Coulomb force between two Dirac monopoles

The quest for magnetic monopoles has captivated theoretical physicists due to their potential to reshape our understanding of electromagnetism. Despite extensive research, definitive mathematical proof of their existence remains elusive. This paper presents a rigorous mathematical framework supporting the existence of magnetic monopoles within the covariant formulation of classical electromagnetism and special relativity. Utilizing tensor calculus and the covariance form of Maxwell's equations, we derive expressions for magnetic charge density and magnetic current density through Lorentz transformations. Our approach emphasizes the expansion of tensorial equations, particularly focusing on the covariant derivatives of the electromagnetic field tensor and its dual. The results demonstrate the recovery of magnetic charge and current densities in Maxwell's equations, providing a theoretical foundation for the existence of magnetic monopoles. A magnetic monopole model is introduced, establishing the relationship between the electric field, the monopole’s magnetic charge, and their dependence on the observer's relative velocity as a direct consequence of the modified Gauss's law for magnetism. This model suggests the existence of dipole bosons—electric or magnetic—which function analogously to the Higgs boson. These dipole bosons are proposed to confer mass, charge, and spin to charged particles, whether they are electric or magnetic in nature.

Gauge formulation for magnetic monopoles from generalized gauge fields

Monopoles, duality, triality

A non-Ohmic current that grows exponentially with the square root of applied electric field is well known from thermionic field emission (the Schottky effect), electrolytes (the second Wien effect) and semiconductors (the Poole-Frenkel effect). It is a universal signature of the attractive Coulomb force between positive and negative electrical charges, which is revealed as the charges are driven in opposite directions by the force of an applied electric field. Here we apply thermal quenches to spin ice to prepare metastable populations of bound pairs of positive and negative emergent magnetic monopoles at millikelvin temperatures. We find that the application of a magnetic field results in a universal exponential-root field growth of magnetic current, thus confirming the microscopic Coulomb force between the magnetic monopole quasiparticles and establishing a magnetic analogue of the Poole-Frenkel effect. At temperatures above 300 mK, gradual restoration of kinetic monopole equilibria causes the non-Ohmic current to smoothly evolve into the high field Wien effect for magnetic monopoles, as confirmed by comparison to a recent and rigorous theory of the Wien effect in spin ice. Our results extend the universality of the exponential-root field form into magnetism and illustrate the power of emergent particle kinetics to describe far-from equilibrium response in complex systems.

The flow of magnetic charge carriers (dubbed magnetic monopoles) through frustrated spin ice lattices, governed simply by Coulombic forces, represents a new direction in electromagnetism. Artificial spin ice nanoarrays realise this effect at room temperature, where the magnetic charge is carried by domain walls. Control of domain wall path is one important element of utilizing this new medium. By imaging the transit of domain walls across different connected 2D honeycomb structures we contribute an important aspect which will enable that control to be realized. Although apparently equivalent paths are presented to a domain wall as it approaches a Y-shaped vertex from a bar parallel to the field, we observe a stark non-random path distribution, which we attribute to the chirality of the magnetic charges. These observations are supported by detailed statistical modelling and micromagnetic simulations. The identification of chiral control to magnetic charge path selectivity invites analogy with spintronics.

Unified theories of strong, weak and electromagnetic interactions which have electric charge quantization predict the existence of topologically stable magnetic monopoles. Intermediate scale monopoles are comparable with detection energies of cosmic ray monopoles at IceCube and other cosmic ray experiments. Magnetic monopoles in some models can be significantly lighter and carry two, three or possibly even higher quanta of the Dirac magnetic charge. They could be light enough for their effects to be detected at the LHC either directly or indirectly. An example based on a D-brane inspired SU(3)C × SU(3)L × SU(3)R (trinification) model with the monopole carrying three quanta of Dirac magnetic charge is presented. These theories also predict the existence of color singlet states with fractional electric charge which may be accessible at the LHC.

We present a proposal for applying nanoscale magnetometry to the search for magnetic monopoles in the spin ice materials holmium and dysprosium titanate. Employing Monte Carlo simulations of the dipolar spin ice model, we find that when cooled to below $1.5\,$K these materials exhibit a sufficiently low monopole density to enable the direct observation of magnetic fields from individual monopoles. At these temperatures we demonstrate that noise spectroscopy can capture the intrinsic fluctuations associated with monopole dynamics, allowing one to isolate the qualitative effects associated with both the Coulomb interaction between monopoles and the topological constraints implied by Dirac strings. We describe in detail three different nanoscale magnetometry platforms (muon spin rotation, nitrogen vacancy defects, and nanoSQUID arrays) that can be used to detect monopoles in these experiments, and analyze the advantages of each.

The transmission and reflection coefficients of shear Alfven waves in an open magnetic structure emerging from the Sun are determined. Both two-dimensional slab and axisymmetric static equilibrium configurations are solved numerically on a curvilinear coordinate grid adapted to the magnetic field geometry and to the density stratification. The equilibrium structure obtained numerically extends from the photosphere up to an altitude of 20,000 km, spanning 14 density scale heights and allowing for a magnetic flux tube expansion by a factor 40-400, corresponding to a filling factor of 2.5% to 0.25%. From 20,000 km up to 4 R ⊙ a magnetic monopole model is adopted

It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ?a = φa (φbφb)−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.

