Charge-monopole System Research Articles

Perfectly spherical Bloch hyper-spheres from quantum matrix geometry

Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with exact spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties, such as von Neumann entropies and Bures quantum metrics.

Form-invariant solution to quantum state on the sphere

This paper investigates the quantum states that emerge from the transformation design of conformal mapping on the two-dimensional sphere. Three results are reported. First, the construction of form-invariant spherical harmonics labelled by the fractional quantum number through a scalar potential interaction is given. Second, the form-invariant equation of the charge-monopole system is studied. Rather than the half-integer classification of the monopole harmonics, the quantization of the monopole harmonics here can be any fractional number specified by the conformal index. The gauge equivalent condition of the vector potentials which result in the invariant equation shows that the monopole field and the quantization condition of the pole strength due to Dirac can be generalized to more general vector fields and values in the conformal space. Finally, we explore the quadratic conformal image of the charged particle coupling to the constant monopole field on the sphere. It is shown that the lowest order approximation of the image is the magnetic Hooke-Newton transmutation.

Mapping between charge-monopole and position-dependent mass systems

We study the non-relativistic charge-monopole system when the charged particle has a position-dependent mass written as M(r) = m0rw. The angular wave functions are the well-known monopole harmonics, and the radial ones are ordinary Bessel functions which depend on the magnetic and electric charge product as well as on the w parameter. We investigate mappings—approximate and exact—between the charge-monopole system with constant mass and the charge with a position-dependent mass solving the position-dependent mass Schrödinger equation for the mass distribution.

Dirac potential in the Doebner–Goldin equation

We study a dissipative quantum system which is described by the Doebner–Goldin equation (DGE) model. For time-independent states, the new three-dimensional analytical solutions of the DGE are obtained by binding the vertical relation of velocity and the gradient of density in the system, when the form of a central potential such as hard core or harmonic oscillator is suggested. Through the gauge-invariant parameters which characterize the physical nature of the dissipation, we find a novel set of gauge-invariant parameters which show that the Galilean invariance is broken in this system. Moreover, a subfamily of the DGE can be obtained after a gauge transformation, which describes a dissipative quantum system with the conserved Galilean invariance. It is interesting that this dissipative quantum system is completely equivalent to a charge-monopole system, in which the Dirac potential is supplied with the nonlinear terms and two cases of the velocity potential. Especially, the two gauge potentials given by Wu and Yang emerge from solving the DGE as two cases in our approach. The results not only present some new physical comprehension of the dissipative quantum system, but also might shed light on the Dirac monopole potential, in the sense that the partition into south and north hemisphere is avoided in our new solutions.

DISAPPEARANCE OF SCHWINGER'S STRING AT THE CHARGE–MONOPOLE "MOLECULE"

An equivalence of total angular momentum operator of charge–monopole system to the momentum operator of a symmetrical quantum top is observed. This explicitly shows the string independence of Dirac's quantization condition leading to disappearance of Schwinger's string and reveals some properties of diatomic molecule for this system.

A monopole near a black hole

A striking property of an electric charge near a magnetic pole is that the system possesses angular momentum even when both the electric and the magnetic charges are at rest. The angular momentum is proportional to the product of the charges and independent of their distance. We analyze the effect of bringing gravitation into this remarkable system. To this end, we study an electric charge held at rest outside a magnetically charged black hole. We find that even if the electric charge is treated as a perturbation on a spherically symmetric magnetic Reissner-Nordstrom hole, the geometry at large distances is that of a magnetic Kerr-Newman black hole. When the charge approaches the horizon and crosses it, the exterior geometry becomes that of a Kerr-Newman hole, with electric and magnetic charges and with total angular momentum given by the standard value for a charged monopole pair. Thus, in accordance with the "no-hair theorem," once the charge is captured by the black hole, the angular momentum associated with the charge monopole system loses all traces of its exotic origin and is perceived from the outside as common rotation. It is argued that a similar analysis performed on Taub-NUT space should give the same result.

Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence

The nonlinear realization of conformal so(2, d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS d+1 of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS 2 × S d−1 . The list of nonrelativistic systems “unified” by such a correspondence comprises the conformal mechanics model, the planar charge–vortex and three-dimensional charge–monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner–Nordström black hole.

Monopole Chern–Simons term: charge–monopole system as a particle with spin

The topological nature of Chern-Simons term describing the interaction of a charge with magnetic monopole is manifested in two ways: it changes the plane dynamical geometry of a free particle for the cone dynamical geometry without distorting the free (geodesic) character of the motion, and in the limit of zero charge's mass it describes a spin system. This observation allows us to interpret the charge-monopole system alternatively as a free particle of fixed spin with translational and spin degrees of freedom interacting via the helicity constraint, or as a symmetric spinning top with dynamical moment of inertia and "isospin" U(1) gauge symmetry, or as a system with higher derivatives. The last interpretation is used to get the twistor formulation of the system. We show that the reparametrization and scale invariant monopole Chern-Simons term supplied with the kinetic term of the same invariance gives rise to the alternative description for the spin, which is related to the charge-monopole system in a spherical geometry. The relationship between the charge-monopole system and (2+1)-dimensional anyon is discussed in the light of the obtained results.

Parity nonconservation in electromagnetic systems

It is shown that a correlated motion of a charge-monopole system emits radiation that violates parity conservation.

Path integral evaluation of a charged particle interacting with a Kaluza-Klein monopole

The propagator for the charge-monopole system in Kaluza-Klein theory is evaluated exactly by path integration. The monopole harmonics are obtained and the radial path integral reduces to that of an isotropic harmonic oscillator in three dimensions. For distances very much greater than the radius of the compactified fifth dimension, the radial Green function is found in closed form and an energy spectrum is obtained.

Comments on the charge-monopole canonical formalism

A recently published canonical formalism of a charge-monopole system written by means of Clifford algebras is discussed. It is shown that the introduction of the Lorentz force must be accompanied by the removal of the pseudo-scalar terms from the lagrangian. Several conclusions follow.

MASSLESS SPINNING PARTICLES IN ALL DIMENSIONS AND NOVEL MAGNETIC MONOPOLES

There is a well-known description of a massless particle with spin in four dimensions in terms of position and momentum operators. The charge-monopole system can be obtained therefrom by a simple canonical transformation and a suitable choice of the Hamiltonian. In this paper, we generalize this description of a massless spinning particle to higher dimensions and then use the aforementioned transformation to define novel charge-monopole systems which are non-Abelian generalizations of the Abelian system of Dirac. The Lagrangian description of all these systems and their canonical quantization are also discussed. The analysis reveals the existence of symplectic structures which are likely to be original in the context of particle physics and which incidentally are capable of generalization to string theories. It further gives rise to new monopole fields which may be of use in the context of dimensional compactification.

Geometry and charge-monopole systems

A covariant formulation of a charge-monopole system in a general metric is discussed. It is shown that electromagnetic fields measured by means of a charge are not identical to the corresponding fields obtained by a monopole. Related problems of the canonical formulation of the system are pointed out.

Phase factor for charge-monopole system

Calculates the Green function for a charge-monopole system and shows that a non-integrable phase factor emerges without using the classical notion of path.

Quantum symmetries from quantum phases, fermions from bosons, A [formula omitted]2 anomally and Galilean invariance

In the presence of a topologically nontrivial Wess-Zumino term, the wave functions are not functions on the configuration space Q of the system; instead they are functions on a U(1) bundle Q̂ over this configuration space. The space Q̂ is obtained by associating circles to points of Q in a suitable way. If a group G operates on Q and is a symmetry of the action, then it can happen that G does not operate on Q̂ and on wave functions. Instead, the “quantum mechanical” symmetry group G QM which does act on Q̂, and thereby wave functions, is in general a central extension of G; it may not even contain G as a subgroup. In this paper, we give an explicit and elementary construction of G QM. The construction is applied to the charge-monopole system and the Skyrme solitons, and used to explain why the states of these systems may be spinorial in nature even though the action contains only tensorial (integral spin) variables. The well known Z 2 anomaly of certain SU(2) gauge theories is also established using a modification of this construction. When a classical system with Galilei invariance is quantized, the quantum theoretic Galilei group G QM is different from the classical Galilei group G. Again the former is a central extension of the latter even though there is no Wess-Zumino term in this case. We present a novel explanation of this fact as well by constructing the lift of G from (galilean) spacetime Q to the appropriate U(1) bundle Q̂.

