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Quantum mechanics in the E( 1 N2) superspace (*)

M. DAUMENSand Y. NOIROT

Centre de Physique Théorique et Modélisation, Unité associée au CNRS, URA 1537 Université de Bordeaux I - Rue du Solarium, 33175 Gradignan Cedex, France (ricevuto il 7 Aprile 1997; revisionato il 18 Giugno 1997; approvato il 24 Giugno 1997)

Summary. — Quantum mechanics in the E( 1 N2) superspace is presented and its differences with the supersymmetric quantum mechanics, proposed by Witten, are exhibited. For central potential, the problem has an osp( 1 N2) symmetry which allows us to separate odd Grassmann variables by means of the dispin basis, and to derive a radial equation. This equation is solved in two cases: for the Coulomb potential bound states, where the energy levels are the same as for the hydrogen atom, and for the isotropic harmonic oscillator, where they are translated. In both cases the energy levels are degenerated, but the states are labelled by the eigenvalues of the superspin and of the spin. The Fock space of the oscillator is constructed.

PACS.11.30.Pb – Supersymmetry. PACS. 02.20 – Group theory.

1. – Introduction

Anticommuting variables appeared in physics many years ago. They belong to a Clifford algebra where their square is one, or to a Grassmann algebra where their square is zero. The Clifford variables are very often realized by matrices, for example Pauli matrices or Dirac matrices. A Grassmann algebra contains even variables that commute and odd variables that anticommute. From the analysis point of view, one can consider the even and odd variables in the same way, i.e. derivation and integration over the odd variables are defined. On the other hand, if the square of an odd variable vanishes the question remains, what quantitative calculations can be performed with these odd variables? What is their concrete physical meaning?

To try to give an answer to these questions, we consider the simplest superspace

E( 1 N2); only three variables of a Grassmann algebra are involved: one is even and the

other two are odd. Then, by treating the three variables on equal footing, we define quantum mechanics in this superspace by means of three conjugate momenta, one for each commuting or anticommuting variable. The three momenta are realized as

(*) The authors of this paper have agreed to not receive the proofs for correction.

differential operators acting on functions defined on this superspace. There are two crucial differences between our work and the supersymmetric quantum mechanics proposed by Witten [1-3] for the same dimensions. There, only the commuting variable has a conjugate momentum and the anticommuting variables belong to a Clifford algebra (their square is one), instead of a Grassmann algebra in our case.

Besides E( 1 N2) is the fundamental representation space of the super-rotation algebra osp( 1 N2). This superalgebra, also denoted by gsu(2) or by B(0, 1), has been thoroughly studied in the literature (see, for instance, [4-7]). Its finite-dimensional irreducible representations are characterized by a superspin j which is either integer or half-integer and by a parity l taking the two values 0 or 1. When the potential depends only on the pseudonorm of the position super-vector, the Hamiltonian is invariant under the super-rotations and the problem factorises into a radial part and superspin eigenfunctions. These eigenfunctions were obtained in a previous article [8]. In this simpler case, only the trivial ( j 40) and fundamental ( j41/2) representations appear; they constitute the dispin realization [9], and we derive the differential equation of the radial part analogous to that obtained in usual quantum mechanics.

To get quantitative results, this radial equation is solved here for two problems: the bound states of the Coulomb potential and the isotropic harmonic oscillator. In both cases the energy levels are degenerate, but the states can be labelled completely by the eigenvalues of the superspin ( j 40, or 1/2) and of the spin; the corresponding eigenfunctions are expressed in terms of the dispin basis and they form a pseudo-orthonormal set. For the Coulomb problem the energy levels are the same as those of the hydrogen atom in the ordinary E( 3 ) space, while for the «isotropic» harmonic oscillator the energy levels are found to be En4 n 2 1 /2 (n non-negative

integer); the usual spectrum is translated because the fundamental state has the negative energy: E04 21 /2. Moreover, the realization of the osp( 1 N2 ) super-algebra in

terms of one boson and two fermion pairs of creation annihilation operators, given in [10], allows us to construct the Fock space of the harmonic oscillator and to confirm the previous results.

This article is organized as follows. In sect. 2 we recall the N 42 supersymmetric quantum mechanics in one dimension and in sect. 3 we present the quantum mechanics in E( 1 N2) superspace. Section 4 is devoted to the central potential case and its osp(1N2) symmetry. The Coulomb potential and the harmonic oscillator are studied in sect. 5 and 6, respectively.

