International Journal of Geometric Methods in Modern Physics

With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces.

We discuss linear in momenta Poisson structure for the generalized nonholonomic Chaplygin sphere problem and prove that it is nontrivial deformation of the canonical Poisson structure on e*(3).

We present here a range of considerations relating to the problem of Dirac for a quantifiable manifold (M, ω) to the dual a Lie algebra with a Poisson structure that is a Lie–Poisson structure minus.

The purpose of this paper is to present, in a covariant form and in their full generality, the equations of motion for space-time-matter (STM) theory. The whole study is based on the new approach of STM theory developed in our first paper [1] of this series. We show that the theory of geodesics in a general Kaluza–Klein space is best presented and explained by splitting the set of all geodesics into horizontal and non-horizontal geodesics. It is noteworthy that the horizontal geodesics (respectively, non-horizontal geodesics) project on the base manifold on motions which generalize the motions from general relativity theory (respectively, motions from Lorentz force equations).

A rigorous construction of Clifford-space (C-space) gravity is presented which is compatible with the Clifford algebraic structure and permits the derivation of the expressions for the connections with torsion in C-spaces. The C-space generalized gravitational field equations are derived from a variational principle based on the extension of the Einstein–Hilbert–Cartan action. We continue by arguing how Lanczos–Lovelock–Cartan (LLC) higher curvature gravity with torsion can be embedded into gravity in C-spaces and suggest how this might also occur for extended gravitational theories based on f(R), f(R_μν), … actions, for polynomial-valued functions. In essence, the LLC curvature tensors appear as Ricci-like traces of certain components of the C-space curvatures. Torsional gravity is related to higher-order corrections of the bosonic string-effective action. In the torsionless case, black-strings and black-brane metric solutions in higher dimensions D > 4 play an important role in finding specific examples of solutions to LL gravity.

We develop a new description of curved manifolds that represent a topological phase for the usual Levi-Civita connection and a dynamical phase for the torsion, being the only part of the connection that generates curvature. The holonomy group structure of such manifolds is studied, and different expressions of the torsion in terms of gauge fields are considered with the purpose to describe different dynamical scenarios. Additionally some models of torsion interacting with fermions are constructed; specifically a four-dimensional model that mimics an usual fermion-gauge fields system is considered; this model undergoes parity symmetry breaking with a dynamical background. Since these manifolds represent new gravitational backgrounds, we speculate on possible applications in the study of qualitative and quantitative aspects of the gauge/gravity duality.

A standard model is formulated in a Weyl space, W₄, yielding a Weyl covariant dynamics of massless chiral Dirac fermion fields for leptons and quarks as well as the gauge fields involved for the groups D(1) (Weyl), U(1)_Y × SU(2)_W (electroweak), SU(3)_c (color), SO(3, 1) (gravity) and SO(4, 1) (strong interaction, symmetry breaking). The dynamics is based on a gauge and Weyl invariant Lagrangian density . Gravitation is included from the beginning as the gauge aspect of the Lorentz group which is here extended in the hadronic sector of the model to the ten-parameter SO(4, 1) de Sitter group. A part of the dynamics is, as usual, a scalar isospinor field ϕ being a section on a bundle related to the electroweak gauge group and to symmetry breaking. In parallel to ϕ on the leptonic side a section on the hadronic side is considered as part of the dynamics, governing the symmetry breaking SO(4, 1) → SO(3, 1) and recovering gravitation in the symmetry breaking limit outside the regions in space–time where strong interactions persist. Besides spin, isospin and helicity the Weyl weights determine the form of the contributions of fields in . Of particular interest is the appearance of a current–current self-interaction of quark fields allowed by the Weyl weight changing the debate about quark masses. In a second step the D(1)-Weyl symmetry is explicitly broken and a universal mass scale is established through the mass of the ϕ-field appearing in the symmetry breaking Lagrangian . The Weyl symmetry breaking is governed by the relation D_μΦ² = 0, where Φ is the norm of ϕ. After D(1) symmetry breaking the masses of the weak bosons and of the electron appear on the scene through the energy–momentum tensor of the ϕ-field.

In this paper the Seiberg–Witten map is first analyzed for non-commutative Yang–Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg–Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space–time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, when the Seiberg–Witten map is eventually inserted. In the attempt of understanding whether or not this is a peculiar property of gravity, in the second part of the paper, the Seiberg–Witten map is considered in the simpler case of Maxwell theory in vacuum in the absence of charges and currents. Once more, no obvious solution of the non-commutative field equations is found, unless the electromagnetic potential depends in a very special way on the wave vector.

Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP-manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise to structures of Leibniz/Loday algebroids, which are characterized by QP-structures. Especially, in three dimensions, a current algebra has a structure of a Lie algebroid up to homotopy introduced by Uchino and one of the authors, which has a bracket of a generalization of the Courant–Dorfman bracket. Anomaly cancellation conditions are reinterpreted as generalizations of the Dirac structure.

We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler–Lagrange equations associated to the Chern–Gauss–Bonnet formula and show that they are given solely in terms of curvature and the second fundamental form and do not involve covariant derivatives, thus generalizing a conjecture of Berger to this context.

In this paper, we treat the so-called dynamical Casimir effect in a cavity with a two-level atom and give an analytic approximate solution under the general setting. The aim of the paper is to show another approach based on Mathematical Physics to the paper [Approximate analytical results on the cavity Casimir effect in the presence of a two-level atom, Phys. Rev. A85 (2012) 063804, arXiv: 1112.0523 [quant-ph]] by A. V. Dodonov and V. V. Dodonov. We believe that our method is simple and beautiful.