International Journal of Geometric Methods in Modern Physics

In this paper we calculate the effect of the inclusion of exotic smooth structures on typical observables in Euclidean quantum gravity. We do this in the semiclassical regime for several gravitational free-field actions and find that the results are similar, independent of the particular action that is chosen. These are the first results of their kind in dimension four, which we extend to include one-loop contributions as well. We find these topological features can have physically significant results without the need for additional exotic physics.

C. Park et al. proved the stability of homomorphisms and derivations in Banach algebras, Banach ternary algebras, C*-algebras, Lie C*-algebras and C*-ternary algebras. In this paper, we improve and generalize some results concerning derivations. We first introduce the following generalized Jensen functional equation and using fixed point methods, we prove the stability of n-ary derivations and n-ary Jordan derivations in n-ary Banach algebras. Secondly, we study this functional equation with *-n-ary derivations in C*-n-ary algebras.

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

The relativistic symmetric tensor fields are, in four dimensions, the right candidates to describe Higher Spin Fields. Their highest spin content is isolated with the aid of covariant conditions, discussed within a group theory framework, in which auxiliary fields remove the lower intrinsic angular momenta sectors. These conditions are embedded within a Lagrangian Quantum Field theory which describes an Higher Spin Field interacting with a Classical background. The model is invariant under a (B.R.S.) symmetric unconstrained tensor extension of the reparametrization symmetry, which include the Fang–Fronsdal algebra in a well defined limit. However, the symmetry setting reveals that the compensator field, which restore the Fang–Fronsdal symmetry of the free equations of motion, is in the existing in the framework and has a relevant geometrical meaning. The Ward identities coming from this symmetry are discussed. Our constraints give the result that the space of the invariant observables is restricted to the ones constructed with the Highest Spin Field content. The quantum extension of the symmetry reveals that no new anomaly is present. The role of the compensator field in this result is fundamental.

We propose a matrix model to describe a class of Fractional Quantum Hall (FQH), states for a system of (N₁ + N₂) electrons with filling factor more general than in the Laughlin case. Our model, which is developed for FQH states with filling factor of the form (k₁ and k₂ odd integers), has a U(N₁) × U(N₂) gauge invariance. Assumes that FQH fluids are composed of coupled branches of the Laughlin type, and uses ideas borrowed from hierarchy scenarios. Interactions are carried, amongst others, by fields in the bi-fundamentals of the gauge group. They simultaneously play the role of a regulator, exactly as does the Polychronakos field. We build the vacuum configurations for FQH states with filling factors given by the series , p₁ and p₂ integers. Electrons are interpreted as a condensate of fractional D0-branes and the usual degeneracy of the fundamental state is shown to be lifted by the non-commutative geometry behavior of the plane. The formalism is illustrated for the state at .

We revise the use of eight-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the eight-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.

We study the Eikonal approximation to quantum mechanics on the Moyal plane. Instead of using a star product, the analysis is carried out in terms of operator-valued wavefunctions depending on noncommuting, operator-valued coordinates.

A theorem due to Lichnerowicz which establishes a lower bound on the lowest nonzero eigenvalue of the Laplacian acting on functions on a compact, closed manifold is reviewed. It is shown how this theorem can be extended to the case of a manifold with nonempty boundary. Lower bounds for different boundary conditions, analogous to the empty boundary case, are formulated and some novel proofs are presented.

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on SU(3) and its dimension is eight, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.

This paper applies the first-order Seiberg–Witten map to evaluate the first-order non-commutative Kerr tetrad. The classical tetrad is taken to follow the locally non-rotating frame prescription. We also evaluate the tiny effect of non-commutativity on the efficiency of the Penrose process of rotational energy extraction from a black hole.

We use affine duals of jet bundles to describe how Legendre maps may be used to provide Hamiltonian representations of variational problems in a single independent variable. Such a problem may be given as a Lagrangian (of first-order or of higher-order), or alternatively as a locally variational form on a jet bundle of arbitrary order with no preferred Lagrangian.