=

22

1

= √ (Q + Q+ )Hp

2

= Q1 Hp (8.30)

© 2001 by Chapman & Hall/CRC

and similarly

Q3 = Q2 Hp (8.31)

2

Further relations, (8.9) and (8.10), can be exploited to yield for the

real and imaginary parts, the conditions

Q1 Q2 + Q2 Q1 Q2 + Q2 Q1 = Q1 Hp (8.32)

2 2

Q2 Q2 + Q1 Q2 Q1 + Q2 Q2 = Q2 Hp (8.33)

1 1

To see a connection to SUSY we write, say (8.33), in the following

manner

[{Q1 , Q1 } ’ 2Hp ] Q2 + [{Q1 , Q2 } + {Q2 , Q1 }] Q1 = 0 (8.34)

This suggests a combined relation (i, j, k = 1, 2)

[{Qi , Qj } ’ 2Hp δij ] Qk + [{Qj , Qk } ’ 2Hp δjk ] Qi

+ [{Qk , Qi } ’ 2Hp δki ] Qj = 0 (8.35)

to be compared with the SUSY formula (2.44).

(b) The scheme of Beckers and Debergh

Beckers and Debergh [2] made an interesting observation that the

choice of the Hamiltonian in de¬ning a PSUSY system is not unique.

They constructed a new Hamiltonian for PSUSY by requiring Hp to

obey the following double commutator

Q, Q+ , Q = QHp (8.36)

Q+ , Q, Q+ = Q+ Hp (8.37)

in addition to the obvious properties (8.7) and (8.11).

One is easily convinced that (8.36) and (8.37) are inequivalent

to the corresponding ones (8.9) and (8.10) of Rubakov and Spiri-

donov. This follows from the fact that an equivalence results in the

conditions

QQ+ Q = QHp (8.38)

Q2 Q+ + Q+ Q2 = QHp (8.39)

a feature which is not present in the model of Rubakov and Spiri-

donov. As such the parasuper Hamiltonian dictated by (8.36) and

(8.37) is nontrivial and o¬ers a new scheme of PSUSYQM.

© 2001 by Chapman & Hall/CRC

To obtain a plausible representation of Hp such as in (8.14) we

assume the parasupercharges Q and Q+ to be given by the matrices

(8.12b) and (8.12d) but controlled by the constraint.

2 2

W2 ’ W1 + (W1 + W2 ) = 0 (8.40)

in place of (8.18). Note that the latter di¬ers from (8.40) in having

just c = 0. However, the components of Hp are signi¬cantly di¬erent

from those in (8.17). Here H1 , H2 , and H3 are

d2

1 2 2

H1 = ’ 2 + 2W1 ’ W2 ’ W2

2 dx

d2

1 2 2

H2 = ’ 2 + 2W2 ’ W1 + 2W2 + W1

2 dx

d2

1 2 2

H3 = ’ 2 + 2W2 ’ W1 + W1 (8.41)

2 dx

An interesting particular case stands for

W1 = ’W2 = ωx (8.42)

(ω is a constant) which is consistent with (8.40). It renders Hp to

the form

d2

1 ω

’ 2 + ω 2 x2 •

Hp = 3+ diag(1, ’1, 1) (8.43)

2 dx 2

which can be looked upon as a natural generalization of the super-

symmetric oscillator Hamiltonian.

In terms of the hermitean quantities Q1 and Q2 , here one can

derive the relations

Q3 = Q1 Hp

1

Q3 = Q2 Hp

2

Q1 Q2 Q1 = Q2 Q1 Q2 = 0

Q2 Q2 + Q2 Q2 = Q2 Hp

1 1

Q2 Q1 + Q1 Q2 = Q1 Hp (8.44)

2 2

Although (8.44) are consistent with (8.30), (8.31), (8.32), and (8.33)

of the Rubakov-Spiridonov scheme, the converse is not true.

© 2001 by Chapman & Hall/CRC

Since c = 0 in the Beckers-Debergh model, the constraint equa-

tion (8.40) is an outcome of the operator relation

A’ A+ = A+ A’ (8.45)

11 22

Because of (8.45), the component Hamiltonians in (8.41) are essen-

tially the result of the following factorizations

1+’

H1 = AA

211

1’+

H2 = AA

211

1’+

H3 = AA (8.46)

222

We can thus associate with

«

A+ A’ 0 0

1 1 1 ’+

0

Hp = 0 A1 A1 (8.47)

2

A’ A+

0 0 22

two distinct supersymmetric Hamiltonians given by

A+ A’

1 0

(1) 11

Hs = (8.48a)

A’ A+

0

2 11

A+ A’

1 0

(2) 22

Hs = (8.48b)

A’ A+

0

2 22

This exposes the supersymmetric connection of the Beckers-Debergh

Hamiltonian.