In this talk we review the subject of electromagnetic duality, and related ones, as "normal" electric and magnetic charges, being different from monopoles (magnetic or electric), the topological content of monopoles, surface terms, Dirac strings and chiral structure in the inclusion of massless spinor field and its relation to torsion. We limit our discussion to the simplest, Heaviside electromagnetic duality, include some links with the nonabelian gauge theories and make no comment on the use of this concept in string theories. Old and recent contributions on the subject are presented and discussed, and possible lines of research will hopefully be indicated.

The microscopic Maxwell-Lorentz equations show an intrinsic symmetry if magnetic charge and current densities are included. Starting from the symmetrized set of field equations a propagator approach is used to describe the transverse (photon) electrodynamics in space-time. Two dyadic propagators are needed: (i) a transverse electric propagator with a near-field part and (ii) a magnetic propagator with a midfield part. The first quantized photon wave mechanical theory, based on the analytical parts of the two transverse Riemann-Silberstein vectors, is extended to include magnetic monopole dynamics. The dynamical equations for the photon helicity eigenvectors and the local energy conservation in the photon field are discussed. The Dirac string concept and the fiber bundle approach are avoided using the double-potential formalism of Cabibbo and Ferrari. The transverse ($T$) parts of the electric (${\mathbf{A}}_{T}^{e}$) and magnetic (${\mathbf{A}}_{T}^{m}$) vector potentials relate in a simple manner to our propagator theory if this is formulated in terms of the Huygens scalar propagator and the transverse electric and magnetic current densities. A transformation of the transverse vector potentials, which is nonlocal in space and time, allows one to express the transverse parts of the electric and magnetic fields as curls of combinations of the original (${\mathbf{A}}_{T}^{e},\phantom{\rule{0.16em}{0ex}}{\mathbf{A}}_{T}^{m}$) and the transformed (${\mathsc{A}}_{T}^{e},\phantom{\rule{0.16em}{0ex}}{\mathsc{A}}_{T}^{m}$) transverse vector potentials. The momentum of the particle-photon system is discussed, and it is shown that the electromagnetic parts of the canonical electric (charge $e$) and magnetic (charge $g$) particle momenta are given by $e({\mathbit{A}}_{T}^{e}+{\mathsc{A}}_{T}^{m})$ and $g({\mathbit{A}}_{T}^{m}\ensuremath{-}{\mathsc{A}}_{T}^{e})$, respectively. The angular momentum of the particle-photon system is analyzed and contact is made to the well-known nonretarded Saha-Wilson orbital angular momentum for an ($e,g$) pair. The near field of a magnetic monopole is studied based on the overlooked fact that the magnetic near field contains both longitudinal ($L$) (with $\mathbf{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{B}}_{L}=0$) and transverse vector field components. The sum of the two parts always is Einstein retarded, with proper account of the limitation caused by the lack of complete spatial photon localization. Finally, the relativistic electron-photon Hamiltonian in an external (prescribed) magnetic monopole field is discussed, paying particular attention to a determination of the transformed transverse magnetic vector potential, ${\mathsc{A}}_{T}^{m}$, and its positive-frequency part.

An array of nanomagnets in a kagome lattice structure should support the creation and separation of oppositely charged monopoles, which are connected by Dirac strings of flipped dipoles. And indeed, such monopoles and their Dirac strings can be observed at room temperature.

The magnetic monopole is a hypothetical particle, which is an important field configuration in many Grand Unified Theories, and whose mass may vary from 104 to 1018 GeV. The quantization of magnetic charge derived by Dirac in 1931 suggests the heavy ionization nature of magnetic monopoles. The NO$\nu$A collaboration is using the NO$\nu$A far detector - the largest scintillator detector in the world - to search for signals of relic monopoles that might have been produced in the early universe. To achieve this, we have simulated the propagation of non-relativistic monopoles with a uniform distribution of velocity within the range 10-4≲β≲ 0.2 , and the corresponding detector response. To record potential monopole-like events, we have developed a data-driven trigger system with two monopole triggers (the fast monopole trigger and the slow monopole trigger). This analysis focuses on the data recorded by the fast monopole trigger, which is sensitive to monopoles with velocity β ≳ 10-3. We have performed a blind analysis on a total live time of 1628.2 hours of data recorded by the fast monopole trigger. We present the updated flux limits with 90% CL (confidence level) as a function of monopole mass and velocity.

A quantum field theory of magnetic and electric charge is constructed. It is verified to be relativistically invariant in consequence of the charge quantization condition $\frac{\mathrm{eg}}{\ensuremath{\hbar}c}=n$, an integer. This is more restrictive than Dirac's condition, which would also allow half-integral values.

Magnetic monopoles have a long history of theoretical predictions and experimental searches, carrying direct implications for fundamental concepts such as electric charge quantization. We analyze in detail for the first time magnetic monopole production from collisions of cosmic rays bombarding the atmosphere. This source of monopoles is independent of cosmology, has been active throughout Earth's history, and supplies an irreducible monopole flux for all terrestrial experiments. Using results for robust atmospheric fixed target experiment flux of monopoles, we systematically establish direct comparisons of previous ambient monopole searches with monopole searches at particle colliders and set leading limits on magnetic monopole production in the ∼5-100 TeV mass range.