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c1 + c2 + . . . + cp’1 + cp = 0 (8.61)

Let us verify (8.58)-(8.60) for the case p = 2 which we have al-
ready addressed to in the Rubakov-Spiridonov scheme. First of all we
notice that on taking derivatives of both sides, (8.60) matches with
the corresponding derivative version of (8.18) for p = 2. Secondly,


© 2001 by Chapman & Hall/CRC
on using (8.60) and (8.61) we can express again (8.58) and (8.59) as
1 d2 1 1
2 2 2
H1 =’ + W1 + W2 + . . . + Wp + 1 ’ W1
2 dx2 2p 2p
3 3 1
+ 1’ W2 + . . . + Wp’1 + Wp ,
2p 2p 2p
H2 = H1 ’ W 1
... ......
Hr+1 = Hr ’ Wr
... ......
Hp+1 = Hp ’ Wp (8.62)
Now it is evident that (8.62) yields (8.17) on putting p = 2. Likewise,
the Hamiltonians for higher order cases can be obtained on putting
p = 3, 4, . . . etc.
The particular case when
W1 = W2 = . . . = Wp = ωx (8.63)
yields the PSUSY oscillator Hamiltonian which, of course, is realized
in terms of bosons and parafermions of order p. This Hamiltonian
can be written as
1 d2 1
+ ω 2 x2 •
H= ’
2 dx2 2
pp p p
’ω diag , ’ 1, . . . , ’ + 1, ’ ,p ≥ 2 (8.64)
22 2 2
whose spectrum is
1
En,ν = n + ’ν ω (8.65)
2
where
n = 0, 1, 2, . . .
pp p
ν= , ’ 1, . . . , ’ (8.66)
22 2
From (8.65) one can see that the ground state is nondegenerate
with its energy given by
1p
E0, p = ’ ω (8.67)
22
2




© 2001 by Chapman & Hall/CRC
It is clearly negative. Further, the pth excited state is (p + 1)-fold
degenerate.
To conclude, there have been several variants of PSUSYQM sug-
gested by di¬erent authors. In [27] a generalization of SUSYQM was
considered leading to fractional SUSYQM having a structure similar
to PSUSYQM. Durand et al. [16] discussed a conformally invariant
PSUSY whose algebra involves the dilatation operator, the confor-
mal operator, the hypercharge, and the superconformal charge. A
bosonization of PSUSY of order two has also been explored [28]. In
this regard, a realization of C» extended oscillator algebra was shown
[29] to provide a bosonization of PSUSYQM of order p = » ’ 1 for
any ».


8.4 Truncated Oscillator and PSUSYQM
The annihilation and creation operators of a normal bosonic oscilla-
tor subjected to the quantum condition [b, b+ ] = 1 are well known to
possess in¬nite dimensional matrix representations

« 
0 1√ 0 0 ...
¬0 0 ...·
0 2√
¬ ·
b = ¬0 0 3 ...·
0
 
. . . . ...
. . . . ...
. . . . ...
« 
0 0 00 ...

¬1 ...·
0 00

¬ ·
¬ ...·
= ¬0 2√00
b+ (8.68)
·
¬
...·
0 0 30 
. . . . ...
. . . . ...
. . . . ...

A truncated oscillator is the one characterized [30,31] by some
¬nite dimensional representations of (8.68). The interest in ¬nite di-
mensional Hilbert space (FHS) comes from the recent developments
[32,33] in quantum phase theory which deals with a quantized har-
monic oscillator in a FHS and which ¬nds important applications
[34-36] in problems of quantum optics. In this section we discuss
PSUSYQM when the PSUSY is between the normal bosons [de-
scribed by the annihilation and creation operators b and b+ obeying
(8.68)] and those corresponding to a truncated harmonic oscillator


© 2001 by Chapman & Hall/CRC
which behave, as we shall presently see, like an exotic para Fermi
oscillator.
The Hamiltonian of a truncated oscillator is given by

