Quantum Poincare Research Articles

On the Structure of Inhomogeneous Quantum Groups

We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincare groups [12]) we prove that our construction has correct `size', find the R-matrices and the analogues of Minkowski space for G.

Quasitriangularity and enveloping algebras for inhomogeneous quantum groups

Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and quantum Poincar\'e groups are classified. The results extend (under certain assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on those objects are described.

(2+1) NULL-PLANE QUANTUM POINCARÉ GROUP FROM A FACTORIZED UNIVERSAL R-MATRIX

The nonstandard (Jordanian) quantum deformations of so(2, 2) and (2+1) Poincare algebras are constructed by starting from a quantum sl(2, ℝ) basis such that simple factorized expressions for their corresponding universal R-matrices are obtained. As an application, the null-plane quantum (2+1) Poincare Poisson-Lie group is quantized by following the FRT prescription. Matrix and differential representations of this null-plane deformation are presented, and the influence of the choice of the basis in the resultant q-Schrödinger equation governing the deformed null-plane evolution is commented.

On the classification of quantum Poincaré groups

Using the general theory of [10] ( hep-th 9412058 ), quantum Poincar\'e groups (without dilatations) are described and investigated. The description contains a set of numerical parameters which satisfy certain polynomial equations. For most cases we solve them and give the classification of quantum Poincar\'e groups. Each of them corresponds to exactly one quantum Minkowski space. The Poincar\'e series of these objects are the same as in the classical case. We also classify possible $R$-matrices for the fundamental representation of the group.

Band structure and quantum Poincaré sections of a classically chaotic quantum rippled channel.

We obtain the energy band spectra, eigenfunctions, and quantum Poincar\'e sections of a free particle moving in a two-dimensional channel bounded by a periodically varying (ripple) wall and a flat wall. Classical Poincar\'e sections show a generic transition from regular to chaotic motion as the size of the ripple is increased. The energy band structure is obtained for two representative geometries corresponding to a wide and a narrow channel. The comparison of numerical results with the level-splitting predictions of low-order quantum degenerate perturbation theory elucidate some aspects of the classical-quantum correspondence. For larger ripple amplitudes the conduction bands for narrow channels become flat and nearly equidistant at low energies. Quantum-classical correspondence is discussed with the aid of quantum Poincar\'e (Husimi) plots.

New quantum deformations ofD=4 conformal algebra

We consider new class of classicalr-matrices forD=4 conformal Lie algebra. These r-matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebras. In such a way we generalize the lowest order of known nonstandard quantum deformation ofsl(2) to the Lie algebrasl(4)≅so(6). As an exercise we interpret nonstandard deformation ofsl(2) as describing quantumD=1 conformal algebra with fundamental mass parameter. Further we describeD=4 conformal bialgebras with deformation parameters equal to the inverse of fundamental masses. It appears that forD=4 the deformation of the Poincare algebra sector coincides with “null plane” quantum Poincare algebra.

Hamiltonian Chaos IV

First Page

Non-standard quantum (1+1) Poincare group: a T-matrix approach

The Hopf algebra dual form for the non--standard uniparametric deformation of the (1+1) Poincar\'e algebra $iso(1,1)$ is deduced. In this framework, the quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite dimensional Lie algebra. A change in the basis of the dual form is obtained in order to compare this deformation to the standard one. Finally, a non--standard quantum Heisenberg group acting on a quantum Galilean plane is obtained.

Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of the $B_n,C_n,D_n$ type, and find the corresponding bicovariant differential calculus. The method is based on a projection from $B_{n+1}, C_{n+1}, D_{n+1}$. For example we obtain the (bicovariant) inhomogeneous $q$-algebra $ISO_q(N)$ as a consistent projection of the (bicovariant) $q$-algebra $SO_q(N+2)$. This projection works for particular multiparametric deformations of $SO(N+2)$, the so-called ``minimal" deformations. The case of $ISO_q(4)$ is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameter $q$. The quantum Poincar\'e Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains the {\sl classical} Lorentz algebra. Only the commutation relations involving the momenta depend on $q$. Finally, we discuss a $q$-deformation of gravity based on the ``gauging" of this $q$-Poincar\'e algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.

