Related papers: $\hat{Q}$ operator for canonical quantum gravity
We study the holographic properties of a class of quantum geometry states characterized by a superposition of discrete geometric data, in the form of generalised tensor networks. This class specifically includes spin networks, the kinematic…
We define a second-order differential operator $\hat{C}$ on the Hilbert space $L^2([-v_c, v_c])$, constructed from a smooth deformation function $C(v)$. The operator is considered on the Sobolev domain $H^2([-v_c, v_c]) \cap H^1_0([-v_c,…
We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $…
We study two classes of quantum spheres and hyperboloids which are $*$-quantum spaces for the quantum orthogonal group $\mathcal{O}(SO_q(3))$. We construct line bundles over the quantum homogeneous space of invariant elements for the…
We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe `excitations moving on the quantum…
We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the…
This is the first of two papers which study the behavior of the SU(2) holonomies of loop quantum gravity (LQG), when they are acted upon by a unidirectional, plane gravity wave. Initially, the LQG flux-holonomy variables are treated as…
In the present paper, we consider the integral operator, which acts in Hilbert space and has sine kernel. This operator generates two operator identities and two corresponding canonical differential systems. We find the asymptotics of the…
We develop two approaches to Quantum (or Non-commutative) Graphs based on arbitrary von Neumann algebras $M\subseteq\mathcal B(H)$: one looking at operator bimodules of Hilbert--Schmidt (instead of bounded) operators, and the second looking…
This paper constructs an approximate sinusoidal wave packet solution to the equations of loop quantum gravity (LQG). There is an SU(2) holonomy on each edge of the LQG simplex, and the goal is to study the behavior of these holonomies under…
We construct a Dirac operator on the quantum sphere $S^2_q$ which is covariant under the action of $SU_q(2)$. It reduces to Watamuras' Dirac operator on the fuzzy sphere when $q\to 1$. We argue that our Dirac operator may be useful in…
The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact…
A string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a ``canonical basis'' for every…
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $\omega$, coupled by means of springs. Such systems have been studied before, and appear in various models. In…
An open U_q(sl_2)-invariant spin chain of spin S and length N with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation…
We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quantized version of a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q}…
We construct Sugawara operators for the quantum affine algebra of type $A$ in an explicit form. The operators are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This generalizes a previous…
In this paper we introduce a new model for the quantum-mechanical system of the hydrogen atom. We start with a four-dimensional Lorentzian quadratic space $(V,q)$ and let $C \subset V$ be the corresponding cone. The Hilbert space of our…
In this article, we introduce an analogue of Kenig and Stein's bilinear fractional integral operator on the Heisenberg group $\mathbb{H}^n$. We completely characterize exponents $\alpha, \beta$ and $\gamma$ such that the operator is bounded…
We investigate the Hilbert space in the Lorentz covariant approach to loop quantum gravity. We restrict ourselves to the space where all area operators are simultaneously diagonalizable, assuming that it exists. In this sector quantum…