Related papers: The 3D Spin Geometry of the Quantum Two-Sphere
We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We…
The quantum free particle on the sphere $S_\kappa^2$ ($\kappa>0$) and on the hyperbolic plane $H_\kappa^2$ ($\kappa<0$) is studied using a formalism that considers the curvature $\k$ as a parameter. The first part is mainly concerned with…
We briefly report our application of a version of noncommutative geometry to the quantum Euclidean space $R^N_q$, for any $N \ge 3$; this space is covariant under the action of the quantum group $SO_q(N)$, and two covariant differential…
In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of 3-manifolds from a graphical calculus and show how to evaluate…
A geometric construction of Z_2-graded orthogonal modular categories is given. Their 0-graded parts coincide with categories previously obtained by Blanchet and the author from the category of tangles modulo the Kauffman skein relations.…
Part I: The geometric algebra of space is derived by extending the real number system to include three mutually anticommuting square roots of plus one. The resulting geometric algebra is isomorphic to the algebra of complex 2x2 matrices,…
We consider integrable open spin chains related to the quantum affine algebras U_q(o(3)) and U_q(A_2^{(2)}). We discuss the symmetry algebras of these chains with the local C^3 space related to the Birman-Wenzl-Murakami algebra. The…
We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-covariant differential calculi. For each we find a frame, a metric and…
This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in…
It is well known that the quantum double structure plays an important role in three dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three dimensional…
We formulate the two-dimensional principal chiral model as a quantum spin model, replacing the classical fields by quantum operators acting in a Hilbert space, and introducing an additional, Euclidean time dimension. Using coherent state…
In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with…
Spin$^h$ manifolds are the quaternionic analogue to Spin$^c$ manifolds. We compute the spin$^h$ bordism groups at the prime 2 by proving a structure theorem for the cohomology of the spin$^h$ bordism spectrum $\mathrm{MSpin}^h$ as a module…
For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity…
To reconstruct a mixed or pure quantum state of a spin s is possible through coherent states: its density matrix is fixed by the probabilities to measure the value s along 4s(s+1) appropriately chosen directions in space. Thus, after…
For $\mu \in (0,1), c> 0,$ we identify the quantum group $SO_\mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural…
Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields,…
Spin-orbital entanglement in the ground state of a one-dimensional SU(2)$\otimes$SU(2) spin-orbital model is analyzed using exact diagonalization of finite chains. For $S=1/2$ spins and $T=1/2$ pseudospins one finds that the quantum…
We study covariant differential calculus on the quantum spheres S_q^{N-1} which are quantum homogeneous spaces with coactions of the quantum groups O_q(N). The first part of the paper is devoted to first order differential calculus. A…
A geometry of superspace corresponding to double field theory is developed, with type II supergravity in D=10 as the main example. The formalism is based on an orthosymplectic extension OSp(d,d|2s) of the continuous T-duality group.…