2. – N 42 supersymmetric quantum mechanics in one dimension

First, we recall the basic facts on N 42 supersymmetric quantum mechanics in one dimension (see ref. [3]), to exhibit the difference between it and quantum mechanics in the E( 1 N2) superspace. Let us consider a system with one bosonic degree of freedom (coordinate x and momentum p) and two fermionic degrees of freedom (c1, c2)

satisfying the algebra

[x , p] 4i , ]ci, cj( 4 2 dij, (i , j 41, 2) ,

(1)

[x , ci] 40 , [p, ci] 40 , x†4 x , p†4 p , ci†4 ci .

(2)

Let us note that the anticommuting variables belong to a Clifford algebra and their square is one.

The most general Hamiltonian depending on those variables H 4H(x, p, ci) is

obtained from two supercharges (Qa, Qb) according to

]Qa, Qb( 4 2 Hda , b, [Qa, H] 40 (a , b 41, 2) .

(3)

In the general case, the supercharges are expressed in terms of two operators (A1, A2)

function of (x , p) as Q14 1 k2(A1c11 A2c2) , Q24 1 k2(2A2c11 A1c2) , (4)

and the corresponding Hamiltonian is

H 4 1

2(A1

2

1 A221 [A1, A2] c1c2) .

(5)

By choosing A14 p and A24 f (x) one gets H 4 1 2p 2 1 1 2f 2 (x) 2if 8(x) c1c2f 1 2p 2 1 V(x , c1c2) . (6)

The potential depends on the product of the two anticommuting variables. The case where these variables are realized by the Pauli matrices is discussed in detail in ref. [1, 2].

3. – Quantum mechanics in the E( 1 N2) superspace

Let G 4G05 G1be a Grassmann algebra, of which we denote by n the Grassmann

degree, and by3_{the complex conjugation}

x , y  Gn, l  C , (xy)34 x3y3, (lx)34 l * x3, (x3)34 (21 )n(x)x .

(7)

The real commuting space E( 1 N2) is the set of the (1 N2)-dimensional supervectors

X 4 (x, x61 /2) such that x  G0, (x)34 x ; xaG1, (xa)34 Gabxb, G 4

u

0 21 1 0

v

. (8)

Let us note that here, in contrast to the previous section, the square of the anti-commuting variables is zero. The scalar product of two supervectors is

(X , Y) 4xy1Gabxahb ,

(9)

and the corresponding square of a super-vector is

X2

4 (X , X) 4 x21 Gabxaxb .

(10)

Let f be a function defined on E( 1 N2). Its value for the supervector X will be f(X) 4

f (x , xa). The scalar product of two functions is

a f , gb 4



f3_{(x , x}

a) g(x , xa) dx dx11 /2dx21 /2 ,

and the super-Hermitian conjugated A‡_{of an operator A is defined by}

a f , A‡

gb 4 (21)n(A) n(f)_{aAf , gb ,}

(12)

with the useful properties (A‡₎‡

4 (21 )n(A)A ; (AB)‡

4 (21 )n(A) n(B)B‡_A‡ _.

(13)

Now, we consider the supervector X as a multiplicative operator on the functions of X, and we define the operators P 4 (p, pb_{), canonically conjugate to X, by differential}

operators acting on the previous functions as

p 4 1 i ¯ ¯x , p a 4 i ¯ ¯xa . (14)

They satisfy the (anti)commutation relations [x , p] 4i , [x, pa

] 4 [xa, p] 40 , ]xa, pb( 4 idab ,

(15)

and the super-Hermitian properties (p)‡

4 p , (pa)‡

4 Gabpb .

(16)

The Hamiltonian depends on the variables X and P. It must be super-Hermitian and an even element of the Grassmann algebra

H 4H(X, P) , H‡_{4 H , (21 )}n(H)_{4 1 .}

(17)

For a particle in a potential V(X) such that

V‡_{4 V , (21 )}n(V)_{4 1 ,}

(18)

the Hamiltonian may be written

H 4 1

2P

2

1 V(X) . (19)

In the E( 1 N2) super-space, the super-spin J4 (JK, Ja), a 461/2, is defined as [8]

