Realizations of SUq(2) Algebra in Terms of q

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Realizations of SUq(2) Algebra in Terms of q-Deformed Multimode Boson Oscillators
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2014 J. Phys.: Conf. Ser. 537 012010
(http://iopscience.iop.org/1742-6596/537/1/012010)
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IWTCP1 & NCTP38
Journal of Physics: Conference Series 537 (2014) 012010
IOP Publishing
doi:10.1088/1742-6596/537/1/012010
Realizations of SUq (2) Algebra in Terms of
q-Deformed Multimode Boson Oscillators
Luu Thi Kim Thanh and Man Van Ngu
Hanoi Pedagogical University No. 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam
E-mail: [email protected]
Abstract. We consider some version of q-deformed multimode boson oscillators. The
realization of SUq (2) algebra in terms of q-deformed multimode boson oscillators which involves
q-interference between oscillators of different modes and the realization of SUq (2) algebra in
terms of q- deformed multimode boson oscillators in which each oscillator mode has its own
defomation parameter are constracted.
1. Introduction
Quantum groups and quantum algebras have been shown to arise in many problems of current
physical and mathematical interest. Much effort is now being devoted to the cosntruction of their
representations and recently many realizations have been usefully devised using q-deformations
of boson and fermion operators [1, 2, 3].
On the other hand, quantum groups are a subject of great activity at present and although their
direct physical interpretation is still lacking, it is of particular importance to study the posible
physical implicatios of these deformation. These structures which first emerged in connection
with the quantum inverse scattering theory [4], the quantum Yang – Baxter equation .... The
algebras may be described as a deformation, depending on one or more parameters of the
ordenary Lie algebras [5].
In this paper, we introduction some versions of q-deformed multimode boson oscillators. The
realization of SUq (2) algebra in terms of q-deformed multimode boson oscillators which involves
q-interference between oscillators of different modes and the realization of SUq (2) algebra in
terms of q-deformed multimode boson oscillators in which each oscillator mode has its own
defomation parameter are constracted.
2. The Realizations of SUq (2) Algebra in Terms of q-Deformed Multimode Boson
Oscillators which Involves q–Interference between Oscillators of Different Modes
In the classical case, the SU (2) generators satisfy the commutation relations
[J0 , J− ] = −J− ,
[J0 , J+ ] = J+ ,
[J+ , J− ] = 2J0 .
(1)
Realization of SU (2) algebra in terms of the (ordinary) boson operators as follows
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1
IWTCP1 & NCTP38
Journal of Physics: Conference Series 537 (2014) 012010
IOP Publishing
doi:10.1088/1742-6596/537/1/012010
J+ = a†1 a2 ,
J− = a†2 a1 ,
J0 = 21 (N1 − N2 ) .
(2)
In this work, we consider a version of deformation which involves some q-interference between
oscillators of diferent modes. The creation and annihilation oscillator operators obeying bosonic
commutation relations
ai a†j − qa†j ai = δij q N ,
(3)
[ai , aj ] = 0, (i 6= j)
(4)
where the deformation parameter q being real, and N is the total oscillator number operators,
N
=
h[Ni , aji] =
Ni , a†j =
k
P
Ni ,
i=1
−δij aj ,
δij a†j .
(5)
Now the oscillators of different mode enter the theory not quite independently but with some
qinterference, which results from the presence of the factor q N in the r. h. s. of 3. Equation 3
gives
m
m
m−1
ai a†i
= q m a†i ai + q m−1 m a†i
qN .
(6)
The q-oscillator algebra 3 can be realised in the Fock space spanned by the orthonormalised
eigenstates of the oscillator number operators
|n1 , n2 , ..., nk i =
−
√q
n(n−1)
4
n1 !n2 !...nk
n1 n2 nk
a†1
a†2
... a†k
|0i
!
(7)
and in this space the following relations hold:
a†i ai = q N −1 Ni ,
ai a†i = q N (Ni + 1),
(8)
and hence
k
P
i=1
k
P
i=1
a†i ai = q N −1 N,
(9)
ai a†i = q N (N + k),
where k is the number of modes.
