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In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to…
We investigate the multi-particle states of the (1+1)-dimensional Ising model using a spectroscopy scheme based on the tensor renormalization group method. We start by computing the finite-volume energy spectrum of the model from the…
We present two different quantum deformations for the (anti)de Sitter algebras and groups. The former is a non-standard (triangular) deformation of SO(4,2) realized as the conformal group of the (3+1)D Minkowskian spacetime, while the…
Entanglement is one of the fundamental properties of a quantum state and is a crucial differentiator between classical and quantum computation. There are many ways to define entanglement and its measure, depending on the problem or…
If M is a manifold with an action of a group G, then the homology group H_1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module V, there exists a…
We show that there exist infinitely many commensurability classes of finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup which is locally free but not free. The main technical tool is the fact that a collection…
A construction of the 2d and 4d fuzzy de Sitter hyperboloids is carried out by using a (vector) coherent state quantization. We get a natural discretization of the dS "time" axis based on the spectrum of Casimir operators of the respective…
This paper is a continuation and elaboration of our work quant-ph/0206057 (Nucl. Phys. B, 1968, 7, 79) where some approach to the variable-mass problem were proposed. Here we have found a concret realization of irreducible representations…
Using geometric quantization procedure, the quantization of algebra of observables for physical system with Ricci-flat phase space is obtained. In the classical case the appointed physical system is reduced to harmonic oscillator when the…
We study space-time symmetries in scalar quantum field theory (including interacting theories) on static space-times. We first consider Euclidean quantum field theory on a static Riemannian manifold, and show that the isometry group is…
A hyperbolic BC(n) Sutherland model involving three independent coupling constants that characterize the interactions of two types of particles moving on the half-line is derived by Hamiltonian reduction of the free geodesic motion on the…
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics…
It is shown how the theory of the fields can be constructed in a consistent way in quantized spaces. All constructions are connected with unitary irreducible representations of real forms of six dimensional rotation algebras O(1,5), O(2,4),…
We review the canonical quantisation of the geometry of the spacetime in the cases of a simply and a non-simply connected manifold. In the former, we analyse the information contained in the solutions of the Wheeler-DeWitt equation and…
As a laboratory for loop quantum gravity, we consider the canonical quantization of the three-dimensional Chern-Simons theory on a noncompact space with the topology of a cylinder. Working within the loop quantization formalism, we define…
In the Chern-Simons formulation of Einstein gravity in 2+1 dimensions the phase space of gravity is the moduli space of flat G-connections, where G is a typically non-compact Lie group which depends on the signature of space-time and the…
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of…
Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view…
The scalar matter and gravity are unified into the geometric scalar matter and quantized. The quantum with a definite 3-metric has definite energy but does not have well-defined momentum. The quantum theory resolves singularities.
We review various aspects of (infinite) quantum group symmetries in 2D massive quantum field theories. We discuss how these symmetries can be used to exactly solve the integrable models. A possible way for generalizing to three dimensions…