The O(3,1) symmetry problem of the charge–monopole interaction

The question of whether there exists a smooth, time-independent conserved vector observable in the classical mechanics of the charge–monopole interaction that spans an O(3,1) Lie algebra together with the angular momentum or not is examined. It turns out that any candidate for such dynamical symmetry algebra has hard singularities at that part of the phase space that corresponds to charge–monopole collisions. In the course of the investigation we use the transformation of Boulware et al. [D. G. Boulware, L. S. Brown, R. N. Cahn, S. D. Ellis, and C. Lee, Phys. Rev. D 14, 2708 (1976)] relating the charge–monopole system to a point mass moving in an inverse square potential. This transformation is shown to be a complete isomorphism between the scattering parts of the related Hamiltonian systems; its global behavior is described in terms of an U(1) principal fiber bundle of nontrivial topology. Several remarks on the symmetries of various monopole problems are made, e.g., the most general O(4) symmetry algebra is given for a special form of the charge–dyon interaction.

Bound states, resonances, and symmetries of a neutral Dirac particle with anomalous magnetic moment, coupled to a fixed monopole

For the system consisting of a neutral Dirac particle with anomalous magnetic moment, interacting with a fixed magnetic monopole, zero-energy bound states are constructed for each possible value of the total angular momentum. Results of Kazama and Yang for the charge–monopole system are used to deduce the existence of other bound states for this system, when the mass of the bound particle is nonzero. In the zero-mass case, there are no other bound states, but there are resonant states, and these are determined exactly. A noncompact, so(3,2) symmetry algebra of the zero-energy bound states is given for the finite-mass case and for the zero-mass case. In each case the infinite number of such states is associated with an irreducible Majorana representation of the algebra.

A duality between massless particles and a charge-monopole system

It is pointed out that the same realisation of a unitary representation of SO(4, 2) from the ladder series is used in the coordinate-space description of a charge-monopole system, (or more generally a dyon-dyon system) and in the canonical momentum-space description of a massless particle. Therefore, in the latter case a momentum-space analogue appears for the monopole vector potential, complete with its Dirac string singularity. Analogues of gauge transformations relate equivalent realisations with different locations of the momentum-space string. Quantisation of helicity replaces quantisation for the product of electric and magnetic charge. The problem of localising a charge on a monopole string is related to recent work by Flato et al. [5] on the localisability of a massless particle in momentum-space. Further, the multi-component form of the generators of SO(4, 2) for a massless particle has a dual, in coordinate space, corresponding to a charge-monopole system for a monopole of the Yang-Mills type. The question of the interpretation of the momentum analogue of the monopole field is raised.

On the dynamical group of the charge-monopole system

The massless Poincare algebra recently discussed is subalgebra of the dynamical algebra O(4,2) of the charge-monopole system. Additional generators are given and the interpretation of mass zero representation is elucidated.

Dirac's monopoles and action-at-a-distance theory: Classical dynamics

We redefine Dirac's action for charge-monopole systems as a certain limit of Wu-Yang's action in a way which is similar to, but different from, the work of Brandt and Primack. The lagrangian theory thus obtained reproduces, in a simple and straightforward manner, all of the Maxwell-Lorentz equations without recourse to Dirac's veto. It also becomes clear that the action-at-a-distance formalism is especially suited for the description of such systems. An action, equations of motion and conserved linear and angular momenta are explicitly given, which do not depend on the string. The existence of a charge-monopole bound state is suggested.