combination $q\Phi_{\rm m}-g\Phi_{\rm e}$10, not by a single-sector field (Azevedo et al., 12 Jul 2026).

Linearizing the same effective action about a purely magnetic background yields effective permittivity and permeability tensors for both sectors, along with refractive indices $q\Phi_{\rm m}-g\Phi_{\rm e}$11 and $q\Phi_{\rm m}-g\Phi_{\rm e}$12 for hybrid propagation modes. The quantum vacuum behaves as an anisotropic medium and exhibits vacuum birefringence. In the symmetric case $q\Phi_{\rm m}-g\Phi_{\rm e}$13, the propagation eigenmodes are hybrid superpositions of photon and “metaphoton” modes, and the birefringence pattern includes one mode with the standard $q\Phi_{\rm m}-g\Phi_{\rm e}$14 scaling and another whose birefringence begins only at $q\Phi_{\rm m}-g\Phi_{\rm e}$15 (Azevedo et al., 12 Jul 2026).

Axion couplings add a different dimension to dQED. In Schwinger–Zwanziger electromagnetodynamics, integrating out heavy PQ-charged fermionic dyons generates not only the familiar $q\Phi_{\rm m}-g\Phi_{\rm e}$16 operator but, more generally, three couplings: $q\Phi_{\rm m}-g\Phi_{\rm e}$17 corresponding respectively to $q\Phi_{\rm m}-g\Phi_{\rm e}$18, $q\Phi_{\rm m}-g\Phi_{\rm e}$19, and mixed $q\Phi_{\rm m}-g\Phi_{\rm e}$20 anomalies. The low-energy axion interaction becomes

$q\Phi_{\rm m}-g\Phi_{\rm e}$21

so dQED generically predicts a CP-odd $q\Phi_{\rm m}-g\Phi_{\rm e}$22-type coupling absent in ordinary axion electrodynamics (Li et al., 2023).

By contrast, axion monodromy in an $q\Phi_{\rm m}-g\Phi_{\rm e}$23-covariant dyonic theory can have severe consequences. A non-standard axion coupling implemented in a duality frame with nontrivial $q\Phi_{\rm m}-g\Phi_{\rm e}$24 monodromy causes electrically charged particles to become dyons when the axion traverses its field range. This “dual Witten effect” acts on the charge lattice by a conjugated $q\Phi_{\rm m}-g\Phi_{\rm e}$25-transformation and implies towers of dyonic partners for electrically charged matter. The analysis argues that such non-standard axion electrodynamics is not phenomenologically viable because it produces dyons near the weak scale, induces large axion masses through Standard Model fermion loops, and strongly distorts Higgs physics (Heidenreich et al., 2023).

7. Scattering, kinematics, and emergent phases

Even in classical or semiclassical regimes, dQED differs sharply from purely electric electrodynamics. In first-order post-Minkowskian scattering of two dyons with charges $q\Phi_{\rm m}-g\Phi_{\rm e}$26, two invariant charge combinations enter separately: $q\Phi_{\rm m}-g\Phi_{\rm e}$27 The first controls the Coulombic part of the force, while the second controls the topological angular momentum associated with electric–magnetic pairing (Yılmaz et al., 5 Jul 2026). On constant-time slices, scattering induces both a shift in the mechanical boost-like angular momentum and, in the genuinely dyonic case, a shift in the mechanical orbital angular momentum,

$q\Phi_{\rm m}-g\Phi_{\rm e}$28

balanced by an opposite field contribution so that total angular momentum is conserved (Yılmaz et al., 5 Jul 2026).

A subtle result is that the “electromagnetic scoot” associated with the boost generator is slice-dependent and disappears at first post-Minkowskian order on hyperboloidal slices, whereas the dyonic angular-momentum scoot persists. This suggests that dQED multiparticle states require pairwise labels beyond single-particle Poincaré quantum numbers. A plausible implication is that a full quantum dyonic S-matrix should be formulated using pairwise dressed states carrying both a helicity-like label proportional to $q\Phi_{\rm m}-g\Phi_{\rm e}$29 and a boost-related label governed by $q\Phi_{\rm m}-g\Phi_{\rm e}$30 (Yılmaz et al., 5 Jul 2026).

