Monopoles without strings: a conflict between the one-photon condition and duality invariance

We prove a new no-go theorem in the Dirac-algebra formulation of generalized electromagnetic theory, which includes magnetic monopoles and uses two potentialsA andM : It is impossible to construct a Lagrangian which is duality invariant and satisfies the one-photon assumption, from which Maxwell's equations and the equations of motion can be derived. Such a Lagrangian can be found only if either duality invariance or the one-photon assumption is sacrificed. These constraints as well as others discussed here are based on recently published results on monopoles without strings in the Dirac algebra, but they do not arise from any artificial restrictions in the Dirac-algebra formulation.     

Magnetic monopoles and non-Abelian gauge theories

We review some properties of magnetic monopoles in non-Abelian gauge theories. Removal of Dirac string singularities and generalizations of the Wu-Yang solution that follow from this procedure are described. A discussion of the possible relevance of monopoles in strong interaction models and their role in quark confinement schemes is given. The magnetic monopole soliton discovered by 't Hooft and Polyakov, the first order formalism developed by Bogomolny, and extensions of these ideas are illustrated.Work supported in part by the U.S. Energy Research and Development Administration under Contract No. EY-76-C-02-2232B^*000.     

Wave equation for a magnetic monopole

We show that there is room, in the Dirac equation, for a massless monopole. The basic idea is that the Dirac equation admits a second electromagnetic minimal coupling associated to the chiral gauge , which is only valid for a massless particle, but satisfies all the symmetry laws of a monopole. In the problem of the diffusion on a central electric field, we find the Poincaré integral and the Dirac relationeg/=n/2. The latter is deduced as a consequence of the fact (which is shown in this paper) thateg/c is the projection of the total angular momentum on the symmetry axis of the system formed by the monopole and the electric charge. Another important property is that a monopole and an antimonopole have opposite helicities (as for the neutrino), but do not have opposite charges: this precludes a vacuum magnetic polarization which would be analogous to the electric one, but allows us to imagine an aether made up of monopole-antimonopole pairs. The theory is then generalized on the basis of a nonlinear equation which is the most general invariant equation under the chiral gauge law. This equation admits solutions corresponding to massive monopoles, among which there are bradyons (i.e., ordinary massive particles) and tachyons. This equation is shown to be closely related to previous works initiated by Hermann Weyl, on Dirac's theory in the framework of general relativity. In conclusion, it is suggested that massless monopoles are perhaps excited states of the neutrino and that they may be produced in some weak interactions. Consequences on the solar activity are considered.     

Monopole charges for arbitrary compact gauge groups and Higgs fields in any representation

The topological invariants of monopoles are described for an arbitrary compact gauge groupG and Higgs field in any representation. The results generalize those obtained recently for compact and simply connectedG and in the adjoint representation. The cases when the residual symmetry group isH=U(1) orH=U(3) are worked out explicitly. This latter is needed to accommodate fractional electric charge with monopoles having one Dirac unit magnetic charge.The general theory is illustrated on the SU(5) monopole.     

电磁学与电动力学中的磁单极-Ⅱ

本系列文章一共4篇,在电磁学和电动力学框架内用尽量科普的方式分别介绍磁单极的若干奇特性质.本篇文章主要介绍狄拉克磁单极是如何展示矢量势的规范变换的.我们首先简要介绍规范变换与规范对称性及狄拉克磁单极与狄拉克弦,然后讨论狄拉克磁单极与规范变换的联系.我们显式演示狄拉克弦摆动产生的规范变换,弦摆动区域对场点所张的立体角正比于规范变换的变换函数.磁偶极子则可以由两个无穷靠近的正反狄拉克磁单极构成.相应两条狄拉克弦位置的变化都对应磁偶极子矢量势的规范变换,特别当两条弦重合时弦效应相互抵消,只剩下纯的磁偶极子.传统的由磁偶极子产生的矢量势的规范变换则可以图像化为组成磁偶极子的正反狄拉克磁单极的狄拉克弦的摆动.我们显式地计算了位于坐标原点弦为直线的狄拉克磁单极,并进一步构造了没有奇异的吴大峻-杨振宁磁单极.     

Singular monopoles from Cheshire bows

Singular monopoles are nonabelian monopoles with prescribed Dirac-type singularities. All of them are delivered by Nahm's construction. In practice, however, the effectiveness of the latter is limited to the cases of one or two singularities. We present an alternative construction of singular monopoles formulated in terms of Cheshire bows. To illustrate the advantages of our bow construction we obtain an explicit expression for one U(2)$U(2)$ gauge group monopole with any given number of singularities of Dirac type.     