J K 4 1 2 ixa(s K )a bpb, Ja4 1 2 i(xGabp b 2 xap) . (20)

These operators have the (anti)commutation rules of the super-rotation algebra

osp( 1 N2). For a general potential V(X), the super-spin commutes neither with V nor

with H

[J , V] c 0 , [J , H] c 0 , (21)

and the problem does not have super-rotation symmetry. In spite of many attemps, we do not succeed in defining super-charges as in the previous section, hence there is not

4. – Central potential and the osp( 1 N2) symmetry

Let us consider the case where the potential is central, i.e. V( X ) depends only on the pseudonorm X of the supervector X

X 4 (X2₎1 /2

4 x 1 1

xx11 /2x21 /2 .

(22)

Then it is straightforward to verify that the Hamiltonian

H 4 1

2P

2

1 V( X ) (23)

commutes with the super-spin

[H , JK_{] 4 [H, J}a] 40 .

(24)

In this case the problem has super-rotation symmetry and the operators ]H , J2, JK2_{, J}

3( form a commuting set, where J24 J K 2

1 GabJaJb is the Casimir of the

super-spin; the eigenvalues of ]J2, JK2, J3( are ]j( j 1 1 /2 ), l(lj 1 1 ), m(, respectively,

with 2 j  N ; l 4j, j21/2; m ]2m, 2l11, . . . , 1l( . The super-spin does not depend on the pseudonorm X, it is explicitly given in [8] in terms of the reduced coordinates X and hadefined by

ha4

1

xxa .

(25)

Let us recall the expressions of J2, that we shall use in the following:

J2 4 1 2 ha¯ a 1 1 2( 1 1h1h2) ¯ 1_¯2 _, (26)

with the abbreviated notations h_{64 h61 /2}, ¯6

4 ¯

¯h_{61 /2}. In this realization only the j 4

0 and j 41/2 representations appear and the corresponding eigenfunctions Ylmj (ha),

that constitute the dispin basis, are simply

Y0

004 1 , Y001 /24 h1h–2 1 , Y1 /2 a1 /2 4 ha .

(27)

They satisfy the pseudo-orthonormality relation



_Yj 3 lm (ha) Y

j

lm(ha)( 1 1h1h2) dh1dh24 (21 )2 j 11djj 8dll 8dmm 8.

(28)

The time-independent Schrödinger equation with central potential in the E( 1 N2) super-space reads HC 4

k

1 2 P 2 1 V( X )

l

C 4EC , (29)

where E is the energy and C is function of X. In terms of the reduced coordinates, the operator P2is expressed in the form

P2_{4 2} ¯ 2 ¯X2 1 2 X ¯ ¯X 2 4 X2J 2 _, (30)

and the Schrödinger equation may be written

y

¯2 ¯X2 2 2 X ¯ ¯X 1 2[E 2 V( X ) ] 1 4 X2J 2

z

C( X , ha) 40 . (31)

Now, the separation of the variables is performed by means of the super-spin eigenfunctions Ylmj for fixed j , l , m

Clmj ( X , ha) 4Fj( X ) Y j lm(ha) ,

(32)

and by noticing that j( j 11/2) 4j for j40, 1/2, the resulting radial equation is

y

d2 d X2 2 2 X d d X 1 2[E 2 V( X ) ] 1 4 j X2

z

Fj( X ) 40 . (33)

The first derivative term can be eliminated through the change of functions Fj( X ) 4 XRj( X ),

(34)

and one gets

y

d2

d X2 1 2[E 2 V( X ) ] 1 2

2 j 21

X2

z

Rj( X ) 40 .

(35)

Let us remark that in the case j 41/2 this equation is the usual one-dimensional Schrödinger equation.

To determine the expression of the scalar product of two functions in terms of the reduced coordinates, one calculates the Jacobian of the change of variables [11]

J 4

u

¯x ¯X ¯xa ¯X 2 ¯x ¯hb ¯xa ¯hb

v

4

u

1 2h1 h₂ ha XGbghg Xdab

v

, (36)

and its super-determinant (Berezinian) is

sdet ( J ) 4 1 1h1h2 X2 .

