On the gauge equivalence of the Landau-Lifshitz and the nonlinear Schrödinger equations on symmetric spaces

The gauge equivalence between a generalized Heisenberg spin chain (G/H) in the classical and continuum limit and the nonlinear Schrödinger equation (NLSE), with special attention to noncompact groups, is established. It has been demonstrated that noncompact groups allow a richer spectrum of possible reductions of the Heisenberg system to the NLSE. Some specialities of the model with nontrivial boundary conditions are discussed. The gauge equivalence between single-axis anisotropic Landau-Lifshitz equations (LLE) and isotropic LLE is briefly discussed.     

Supersymmetric Dirac particles in Riemann-Cartan space-time

A natural extension of the supersymmetric model of Di Vecchia and Ravndal yields a nontrivial coupling of classical spinning particles to torsion in a Riemann-Cartan geometry. The equations of motion implied by this model coincide with a consistent classical limit of the Heisenberg equations derived from the minimally coupled Dirac equation. Conversely, the latter equation is shown to arise from canonical quantization of the classical system. The Heisenberg equations are obtained exact in all powers of and thus complete the partial results of previous WKB calculations. We touch also on such matters of principle as the mathematical realization of anticommuting variables, the physical interpretation of supersymmetry transformations, and the effective variability of rest mass.     

Gauge Formulation for Two Potential Theory of Dyons

Dual electrodynamics and corresponding Maxwell’s equations (in the presence of monopole only) are revisited from the symmetry of duality and gauge invariance. Accordingly, the manifestly covariant, dual symmetric and gauge invariant two potential theory of generalized electromagnetic fields of dyons has been developed consistently from U(1)×U(1) gauge symmetry. Corresponding field equations and equation of motion are derived from Lagrangian formulation adopted for U(1)×U(1) gauge symmetry for the justification of two four potentials of dyons.     

Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein-Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.      

Wigner elementary particle in an external homogeneous field

A covariant Hamiltonian is proposed which permits to describe in the Heisenberg picture the motion of a Wigner elementary particle in a homogeneous electromagnetic field. More precisely, at any time, the elementary particle is in a state associated with a given irreducible representation of the Poincaré group. As a remarkable result, the spin motion is shown to be governed by the Thomas-Bargmann-Michel-Telegdi equation. Also the Galilean limit is discussed.     

A Particle-Picture Approach to Anomalies in Chiral Gauge Theories

The system of a chiral fermion field coupled to a background gauge field is considered. By taking what we call the particle picture and carefully defining the S-matrix in the Heisenberg picture, we investigate anomalous phenomena in this system. It is shown by explicit calculations that the gauge-field configuration with nonvanishing topological-charge causes anomalous production of particles that is directly responsible for the chiral U(1) anomaly. Unlike the chiral U(1) anomaly, the gauge anomaly, that is, gauge non-invariance of the S-matrix is a problem that arises in the phase of the S-matrix. It is shown that this phase is related to the freedom existing in the quantization method, and that a suitably chosen phase which of course is consistent with the equation of motion can remove the gauge anomaly. Finally, a modified form of path-integral quantization for this system is proposed.     

Field theories on a lattice

Lattice field theories are described as a way to regularize continuum quantum field theories. They are obtained by replacing ordinary space time by a lattice, space time derivatives by suitable differences and Minkowski by Euclidean space. The connection between a quantum field theory isd space dimension and classical statistical mechanics in (d+1) dimensions is brought outvia elementary examples. The problem of regaining the continuum limit and of handling nonabelian gauge theories are briefly discussed.     

Propagation of density perturbations in the deDonder gauge on a gravitational background field II

For the problem of propagation of density waves in a preexisting gravitational field, the advantages of the deDonder gauge over the commonly used synchronous gauge are outlined. In a background matter substratum withp as equation of state there are in the deDonder gauge only decaying modes of the perturbation density contrast with arbitrary large spatial extension, whereas in synchronous gauge there is one growing mode (calculated for vanishing spatial divergency of the perturbation in the 4-velocity, i.e.,usk(1),j/j0). The calculations are extended to the case of finite spatial extensions of the density perturbations. This is done by expanding all perturbations in a power series of the inverse square of the speed of light with the result of getting a recursive set of differential equations in both gauges for the equation of motion of the density perturbations. The lowest orders of this equation are the same in both gauges, but only in the deDonder gauge is the correct Newtonian limit of propagation of density waves in an expanding universe obtained. The correction by the next higher orders in the deDonder gauge are dependent explicitly on the spatial extension of the perturbations; whereas in synchronous gauge this is not the case. For attaining the Newtonian limit this dependence is a necessary condition. At appropriately large spatial extensions, however exact, this dependence in deDonder gauge leads ultimately to a decaying of density contrast modes growing in zeroth order (at least forp=0 andp/3 as equations of state for the background matter substratum). Hence, there are upper boundaries in the spatial extensions of instable growing modes of density contrast.     

On the evolution of higher dimensional Heisenberg continuum spin systems

We consider the evolution of a classical Heisenberg ferromagnetic spin chain in its continuum limit in higher spatial dimensions. It is shown that the evolution of a radially symmetric chain could be identified with the motion of a helical space curve as in the linear case. The resulting invariant equations for the curvature (radial energy density) and torsion (related to current density) are shown to be equivalent to a generalized nonlinear Schrödinger equation, similar to the one derived by Ruijgrok and Jurkiewicz recently. Equivalent linear equations as well as special static solutions of point singular type are obtained. Similarity solutions, a class of which belong to Riccati type, are discussed in detail. For general higher dimensions, a potentially useful formulation is presented: Under stereographic projection of the unit sphere of spin, the equation of motion takes a neater form even with the inclusion of anisotropic interactions. Classes of explicit solutions are reported in higher dimensions. Propagating spin waves, static spin waves of point singular nature and of finite energy in some cases are also discussed.     

