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1.
We show that the quantum Heisenberg groupH q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU q (h(1)). We derive left and right regular representations forU q (h(1)) as acting on its dualH q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated. 相似文献
2.
Alexander L. Rosenberg 《Communications in Mathematical Physics》1991,142(3):567-588
The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field. 相似文献
3.
M.A. Rego-Monteiro 《The European Physical Journal C - Particles and Fields》2001,21(4):749-756
We show that the Heisenberg-type algebra describing the first levels of the quantum harmonic oscillator on a circle of large
length L is a deformed Heisenberg algebra. The successive energy levels of this quantum harmonic oscillator on a circle of large length
L are interpreted, similarly to the standard quantum one-dimensional harmonic oscillator on an infinite line, as being obtained
by the creation of a quantum particle of frequency w at very high energies.
Received: 29 March 2001 / Revised version: 17 July 2001 / Published online: 31 August 2001 相似文献
4.
E. H. EL Kinani 《International Journal of Theoretical Physics》2000,39(6):1457-1462
We review the R-deformed Heisenberg algebra and its Fock space representation.We construct the R-deformed quantum mechanics in N dimensions, and proposea new R-deformed Virasoro algebra. 相似文献
5.
We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted pairing. We state a Stone–von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction. 相似文献
6.
Quantum Electrodynamics on a finite lattice is investigated within the hamiltonian approach. First, the structure of the algebra
of lattice observables is analyzed and it is shown that the charge superselection rule holds. Next, for every eigenvalue of
the total charge operator a canonical irreducible representation is constructed and it is proved that all irreducible representations
corresponding to a fixed value of the total charge are unique up to unitary equivalence. The physical Hilbert space is by
definition the direct sum of these superselection sectors. Finally, lattice quantum dynamics in the Heisenberg picture is
formulated and the relation of our approach to gauge fixing procedures is discussed.
Received: 21 October 1996 / Accepted: 10 February 1997 相似文献
7.
F. Bonechi R. Giachetti E. Sorace M. Tarlini 《Communications in Mathematical Physics》1995,169(3):627-633
A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classicalr-matrix satisfying the modified Yang-Baxter equation. The corresponding quantum group is studied and itsR-matrix is explicitly calculated. 相似文献
8.
J. Krause 《International Journal of Theoretical Physics》1998,37(2):759-784
Non-Abelian quantum kinematics is applied to thePoincare group P
+
(1, 1),as an example of the quantization-through-the-symmetryapproach to quantum mechanics. Upon quantizing thegroup, generalized Heisenberg commutation relations are obtained, and aclosed Heisenberg–Weyl algebra follows. Then,according to the general theory, the three basicquantum-kinematic invariant operators are calculated;these afford the superselection rules for diagonalizing theincoherent rigged Hilbert space H(P
+
) of the regularrepresentation. This paper examines only one of thesediagonalization schemes, while introducing a irreducible spacetime representation carried by isotopicplane-wave eigenvectors of two compatible superselectionoperators (which define a Poincare-invariant linear2-momentum). Thereafter, the principle of microcausality produces massive 2-spinor isotopic states in 1+ 1 Minkowski space. The Dirac equation is thus deducedwithin the quantum kinematic formalism, and the familiarJordan–Pauli propagation kernel in 2-dimensional spacetime is also obtained as a Hurwitzinvariant integral over the group manifold. The maininterest of this approach lies in the adoptedgroup-quantization technique, which is a strictlydeductive method and uses exclusively the assumed Poincaresymmetry. 相似文献
9.
Nonstandard deformations of the Poincaré group Fun(P(1+1)) and its dual enveloping algebra U (p(1+1)) are obtained as a contraction of the h-deformed (Jordanian) quantum group Fun( SL
h
(2)) and its dual. A nonstandard quantization of the Heisenberg algebra U(h(1)) is also investigated. 相似文献
10.