The realisation of Lie algebra in terms of (ordinary) bosons are useful not only as a convenient
mathematical tool, but also because of their applications in physics. In the case of quantum
algebras it turns out that boson realizations are possible in terms the q-deformed boson operators
already introduced above. In the case of SUq (2) algebra for two mode oscillator eqs. 3 and 4
read:
2
IWTCP1 & NCTP38
Journal of Physics: Conference Series 537 (2014) 012010
IOP Publishing
doi:10.1088/1742-6596/537/1/012010
ai a†i − qa†i ai = q N , (i = 1, 2)
a1 a†2 = qa†2 a1 ,
a1 a2 = a2 a1 .
(10)
The realisation of SUq (2) algebra besed on the q-oscillator algebra 10 can be performed in the
Fock space spanned by the orthonormalized of Ni defined as
√
|j, mi =
−j j− 1
q ( 2)
(j+m)!(j−m)!
j+m j−m
a†1
a†2
|0i ,
(11)
and the generators can be mapped onto q deformed bosons as follows
J+ = q 1−N a†1 a2 ,
J− = q 1−N a†2 a1 ,
J0 = 21 (N1 − N2 ) .
(12)
In fact, using 10, we can check that these generators satisfy the algebras 1.
3. The Realizations of SUq (2) Algebra in Terms of q-Deformed Multimode Boson
Oscillators in which each Oscillator Mode Has Its Own Deformation Parameter
Now, we consider a version of q-deformed multimode boson oscillators in which each oscillator
mode has its own deformation parameter of form:
ai a†j − qi qj a†j ai = δij qi2N ,
qj−1 ai aj − qi−1 ai aj
= 0,
[N, ai ] = −ai ,
(13)
(14)
(15)
where N is the total oscillator number operators,
N
=
h[Ni , aji] =
Ni , a†j =
k
P
Ni ,
i=1
−δij aj ,
δij a†j .
(16)
The basic of the Fock space is defined by repeated action of the creation operators on the vacuum
state:
|n1 , n2 , ..., nk i =
1
n1 !n2 !...nk !
P
n (n −1)
nj − i 2i
k −ni
√
Q
i=1
qi
j>i
a†1
n1 n2 nk
a†2
... a†k
|0i ,
(17)
where k is the number of the modes. In this space the following relations hold:
2(N −1)
a†i ai = qi
Ni ,
ai a†i = qi2N (Ni + 1) .
(18)
In particular, for two mode oscillator equations 13 and 14 read:
ai a†i − qi2 a†i = qi2N , (i = 1, 2)
3
(19)
IWTCP1 & NCTP38
Journal of Physics: Conference Series 537 (2014) 012010
IOP Publishing
doi:10.1088/1742-6596/537/1/012010
a1 a†2 = q1 q2 a†2 a1
(20)
q2−1 a1 a2 = q1−1 a2 a1 .
(21)
The realisation of SUq (2) algebra based on the qdeformed boson oscillator can be performed in
the Fock spase spanned by the orthonormalized eigenstates of Ni defined as
|jmi =
h
i
(j+m)(j+m−1)
− (j+m)(j−m)+
2
1
√
q
(j+m)!(j−m)! 1
j+m j−m
a†1
a†2
|0i
−[(j−m)(j−m−1)]
2
q2
(22)
with the identification
J+ = q11−N q21−N a†1 a2 ,
1−N †
J− = q11−N
h q2 a2 a1 ,
i
2(1−N ) †
2(1−N ) †
J0 = 12 q1
a 1 a 1 − q2
a2 a2
(23)
Using algebra 20, we can check that these generators satisfy the algebras 13.
4. Conclusions
In this paper we have constructed generators of of SUq (2) algebra in terms of q-deformed
multimode boson oscillators which involves q- interference between oscillators of different modes
(2.11) and in terms of q- deformed multimode boson oscillators in which each oscillator mode
has its own deformation parameter (3.9). These realisations of SUq (2) algebra in terms of
q- deformed multimode boson oscillators have subsequently found applications in the SUq (2)
rotator model, as for example, rotational spectra of diatomic molecules.
Acknowledgment
The authors would like to thank Prof. Dao Vong Duc for helpful discussions.
References
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O.W. Greenberg, Phys. Rev. Lett. 64, 705.
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