General quantum surface-of-section method

A new method for exact quantization of general bound Hamiltonian systems is presented. It is the quantum analogue of the classical Poincare surface-of-section (SOS) reduction of classical dynamics. The quantum Poincare mapping is shown to be the product of the two generalized (non-unitary but compact) on-shell scattering operators of the two scattering Hamiltonians which are obtained from the original bound one by cutting the f-dimensional configuration space (CS) the along the (f-1)-dimensional configurational SOS and attaching the flat quasi-one-dimensional waveguides instead. The quantum Poincare mapping has fixed points at the eigenenergies of the original bound Hamiltonian. The energy-dependent quantum propagator (E-H)-1 can be decomposed in terms of the four energy-dependent propagators which propagate from and/or to CS to and/or from configurational SOS (which may generally be composed of many disconnected parts). I show that in the semiclassical limit (h(cross) to 0) the quantum Poincare mapping converges to the Bogomolny propagator and explain how the higher-order semiclassical corrections can be obtained systematically.

The gamma -Poincare quantum group from quantum group contraction

We propose a contraction of the de Sitter quantum group leading to the quantum Poincare group in any dimensions. The method relies on the coaction of the de Sitter quantum group on a non--commutative space, and the deformation parameter $q$ is sent to one. The bicrossproduct structure of the quantum Poincar\'e group is exhibited and shown to be dual to the one of the $\kappa$--Poincar\'e Hopf algebra, at least in two dimensions.

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutations $RTT=TTR$ and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type are found in terms of the $R$-matrix of $B_{n+1},C_{n+1},D_{n+1}$. A consistent Hopf structure on these inhomogeneous $q$-groups is constructed by means of a projection from $B_{n+1},C_{n+1},D_{n+1}$. Real forms are discussed: in particular we obtain the $q$-groups $ISO_{q,r}(n+1,n-1)$, including the quantum Poincar\'e group.

Quantum Poincare group related to the kappa -Poincare algebra

The classical r-matrix implied by the quantum kappa -Poincare algebra of Lukierski, Nowicki and Ruegg is used to generate a Poisson structure on the Poincare group. A quantum deformation of the Poincare group (on the Hopf *-algebra level) is obtained by a trivial quantization.

Quantum poincaré algebra with standard real structure

A new real quantum Poincare algebra which is a standard ∗-Hopf algebra is obtained by the construction of Uq (O(3,2)) (q real). The deformation parameter K is mass-like, and the classical Poincare algebra is obtained in the limit K → ∞. For our K-Poincare algebra both Casimirs are given.

Quantum Poincaré sections for two-dimensional billiards.

We show a method to extract the quantum Poincar\'e section corresponding to an eigenstate of a two-dimensional billiard. This quantum Poincar\'e section is given in terms of the Birkhoff variables of the problem.

Phase spaces for quantum elementary systems in anti-de Sitter and Minkowski spacetimes

The authors give a phase space description of a massive, spin s, quantum elementary system on the anti-de Sitter spacetime. The latter is associated with a discrete series representation of the kinematical group SO0(3,2) approximately=Sp(4,R)/A2, taken in its Fock-Bargmann realization. When the zero curvature limit (contraction) is carried out, they obtain a Poincare quantum elementary system in its momentum representation (i.e. the usual Wigner representation), at the expense of imposing a polarization condition. This polarization appears as a consequence of the contraction procedure, and it is imposed in order to avoid the appearance of singular terms in the contracted generators.

Classical and quantum phase-space behavior of a spin-boson system.

We study the phase-space behavior of the standard two-level system (spin 1/2) coupled to a harmonic oscillator (without the rotating-wave approximation), the classical counterpart of which is known to display deterministic chaos. We study the quantum-mechanical phase-space behavior by means of harmonic-oscillator coherent states (Husimi representation). Stationary quantum Poincar\'e sections may be defined; they show many parallels to their classical counterparts. We also study the time development of initially coherent states and observe a tendency towards a decay into many small wave packets in phase space for parameters in the chaotic regime.