Let us now comment on the relevance of the parasupersymmet-

ric matrix Hamiltonian in higher derivative supersymmetric schemes

[22-24]. Indeed the representations (8.46) are strongly reminiscent

of the components h’ , h0 , and h+ given in Section 4.9 of Chapter 4

[see the remarks following (4.109b)]. Recall that we had expressed

the quasi-Hamiltonian K as the square of the Schroedinger type op-

erator h by setting the parameters µ = » = 0. The resulting com-

ponents of h, namely, h’ and h+ , can be viewed as being derivable

from the second-order PSUSY Hamiltonian Hp given by (8.47) by

deleting its intermediate piece. Conversely we could get the p = 2

PSUSY form of the Hamiltonian from the components h’ and h+

by glueing these together to form the (3 — 3) system (8.47). Thus

© 2001 by Chapman & Hall/CRC

SSUSY Hamiltonian can be interpreted as being built up from two

ordinary supersymmetric Hamiltonians such as of the types (8.48a)

and (8.48b).

Consider the case when c = 0 (which is consistent with the

Rubakov-Spiridonov scheme) corresponds to the situation (4.112)

where ν = 0. Here also glueing of supersymmetric Hamiltonians

can be done to arrive at a 3 — 3 matrix structure. Truncation of the

intermediate component then yields the SSUSY Hamiltonian in its

usual 2 — 2 form.

8.3 PSUSY of Arbitrary Order p

Khare [25,26] has shown that a PSUSY model of arbitrary order p

can be developed by generalizing the fundamental equations (8.7)

and (8.9)-(8.10) to the forms (p ≥ 2)

Qp+1 = 0 (8.49)

[Hp , Q] = 0 (8.50)

Qp Q+ + Qp’1 Q+ Q + . . . + Q+ Qp = pQp’1 Hp (8.51)

along with their hermitean conjugated relations.

The parasupercharges Q and Q+ can be chosen to be (p + 1) —

(p + 1) matrices as natural extensions of the p = 2 scheme

(Q)±β = A+ δ±+1,β (8.52)

±

(Q+ )±β = A’ δ±,β+1 (8.53)

β

where ±, β = 1, 2, . . . p + 1. The generalized matrices for c and c+

read

«

0

0 0 ... 0

√

¬p 0·

0 ... 0

¬ ·

¬ 0·

c = ¬0 2(p ’ 1) . . . 0 ·

¬. .·

. . .

. . . . .

. . . . .

√

0 0 ... p 0

© 2001 by Chapman & Hall/CRC

√

«

0 p 0 ... 0

¬0 2(p ’ 1) . . . 0 ·

0

¬ ·

¬... ... ... ·

... ...

¬ ·

c+ =¬ (8.54)

·

¬... ... ... ... ... ·

¬ √·

0 p

0 ... ...

0 0 ... ... 0

In (8.52) and (8.53), A+ and A’ can be written explicitly as

± ±

d

A+ = + W± (x) (8.55)

±

dx

d

A’ =’ + W± (x), ± = 1, 2, . . . , p (8.56)

±

dx

The Hamiltonian Hp being diagonal is given by

Hp = diag(H1 , H2 , . . . , Hp+1 ) (8.57)

where

1 d2 1 1

2

Hr =’ + Wr + Wr + cr , r = 1, 2, . . . p (8.58)

2 dx2 2 2

1 d2 1 1

2

Hp+1 =’ + Wp ’ Wp + cp (8.59)

2 dx2 2 2

subject to the constraint

2 2

Wr’1 ’ Wr’1 + cr’1 = Wr + Wr + cr , r = 2, 3, . . . p (8.60)

The constants c1 , c2 , . . . , cp are arbitrary and have the dimension of

energy. However, if one explicitly works out the quantities Qp Q+ , Qp’1

Q+ Q, etc. appearing in (8.51), it turns out that the constants

c1 , c2 , . . . cp are not independent