<< Предыдущая стр. 35(из 42 стр.)ОГЛАВЛЕНИЕ Следующая >>
в€љ 2Q+ Hp + 2QHp
=
22
1
= в€љ (Q + Q+ )Hp
2
= Q1 Hp (8.30)

В© 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 } в€’ 2Hp ] Q2 + [{Q1 , Q2 } + {Q2 , Q1 }] Q1 = 0 (8.34)
This suggests a combined relation (i, j, k = 1, 2)
[{Qi , Qj } в€’ 2Hp Оґij ] Qk + [{Qj , Qk } в€’ 2Hp Оґjk ] Qi
+ [{Qk , Qi } в€’ 2Hp Оґki ] Qj = 0 (8.35)
to be compared with the SUSY formula (2.44).

(b) The scheme of Beckers and Debergh

Beckers and Debergh  made an interesting observation that the
choice of the Hamiltonian in deп¬Ѓning 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 oп¬Ђers a new scheme of PSUSYQM.

В© 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 в€’ W1 + (W1 + W2 ) = 0 (8.40)

in place of (8.18). Note that the latter diп¬Ђers from (8.40) in having
just c = 0. However, the components of Hp are signiп¬Ѓcantly diп¬Ђerent
from those in (8.17). Here H1 , H2 , and H3 are

d2
1 2 2
H1 = в€’ 2 + 2W1 в€’ W2 в€’ W2
2 dx
d2
1 2 2
H2 = в€’ 2 + 2W2 в€’ W1 + 2W2 + W1
2 dx
d2
1 2 2
H3 = в€’ 2 + 2W2 в€’ W1 + W1 (8.41)
2 dx

An interesting particular case stands for

W1 = в€’W2 = П‰x (8.42)

(П‰ is a constant) which is consistent with (8.40). It renders Hp to
the form
d2
1 П‰
в€’ 2 + П‰ 2 x2 вЂў
Hp = 3+ diag(1, в€’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.

В© 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

Aв€’ A+ = A+ Aв€’ (8.45)
11 22

Because of (8.45), the component Hamiltonians in (8.41) are essen-
tially the result of the following factorizations
1+в€’
H1 = AA
211
1в€’+
H2 = AA
211
1в€’+
H3 = AA (8.46)
222
We can thus associate with
пЈ« пЈ¶
A+ Aв€’ 0 0
1пЈ­ 1 1 в€’+
0пЈё
Hp = 0 A1 A1 (8.47)
2
Aв€’ A+
0 0 22

two distinct supersymmetric Hamiltonians given by
A+ Aв€’
1 0
(1) 11
Hs = (8.48a)
Aв€’ A+
0
2 11

A+ Aв€’
1 0
(2) 22
Hs = (8.48b)
Aв€’ 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 hв€’ , 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 Вµ = О» = 0. The resulting com-
ponents of h, namely, hв€’ 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 hв€’ and h+
by glueing these together to form the (3 Г— 3) system (8.47). Thus

В© 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 ОЅ = 0. Here also glueing of supersymmetric Hamiltonians
can be done to arrive at a 3 Г— 3 matrix structure. Truncation of the
intermediate component then yields the SSUSY Hamiltonian in its
usual 2 Г— 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 в‰Ґ 2)

Qp+1 = 0 (8.49)
[Hp , Q] = 0 (8.50)

Qp Q+ + Qpв€’1 Q+ Q + . . . + Q+ Qp = pQpв€’1 Hp (8.51)

along with their hermitean conjugated relations.
The parasupercharges Q and Q+ can be chosen to be (p + 1) Г—
(p + 1) matrices as natural extensions of the p = 2 scheme

(Q)О±ОІ = A+ ОґО±+1,ОІ (8.52)
О±
(Q+ )О±ОІ = Aв€’ ОґО±,ОІ+1 (8.53)
ОІ

where О±, ОІ = 1, 2, . . . p + 1. The generalized matrices for c and c+
пЈ«
0пЈ¶
0 0 ... 0
в€љ
пЈ¬p 0пЈ·
0 ... 0
пЈ¬ пЈ·
пЈ¬ 0пЈ·
c = пЈ¬0 2(p в€’ 1) . . . 0 пЈ·
пЈ¬. .пЈ·
. . .
пЈ­. . . . .пЈё
. . . . .
в€љ
0 0 ... p 0

В© 2001 by Chapman & Hall/CRC
в€љ
пЈ« пЈ¶
0 p 0 ... 0
пЈ¬0 2(p в€’ 1) . . . 0 пЈ·
0
пЈ¬ пЈ·
пЈ¬... ... ... пЈ·
... ...
пЈ¬ пЈ·
c+ =пЈ¬ (8.54)
пЈ·
пЈ¬... ... ... ... ... пЈ·
пЈ¬ в€љпЈ·
пЈ­0 pпЈё
0 ... ...
0 0 ... ... 0

In (8.52) and (8.53), A+ and Aв€’ can be written explicitly as
О± О±

d
A+ = + WО± (x) (8.55)
О±
dx
d
Aв€’ =в€’ + WО± (x), О± = 1, 2, . . . , p (8.56)
О±
dx

The Hamiltonian Hp being diagonal is given by

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

where

1 d2 1 1
2
Hr =в€’ + Wr + Wr + cr , r = 1, 2, . . . p (8.58)
2 dx2 2 2
1 d2 1 1
2
Hp+1 =в€’ + Wp в€’ Wp + cp (8.59)
2 dx2 2 2

subject to the constraint

2 2
Wrв€’1 в€’ Wrв€’1 + crв€’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+ , Qpв€’1
Q+ Q, etc. appearing in (8.51), it turns out that the constants
c1 , c2 , . . . cp are not independent
 << Предыдущая стр. 35(из 42 стр.)ОГЛАВЛЕНИЕ Следующая >>