Gauge formulation for magnetic monopoles from generalized gauge fields

The model of magnetic monopoles that was proposed by Paul Dirac in 1931 has long been a subject of theoretical interest in physics because of its potential to explain the quantization of electric charge. While much attention has been given to non-Dirac monopoles, Dirac's model, which involves an infinitely thin solenoid known as a Dirac string, presents subtleties in the interaction between monopoles. In this paper, we show that the force between two Dirac monopoles obeys a Coulomb-like interaction law. This derivation offers an instructive exercise in fundamental electromagnetism concepts and is appropriate for undergraduate and early graduate-level students.

Monopoles, duality, triality

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.

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.

We construct a symplectic realisation of the twisted Poisson structure on the phase space of an electric charge in the background of an arbitrary smooth magnetic monopole density in three dimensions. We use the extended phase space variables to study the classical and quantum dynamics of charged particles in arbitrary magnetic fields by constructing a suitable Hamiltonian that reproduces the Lorentz force law for the physical degrees of freedom. In the source-free case the auxiliary variables can be eliminated via Hamiltonian reduction, while for non-zero monopole densities they are necessary for a consistent formulation and are related to the extra degrees of freedom usually required in the Hamiltonian description of dissipative systems. We obtain new perspectives on the dynamics of dyons and motion in the field of a Dirac monopole, which can be formulated without Dirac strings. We compare our associative phase space formalism with the approach based on nonassociative quantum mechanics, reproducing extended versions of the characteristic translation group three-cocycles and minimal momentum space volumes, and prove that the two approaches are formally equivalent. We also comment on the implications of our symplectic realisation in the dual framework of non-geometric string theory and double field theory.

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.

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.

Triality monopoles, hadronic vortices and the Yang-Mills connection

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.

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.

By using an appropriate transformation, it was shown that the quantum system of four-dimensional (4D) simple harmonic oscillator can describe the motion of a charged particle in the presence of a magnetic monopole field. It was shown that the Dirac magnetic monopole has the hidden algebra of U(1) symmetry and by reducing the dimensions of space, the U(1) × U(1) dynamical group for 4D harmonic oscillator quantum system was obtained. Using the group representation and based on explicit solution of the obtained differential equation, the spectrum of system was calculated.

It is well known that the Dirac monopole solution with the U(1) gauge group embedded into the group SU(2) is equivalent to the SU(2) Wu–Yang point monopole solution having no Dirac string singularity. We consider a multicenter configuration of m Dirac monopoles and n antimonopoles and its embedding into SU(2) gauge theory. Using geometric methods, we construct an explicit solution of the SU(2) Yang–Mills equations which generalizes the Wu–Yang solution to the case of m monopoles and n antimonopoles located at arbitrary points in R3.