From Heisenberg algebra to conformal dynamical group

2008 ◽  
pp. 3-21 ◽  
Author(s):  
A. O. Barut
Keyword(s):  
Heisenberg Algebra ◽  
Dynamical Group

q-ANALOGUE OF BOSON COMMUTATOR AND THE QUANTUM GROUPS SUq(2) AND SUq(1, 1)

1990 ◽  
Vol 05 (04) ◽  
pp. 237-242 ◽  
Author(s):  
HARUO UI ◽  
N. AIZAWA
Keyword(s):  
Harmonic Oscillator ◽  
Quantum Group ◽  
Symmetry Group ◽  
Quantum Groups ◽  
Heisenberg Algebra ◽  
Positive Definiteness ◽  
Two Dimensional ◽  
Number Operator ◽  
Boson Operator ◽  
Dynamical Group

We propose a defining set of commutation relations to a q-analogue of boson operator; [Formula: see text], [Formula: see text] and [N, aq]=−aq, which contracts to the Heisenberg algebra of boson operators in the limit of q=1. Here, N is the number operator, [N]q being its q-analogue operator. By making use of this set, we construct a new realization of the “noncompact” quantum group SUq(1, 1) in addition to that of the SUq(2) recently proposed by Biedenharn. The explicit form of the number operator is given in terms of aq and [Formula: see text] and its positive definiteness is proved. A uniqueness of our commutators is also discussed. It is shown that the quantum group SUq(2) appears as a true symmetry group of a q-analogue of the two-dimensional harmonic oscillator and the SUq(1, 1) as its dynamical group.

Bialgebra structures on the Heisenberg algebra

1989 ◽  
Vol 75 (1) ◽  
pp. 315-321
Author(s):  
Michel Cahen ◽  
Christian Ohn
Keyword(s):  
Heisenberg Algebra

scholarly journals Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1309
Author(s):  
Jerzy Lukierski
Keyword(s):  
Phase Space ◽  
Hopf Algebras ◽  
Deformation Parameter ◽  
Heisenberg Algebra ◽  
Planck Length ◽  
Covariant Quantum ◽  
Drinfeld Twist ◽  
Quantum Deformations ◽  
Heisenberg Double ◽  
Conformal Symmetries

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

scholarly journals Gauging the Heisenberg algebra of special quaternionic manifolds

2005 ◽  
Vol 610 (1-2) ◽  
pp. 147-151 ◽  
Author(s):  
R. D'Auria ◽  
S. Ferrara ◽  
M. Trigiante ◽  
S. Vaulà
Keyword(s):  
Heisenberg Algebra ◽  
Quaternionic Manifolds

Dynamical-group formulation of the scaling variational method

1984 ◽  
Vol 29 (3) ◽  
pp. 1574-1577 ◽  
Author(s):  
Christopher C. Gerry ◽  
Steven J. Silverman
Keyword(s):  
Variational Method ◽  
Dynamical Group

The Aharonov–Bohm effect and its stationary scattering states from the flat circular annulus U(1) holonomy

2014 ◽  
Vol 11 (10) ◽  
pp. 1450084
Author(s):  
Gabriel Y. H. Avossevou ◽  
Bernadin D. Ahounou
Keyword(s):  
Scattering Problem ◽  
Heisenberg Algebra ◽  
Scattering Data ◽  
Gauge Potential ◽  
Circular Annulus ◽  
Topological Quantum ◽  
Vector Gauge ◽  
Simply Connected ◽  
Aharonov Bohm ◽  
Stationary Scattering

In this paper we study the stationary scattering problem of the Aharonov–Bohm (AB) effect. To achieve this goal we construct a Hamiltonian from the most general representations of the Heisenberg algebra. Such representations are defined on a non-simply-connected manifold which we set as the flat circular annulus. By means of the von Neumann's self-adjoint extensions formalism, the scattering data are then provided. No solenoid is considered in this paper. The corresponding Hamiltonian is based on a topological quantum degree of freedom inherent in such representations. This variable stands for the magnetic vector gauge potential at quantum level. Our outcomes confirm the topological nature of this effect.

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