(37)

An artificial singularity appears at X 40. To remove it, the previous expression must be considered in the distribution sense, i.e. the scalar product is defined through the

principal value of the integral

a f , gb 4P



1 X2 f

3_{( X , h}

a) g( X , ha)( 1 1h1h2) d X dh1dh2 ,

where the principal value is defined as P



2Q 1Q f (x) x2 dx 4 lim_{e K0}

y



2Q 2e f (x) x2 dx 1



1e 1Q f (x) x2 dx 22 f ( 0 ) e

z

. (39)

After integration over the Grassmann variables ha, the pseudonorm of a wave function

is given by aCjlm, C j lmb 4 (21)2 j 11P



1 X2NFj( X ) N 2 d X 4 (21)2 j 11P



_NRj( X ) N2d X , (40)

and in most cases this last integral is regular. However, the above regularization will be necessary for the ground state of the harmonic oscillator. Hence the Rj functions must

satisfy the boundary conditions lim

X K0XRj( X ) finite , N X N K Qlim

Rj( X ) 40 .

(41)

5. – The Coulomb potential

For the Coulomb potential V( X ) 42 1

X the radial equation on the Rjfunctions is

R 9j ( X ) 1

k

2 E 1 2 X1 2 2 J 21 X2

l

Rj( X ) 40 . (42)

For X going to infinity, to obtain bound states the energy must be negative E E0 so that the asymptotic behaviour of the solution be e2k22 E X_{. By analogy with the}

hydrogen atom, one defines a parameter n as

n 4 1

k_{22 E}

` _{E 42} 1 2 n2 .

(43)

Then one carries out the change of variable and function

r 4 2 X

n , Fj(r) 4Rj( nr

2 ) , (44)

and the Fjfunctions satisfy

Fj9 (r) 1

k

[ 2 2 j 21 r2 1 n r 2 1 4

l

Fj(r) 40 . (45)

This is the Whittaker equation [12] with parameters n and m 4 (1/2)k_{9 216j. In}

general, the solution of this equation is a confluent hypergeometric function, to get solutions with the correct behaviour at infinity, the series must reduce to a polynomial. Then the parameter n must be a positive integer n such that n F1 for j41/2 and nF2 for j 40. Hence the energy levels are the same as those of the hydrogen atom

En4 2

1

2 n2 , n F1 .

The corresponding orthonormalized radial functions Fnjare simply Fn1 /2( X ) 4 1 k2 XRn0( X ) , n F1 , Fn0( X ) 4 1 k2 XRn1( X ) , n F2 , (47)

where the functions Rnl are the normed solutions of the hydrogen atom [13]; they

involve the Sonine-Laguerre polynomials

Rnl( X ) 4Cnl XlLn 2l212 l 11

g

2 X n

h

e 2X n (48) with normalization Cnl4 2l 11 nl 12

o

(n 2l21)! [ (n 1l)! ]3 . (49)

The complete eigenfunctions are Cj

nlm( X , ha) 4Fnj( X ) Y j lm(ha) ,

(50)

they satisfy the pseudo-orthonormality condition

aCjnlmNCj 8n 8 l 8 m 8b 4 (21)2 j 11dnn 8djj 8dll 8dmm 8 .

(51)

Finally, we note that the ground state n 41 has super-spin j41/2 and is threefold degenerate that the other levels n F2 have super-spin either 0 or 1/2 and are fourfold degenerate, while the energy levels of the hydrogen atom are n2_{-fold degenerate.}

6. – The harmonic oscillator

For the harmonic oscillator, when the potential is V( X ) 4 (1/2) X2

4 ( 1 /2 ) X2 we call it «isotropic» although one dimension is bosonic and the other two are fermionic. The radial equation on the Rjfunctions reads

R 9j ( X ) 1

k

2 E 2 X21 2 2 j 21

X2

l

Rj( X ) 40 . (52)

In the case j 41/2, this equation reduces to the radial equation of the usual one-dimensional harmonic oscillator. The energy levels are En4 n 2 1 /2 , n  N* and the

corresponding orthonormalized radial functions read

Fn1 /2( X ) 4 Xfn 21( X ) , n F1 ,

(53)

where the functions fn are the solutions of the one-dimensional quantum

oscillator [13]; they involve Hermite polynomials

fn( X ) 4 [kp n! ]21 /2Hn( X ) e2 X

2

O2 _.

