Related papers: Quantum groups and q-lattices in phase space
We interpret quantum computing as a geometric evolution process by reformulating finite quantum systems via Connes' noncommutative geometry. In this formulation, quantum states are represented as noncommutative connections, while gauge…
Phase space dynamics in classical mechanics is described by transport along trajectories. Anharmonic quantum mechanical systems do not allow for a trajectory-based description of their phase space dynamics. This invalidates some approaches…
First a description of 2+1 dimensional non-commutative(NC) phase space is presented, where the deformation of the planck constant is given. We find that in this new formulation, generalized Bopp's shift has a symmetric representation and…
We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our…
The central notion in Connes' formulation of non commutative geometry is that of a spectral triple. Given a homogeneous space of a compact quantum group, restricting our attention to all spectral triples that are `well behaved' with respect…
Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is…
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…
The relativistic quantum phase space (QPS) formalism extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, thereby providing a unified setting where the uncertainty principle and…
We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties…
We consider a quantum space with rotationally invariant noncommutative algebra of coordinates and momenta. The algebra contains tensors of noncommutativity constructed involving additional coordinates and momenta. In the rotationally…
We review briefly the motivations for introducing additional group-theoretic data in tensor models, leading to the richer framework of group field theories, themselves a field theory formulation of loop quantum gravity. We discuss how these…
A quantum version of the action principle in a simple covariant dynamical theory of two relativistic particles is formulated. The central object of this new formulation of quantum theory is a stationary eigenvalue of the quantum action.…
Discussed are quantized dynamical systems on orthogonal and affine groups. The special stress is laid on geodetic systems with affinely-invariant kinetic energy operators. The resulting formulas show that such models may be useful in…
Non-commutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on non-commutative configuration space. Within this framework an unambiguous definition can be given for the…
A formulation of quantum mechanics based on replacing the general unitary group by finite groups is considered. To solve problems arising in the context of this formulation, we use computer algebra and computational group theory methods.
I present a selective survey of the phases of quantum matter with varieties of many-particle quantum entanglement. I classify the phases as gapped, conformal, or compressible quantum matter. Gapped quantum matter is illustrated by a simple…
The algebraic formulation of the quantum group gauge models in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields…
The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with…
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose-Einstein statistics the commutation relations of phase space variables should simultaneously include…
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a…