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From N = 2 supersymmetry to quantum deformations
Physics Letters B 286 (1992) 290-292 North-Holland
PHYSICS LETTERS B
From N = 2 supersymmetry to quantum deformations J. Beckers a n d N. D e b e r g h 1 Theoretical and Mathematical Physics, Institute of Physics, B.5, University of Libge, B-4000 Liege 1, Belgium
Received 8 April 1992; revised manuscript received 14 May 1992
A remarkable link between N=2-supersymmetry and quantum deformations is pointed out. It already appears in the onedimensional context of supersymmetric quantum mechanics when generalized expressions of the supercharges are constructed. In particular, these results are illustrated through a simple application to harmonic oscillatorlike systems hut they can be applied to arbitrary superpotentials.
Let us consider the standard Witten supersymmetrization procedure [1 ] in N = 2 - s u p e r s y m m e t r i c q u a n t u m mechanics but by generalizing the two hermitian (odd) supercharges Qt and Qz which, with the supersymmetric (even) hamiltonian H ss, generate the well known (lie) algebra s q m ( 2 ) characterized by the structure relations: Q2 = Q 2 = H S S ,
MR=NS
[QI, H ss ] = [Q2, H ss ] = 0 .
( 1)
RM=SN,
MS= - NR ,SM = - RN .
(3b)
The matrices (M, N) and (R, S) have thus to generate at least two Clifford algebras ~ 2 : they can be realized in terms o f Pauli matrices as follows: M =~r~,
{Q,, Q 2 } = 0 ,
,
N =or2,
R = aa~ + brr2 ,S = arr2 - ba2 ,
(4)
where the parameters a and b are constrained by Let us indeed propose the following general forms in the one-dimensional spatial context: 1 Q, = ~7~ [Mpx + R Ox W ( x ) ] , 1
Q2 = ~7~ [ N p y + S O x W ( x ) ]
(2a)
(2b)
,
where M, R, N, S are constant matrices and W ( x ) is the so-called superpotential, while p x - - iOx as usual. It is easy to convince ourselves that the relations ( 1 ) imply on M, R, N and S that {M,N}=0, M2=N2=I2,
{R,S}=0, (3a)
R2=$2=I2
supplemented by the "crossed" conditions Chercheur, Institut Interuniversitaire des Sciences nucl6aires, Brussels, Belgium. 290
aZ+b2= 1 ,
a, bEgq.
(5)
We notice that, if b = + 1 ( a = 0 ) , we immediately recover the standard context [ 1 ]. For all other sets of values o f the parameters a and b, we get here unexploited supersymmetrization procedures (up to the de Crombrugghe-Rittenberg discussion [2 ] ) leading to the following respective hamiltonians: HSS__l 2 ]2 +½ha30~W(x) _~p,+½[O,.W(x) +a{ [Ox W ( x ) ] P x - ½io2 W ( x ) } .
(6)
The above a terms and the values b• _+ 1 characterize the differences with respect to the usual hamiltonian [ 1 ] SS I 2 Hwiue. = ~px + 1 [O,. W(X) ]2 + ~cr3 I G2 W(x),
(7)
corresponding to an arbitrary superpotential W ( x ) . Let us here restrict ourselves to harmonic oscillatorlike systems by dealing with the superpotential
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 286, number 3,4
W(x)
PHYSICS LETTERS B
(8)
= ½tox 2 ,
to being the corresponding angular frequency. The operator (6) then becomes HSS t . 2 31_ltoZx2.~_ ½toba3 b ~ ~Px
formed bosonic operators acting in a Hilbert space { I n ) , n = 0 , 1 , 2 .... }suchthat a b l n ) = [x/-~b I n - l ) , a~ln)=~l]b
In+l),
(15)
where the new bracket [ n ] b is defined according to
+to( 1 - b 2) ~/2(Xpx - li) .
(9)
Its eigenvalues and eigenfunctions can then be obtained by the polynomial method largely used in quantum mechanics. We easily get
E,=½to[b(2n+l)+be],
30 July 1992
n=0,1,2,...,
(10)
[n]b=bn,
(16)
in order to ensure that H(b . . . . ic) ln ) = ½tob(2n + 1 )1 n) .
