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We show that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. We propose a construction of quantum fibre bundles on such spaces. The…
Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no…
We provide a mathematical framework for PT-symmetric quantum theory, which is applicable irrespective of whether a system is defined on R or a complex contour, whether PT symmetry is unbroken, and so on. The linear space in which…
The phase space of a particle on a group manifold can be split in left and right sectors, in close analogy with the chiral sectors in Wess Zumino Witten models. We perform a classical analysis of the sectors, and the geometric quantization…
A quantum theory in a finite-dimensional Hilbert space can be geometrically formulated as a proper Hamiltonian theory as explained in [2, 3, 7, 8]. From this point of view a quantum system can be described in a classical-like framework…
The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility,…
We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. This Hamiltonian has a clear interpretation in terms of discrete geometries: it computes the…
In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which…
In this paper we establish the existence of the non-perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of the QHD(M) algebra, which is an algebra generated by…
A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and…
Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two…
$SU(\infty)$-QGR is a quantum approach to Universe and gravity. Its main assumption is infinite mutually commuting observables in the Universe, leading to representation of $SU(\infty)$ by its Hilbert spaces and those of its subsystems. The…
A 4-dimensional Lorentzian static space-time is equivalent to 3-dimensional Euclidean gravity coupled to a massless Klein-field. By canonically quantizing the coupling model in the framework of loop quantum gravity, we obtain a quantum…
Loop quantum gravity corrections, in the presence of inhomogeneities, can lead to a deformed constraint algebra. Such a deformation implies that the effective theory is no longer generally covariant. As a consequence, the geometrical…
Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the…
Group field theory is a background-independent approach to quantum gravity whose starting point is the definition of a quantum field theory on an auxiliary group manifold (not interpreted as spacetime, but rather as the finite-dimensional…
We start with the Hamiltonian formulation of the first order action of pure gravity with a full $\mathfrak{sl}(2,\mathbb C)$ internal gauge symmetry. We make a partial gauge-fixing which reduces $\mathfrak{sl}(2,\mathbb C)$ to its…
We study the quantum mechanics of a system of topologically interacting particles in 2+1 dimensions, which is described by coupling the particles to a Chern-Simons gauge field of an inhomogeneous group. Analysis of the phase space shows…
In this article, the quantum representation of the algebra among reduced twisted geometries (with respect to the Gauss constraint) is constructed in the gauge invariant Hilbert space of loop quantum gravity. It is shown that the reduced…
Loop Quantum Gravity (LQG) is a promising approach to quantum gravity, in particular because it is based on a rigorous quantization of the kinematics of gravity. A difficult and still open problem in the LQG program is the construction of…