Self-dual dQED also admits nontrivial gapped descendants. In the boundary $q\Phi_{\rm m}-g\Phi_{\rm e}$31 realization, condensing $q\Phi_{\rm m}-g\Phi_{\rm e}$32 or $q\Phi_{\rm m}-g\Phi_{\rm e}$33 would break time reversal because $q\Phi_{\rm m}-g\Phi_{\rm e}$34 and $q\Phi_{\rm m}-g\Phi_{\rm e}$35 are exchanged by $q\Phi_{\rm m}-g\Phi_{\rm e}$36. A symmetric gapped phase is instead obtained by condensing Cooper pairs of dyons $q\Phi_{\rm m}-g\Phi_{\rm e}$37, leading to a 3D $q\Phi_{\rm m}-g\Phi_{\rm e}$38 topological order whose point excitation is the fermionic dyon $q\Phi_{\rm m}-g\Phi_{\rm e}$39 and whose flux excitation is a vison loop. Because the vison loop is gapless or degenerate and the gauge charge is fermionic, this topological order cannot be trivially confined while preserving symmetry (Bi et al., 2015).

When the fourth spatial dimension is finite, one boundary can host self-dual photon dQED and the opposite boundary the $q\Phi_{\rm m}-g\Phi_{\rm e}$40 topological order just described. The combined 3D slab then behaves as an exotic self-dual “topological photon phase,” in which the low-energy degrees of freedom include a gapless photon but the realization is impossible in a strictly 3D short-range-entangled bosonic system with the same symmetry (Bi et al., 2015).

8. Misconceptions, tensions, and present status

A common misconception is that dQED is simply ordinary QED with a $q\Phi_{\rm m}-g\Phi_{\rm e}$41 term. The $q\Phi_{\rm m}-g\Phi_{\rm e}$42-term and Witten effect do produce dyons in conventional electrodynamics, but several dQED constructions differ qualitatively from that picture. In self-dual boundary QED, the nontrivial dyon spectrum and exact electric–magnetic interchange are enforced by a bulk WZW term and symmetry, not by a tunable emergent $q\Phi_{\rm m}-g\Phi_{\rm e}$43-angle (Bi et al., 2015). In doubled $q\Phi_{\rm m}-g\Phi_{\rm e}$44 dQED, the existence of a second gauge potential is structural rather than a consequence of a $q\Phi_{\rm m}-g\Phi_{\rm e}$45-term (Azevedo et al., 12 Jul 2026). In higher-dimensional gauge–Higgs unification, discrete dyon charges arise from a quantized Wilson line and induced Chern–Simons term rather than from an arbitrary 4D $q\Phi_{\rm m}-g\Phi_{\rm e}$46-parameter (Adachi et al., 27 May 2025).

Another misconception is that exact electric–magnetic duality can always be realized in a microscopic 3+1D theory. The boundary and anomaly literature shows that some of the most symmetric dQEDs are intrinsically anomalous. All-fermion electrodynamics, for example, is internally consistent as a low-energy 3+1D theory but cannot be UV-completed with only bosonic microscopic degrees of freedom in 3+1D; it requires a 4+1D bulk (Kravec et al., 2014). The self-dual bosonic SPT boundary construction similarly implies that specific projective symmetry assignments of charges and monopoles are not realizable in purely 3+1D bosonic systems (Bi et al., 2015).

There is also a formal tension between locality and duality symmetry. One-particle dyon mechanics in prescribed electromagnetic backgrounds generally requires nonlocal Lagrangians, while local doubled-potential field theories trade manifest single-potential simplicity for an enlarged gauge sector (Heras, 2019, Azevedo et al., 12 Jul 2026). Quaternionic and split-octonionic formalisms provide compact duality-symmetric equations, but they do not yet constitute full renormalizable dQEDs in the modern QFT sense (Chanyal, 2017, Chanyal, 2017, Chanyal et al., 2010).

The current status is therefore plural rather than unified. The best-controlled sectors of dQED are: the kinematics of dyonic phases and scattering in external fields (Heras, 2019), the nonlinear effective action of local doubled-$q\Phi_{\rm m}-g\Phi_{\rm e}$47 theories (Azevedo et al., 12 Jul 2026), the topological and anomalous structure of boundary dQEDs (Bi et al., 2015, Kravec et al., 2014), and the ultraviolet origin of dyon lattices from non-Abelian gauge theory or higher dimensions (Olszewski, 2010, Olszewski, 2015, Adachi et al., 27 May 2025). A fully satisfactory microscopic, local, renormalizable, manifestly duality-symmetric dQED with dynamical dyons remains a nontrivial open problem. A plausible synthesis is that different formulations capture different universal aspects: charge quantization and duality, anomaly structure, nonlinear response, or ultraviolet origin, but no single presently available framework exhausts all of them simultaneously.