Related papers: The quantum metric structure of quantum SU(2)
The method of group quantization described in the preceeding paper I is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface. A simple completely solvable toy system is studied…
We investigate a one-parameter family of quantum Harish-Chandra modules of U_q sl(2n). This family is an analog of the holomorphic discrete series of representations of the group SU(n,n) for the quantum group U_q su(n, n). We introduce a…
We perform the momentum-space quantization of a spin-less particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using a non-canonical method entirely based on symmetry grounds. To achieve this…
We consider the coupling between massive and spinning particles and three dimensional gravity. This allows us to construct geometric operators (distances between particles) as Dirac observables. We quantize the system a la loop quantum…
Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that: i) While local operators such as the metric at a point may not be…
This thesis explores the application of differential geometric and general relativistic techniques to deepen our understanding of quantum mechanical systems. We focus on three systems, employing these mathematical frameworks to uncover…
In the high-energy quantum-physics literature one finds statements such as "matrix algebras converge to the sphere". Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact…
Based on a recent purely geometric construction of observables for the spatial diffeomorphism constraint, we propose two distinct quantum reductions to spherical symmetry within full 3+1-dimensional loop quantum gravity. The construction of…
We demonstrate that the matrix quantum group $SL_q(2)$ gives rise to nontrivial matrix product operator representations of the Lie group $SL(2)$, providing an explicit characterization of the nontrivial global $SU(2)$ symmetry of the XXZ…
We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger-Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a…
We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary,…
Global gauge symmetry becomes more intricate in low dimensional QFT. We survey the mathematical concepts leading to the relevant analogues of the (D=4) Doplicher-Haag-Roberts theory of superselection sectors and internal symmetry. We also…
Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play…
There were many attempts to geometrize electromagnetic field and find out new interpretation for quantum mechanics formalism. The distinctive feature of this work is that it combines geometrization of electromagnetic field and…
We modify the construction of the spectral triple over an algebra of holonomy loops by introducing additional parameters in form of families of matrices. These matrices generalize the already constructed Euler-Dirac type operator over a…
The coadjoint orbits of compact Lie groups carry many K\"ahler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi-Civita connection of the Riemannian metric, and we use…
Canonical quantization of spherically symmetric space-times is carried out, using real-valued densitized triads and extrinsic curvature components, with specific factor ordering choices ensuring in an anomaly free quantum constraint…
In this paper we employ the construction of Dirac bracket for the remaining current of $sl(2)_q$ deformed Kac-Moody algebra when constraints similar to those connecting the $sl(2)$-WZW model and the Liouville theory are imposed and show…
For the q-deformation G_q, 0<q<1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator…
By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also…