(17)
Moreover, we can deal in eq. (13) with the purely fermionic part
and n(fermionic ) ~-- l(D[f~,fb ] ,
7~,(x) = Y exp[ - ½(b+ia)cox2]H,(x/~ x ) , (11) where e takes the cr3-eigenvalues + 1 or - l, Y is an ad-hoc normalization factor and H . (y) are the usual Hermite polynomials. Let us point out that the polynomial method also requires the condition (5) and the specific demand b>~0.
(12)
These results coincide with the expected characteristics for a supersymmetric harmonic oscillatorlike system with double degeneracies except for the fundamental level Eo = 0 (e = - 1 ) which is not degenerate. For b-~l ( a - , 0 ) , the usual supersymmetric harmonic oscillator is immediately recovered while, for b--,0, the spectrum completely reduces to the zero eigenvalue. The above characteristics can now be reexamined in the context of quantum deformations associated with specific q-harmonic oscillatorlike systems, where the parameter b could play the role of the deformation parameter q. In fact, let us recall that some qdeformations of the (bosonic) harmonic oscillator have been proposed [ 3-5 ], so that we can rewrite the hamiltonian (9) as a supersymmetric deformed one written as H ss = ½to{ab, aJ;}+ ½to[f ;,fbl •
(13)
Here we point out the purely bosonic part H(b . . . . ic) = ½to{ab, a;},
(14)
function of annihilation (ab) and creation (a~) de-
(18)
function of annihilation (fb) and creation (f~) deformed fermionic operators acting only on the states I0) or I 1 ) and ensuring that, with the same bracket [n]b, we have H(fermi°nic) ( [ 1 ) ) = 2 to
~[1))"
(19)
We thus observe that the energy eigenvalues (10) are also issued from the supersymmetric deformed hamiltonian (13), so that we have here an example of the link between n = 2 supersymmetry and quantum deformations. Our b-deformation is different from the already proposed q-deformations [ 3-5 ] but it can be analysed in connection with them. Let us, for example, point out that, with respect to the Biedenharn choice [ 4 ] where
qn/2q-n/2 [nlq_qj--fT~_q~77 ,
(20)
we find that the definitions (16) and (20) lead to b---
In q
q~/Z_q-l/2
, =q~/2[1-½(q-1)+](q-1)2+...],
(21)
if we accept that (ln q ) ~ ~ f i ~ ~ 0 ,
¥a>~3.
(22)
The last condition has to be imposed in order to limit, in accordance with eq. (16), the series included in eq. (20). When b-~ 1 or q--, 1, we recover the usual bosonic oscillator and all the above relations are compatible, 291
Volume 286, number 3,4
PHYSICS LETTERS B
30 July 1992
if we require that In q is an infinitesimal q u a n t i t y such that ( 2 2 ) is valid, we clearly satisfy
cations being actually the most developed ones in sup e r s y m m e t r i c q u a n t u m mechanics.
0~
References
exp(½6)-exp(-
½6)
~<1,
(23)
according to the constraint (12). F o r brevity, let us only m e n t i o n here that we have also applied the a b o v e d e v e l o p m e n t s to other physical applications [ 6 ] such as the s u p e r s y m m e t r i c hydrogen a t o m and the Calogero context a d m i t t i n g the largest numbers o f s u p e r s y m m e t r i e s [ 7 ], besides the case o f the superpotential (21), these typical appli-
292
[ 1] E. Witten, Nucl. Phys. B 188 ( 1981 ) 513. [ 2 ] M. de Crombrugghe and V. Rittenberg, Ann. Phys. (NY) 151 (1983) 99. 13] A.J. Macfarlane, J. Phys. A 22 (1989) 458l. [4] L.C. Biedenharn, J. Phys. A 22 (1989) L873. [5] Y.J. Ng, J. Phys. A 23 (1990) 1023. [6] J. Beckers and N. Debergh, University of Liege preprint (April 1992), to be published. [7] J. Beckers, N. Debergh and A.G. Nikitin, J. Math. Phys. 33 (1992) 152.
