Related papers: Dynamical Aspects of Lie--Poisson Structures
We canonically quantize a Poisson manifold to a Lie 2-groupoid, complete with a quantization map, and show that it relates geometric and deformation quantization: the perturbative expansion in $\hbar$ of the (formal) convolution of two…
We propose a new method of quantization of a wide class of dynamical systems that originates directly from the equations of motion. The method is based on the correspondence between the classical and the quantum Poisson brackets, postulated…
In a previous work we have introduced the concept of quasi-integrable quantum system. In the present one we determine sufficient conditions under which, given an integrable classical system, it is possible to construct a quasi-integrable…
Quantum Algebras (q-algebras) are used to describe interactions between fermions and bosons. Particularly, the concept of a su_q(2) dynamical symmetry is invoked in order to reproduce the ground state properties of systems of fermions and…
We consider the space of bilinear forms on a complex N-dimensional vector space endowed with the quadratic Poisson bracket studied in our previous paper arXiv:1012.5251. We classify all possible quadratic brackets on the set of pairs of…
The new method of q-bosonization for quantum groups based on the Gauss decomposition of a transfer matrix of generators is suggested. The simplest example of the quantum group $GL_q(2)$ is considered in some details.
We discuss the process to obtain Poisson brackets among the phase-space variables of a system of a charged particle on a Poincar\'e hyperboloid in the presence of a uniform magnetic field. We show that after quantization the Dirac bracket…
We describe an $p$-mechanical (see funct-an/9405002 and quant-ph/9610016) brackets which generate quantum (commutator) and classic (Poisson) brackets in corresponding representations of the Heisenberg group. We \emph{do not} use any kind of…
We construct quantum deformation of Poincar\'e group using as a starting point $SU(2,2)$ conformal group and twistor-like definition of the Minkowski space. We obtain quantum deformation of $SU(2,2)$ as a real form of multiparametric…
We construct a family of $q$ deformations of $E(2)$ group for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle group $\mathbb{T}$ viewed as a quasitriangular quantum group with respect to the…
Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed…
In quantum physics, the operators associated with the position and the momentum of a particle are unbounded operators and $C^*$-algebraic quantisation does therefore not deal with such operators. In the present article, I propose a…
Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).
We describe a decomposition of the Lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions. The decomposition is based on a recursive procedure which systematically uses the Cartan classification of the…
We consider Hamiltonian formulation of a dynamical system forced to move on a submanifold $G_\alpha(q^A)=0$. If for some reasons we are interested in knowing the dynamics of all original variables $q^A(t)$, the most economical would be a…
At the heart of quantum technology development is the control of quantum systems at the level of individual quanta. Mathematically, this is realised through the study of Hamiltonians and the use of methods to solve the dynamics of quantum…
Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding…
In quantum field theory the creation and annihilation operators that are located at the points in 3-momentum space have commutation relations that are conserved under the action of a $U({\infty})$ group. Here it is shown how to define an…
We study compound systems with a classical sector and a quantum sector. Among other consistency conditions we require a canonical structure, that is, a Lie bracket for the dynamical evolution of hybrid observables in the Heisenberg picture,…
We study Poisson structures of dynamical systems with three degrees of freedom which are known for their chaotic properties, namely L\"u, modified L\"u, Chen, $T$ and Qi systems. We show that all these flows admit bi-Hamiltonian structures…