(54)

In the other case j 40, it is more judicious to consider eq. (33) on the F functions F90( X ) 2 2 XF08 ( X ) 1 ( 2 E 2 X 2_{) F} 0( X ) 40 , (55)

and the asymptotic behaviour for X going to infinity suggests the substitution F0( X ) 4z( X ) e2 X

2_O2

. (56)

Then, one gets

z 9( X )22

g

X 1 1

X

h

z 8( X )1 (2E11)z( X ) 40 . (57)

This equation has an obvious constant solution corresponding to the fundamental energy level E04 21 /2. By using the regularization process (38), one deduces the

normalized fundamental eigenfunction

C0( X ) 4F0( X ) 4 [2kp]21 /2e2 X

2

O2 _,

(58)

where one remarks that

aC0, C0b 42[2kp]21P



1 X2e

2 X2_{d X 41 .}

(59)

Besides, for 2 E 11 D0, eq. (57) for the z-functions can be solved by the Frobenius series expansion methods. To get a polynomial solution, the following condition is required for the energy En4 n 2 1 /2 , n F 2, and the corresponding solution reads in

terms of Hermite polynomials Hn

zn( X ) 4Cn[Hn( X ) 12nHn 22] , n F2 ,

(60)

where Cnis a normalization constant. Then, one gets the normalized solutions

Fn0( X ) 4

o

n 21 2 fn( X ) 1

o

n 2 fn 22( X ) , n F2 , (61)

where the functions fnare defined in eq. (6.3).

Let us sum up the above results. The energy levels and the corresponding eigenfunctions are

n F0 , En4 n 2 1 /2 , Cjnlm( X , ha) 4Fnj( X ) Y j lm(ha) ,

(62)

and these functions satisfy the pseudo-orthonormality relation aCj

nlmNCj 8n 8 l 8 m 8b 4e(n, j) dnn 8djj 8dll 8dmm 8 ,

(63)

with e( 0 , 0 ) 41, e(n, j) 4 (21)2 j 11_{for n F2.}

The ground state (n 40) has a super-spin j40 and it is not degenerate; the first level (n 41) has j41/2 and it is three-fold degenerate, while the general levels for

n F2 are four-fold degenerate with either j40 or j41/2.

As for the usual quantum oscillator, an alternative way to study this oscillator is to construct its Fock space. We define the creation and annihilation operators

b†, b , aa†, aa(a 461/2) as follows: b 4 1 k2(x 1ip) , b † 4 b‡4 1 k2(x 2ip) , (64) aa4 1 k2(Gabxb2 ip a_{) , a}† 4 2aa‡4 1 k2(xa2 iGabp b_{) .} (65)

These operators satisfy the usual (anti)commutation relations [b , b†_{] 41, ]a}a, ab†( 4 dab ,

(66)

[b , aa] 4 [b, aa†] 4 [b†, aa] 4 [b†, aa†] 4 ]aa, ab( 4 ]aa†, ab†( 4 0 ,

(67)

so that b†_{, b are called boson operators and a}

a†, aa fermion operators. The total

occupation number operator N is

N 4b† b 1a11 /2 † _a 11 /21 a21 /2 † _a 21 /2 . (68)

Then, the Hamiltonian of the harmonic oscillator and the energy levels can be written as

H 4N21/2 , En4 n 2 1 /2 (n non-negative integer) ,

(69)

where n is the eigenvalue of the operator N. Of course, these levels are degenerate; to label the states completely, we shall use irreducible representations of the super-rotation algebra. We get a realization of this super-algebra by

J K 4 1 2aa †_(s)a bab, Ja4 1 2(b †_G abab1 aa†b) . (70)

This realization of osp( 1 N2) by one boson and two fermions pairs has been analyzed in [10] and the results are the following. The commuting operator set is ]N, J2_{, J}K2_{, J}