G2-Monopoles with Singularities (Examples)

G₂-Monopoles are solutions to gauge theoretical equations on G₂-manifolds. If the G₂-manifolds under consideration are compact, then any irreducible G₂-monopole must have singularities. It is then important to understand which kind of singularities G₂-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.     

Coulomb scattering of an electron by a monopole

The Coulomb scattering of an electron by a magnetic monopole is analyzed using a lowest-order quantum perturbation approximation suggested by a two-potential Lagrangian form for classical electromagnetism, generalized through the use of spacetime algebra to include magnetic monopoles. Good agreement with existing conventional analyses of this problem is demonstrated.1. Work supported by Department of Energy contract DE-AC03-76SF00515.2. The idea to employ spacetime algebra (sometimes called Dirac algebra) to incorporate magnetic monopoles into classical electromagnetic theory was proposed by de Faria-Rosaet al. [3].3. This is a factori difference between the definition of 5 by Eq. (3) and that by Bjorken and Drell [6]. Since a cross section (without interference terms) is being calculated, we can ignore this distinction.     

Heavy and light monopoles in magnetic reversion in artificial spin ice

This work makes a theoretical study of the dynamics of emergent elemental excitations in artificial spin ice systems with hexagonal geometry during the magnetic reversion of the system. The magnetic and physical parameters of the nanoislands that form the array are considered as variables in the study. The parameters considered are: the energy barrier for the inversion of each nanoisland, the magnetic moment of the nanomagnets and the possible disorder in the sample. Our results show that the reversion dynamic presents two distinct mechanisms of magnetic reversion, with different elemental excitations for each mechanism. The first mechanism presents a reversion with the appearance of magnetic monopoles that do not move in the samples (heavy monopoles) and the absence of Dirac chains. In the other mechanism elemental magnetic excitations (light monopoles) appear that move great distances in the sample, giving rise to extensive Dirac chains during the magnetic reversion.     

Geometrical aspects of magnetic monopoles

The possible topological structures of elementary particles have been investigated to explore the possibility of the existence of magnetic monopoles. It is pointed out that when an elementary charged particle is depicted as an extended body such that the orientation of the internal space (internal helicity) defines the fermion number, the global conservation of this does not allow the existence of a magnetic monopole. Again it is argued that when anisotropy is introduced in the microlocal space-time depicting the internal space of hadrons, this gives rise to the internal symmetry algebra and no non-Abelian gauge fields and Higgs scalars are necessary to have a grand unified scheme of interactions. This avoids theSU ₂ and GUT monopoles. Besides, in this formalism, baryon number corresponds to the orientation or internal helicity of the composite system and the global conservation of this quantum number is found to be a consequence of Lorentz invariance. This forbids the existence of any sort of cosmological monopole in this Lorentz invariant Universe.     

Dirac and Maxwell equations in the Clifford and spin-Clifford bundles

We show how to write the Dirac and the generalized Maxwell equations (including monopoles) in the Clifford and spin-Clifford bundles (of differential forms) over space-time (either of signaturep=1,q=3 orp=3,q=1). In our approach Dirac and Maxwell fields are represented by objects of the same mathematical nature and the Dirac and Maxwell equations can then be directly compared. We show also that all presentations of the Maxwell equations in (matrix) Dirac-like spinor form appearing in the literature can be obtained by choosing particular global idempotents in the bundles referred to above. We investigate also the transformation laws under the action of the Lorentz group of Dirac and Maxwell fields (defined as algebraic spinor sections of the Clifford or spin-Clifford bundles), clearing up several misunderstandings and misconceptions found in the literature. Among the many new results, we exhibit a factorization of the Maxwell field into two-component spinor fields (Weyl spinors), which is important.     

Magnetic charges,inertia, and arrow of time

The prerelativistic concept of inertial mass (as opposed to gravitational mass) is reconsidered in view of a possible relationship between inertia and magnetic (mass) monopoles. Assuming that such fictitious (topological) charges could have developed in the chaotic early cosmology, a physical principle is suggested, based on dissipation of topological charges and decoupling of interactions, which could have governed the onset of inertia and of the arrow of time, and controlled the critical balance between mass density and expansion rate in the FRW universe. In view of the recent accomplishments in the detection of Dirac monopoles, a generalization of the Eötvos experiment is proposed which could shed light on the grand unification regime. A comment is given on the issue of relating the psychological and the cosmological arrows of time.     

