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√ 2Q+ Hp + 2QHp
=
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

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