PHYSICS LETTERS B
From N = 2 supersymmetry to quantum deformations J. Beckers a n d N. D e b e r g h 1 Theoretical and Mathematical Physics, Institute of Physics, B.5, University of Libge, B-4000 Liege 1, Belgium
Received 8 April 1992; revised manuscript received 14 May 1992
A remarkable link between N=2-supersymmetry and quantum deformations is pointed out. It already appears in the onedimensional context of supersymmetric quantum mechanics when generalized expressions of the supercharges are constructed. In particular, these results are illustrated through a simple application to harmonic oscillatorlike systems hut they can be applied to arbitrary superpotentials.
Let us consider the standard Witten supersymmetrization procedure [1 ] in N = 2 - s u p e r s y m m e t r i c q u a n t u m mechanics but by generalizing the two hermitian (odd) supercharges Qt and Qz which, with the supersymmetric (even) hamiltonian H ss, generate the well known (lie) algebra s q m ( 2 ) characterized by the structure relations: Q2 = Q 2 = H S S ,
MR=NS
[QI, H ss ] = [Q2, H ss ] = 0 .
( 1)
RM=SN,
MS= - NR ,SM = - RN .
(3b)
The matrices (M, N) and (R, S) have thus to generate at least two Clifford algebras ~ 2 : they can be realized in terms o f Pauli matrices as follows: M =~r~,
{Q,, Q 2 } = 0 ,
,
N =or2,
R = aa~ + brr2 ,S = arr2 - ba2 ,
(4)
where the parameters a and b are constrained by Let us indeed propose the following general forms in the one-dimensional spatial context: 1 Q, = ~7~ [Mpx + R Ox W ( x ) ] , 1
Q2 = ~7~ [ N p y + S O x W ( x ) ]
(2a)
(2b)
,
where M, R, N, S are constant matrices and W ( x ) is the so-called superpotential, while p x - - iOx as usual. It is easy to convince ourselves that the relations ( 1 ) imply on M, R, N and S that {M,N}=0, M2=N2=I2,
{R,S}=0, (3a)
R2=$2=I2
supplemented by the "crossed" conditions Chercheur, Institut Interuniversitaire des Sciences nucl6aires, Brussels, Belgium. 290
aZ+b2= 1 ,
a, bEgq.
(5)
We notice that, if b = + 1 ( a = 0 ) , we immediately recover the standard context [ 1 ]. For all other sets of values o f the parameters a and b, we get here unexploited supersymmetrization procedures (up to the de Crombrugghe-Rittenberg discussion [2 ] ) leading to the following respective hamiltonians: HSS__l 2 ]2 +½ha30~W(x) _~p,+½[O,.W(x) +a{ [Ox W ( x ) ] P x - ½io2 W ( x ) } .
(6)
The above a terms and the values b• _+ 1 characterize the differences with respect to the usual hamiltonian [ 1 ] SS I 2 Hwiue. = ~px + 1 [O,. W(X) ]2 + ~cr3 I G2 W(x),
(7)
corresponding to an arbitrary superpotential W ( x ) . Let us here restrict ourselves to harmonic oscillatorlike systems by dealing with the superpotential
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 286, number 3,4
W(x)
PHYSICS LETTERS B
(8)
= ½tox 2 ,
to being the corresponding angular frequency. The operator (6) then becomes HSS t . 2 31_ltoZx2.~_ ½toba3 b ~ ~Px
formed bosonic operators acting in a Hilbert space { I n ) , n = 0 , 1 , 2 .... }suchthat a b l n ) = [x/-~b I n - l ) , a~ln)=~l]b
In+l),
(15)
where the new bracket [ n ] b is defined according to
+to( 1 - b 2) ~/2(Xpx - li) .
(9)
Its eigenvalues and eigenfunctions can then be obtained by the polynomial method largely used in quantum mechanics. We easily get
E,=½to[b(2n+l)+be],
30 July 1992
n=0,1,2,...,
(10)
[n]b=bn,
(16)
in order to ensure that H(b . . . . ic) ln ) = ½tob(2n + 1 )1 n) .
