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)

© 2001 by Chapman & Hall/CRC

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)