1
P 2 + Q2
H= (8.69)
2

where P and Q are the corresponding truncated versions of the
canonical observables p and q of the standard harmonic oscillator.
The variables P and Q, however, are not canonical in that their
commutation has the form

QP ’ P Q = i(I ’ N K) (8.70)

where I and K are N dimensional matrices, I being the unit matrix
and K having the form [37]
« 
0 0 ... 0
¬0 0 ... 0·
¬ ·
¬ . . ... . ·
. . ... . ·
K = ¬ . . ... . (8.71)
¬ ·
0 0 ... 0
0 0 ... 1

In other words, K is a diagonal matrix with the last element unity
as the only nonzero element. In (8.70), N (> 1) is a parameter and
signi¬es that at the N -th now and column the matrices p and q have
been truncated.
We ¬rst show that K plays the role of a projection operator
|t >< t| in the FHS which for concreteness is taken to be a (t + 1)-
dimensional Fock space

T = {|0 >, |1 >, . . . |t >} (8.72)

t being a positive interger. This enables us to connect Buchdahl™s
work [30] on the truncated oscillator and some recent works which
have tried to explain [32,33] the existence of a hermitean phase op-
erator. We also bring to surface the remarkable similarities between
the rules obeyed by the representative matrices of the truncated os-
cillator and those of the parafermionic operators.


© 2001 by Chapman & Hall/CRC
(a) Truncated oscillator algebra

Let us begin by writing down the Hamiltonian for the linear
harmonic oscillator
12
p + q2
H= (8.73)
2
where the observables q and p satisfy [q, p] = i¯ .
h
The associated lowering and raising operators b and b+ are de-
¬ned by (2.2). These obey, along with the bosonic number operator
NB , the relations

[NB , b] = ’b
NB , b+ = b+ (8.74)

where NB = b+ b. The essential properties of b and b+ may be sum-
marized in terms of the following nonvanishing matrix elements

< n |b|n > = n δn ,n’1

< n |b+ |n > = n + 1 δn ,n+1 (8.75)

where n, n = 0, 1, 2, . . .. The above relations re¬‚ect the consequences
of seeking matrix representations of b and b+ or equivalently p and
q.
Let us consider the truncation of the matrices (8.68) for b and b+
at the N th row and column and call the corresponding new operators
to be B and B + where
1
√ (Q + iP )
B=
2
1
B+ = √ (Q ’ iP ) (8.76)
2

Using (8.70) the modi¬ed commutation relation for B and B + reads

[B, B + ] = 1 ’ N K (8.77)

where

KB = 0
K2 = K = 0 (8.78)


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When N = 2, a convenient set of representations for (8.77) is
1
B= σ+
2
1
B+ = σ’
2
1
K= (1 ’ σ3 ) (8.79)
2
where σ± have been already de¬ned in Chapter 2.
Let us now suppose that a FHS is generated by an orthonormal
set of kets |i >, i = 0, 1, . . . t along with a completeness condition,
namely [38]

< j|k > = δj,k
t
< j|j > = I (8.80)
j=0

The operators B and B + may be introduced through
t

B= m|m ’ 1 >< m|
m=0
t

B+ = m|m >< m ’ 1| (8.81)
m=0

These ensure that

B|m > = m|m ’ 1 >
B|0 > = 0

B + |m > = m + 1|m + 1 >

B + |t >= 0 (8.82a, b, c, d)
where m = 0, 1, 2, . . . t.
Any ket |j > can be derived from the vacuum by applying B +
on |0 > j times
1
|j >= √ (B + )j |0 > (8.83)
j!
where j = 0, 1, 2, . . . t. However since the dimension of the Hilbert
space is ¬nite, the ket |t > cannot be pushed up to a higher position.


© 2001 by Chapman & Hall/CRC
This is expressed by (8.82d). Note that the Fredholm index δ vanishes
in this case [see (4.41b)].
In view of the conditions (8.80), the expansions (8.81) imply that

[B, B + ] = 1 ’ (t + 1)|t >< t| (8.84)

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