International Journal of Geometric Methods in Modern Physics

Normal Gaussian prescription breaks down when treating null boundaries since the normal to a null boundary declines into tangency to the boundary. The normal extrinsic curvature is therefore disabled as a carrier of transverse geometrical information for a null boundary. In this paper we will construct a distributional approach based on admissible coordinates to find junction conditions and dynamics of null boundaries. Since a general null boundary can consist of matter and stress-energy distribution, we consider the general case of a null "shell" or "boundary layer". As a simple example, junction conditions and dynamics of a spherical null shell with non-vanishing energy–momentum content are derived.

In the present work, it is shown that the application of the Bazanski approach to Lagrangians, written in AP-geometry and including the basic vector of the space, gives rise to a new class of path equations. The general equation representing this class contains four extra terms, when vanished reduces this equation to the geodesic one. If the basic vector of the AP-geometry is considered as playing the role of the electromagnetic potential, as done in a previous work, then the second term (of the extra terms) will represent Lorentz force while the fourth term gives a direct effect of the electromagnetic potential on the motion of the charged particle. This last term may give rise to an effect similar to the Aharanov–Bohm effect. It is to be considered that all extra terms will vanish if the spacetime used is torsion-less.

We introduce a suitable generalization of the momentum map for nonholonomic field theories and prove a covariant form of the nonholonomic momentum equation. We show that these covariant objects coincide with their counterparts in mechanics by making the transition to the Cauchy formalism.

In this paper we propose a "quantum reduction procedure" based on the reduction of algebras of differential operators on a manifold. We use these techniques to show, in a systematic way, how to relate the hydrogen atom to a family of quantum harmonic oscillators, by the means of the Kustaahneimo–Stiefel fibration.

Given a couple of Yang–Baxter operators 𝖱^[k] and 𝖱^[l] corresponding to integrable anisotropic vertex models of A_k-1 and A_l-1 type, respectively, we construct and study a class of related lattice models whose monodromy matrices alternate between the mentioned operators. In order to do that, we use a natural generalization of the idea of coproduct in a bialgebra, that appears in the scenario of non-commutative algebraic geometry, related to the notion of internal homomorphisms of quantum spaces. We build up all eigenstates and eigenvalues of the transfer matrix by means of algebraic Bethe ansatz technics, where not only one vector, but a pseudovacuum subspace is needed for the process of diagonalization.

In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations — and solution-preserving maps — we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physical relevance, with scaling and translation symmetries of scalar equations. We apply the general discussion to a class of "Richardson-like" anomalous diffusion and reaction-diffusion equations, whose solution are known by numerical experiments to be asymptotically scale invariant; we obtain an analytical explanation of the numerically observed asymptotic scaling properties. We also apply our method to a different class of anomalous diffusion equations, relevant in optical lattices. The methods developed here can be applied to more general equations, as shown by their geometrical construction.

The Hamiltonian formulation of the theory proposed in [1, 2] is given both in the Hamilton–De Donder and in the multimomentum Hamiltonian geometrical approaches. (3 + 3) Yang–Mills gauge theories are dealt with explicitly in order to restate them in terms of Einstein–Cartan like field theories.

Let A be an algebra over an operad in a cocomplete closed symmetric monoidal category. We study the category of A-modules. We define certain symmetric product functors of such modules generalizing the tensor product of modules over commutative algebras, which we use to define the notion of a jet module. This in turn generalizes the notion of a jet module over a module over a classical commutative algebra. We are able to define Atiyah classes (i.e., obstructions to the existence of connections) in this generalized context. We use certain model structures on the category of A-modules to study the properties of these Atiyah classes.

The purpose of the paper is not to present any really deep theorem. It is more about the right concepts when dealing with modules over an algebra that is defined over an arbitrary operad, i.e., the aim is to show how to generalize various classical constructions, including modules of jets, the Atiyah class and the curvature, to the operadic context.

For convenience of the reader and for the purpose of defining the notations, the basic definitions of the theory of operads and model categories are included.