Monopoles and family-replicated unification

The present theory is based on the assumption that, at very small (Planck scale) distances our spacetime is discrete, and this discreteness influences the Planck scale physics. Considering our (3+1)-dimensional spacetime as a regular hypercubic lattice with a parameter a=λ_Pl, where λ_Pl is the Planck length, we have investigated a role of lattice artifact monopoles, which is essential near the Planck scale if the family-replicated gauge group model (FRGGM) is an extension of the Standard Model (SM) at high energies. It was shown that monopoles have N times smaller magnetic charge in the FRGGM than in the SM (N is the number of families in the FRGGM). These monopoles can give an additional contribution to β functions of the renormalization-group equations for the running fine structure constants α_i(μ) (i=1, 2, 3 correspond to the U(1), SU(2), and SU(3) gauge groups of the SM). We have used the Dirac relation for renormalized electric and magnetic charges. Also, we have estimated the enlargement of a number of fermions in the FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. The different role of monopoles in the vicinity of the Planck scale gives rise either to anti-GUT or to the new possibility of unification of gauge interactions (including gravity) at the scale μ_GUT≈10^18.4 GeV. We discussed the possibility of the [SU(5)]³ SUSY or [SO(10)]³ SUSY unifications.     

Monopolium production from photon fusion at the Large Hadron Collider

Magnetic monopoles have attracted the attention of physicists since the founding of the electromagnetic theory. Their search has been a constant endeavor which was intensified when Dirac established the relation between the existence of monopoles and charge quantization. However, these searches have been unsuccessful. We have recently proposed that monopolium, a monopole–antimonopole bound state, so strongly bound that it has a relatively small mass, could be easier to find and become an indirect but clear signature for the existence of magnetic monopoles. Here we extend our previous analysis for its production to two photon fusion at LHC energies.     

Magnetic monopoles with Zn charges

Under some circumstances magnetic monopoles can be formed with charges which are additive modulo n. We speak of these as monopoles with Z_n charges. The conditions under which such monopoles arise are examined, and illustrated with some specific examples. These monopoles can be produced at intermediate stages of symmetry breakdown and converted at subsequent stages to ones with ordinary additive charges, specified by the group Z ≡ Z_∞. Simultaneously, new light monopoles with n times the Dirac charge can be produced. It is argued that such monopoles may be exempt from the most stringent cosmological bounds, and hence may be easiest to observe.     

Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form.     

Fermions and spherically symmetric monopoles

The interaction of fermions with monopoles is analyzed for general spherically symmetric monopoles of arbitrary strength and massless fermions in arbitrary representations of the gauge group. The results are based on a solution to the Dirac equation in the field of the monopole, valid in the point limit and in all partial waves. The partial waves in which non-conservation of global charges may occur are identified. A two-dimensional lagrangian is derived, which includes the Coulomb interactions generated by the complete abelian subgroup of the unbroken gauge group. The fermions are represented by a set of doublets with a boundary condition that reflects the structure of the core. We derive a simple formula for the conservation laws of this model which determine all scattering processes completely. The results have a consistent interpretation in terms of anomalies and field configurations with non-trivial winding number. We give an explicit construction of the zero modes for all helicity-violating multi-fermion condensates. The formalism is applied to several higher strength monopoles in SO(10) based models.     

Dirac monopoles for general gauge theories

This paper develops a non-local potential formalism for general gauge theories. With the help of this mathematical apparatus an argument for quantisation of the generalised charge is given, assuming that the Dirac monopoles are present.     

Dirac monopoles embedded in <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) gauge theory with the θ term

Dirac monopoles embedded in SU(N) gauge theory with the θ term are considered. For θ=4πM (where M is half-integer and integer for N=2 and N>2, respectively), these monopoles acquire an SU(N) charge due to the θ term and become dyons. They belong to various (but not any) irreducible representations of the SU(N) group. The admissible representations are listed. Their minimum dimension increases with N. The basic result of the study is the representation of the partition function of any SU(N) model involving the θ term and complemented by singular gauge fields corresponding to the indicated monopoles in the form of a vacuum average of the product of Wilson loops viewed along the world lines of the monopoles. This vacuum average must be calculated in the corresponding model without the θ term.