Action principle and equations of motion for nonabelian monopoles

Monopoles in gauge theories have an intrinsic interaction with the gauge field. The definition of a monopole as a topological charge implies a certain constraint coupling the gauge field to the coordinates of a particle carrying that charge. Hence, even starting with the free action, the constraint will give equations of motion in which field and particle interact. Applied to electromagnetism, this procedure gives the Maxwell and Lorentz equations. In this paper, we apply the same idea to nonabelian monopoles to deduce their equations of motion which are otherwise unknown. To surmount certain technical difficulties connected with patching, loop space techniques are developed to solve the variational problem. A closed set of equations are obtained, which are analogous to the Maxwell and Lorentz equations, and bear also a formal resemblance to the Wong equations for a “classical” point source of Yang-Mills fields.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.     

Dynamical response functions due to pulse solitons in the isotropic ferromagnetic Heisenberg chain in an external magnetic field

Dynamical correlation functions are evaluated for the classical, isotropic, ferromagnetic Heisenberg chain in an external magnetic field. The calculation is approximate, using as its basis an ideal gas of ballistic pulse-soliton excitations which are known to be exact solutions to the spin equations of motion in the continuum limit. For the longitudinal correlation function (i.e. parallel to the applied field) a central peak structure is found which becomessplit at sufficientlylow temperatures and smears to higher frequence with increasing temperatures. This behaviour is contrasted with that found in the sine-Gordon and related solition systems. Comparisons with the results of numerical simulations of the Heisenberg chain are made, and the possible relevance to magnetic chain materials such as CsNiF₃ is discussed.This work has been performed under the auspices of the US DOE     

Field theory onR×S 3 topology. V:SU 2 gauge theory

A gauge theory on R×S ³ topology is developed. It is a generalization to the previously obtained field theory on R×S ³ topology and in which equations of motion were obtained for a scalar particle, a spin one-half particle, the electromagnetic field of magnetic moments, and a Shrödinger-type equation, as compared to ordinary field equations defined on a Minkowskian manifold. The new gauge field equations are presented and compared to the ordinary Yang-Mills field equations, and the mathematical and physical differences between them are discussed.On leave from Center for Theoretical Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.     

Inverse Variational Problem for Nonlinear Evolution Equations

The Helmholtz solution of the inverse problem for the variational calculus is used to study the analytic or Lagrangian structure of a number of nonlinear evolution equations. The quasilinear equations in the KdV hierarchy constitute a Lagrangian system. On the other hand, evolution equations with nonlinear dispersive terms (FNE) are non-Lagrangian. However, the method of Helmholtz can be judiciously exploited to construct Lagrangian system of such equations. In all cases the derived Lagrangians are gauge equivalent to those obtained earlier by the use of Hamilton’s variational principle supplemented by the methodology of integer-programming problem. The free Hamiltonian densities associated with the so-called gauge equivalent Lagrangians yield the equation of motion via a new canonical equation similar to that of Zakharov, Faddeev and Gardner. It is demonstrated that the Lagrangian system of FNE equations supports compacton solutions.PACS: 47.20.Ky; 42.81.Dp     

Extended Einstein–Cartan theory à la Diakonov: The field equations

Diakonov formulated a model of a primordial Dirac spinor field interacting gravitationally within the geometric framework of the Poincaré gauge theory (PGT). Thus, the gravitational field variables are the orthonormal coframe (tetrad) and the Lorentz connection. A simple gravitational gauge Lagrangian is the Einstein–Cartan choice proportional to the curvature scalar plus a cosmological term. In Diakonov?s model the coframe is eliminated by expressing it in terms of the primordial spinor. We derive the corresponding field equations for the first time. We extend the Diakonov model by additionally eliminating the Lorentz connection, but keeping local Lorentz covariance intact. Then, if we drop the Einstein–Cartan term in the Lagrangian, a nonlinear Heisenberg type spinor equation is recovered in the lowest approximation.     

On a integrable deformations of Heisenberg supermagnetic model

The Heisenberg supermagnet model which is the supersymmetric generalization of the Heisenberg ferromagnet model is an important integrable system. We consider the deformations of Heisenberg supermagnet model under the two constraint 1. S² = S for S ∈ USPL(2/1)/S(L(1/1) × U(1)) and 2. S² = 3S ? 2I S ∈ USPL(2/1)/S(U(2) × U(1)). By means of the gauge transformation, we construct the gauge equivalent counterparts, i.e., the super generalized Hirota equation and Gramman odd nonlinear Schrödinger equation.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

On the dynamics of a continuum spin system

For a one-dimensional system of classical spins with nearest neighbour Heisenberg interaction we derive the equation of motion for each three-dimensional spin vector. In the continuum limit where the spins lie dense on a line this set of equations reduces to a nonlinear partial differential equation. In addition to spin-wave solutions we obtain some other special solutions of this equation. In particular we find solitary waves having total energy localised in a finite region, with velocity of propagation inversely proportional to the width of this region. Solutions of still another type are shown to have a diffusive character. The stability of such solutions and the possibility of interaction of two or more solitary waves have not yet been studied.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.