Related papers: Quantum and braided group Riemannian geometry
GL_q(N)- and SO_q(N)-covariant deformations of the completely symmetric/antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices. The precise relation between the…
The role of quantum groups and braid groups in the description of Standard Model particles is discussed. Some recent results on the use of the quantum group $SU_q(3)$ as a flavour symmetry are reviewed and a connection between two…
This paper is the first of two papers devoted to formulation of quantum mechanics of a particle in a normal geodesic frame of reference in the general Riemannian space-time. Here canonical quantization of geodesic motion in the…
Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows \Omega$ associated with a…
We construct a graded Lie algebra in which a solution to the vacuum Einstein equations is any element of degree 1 whose bracket with itself is zero. Each solution generates a cochain complex, whose first cohomology is linearized gravity…
A generalized Noether's theorem and the operational determination of a physical geometry in quantum physics are used to motivate a quantum geometry consisting of relations between quantum states that are defined by a universal group. Making…
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution…
The third modification of the space-time geometry is considered. (The first modification is the spacial relativity, the second one is the general relativity.) After the third modification of the space-time geometry the motion of free…
We present a new, general approach to gauge theory on principal $G$-spectral triples, where $G$ is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for $G$-$C^\ast$-algebras and prove that the resulting…
We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a…
The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physical models. The R-matrix approach to the theory of quantum groups is discussed in…
The purpose of this essay is to trace the historical development of geometry while focusing on how we acquired mathematical tools for describing the "shape of the universe." More specifically, our aim is to consider, without a claim to…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the…
We examine the problem of determining which representations of the braid group on a Riemann surface are carried by the wave function of a quantized Abelian Chern-Simons theory interacting with non-dynamical matter. We generalize the…
We discuss a possible framework for the construction of a quantum gravity theory where the principles of QFT and general relativity can coexist harmonically. Moreover, in order to fix the correct gauge group of the theory we study the most…
In this follow-up of the article: Quantum Group of Isometries in Classical and Noncommutative Geometry(arXiv:0704.0041) by Goswami, where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such…
We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any…
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of…
We show that the group of smooth isometries of a complemented sub-Riemannian manifold form a Lie group and establish dimension estimates based on the torsion of the canonical connection. We explore the interaction of curvature and the…
The gravitational field and the source-free electromagnetic field can be unified preliminarily by the equations in the Riemannian geometry, both are contractions of im and ik, respectively. So it will be equivalent to the Yang gravitational…