Vahid Karimipour 《Letters in Mathematical Physics》1994,30(2):87-98
Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the nonstandard deformation of Fun(SL(2)). It is shown that in contrast to the standardq-deformation of SL(2), the above complexes are unique for SL
h
(2). Also, as a byproduct, a new deformation of the two-dimensional Heisenberg algebra is obtained which can be used to construct models ofh-deformed quantum mechanics. 相似文献
11.
A basis for an irreducible representation of the quantum algebraE
q
(2) is given,consisting of eigenfunctions of the q-differential representationof the Casimiroperator of the quantum algebra E
q
(2). 相似文献
12.
Coherent states are introduced and their properties are discussed for simple quantum compact groupsA
l, Bl, Cl andD
l. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates
on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this orbit with values
in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss
decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and
given in a compactR-matrix formulation (generalizing this way theq-deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum
group (the analogue of the Borel-Weil construction) is described using the concept of coherent state. The relation between
representation theory and non-commutative differential geometry is suggested.
Dedicated to Professor L.D. Faddeev on his 60th birthday 相似文献
13.
By introducing a pair of canonical conjugate two-parameter deformed operators Dqs, Xqs,we can naturally obtain the form of qs-analogous Taylor series for an arbitrary analytic function, and explicitly construct the realizations of Heisenberg and two-parameter deformed quantum Heisenberg algebra by means of the operators Dqs and Xqs, and it is shown that the qs-analogous Hermite polynomials are the representations of Heisenberg and the quantum Heisen berg algebra. 相似文献
14.
A general analysis of bilinear algebras of creation and destruction operators is performed. Generalizing the earlier work
on the single-parameterq-deformation of the Heisenberg algebra, we study two-parameter and four-parameter algebras. Two new forms of quantum statistics
called orthofermi and orthobose statistics and aq-deformation interpolating between them have been found. In the Fock representation, quadratic relations among destruction
operators, wherever they are allowed, are shown to follow from the bilinear algebra of creation and destruction operators.
Postitivity of the Hilbert space for the four-parameter algebra has been studied in the two-particle sector, but for the two-parameter
algebra, results are presented up to the four-particle sector. 相似文献
15.
F. Bonechi 《Czechoslovak Journal of Physics》1994,44(11-12):993-999
After reviewing the way the quantization of Poisson Lie Groups naturally leads to Quantum Groups we use the existing quantum versionH(1)q of the Heisenberg algebra to give an explicit example of this quantization on the Heisenberg group. 相似文献
16.
We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate
(4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential
or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator.
We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable
system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials. 相似文献
17.
In Thermal Field Dynamics, thermal states are obtained from restrictions of vacuum states on a doubled field algebra. It is shown that the suitably doubled Fock representations of the Heisenberg algebra do not need to be introduced by hand but can be canonically handed down from deformations of the extended Heisenberg bialgebra. The relationship between quantum symmetries and doublings is discussed. 相似文献
18.
Zu-Rong YU 《理论物理通讯》1992,17(1):115-118
The explicit forms of the irreducible representation matricea for the quantum SLq(3) enveloping algebra are computed in analogy to the irreducible tensor technique in the classical SU(3) algebra. 相似文献
19.
Antun Milas 《Communications in Mathematical Physics》2008,277(2):497-529
This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of
Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one
Heisenberg vertex operator algebra M(1)
a
, of central charge 1 − 12a
2. We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators
resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)0 and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct
explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine
logarithmic L(1, 0)-module. 相似文献
20.
For the 3: (?1) resonance Penning trap, we describe the algebra of symmetries which turns out to be a non-Lie algebra with cubic commutation relations. The irreducible representations and coherent states of this algebra are constructed explicitly. The perturbing inhomogeneous magnetic field of Ioffe type, after double quantum averaging, generates an effective Hamiltonian of the trap. In the irreducible representation, this Hamiltonian becomes a second-order ordinary differential operator of the Heun type. 相似文献