International Journal of Geometric Methods in Modern Physics

In this paper we present a new approach on the study of dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have investigated the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generalized fractional Liu process and the combination between them, generalized fractional hybrid process. Corresponding generalized fractional stochastic, respectively fuzzy, respectively hybrid dynamical systems were defined. We have applied the theory for generalized fractional hybrid Hamilton–Pontryagin (HP) equation and generalized fractional Hamiltonian equations. We have found fractional Langevin equations from the general fractional hybrid Hamiltonian equations. For these cases and specific parameters, numerical simulations were done.

Following the formalism derived from one method of constructing common projective spaces along with using a special kind of odd module homomorphisms, denoted by ν, a novel supergeometric generalization of projective spaces is constructed. Existence of canonical line bundles over these spaces and their Chern classes are discussed.

Of those gauge theories of gravity known to be equivalent to general relativity, only the biconformal gauging introduces new structures — the quotient of the conformal group of any pseudo-Euclidean space by its Weyl subgroup always has natural symplectic and metric structures. Using this metric and symplectic form, we show that there exist canonically conjugate, orthogonal, metric submanifolds if and only if the original gauged space is Euclidean or signature 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. Therefore, in this context, time may be viewed as a derived property of general relativity.

Four-level quantum systems, known as quartits, and their relation to two-qubit systems are investigated group theoretically. Following the spirit of Klein's lectures on the icosahedron and their relation to Hopf sphere fibrations, invariants of complex reflection groups occurring in the theory of qubits and quartits are displayed. Then, real gates over octits leading to the Weyl group of E₈ and its invariants are derived. Even multilevel systems are of interest in the context of solid state nuclear magnetic resonance.

By using the two-component spinor formalism and the symplectic geometry of the phase space, it is shown that the Kodama state actually exist within a topological phase where the Weyl spinor does not vanish. This topological phase, belonging to the complete topological sector of general relativity (GR), is bigger than the one associated with gravitational instantons. The conditions of projection onto the sector of gravitational instantons are discussed as well.

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and momentum-Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie–Poisson reductions and with the techniques of present framework.

On Spin^c manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spin^c manifold. Using the notion of generalized cylinders, we derive the variational formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spin^c Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.

There are two families of geometric structures associated to a surface, with both structures related to representations of the fundamental group of the surface into SL(2, ℂ). These are projective structures on the surface, and Higgs bundles for a given conformal structure of the surface. This note discusses the links between the two.

Assuming that the natural gauge group of gravity is given by the group of isometries of a given space, for a maximally symmetric space we derive a model in which gravity is essentially a gauge theory of translations. Starting from first principles we verify that a nonlinear realization of the symmetry provides the general structure of this gauge theory, leading to a simple choice of dynamical variables of the gravity field corresponding, at first-order, to a diagonal matrix, whereas the non-diagonal elements contribute only to higher orders.

The so-called Gell–Mann or decontraction formula is proposed as an algebraic expression inverse to the Inönü–Wigner Lie algebra contraction. It is tailored to express the Lie algebra elements in terms of the corresponding contracted ones. In the case of sl(n,ℝ) and su(n) algebras, contracted w.r.t. so(n) subalgebras, this formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell–Mann formula for sl(n,ℝ) and su(n) algebras, that is valid for all tensorial, spinorial, (non)unitary representations, is obtained in a group manifold framework of the SO(n) and/or Spin(n) group. The generalized formula is simple, concise and of ample application potentiality. The matrix elements of the , i.e. SU(n)/SO(n), generators are determined, by making use of the generalized formula, in a closed form for all irreducible representations.

It is shown how Seiberg–Witten equations can be obtained by means of Fedosov deformation quantization of endomorphism bundle and the corresponding theory of equivalences of star products. In such setting, Seiberg–Witten map can be iteratively computed for arbitrary gauge group up to any given degree with recursive methods of Fedosov construction. Presented approach can be also considered as a generalization of Seiberg–Witten equations to Fedosov type of noncommutativity.

The present paper focuses on a certain class of Banach manifolds we call Rogers supermanifolds since they are indeed supermanifolds modeled on graded Banach spaces. Although the subject of holonomy is well-developed for superanalytic supermanifolds utilizing local ring formulations of supermanifolds this seems not to be the case for supermanifolds modeled on graded Banach manifolds in the sense of Rogers. The proof of our main result requires a partial development of these concepts for such supermanifolds. Our main result determines conditions under which a super connection on a superprincipal bundle induces a connection on a quotient superprincipal bundle where is a foliation of and is the induced foliation on . We also show how such a quotient formulation may be used to describe in a fully geometric fashion the so-called "conventional constraints" of super Yang–Mills theory. One consequence of our development is that instead of requiring two superconnections to describe Yang–Mills theory as is the case in some formulations, we describe the relevant concepts using a single superconnection and moreover we show that the "pregauge transformations" are simply ordinary gauge transformations on the appropriate quotient bundles.