√

c+ 20 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