International Journal of Geometric Methods in Modern Physics

Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

In this paper, we investigate revival and super revivals of nonlinear coherent states while generating these states through the interaction of coherent states of a generalized isotonic oscillator with the nonlinear media during time evolution. We construct the f-deformed generalized isotonic oscillator which is a non-isochronous partner of the generalized isotonic oscillator. We connect these two nonlinear oscillators through deformed ladder operators. The generalized isotonic oscillator possesses linear energy spectrum whereas f-deformed generalized isotonic oscillator exhibits nonlinear energy spectrum. The presence of the cubic nonlinearity in the f-deformed oscillator motivates us to study revivals, super and fractional revivals of coherent states which are nonlinearly evolved. We also investigate time-dependent expectation values of uncertainties in certain canonically conjugate variables and demonstrate that at revival and super revival times the uncertainty relation attains its minimum value.

In this paper, we discuss the dynamical behavior of a family of fractional differential systems associated to 3D Maxwell–Bloch top system. The stability analysis of equilibrium states of the controlled fractional 3D Maxwell–Bloch top system is studied.

The use of geometrical quantities and geometrical methods in the calculation and interpretation of physical systems have been very successful. Among others it has been shown how non-relativistic spin one half and massless relativistic spin one half with the corresponding Pauli matrices can be phrased geometrically. An analogous formulation for massive and relativistic spin one half was however missing. By the use of geometrical angular coordinates we construct a concrete realization of relativistic and massive spin one half. A Lagrange formulation is found, equations of motion are derived, and Lorentz invariance is discussed.

The object of this paper is to study spacetimes admitting W₂-curvature tensor. At first we prove that a W₂-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W₂-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W₂-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher-dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I₂(h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group D_h of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A₄ onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine W_a(A₄) can be decomposed into two orthogonal spaces whose point group is the dihedral group D₅ which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group W_a(A₃), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group W_a(A_n), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of W_a(A₄) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

The fundamental field equations in modified gravity (including general relativity; massive and bimetric theories; Hořava–Lifshitz (HL); Einstein–Finsler gravity extensions etc.) possess an important decoupling property with respect to nonholonomic frames with 2 (or 3) + 2 + 2 + ⋯ spacetime decompositions. This allows us to construct exact solutions with generic off-diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un-)broken symmetry parameters. Such nonholonomic configurations/models have a nice ultraviolet behavior and seem to be ghost-free and (super-)renormalizable in a sense of covariant and/or massive modifications of HL gravity. The apparent noncommutativity and breaking of Lorentz invariance by quantum effects can be encoded into fibers of noncommutative tangent Lorentz bundles for corresponding "partner" anisotropically induced theories. We show how the constructions can be extended to include conjectured covariant renormalizable models with massive graviton fields and effective Einstein fields with (non)commutative variables.

Considering a twisted version of the gravitational Chern–Simons action for three-manifolds as a Perelman entropy functional, a generalization of the Cotton flow for the metric, in co-evolution with torsion is developed. In the case of manifolds with boundary, there exists an entropy functional induced on it, and hence metric and torsional flows can be defined. The integrability of these boundary flows is studied in detail. In a particular case, the solutions of the two-dimensional logarithmic diffusion equation determine completely the dynamics of the fields in co-evolution. In this scheme of three-manifolds with evolving boundaries, the orthogonality of the twisted flows to the Yamabe-like flows is established; additionally, the evolution of the holonomy groups of the three-manifold and of its boundary is studied, showing in some cases an oscillatory behavior around orthogonal groups. In this approach, the twisted canonical geometries (TCG) are defined as manifolds with a vanishing contribution of the torsion to the curvature, which will be generated fully by the metric; these geometries evolve to manifolds with curvature generated by both metric and torsion, and cannot smoothly evolve into manifolds with a vanishing torsion.

In this paper, we consider an intrinsic point of view to describe the equations of motion for higher-order variational problems with constraints on higher-order trivial principal bundles. Our techniques are an adaptation of the classical Skinner–Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics. As an interesting application we deduce the equations of motion for optimal control of underactuated mechanical systems defined on principal bundles.

The coherent state representation of the Jacobi group is indexed with two parameters, , describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of the Siegel–Jacobi disk are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel–Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding and the Cauchy formula for the Siegel–Jacobi disk are presented.

We present a systematic approach for the separation of variables for the two-dimensional (2D) Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential are chosen to have angular dependence which emanate the Dirac equation to complete separation of variables. Exact solutions are obtained for a class of solvable potentials along with their relativistic spinor wavefunctions. Particular attention is paid to the situation where the potentials are confined to a quantum dot region and are of scalar, vector and pseudo-scalar type. The study of a single charged impurity embedded in a 2D Dirac equation in the presence of a uniform magnetic field was treated as a particular case of our general study.

We determine the Lie point symmetries of the Schrödinger and the Klein–Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the space, respectively. We consider the kinematic metric defined by the classical Lagrangian and show how the Lie point symmetries of the Schrödinger equation and the Klein–Gordon equation are related with the Noether point symmetries of this Lagrangian. The general results are applied to two practical problems: (a) The classification of all two- and three-dimensional potentials in a Euclidean space for which the Schrödinger equation and the Klein–Gordon equation admit Lie point symmetries; and (b) The application of Lie point symmetries of the Klein–Gordon equation in the exterior Schwarzschild spacetime and the determination of the metric by means of conformally related Lagrangians.