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number operator NF . Writing in (2.126) ψψ as 1 (ψψ ’ ψψ) and
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
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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
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