scholarly journals A Schwinger Term in q-Deformed su(2) Algebra

1997 ◽  
Vol 12 (06) ◽  
pp. 403-409 ◽  
Author(s):  
Kazuo Fujikawa ◽  
Harunobu Kubo ◽  
C. H. Oh
Keyword(s):  
Current Algebra ◽  
Deformation Parameter ◽  
Oscillator Algebra ◽  
Negative Norm ◽  
Bloch Electron ◽  
Extra Term ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

An extra term generally appears in the q-deformed su(2) algebra for the deformation parameter q= exp 2π iθ, if one combines the Biedenharn–Macfarlane construction of q-deformed su(2), which is a generalization of Schwinger's construction of conventional su(2), with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra. Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed.

scholarly journals Q-DEFORMED OSCILLATOR ALGEBRA AND AN INDEX THEOREM FOR THE PHOTON PHASE OPERATOR

1995 ◽  
Vol 10 (33) ◽  
pp. 2543-2551 ◽  
Author(s):  
KAZUO FUJIKAWA ◽  
L.C. KWEK ◽  
C.H. OH
Keyword(s):  
Deformation Parameter ◽  
Matrix Representation ◽  
Index Theorem ◽  
Oscillator Algebra ◽  
Quantum Deformation ◽  
Phase Operator ◽  
Negative Norm ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Index Condition

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a viewpoint of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q=exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a–dim ker a†=1 which allows only a nonhermitian phase operator with dim ker eiφ–dim ker(eiφ)†=1. For q=exp(2πiθ) with a rational θ, one formally obtains the singular situation dim ker a=∞ and dim ker a†=∞, which allows a hermitian phase operator with dim ker eiΦ–dim ker(eiΦ)†=0 as well as the nonhermitian one with dim ker eiφ– dim ker(eiφ)†=1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q=exp(2πiθ).

scholarly journals THREE-PARAMETER (TWO-SIDED) DEFORMATION OF HEISENBERG ALGEBRA

2012 ◽  
Vol 27 (21) ◽  
pp. 1250114 ◽  
Author(s):  
A. M. GAVRILIK ◽  
I. I. KACHURIK
Keyword(s):  
Deformation Parameter ◽  
Particle Number ◽  
Heisenberg Algebra ◽  
Unusual Property ◽  
Oscillator Algebra ◽  
Number Operator ◽  
The Third ◽  
Defining Relation ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

A three-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the L.H.S. of basic defining relation and certain deformation of its R.H.S., is introduced and studied. The third deformation parameter μ appears in an extra term in the R.H.S. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, μ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor μ in the considered deformed HA inevitably depends explicitly on particle number operator N. Such a novel N-dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

scholarly journals Oscillator Quantum Algebra and Deformed su(2) Algebra

1997 ◽  
Vol 12 (18) ◽  
pp. 1335-1341 ◽  
Author(s):  
Harunobu Kubo
Keyword(s):  
Casimir Operator ◽  
Difference Operator ◽  
Nonlinear Mapping ◽  
Oscillator Algebra ◽  
Bloch Electron ◽  
Cyclic Representation ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Operator Realization ◽  
Oscillator Quantum

A difference operator realization of quantum deformed oscillator algebra ℋq(1) with a Casimir operator freedom is introduced. We show that this ℋq(1) has a nonlinear mapping to the deformed quantum su(2) which was introduced by Fujikawa et al. We also examine the cyclic representation obtained by this difference operator realization and the possibility to analyze a Bloch electron problem by ℋq(1).

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

Representations of the deformed oscillator algebra for different choices of generators

2000 ◽  
Vol 100 (2) ◽  
pp. 2061-2076 ◽  
Author(s):  
V. V. Borzov ◽  
E. V. Damaskinskii ◽  
S. B. Yegorov
Keyword(s):  
Oscillator Algebra ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra
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