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0 10

c+ 20 0 1
= (8.3)
0 00
From the above it is clear that the nature of the operators c and
c+ is parafermionic [17-21] of order 2
3
c3 = c+ = 0
cc+ c = 2c

c2 c+ + c+ c2 = 2c (8.4a, b, c)
Note that (8.4b) and (8.4c) are also consistent with the alternative
algebra in terms of double commutators

c, c, c+ = ’2c,
c+ , c, c+ = 2c+ (8.5)

One is thus motivated into de¬ning a set of parasupersymmet-
ric charges by combining the usual bosonic ones with parafermionic
operators. At the level of order 2 these are given by

Q = b — c+
Q+ = b+ — c (8.6)

which generalize the supersymmetric forms (2.21). In the following,
we shall assume the same notations for the parasupercharges as for
the supercharges.
With (8.4a) holding, it is obvious that
3
Q3 = 0 = Q+ (8.7)

More generally one can construct parasupercharges of order p with
the properties

(Q)p+1 = (Q+ )p+1 = 0, p = 1, 2, . . . (8.8)

When p = 1, (8.8) implies the usual nilpotency conditions of the su-
percharges in SUSYQM. Note that for higher order (p > 2) PSUSY,
the diagonal term in the right-hand-side of (8.2) needs to be replaced
by the generalized matrix of the type diag p , p ’ 1, . . . , ’ p + 1, ’ p .
22 2 2



© 2001 by Chapman & Hall/CRC
8.2 Models of PSUSYQM
(a) The scheme of Rubakov and Spiridonov

Rubakov and Spiridonov [1] were the ¬rst to propose a gen-
eralization of the Witten supersymmetric Hamiltonian (2.22) to a
PSUSY form. They de¬ned the PSUSY Hamiltonian Hp as arising
from the relations

Q2 Q+ + QQ+ Q + Q+ Q2 = 2QHp (8.9)
+2
Q+2 Q + Q+ QQ+ + QQ = 2Q+ Hp (8.10)

where Q and Q+ apart from obeying (8.7) also commute with Hp

[Hp , Q] = 0 = Hp , Q+ (8.11)

Keeping in mind the transitions (2.34),the parasupercharges Q
and Q+ can be given a matrix representation as follows
1 d
(Q)ij = √ + Wi (x) δi+1,j , i, j = 1, 2, 3 (8.12a)
2 dx
That is « 
0 A+ 0
1
1
A+ 
√ 0 0
Q= (8.12b)
2
200 0
Also
1 d
(Q+ )ij = √ ’ + Wj (x) δi,j+1 , i, j = 1, 2, 3 (8.12c)
dx
2
That is « 
0 0 0
1 ’
+
0
Q =√ A1 0 (8.12d)
2 A’
0 0
2

In (8.12b) and (8.12d) the notations A± (i = 1, 2) stand for
i

d
A± = Wi (x) ± (8.13)
i
dx
Here the parasupercharges Q and Q+ are de¬ned in terms of a pair
of superpotentials W1 (x) and W2 (x) indicating a switchover from the


© 2001 by Chapman & Hall/CRC
order p = 1 (which is SUSYQM) to p = 2 (which is PSUSYQM of
order 2).
Given (8.12), the PSUSY algebra (8.9) and (8.10) lead to the
following diagonal form for Hp

Hp = diag (H1 , H2 , H3 ) (8.14)

with H1 , H2 , and H3 in terms of A± (i = 1, 2) being
i

1
A’ H1 = A’ A+ + A+ A’ A’
1 11 22 1
4
1
A’ A+ + A+ A’
H2 = 11 22
4
1
A+ H3 = A’ A+ + A+ A’ A+ (8.15)
2 11 23 2
4
Now, the above representations are of little use unless H1 and
H3 , like H2 , are reducible to tractable forms. One way to achieve
this is to impose upon (8.15) the constraint

A’ A+ = A+ A’ + c (8.16)
11 22

where c is a constant. However, the components of Hp can be given
expressions which are independent of c, namely,

d2
1 2 2
H1 = ’2 2 + W1 + W2 + 3W1 + W2
4 dx
d2
1 2 2
H2 = ’2 2 + W1 + W2 ’ W1 + W2
4 dx
d2
1 2 2
H3 = ’2 2 + W1 + W2 ’ W1 ’ 3W2 (8.17)
4 dx

where the functions W1 and W2 , on account of (8.16), are restricted
by
2 2
W2 ’ W1 + W1 + W2 + c = 0 (8.18)
Let us examine the particular case when the derivatives of the
superpotentials W1 and W2 are equal

W1 = W2 (8.19)


© 2001 by Chapman & Hall/CRC
The above proposition leads to a simple form of the PSUSY Hamil-
tonian Hp which a¬ords a straightforward physical meaning

1 d2 •
Hp = ’ + V (x) + B(x)J3 (8.20)
3
2 dx2

where
1 2 2
V (x) = W1 + W2
4
« 
100
• 0 1 0
3=
001
dW1
B(x) =
dx
« 
10 0
J3 =  0 0 0  (8.21)
0 0 ’1

The interpretation of Hp is self-evident, it represents the motion of

a spin 1 particle placed in a magnetic ¬eld B directed along the 3rd
axis.
Two solutions of (8.18) corresponding to (8.19) are of the types
either
W1 = W2 = ω1 x + ω2 (8.22)

or
W1 = ω1 e’kx + ω2 , W2 = ω1 + k (8.23)

while the ¬rst one is for the homogeneous magnetic ¬eld, the second
one is for the inhomogeneous type.
The above PSUSY scheme corresponding to the supercharges
given by (8.12) can also be put in an alternative form by making use
of the fact that at the level of order 2 there are 2 independent para-
supercharges. Actually we can write down Q as a linear combination
of 2 supercharges Q1 and Q2 , namely,

Q = Q1 + Q2
Q+ = Q+ + Q+ (8.24)
1 2



© 2001 by Chapman & Hall/CRC
where
« 
0 A+ 0
1
1
0 0 0

Q1 =
2000
« 
00 0
1
√  0 0 A+ 
Q2 = 2
200 0
« 
0 00
1
Q+ = √  A’ 0 0 
1 1
2 0 00
« 
000
1
Q+ = √ 0 0 0 (8.25)
2
2 0 A’ 0
2
It is easily veri¬ed that Qi and Q+ (i = 1, 2) are in fact super-
i
charges in the sense that they are endowed with the properties
2
Q2 = 0 = Q+ i = 1, 2 (8.26)
i i

for c = 0. Besides, they satisfy
Qi Q+ = Q+ Qj = 0 (i = i = 1, 2
j), (8.27)
j i

A more transparent account of the natural embedding of the
supersymmetric algebra in (8.9) and (8.10) can be brought about by
invoking the hermitean charges
1
Q1 = √ Q+ + Q (8.28)
2
1
Q+ ’ Q
Q2 = √ (8.29)
2i
We may work out
1
Q3 = √ Q+ + Q Q+ + Q Q+ + Q
1
22
1 2 2
√ Q+ Q + Q+ QQ+ + Q+ Q2 + QQ+ + QQ+ Q + Q2 Q+
=
22
1

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