Fock Spaces, Landau Operators and the Regular Solutions of time-harmonic Maxwell equations

We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode $n-$dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schr\"odinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.Comment: Exposition improved; Some typos corrected; Accepted for publication in J.Phys.A (February 2011). http://www.mat.uc.pt/preprints/ps/p1047.pd

Recent progress on the description of relativistic spin: vector model of spinning particle and rotating body with gravimagnetic moment in General Relativity

We review the recent results on development of vector models of spin and apply them to study the influence of spin-field interaction on the trajectory and precession of a spinning particle in external gravitational and electromagnetic fields. The formalism is developed starting from the Lagrangian variational problem, which implies both equations of motion and constraints which should be presented in a model of spinning particle. We present a detailed analysis of the resulting theory and show that it has reasonable properties on both classical and quantum level. We describe a number of applications and show how the vector model clarifies some issues presented in theoretical description of a relativistic spin: A) One-particle relativistic quantum mechanics with positive energies and its relation with the Dirac equation and with relativistic {\it Zitterbewegung}; B) Spin-induced non commutativity and the problem of covariant formalism; C) Three-dimensional acceleration consistent with coordinate-independence of the speed of light in general relativity and rainbow geometry seen by spinning particle; D) Paradoxical behavior of the Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body in ultra relativistic limit, and equations with improved behavior.Comment: Invited review article for the Journal "Advances in Mathematical Physics". Based on the recent works: arXiv:1312.6247, arXiv:1406.6715, arXiv:1409.4756, arXiv:1509.05357, arXiv:1511.00645, arXiv:1609.00043. 61 pages, 3 figure

Quantum mechanics of an electron in a homogeneous magnetic field and a singular magnetic flux tube

The eigenvalue problem of the Hamiltonian of an electron confined to a plane and subjected to a perpendicular time-independent magnetic field which is the sum of a homogeneous field and an additional field contributed by a singular flux tube, i.e. of zero width, is investigated. Since both a direct approach based on distribution-valued operators and a limit process starting from a non-singular flux tube, i.e. of finite size, fail, an alternative method is applied leading to consistent results. An essential feature is quantum mechanical supersymmetry at g=2 which imposes, by proper representation, the correct choice of "boundary conditions". The corresponding representation of the Hilbert space in coordinate space differs from the usual space of square-integrable 2-spinors, entailing other unusual properties. The analysis is extended to $g\ne 2$ so that supersymmetry is explicitly broken. Finally, the singular Aharonov-Bohm system with the same amount of singular flux is analysed by making use of the fact that the Hilbert space must be the same.Comment: 23 pages, LaTeX, minor change

Random Network Models and Quantum Phase Transitions in Two Dimensions

An overview of the random network model invented by Chalker and Coddington, and its generalizations, is provided. After a short introduction into the physics of the Integer Quantum Hall Effect, which historically has been the motivation for introducing the network model, the percolation model for electrons in spatial dimension 2 in a strong perpendicular magnetic field and a spatially correlated random potential is described. Based on this, the network model is established, using the concepts of percolating probability amplitude and tunneling. Its localization properties and its behavior at the critical point are discussed including a short survey on the statistics of energy levels and wave function amplitudes. Magneto-transport is reviewed with emphasis on some new results on conductance distributions. Generalizations are performed by establishing equivalent Hamiltonians. In particular, the significance of mappings to the Dirac model and the two dimensional Ising model are discussed. A description of renormalization group treatments is given. The classification of two dimensional random systems according to their symmetries is outlined. This provides access to the complete set of quantum phase transitions like the thermal Hall transition and the spin quantum Hall transition in two dimension. The supersymmetric effective field theory for the critical properties of network models is formulated. The network model is extended to higher dimensions including remarks on the chiral metal phase at the surface of a multi-layer quantum Hall system.Comment: 176 pages, final version, references correcte

Haldane limits via Lagrangian embeddings

In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(N)/U(1)^N. We discuss possible other future applications of Lagrangian/isotropic embeddings in this context.Comment: 29 pages, 2 figure

Ultra-relativistic spinning particle and a rotating body in external fields

We use the vector model of spinning particle to analyze the influence of spin-field coupling on the particle's trajectory in ultra-relativistic regime. The Lagrangian with minimal spin-gravity interaction yields the equations equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations of a rotating body. We show that they have unsatisfactory behavior in the ultra-relativistic limit. In particular, three-dimensional acceleration of the particle increases with velocity and becomes infinite in the ultra-relativistic limit. The reason is that in the equation for trajectory emerges the term which can be thought as an effective metric generated by the minimal spin-gravity coupling. Therefore we examine the non-minimal interaction through the gravimagnetic moment $\kappa$, and show that the theory with $\kappa=1$ is free of the problems detected in MPTD-equations. Hence the non-minimally interacting theory seem more promising candidate for description of a relativistic rotating body in general relativity. The Lagrangian for the particle in an arbitrary electromagnetic field in Minkowski space leads to generalized Frenkel and Bargmann-Michel-Telegdi equations. The particle with magnetic moment in electromagnetic field and the particle with gravimagnetic moment in gravitational field have very similar structure of equations of motion. In particular, the spin-electromagnetic coupling also produces an effective metric for the particle with anomalous magnetic moment. If we use the usual special-relativity notions for time and distance, then the critical speed, which the particle cannot exceed during its evolution in electromagnetic field, is different from the speed of light. This can be corrected assuming that the three-dimensional geometry should be defined with respect to the effective metric.Comment: 34 pages, close to published version. arXiv admin note: text overlap with arXiv:1509.0492

Ultra-relativistic spinning particle and a rotating body in external fields

We use the vector model of spinning particle to analyze the influence of spin-field coupling on the particle's trajectory in ultra-relativistic regime. The Lagrangian with minimal spin-gravity interaction yields the equations equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations of a rotating body. We show that they have unsatisfactory behavior in the ultra-relativistic limit. In particular, three-dimensional acceleration of the particle increases with velocity and becomes infinite in the ultra-relativistic limit. The reason is that in the equation for trajectory emerges the term which can be thought as an effective metric generated by the minimal spin-gravity coupling. Therefore we examine the non-minimal interaction through the gravimagnetic moment $\kappa$, and show that the theory with $\kappa=1$ is free of the problems detected in MPTD-equations. Hence the non-minimally interacting theory seem more promising candidate for description of a relativistic rotating body in general relativity. The Lagrangian for the particle in an arbitrary electromagnetic field in Minkowski space leads to generalized Frenkel and Bargmann-Michel-Telegdi equations. The particle with magnetic moment in electromagnetic field and the particle with gravimagnetic moment in gravitational field have very similar structure of equations of motion. In particular, the spin-electromagnetic coupling also produces an effective metric for the particle with anomalous magnetic moment. If we use the usual special-relativity notions for time and distance, then the critical speed, which the particle cannot exceed during its evolution in electromagnetic field, is different from the speed of light. This can be corrected assuming that the three-dimensional geometry should be defined with respect to the effective metric.Comment: 34 pages, close to published version. arXiv admin note: text overlap with arXiv:1509.0492