3(

and the corresponding eigenstates are denoted by Nnjlmb. In this realization, only the dispin 0 5 1 /2 representation [9] appears, and according to the value of n the degenerate states are

n 40, the fundamental state: N0000b 4N0b,

n 41, three states: N11/200b 42b†N0 b, N11 /21 /2 ab 4 aa†N0 b, n F2, four states Nn000 b 4

y

k_{n 21}(b †₎n kn! 1 kn (b†₎n 22

k

(n 22)! a† 11 /2a † 21 /2

z

N0 b , (71) Nn1 /200 b 4 2

y

kn(b †₎n kn! 1kn 21 (b†₎n 22

k

(n 22)! a† 11 /2a † 21 /2

z

N0 b , (72) Nn1 /21 /2 ab 4 (b †₎(n 21)

k

(n 21)! a† aN0 b , (73)

where the minus sign was chosen to satisfy the conventions of a standard osp( 1 N2) basis [4]. The super wave functions in the X-representation and in Cartesian coordinates, Cj

nlm(X), can be derived from the fundamental state one C0(X) by the

action of the previous creation operators. The function C0(X) is determined by the fact

that it must vanish through the action of the annihilation operators. The normalized solution is

C0(X) 4 [2kp]21 /2( 1 2x1x2) e2x

2

O2

and, of course, in Cartesian coordinates there is no regularization problem at the origin. The action of the creation operators on this function C0gives

n 41 C1 /2100(X) 42p21 /4( 1 2x1x2) xe2x 2 O2_, (75) C1 /2 11 /2 a(X) 4p21 /4xae2x 2_O2 , (76) n F2 C0n00(X) 4 1 k2[kn 21(12x1x2) fn(x) 1kn(11x1x2) fn 22(x) ] , (77) C1 /2n00(X) 42 1 k2[kn( 1 2x1x2) fn(x) 1kn 21(11x1x2) fn 22(x) ] , (78) C1 /2 n1 /2 a(X) 4xafn 21(x) . (79)

We have verified that these C eigenfunctions, written in terms of the Cartesian coordinates, are the same as that written above in terms of the reduced coordinates. 7. – Conclusion

By treating the three variables of E( 1 N2) on an equal footing, we have defined a quantum mechanics that is quite different of the super quantum mechanics proposed by Witten, in the same dimensions. For a general potential, there is no symmetry, while for a central potential the system has the super-rotation symmetry. This symmetry allows us to factorize the Grassmann variables into the superspin eigenfunctions and to derive a radial equation. To get quantitative results, we have solved this equation by techniques similar to that of the ordinary quantum mechanics in two cases: the Coulomb problem and the «isotropic» harmonic oscillator. In both cases, the levels are degenerate, but the states can be labelled completely by the superspin and spin variables.

For the Coulomb potential, we get the energy levels: En4 2

1

2 n2, n F1, which are

the same as in the ordinary three-dimensional space. This fact can be understood in a more general context, by studying the canonical realizations of the osp(DNd) superalgebra [14]. This analysis gives the following levels for the Coulomb problem in the E(DNd) superspace (D integer F1 and d even integer F0)

D Dd11 , En4 2 1 2 1 [n 11/4(D2d23) ]2 , (80) D 4d11 , En4 2 1 2 1 n2 , (81) D Ed11 , En4 2 1 2 1 [n 11/4(D2d21) ]2 . (82)

We see that, among several other ones, the two cases (D 43, d40) and (D41, d42) yield the same spectrum.

For the harmonic oscillator, the creation-annihilation operators and the corresponding Fock space are constructed, and the energy levels are En4 n 2 1 /2 ,

n F0. Up to a translation, these levels are the same as those obtained in the ordinary

one-dimensional space. This originates from the following fact: as the E( 1 N2) superspace is too restricted, only solutions with superspin ( j 40 or 1/2) appear; the value j 41/2 yields to the radial equation of the usual one-dimensional harmonic oscillator while for j 40, one only gets a peculiar solution associated to the level

E04 21 /2.

To get higher values of j, we must consider functions defined on a wider space. The simplest superspace generalization of the ordinary space E( 3 ) is the superspace

E( 3 N2) and it seems interesting to generalize the previous quantum mechanics to this

superspace.

* * *

We would like to acknowledge Profs. J. T. DONOHUE and P. MINNAERT for useful

discussions.

R E F E R E N C E S

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[5] SCHEUNERTM., NAHMW. and RITTENBERGV., J. Math. Phys., 18 (1977) 146, 155. [6] BEREZINF. A. and TOLSTOYV. N., Commun. Math. Phys., 78 (1981) 409.

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[14] DAUMENSM. and MINNAERTP., Canonical realizations of the osp(DNd) superalgebra and the Coulomb problem in superspaces, in progress.