2

identifying the fermionic operators a and a+ as the quantized ver-

sions of ψ and ψ = (≡ ψ + ), respectively, [in (2.127) the variables ψ

and ψ play the role of classical fermionic variables], one can make a

transition to the Hamiltonian H of the system (2.127). H turns out

to be

1 121

H = p2 + U + U σ 3 (2.129)

2 2 2

where the momentum conjugate to ψ is clearly ’iψ while {ψ, ψ} =

0 = {ψ, ψ}, {ψ, ψ} = 1 and (ψ, ψ) represented by 1 σ“ .

2

This H is of the same form as (2.26) if we make the identi¬cation

U = ’W , the superpotential. Note that NF becomes 1 (1+σ3 ). The

2

above forms of the Lagrangian and Hamiltonian are the ones relevant

to N = 2 supersymmetric quantum mechanics. This completes our

discussion on the construction of the Lagrangians for N = 1 and

2 supersymmetric theories. In this connection it is interesting to

note that the notion of seeking supersymmetric extensions has been

successful in establishing superformulation of various algebras [92].

It has been possible [93] to relate the superconformal algebra to the

supersymmetric extension of integrable systems such as the KdV

equation [94-96]. Moreover a representative for the SO(N ) or U (N )

superconforma algebra has been found possible in terms of a free

boson, N free fermions, and an accompanying current algebra [97].

It is also worth emphasizing that the properties characteristic of

fermionic variables are crucial to the development of the supersym-

metrization procedure. Noting that the fermionic quantities fi , fj

form the basis of a Cli¬ord algebra CL2n given by

{f+j , f’l } = δil I, (2.130)

{f±j , f±l } = 0 (2.131)

if we consider the replacement [98] of the rhs of (2.131) as δjl I ’

+

δjl I ’ 2Ojl with Ojl = ’Olj , Ojl = Ojl , we are led to a Hamilto-

© 2001 by Chapman & Hall/CRC

nian of the type (2.35) but inclusive of a spin-orbit coupling team ∼

(xj pl ’ xl pj ) Ojl . The conformal invariances associated with the su-

persymmetrized harmonic oscillator have been judiciously exploited

in [99-101] and the largest kinematical and dynamical invariance

properties characterizing a higher dimensional harmonic oscillator

system, in the framework of spin-orbit supersymmetrization, have

also been studied. Related works [102] also include the “exotic” su-

persymmetric schemes˜in two-space dimensions arising for each pair

of integers v+ and v’ yielding an N = 2(v+ + v’ ) superalgebra in

nonrelativistic Chem-Simons theory.

2.7 Other Schemes of SUSY

From (2.34) it can be easily veri¬ed that the commutator of the oper-

ators A and A+ is proportional to the derivative of the superpotential

dW

A, A+ = 2 (2.132)

dx

In this section we ask the question, whether we can impose some

group structure upon A and A+ in the framework of SUSY.

Consider the following representations of A and A+ [103]

‚ ‚

A = eiy k(x) ’ ik (x) + U (x)

‚x ‚y

‚ ‚

A+ = e’iy ’k(x) ’ ik (x) + U (x) (2.133)

‚x ‚y

where a prime denotes partial derivative with respect to x and k(x),

U (x) are arbitrary functions of x. It is readily found that if we

introduce an additional operator

‚

A3 = ’i (2.134)

‚y

A, A+ and A3 satisfy the algebra [104]

[A, A+ ] = ’2aA3 ’ bI

[A3 , A] = A

[A3 , A+ ] = ’A+ (2.135a, b, c)

© 2001 by Chapman & Hall/CRC

where I is the identity operator and a, b are appropriate functions of

x

a = [k (x)]2 ’ k(x)k (x)

b = 2[k (x)U (x) ’ k(x)U (x)] (2.136)

The simultaneous presence of the functions a(x) and b(x) in

(2.135a) is of interest. Without a(x), (2.135a) reduces to (2.132).

This is because a = 0 is consistent with k = 1, U = W , and b = ’W .

On the other hand, the case b = 0 is associated with U (x) = 0.

Clearly, the latter is a new direction which does not follow from Wit-

ten™s model. Note that when b = 0, k(x) = sinx, and a = 1 so A, A+ ,

and A3 may be identi¬ed with the generators of the SU (1, 1) group

[105-107].

Given the representations in (2.133), we can work out the mod-

i¬ed components H+ and H’ as follows

1+

H+ = AA

2

2

1 2‚ ‚

= ’k + ikk ’ kU + k U

‚x2

2 ‚y

‚2 2‚

+ U ’ ik ’ ik (2.137)

‚y ‚y

1

AA+

H’ =

2

2

1 2‚ ‚

= ’k ’ ikk + kU ’ k U

‚x2

2 ‚y

2

‚ ‚

2

+ U ’ ik +k (2.138)

‚y ‚y

It should be stressed that the variable y is not to be confused with an

extra spatial dimension and merely serves as an auxiliary parameter.

This means that for a physical eigenvalue problem, the square of the

modules of the eigenfunction must be independent of y.

The above model has been studied in [103] and also by Janus-

sis et al. [108], Chuan [109], Beckers and Ndimubandi [110] and

Samanta [111]. In [108], a two-term energy recurrence relation has

been derived Wittin the Lie admissible formulation of Santilli™s the-

ory [112-113]. In [109] a set of coupled equations has been proposed,

© 2001 by Chapman & Hall/CRC

a particular class of which is in agreement with the results of [103].

In [110] connections of (2.133) have been sought with quantum deor-

mation. Further, in [111], (2.137) and (2.138) have been successfully

applied to a variety of physical systems which include the particle in a

box problem, Morse potential, Coulomb potential, and the isotropic

oscillator potential.

