Narrow Results

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from ${ℂ}^{1+n}$ with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space $ℂ{\mathbb{P}}^n$ and the complex hyperbolic disc ${𝔻}^n$. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on $ℂ{\mathbb{P}}^n$ and ${𝔻}^n$. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the ${}^{\ast }$-involution of pointwise complex conjugation.

In this paper, by applying the direct symmetry method, we obtain the symmetry reductions, group invariant solutions and some new exact solutions of the Bogoyavlenskii equation, which include hyperbolic function solutions, trigonometric function solutions and power series solutions. We also give the conservation laws of the Bogoyavlenskii equation.

With the aid of symbolic computation, we use the Lie group method to find the Lie point symmetries of (3+1)-dimensional Burgers system, and reduce the system with the obtained symmetries. As a result, many kinds of new symmetry reductions have been presented. And some explicit solutions are given due to the Riccati equation method.

The nonlocal symmetry of the new integrable $(2+1)$-dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable $(2+1)$-dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new $(2+1)$-dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants.

In this paper, the symmetry analysis and similarity reduction of the (2+1)-dimensional Bogoyavlensky–Konopelchenko (B–K) equation are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Using the extended Harrison and Estabrook’s differential forms approach, the infinitesimal generators for (2+1)-dimensional B–K equation are obtained. Firstly, the vector field associated with the Lie group of transformation is derived. Then the symmetry reduction and the corresponding explicit exact solution of (2+1)-dimensional B–K equation is obtained.

We analyze cylindrical gravitational waves in vacuo with general polarization and develop a viewpoint complementary to that presented recently by Niedermaier showing that the auxiliary sigma model associated with this family of waves is not renormalizable in the standard perturbative sense.

This is a brief overview of the current status of symmetry reduced models in Loop Quantum Gravity. The goal is to provide an introduction to other more specialized and detailed reviews that follow. Since most of this work is motivated by the physics of the very early universe, I will focus primarily on Loop Quantum Cosmology and discuss quantum aspects of black holes only briefly.

This paper summarizes a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, thereby enabling a detailed comparison of results in loop quantum gravity and loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase-space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. Some additional steps, such as constructing a specific embedding of the Hilbert space of loop quantum cosmology into a space of (distributional) states in the full theory, remain incomplete. However, we also describe, as a proof of concept, a complete analysis of an analogous embedding of homogeneous and isotropic loop quantum cosmology into the quantum Bianchi I model of Ashtekar and Wilson-Ewing. Details will appear in a pair of forthcoming papers.

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. In addition, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.

This paper considers reaction-diffusion equations from a new point of view, by including spatiotemporal dependence in the source terms. We show for the first time that solutions are given in terms of the classical Painlevé transcendents. We consider reaction-diffusion equations with cubic and quadratic source terms. A new feature of our analysis is that the coefficient functions are also solutions of differential equations, including the Painlevé equations. Special cases arise with elliptic functions as solutions. Additional solutions given in terms of equations that are not integrable are also considered. Solutions are constructed using a Lie symmetry approach.

In a recent paper, a variational principle for Liouville dynamics (vector fields preserving a volume form) was introduced. In the present paper, we discuss the relation between symmetries and conservation laws in Liouville dynamics. We recall a result by Crampin relating symmetries and conserved quantities in Liouville dynamics, and discuss how symmetry reduction is implemented in the frame of the variational principle for Liouville dynamics.

This paper analyzes the constraint structure of a class of degenerate second-order particle Lagrangian that includes chiral oscillator, noncommutative oscillator, and two examples from reduced topologically massive gravity. For even-dimensional configuration spaces with maximal nondegeneracy, Dirac bracket is defined solely by coefficient field of highest derivative whereas for odd dimensions almost all fields may contribute. Ostrogradskii’s theorem on energy instability is discussed. Results of Dirac analysis are used to identify ghost degrees of freedom. Translational symmetries are used to construct first-order variational formalisms for oscillator examples, thereby making them ghost-free.