(17)
Moreover, we can deal in eq. (13) with the purely fermionic part
and n(fermionic ) ~-- l(D[f~,fb ] ,
7~,(x) = Y exp[ - ½(b+ia)cox2]H,(x/~ x ) , (11) where e takes the cr3-eigenvalues + 1 or - l, Y is an ad-hoc normalization factor and H . (y) are the usual Hermite polynomials. Let us point out that the polynomial method also requires the condition (5) and the specific demand b>~0.
(12)
These results coincide with the expected characteristics for a supersymmetric harmonic oscillatorlike system with double degeneracies except for the fundamental level Eo = 0 (e = - 1 ) which is not degenerate. For b-~l ( a - , 0 ) , the usual supersymmetric harmonic oscillator is immediately recovered while, for b--,0, the spectrum completely reduces to the zero eigenvalue. The above characteristics can now be reexamined in the context of quantum deformations associated with specific q-harmonic oscillatorlike systems, where the parameter b could play the role of the deformation parameter q. In fact, let us recall that some qdeformations of the (bosonic) harmonic oscillator have been proposed [ 3-5 ], so that we can rewrite the hamiltonian (9) as a supersymmetric deformed one written as H ss = ½to{ab, aJ;}+ ½to[f ;,fbl •
(13)
Here we point out the purely bosonic part H(b . . . . ic) = ½to{ab, a;},
(14)
function of annihilation (ab) and creation (a~) de-
(18)
function of annihilation (fb) and creation (f~) deformed fermionic operators acting only on the states I0) or I 1 ) and ensuring that, with the same bracket [n]b, we have H(fermi°nic) ( [ 1 ) ) = 2 to
~[1))"
(19)
We thus observe that the energy eigenvalues (10) are also issued from the supersymmetric deformed hamiltonian (13), so that we have here an example of the link between n = 2 supersymmetry and quantum deformations. Our b-deformation is different from the already proposed q-deformations [ 3-5 ] but it can be analysed in connection with them. Let us, for example, point out that, with respect to the Biedenharn choice [ 4 ] where
qn/2q-n/2 [nlq_qj--fT~_q~77 ,
(20)
we find that the definitions (16) and (20) lead to b---
In q
q~/Z_q-l/2
, =q~/2[1-½(q-1)+](q-1)2+...],
(21)
if we accept that (ln q ) ~ ~ f i ~ ~ 0 ,
¥a>~3.
(22)
The last condition has to be imposed in order to limit, in accordance with eq. (16), the series included in eq. (20). When b-~ 1 or q--, 1, we recover the usual bosonic oscillator and all the above relations are compatible, 291
Volume 286, number 3,4
PHYSICS LETTERS B
30 July 1992
if we require that In q is an infinitesimal q u a n t i t y such that ( 2 2 ) is valid, we clearly satisfy
cations being actually the most developed ones in sup e r s y m m e t r i c q u a n t u m mechanics.
0~
References
exp(½6)-exp(-
½6)
~<1,
(23)
according to the constraint (12). F o r brevity, let us only m e n t i o n here that we have also applied the a b o v e d e v e l o p m e n t s to other physical applications [ 6 ] such as the s u p e r s y m m e t r i c hydrogen a t o m and the Calogero context a d m i t t i n g the largest numbers o f s u p e r s y m m e t r i e s [ 7 ], besides the case o f the superpotential (21), these typical appli-
292
[ 1] E. Witten, Nucl. Phys. B 188 ( 1981 ) 513. [ 2 ] M. de Crombrugghe and V. Rittenberg, Ann. Phys. (NY) 151 (1983) 99. 13] A.J. Macfarlane, J. Phys. A 22 (1989) 458l. [4] L.C. Biedenharn, J. Phys. A 22 (1989) L873. [5] Y.J. Ng, J. Phys. A 23 (1990) 1023. [6] J. Beckers and N. Debergh, University of Liege preprint (April 1992), to be published. [7] J. Beckers, N. Debergh and A.G. Nikitin, J. Math. Phys. 33 (1992) 152.