Finally, let us remark that other extensions of the algebraic ap-

proach towards SUSYQM have also appeared (see Pashnev [114]) for

N = 2, 3. Moreover, Verbaarschot et al. have calculated the large

order behaviour of N = 4 SUSYQM using a perturbative expansion

[115]. The pattern of behavior has been found to be of the same

form as models with two supersymmetrics. Akulov and Kudinov

[116] have considered the possibility of enlarging SUSYQM to any N

by expressing the Hamiltonian as the sum of irreducible representa-

tions of the symmetry group SN . To set up a working scheme certain

compatibility conditions arise by requiring the representations to be

totally symmetric and to satisfy a superalgebra. Very recently, Znojil

et al. [117] have constructed a scheme of SUSY using nonhermitean

operators. Its representation spere is spanned by bound states with

P T symmetry but yields real energies.

2.8 References

[1] P.A.M. Dirac, The Principles of Quantum Mechanics, 4th ed.,

Clarendon Press, Oxford, 1958.

[2] L.I. Schi¬, Quantum Mechanics, 3rd ed., McGraw-Hill, New

York, 1968.

[3] S. Gasiorowicz, Quantum Physics, John Wiley & Sons, New

York, 1974.

[4] O.L. de Lange and R.E. Raab, Operator Methods in Quantum

Mechanics, Clarendon Press, Oxford, 1991.

[5] R.J. Glauber, in Recent Developments in Quantum Optics, R.

Inguva, Ed., Plenum Press, New York, 1993.

[6] E. Fermi, Notes on Quantum Mechanics, The University of

Chicago Press, Chicago, 1961.

© 2001 by Chapman & Hall/CRC

[7] F. Ravndahl, Proc CERN School of Physics, 302, 1984.

[8] E. Witten, Nucl. Phys., B188, 513, 1981.

[9] E. Witten, Nucl. Phys., B202, 253, 1982.

[10] A.I. Vainshtein, A.V. Smilga, and M.A. Shifman, Sov. Phys.

JETP., 67, 25, 1988.

[11] A. Arai, Lett. Math. Phys., 19, 217, 1990.

[12] P. Salomonson and J.W. van Holten, Nucl. Phys., B196, 509,

1982.

[13] M. Clandson and M. Halpern, Nucl. Phys., B250, 689, 1985.

[14] F. Cooper and B. Freedman, Ann. Phys., 146, 262, 1983.

[15] G. Parisi and N. Sourlas, Nucl. Phys., B206, 321, 1982.

[16] S. Cecotti and L. Girardello, Ann. Phys., 145, 81, 1983.

[17] L.F. Urrutia and E. Hernandez, Phys. Rev. Lett., 51, 755,

1983.

[18] M. Bernstein and L.S. Brown, Phys. Rev. Lett., 52, 1983,

1984.

[19] H.C. Rosu, Phys. Rev., E56, 2269, 1997.

[20] E. Gozzi, Phys. Rev., D30, 1218, 1984.

[21] E. Gozzi, Phys. Rev., D33, 584, 1986.

[22] J. H´rby, Czech J. Phys., B37, 158, 1987.

u

[23] J. Maharana and A. Khare, Nucl. Phys., B244, 409, 1984.

[24] R. Akhoury and A. Comtet, Nucl. Phys., B245, 253, 1984.

[25] D. Boyanovsky and R. Blankenbekler, Phys. Rev., D30, 1821,

1984.

[26] L.E. Gendenshtein, JETP Lett., 38, 356, 1983.

© 2001 by Chapman & Hall/CRC

[27] A.A. Andrianov, N.V. Borisov, and M.V. Io¬e, Phys. Lett.,

A105, 19, 1984.

[28] M.M. Nieto, Phys. Lett., B145, 208, 1984.

[29] C.V. Sukumar, J. Phys. A. Math. Gen., 18, L57, 2917, 2937,

1985.

[30] C.V. Sukumar, J. Phys. A. Math. Gen., A20, 2461, 1987.

[31] V.A. Kostelecky, M.M. Nieto, and D.R. Truax, Phys. Rev.,

D32, 2627, 1985.

[32] V.A. Kosteleeky and M.M. Nieto, Phys. Rev. Lett., 53, 2285,

1984.

[33] V.A. Kosteleeky and M.M. Nieto, Phys. Rev. Lett., 56, 96,

1986.

[34] V.A. Kosteleeky and M.M. Nieto, Phys. Rev. A32, 1293, 1985.

[35] E. D™Hoker and L. Vinet, Phys. Lett., B137, 72, 1984.

[36] C.A. Blockley and G.E. Stedman, Eur. J. Phy., 6, 218, 1985.

[37] P.D. Jarvis and G. Stedman, J. Phys. A. Math. Gen., 17, 757,

1984.

[38] M. Baake, R. Delbourgo, and P.D. Jarvis, Aust. J. Phys., 44,

353, 1991.

[39] de Crombrugghe and V. Rittenberg, Ann. Phys., 151, 99, 1983.

[40] D. Lancaster, Nuovo Cim, A79, 28, 1984.

[41] S. Fubini and E. Rabinovici, Nucl. Phys., B245, 17, 1984.

[42] H. Yamagishi, Phys. Rev., D29, 2975, 1984.

[43] H. Ui and G. Takeda, Prog. Theor. Phys., 72, 266, 1984.

[44] H. Ui, Prog. Theor. Phys., 72, 813, 1984.

[45] D. Sen, Phys. Rev., D46, 1846, 1992.