Just as we worked out the supersymmetric Hamiltonian in Chap-

ter 2 as arising from the superposition of bosonic and fermionic oscil-

lators, here too we can think of a q-deformed SUSY scheme [55-59]

by considering q-superoscillators. Indeed we can write down a q-

q

deformed supersymmetric Hamiltonian Hs in the form

ω

q

A, A+ + F + , F

Hs = (4.51)

2

© 2001 by Chapman & Hall/CRC

with A and A+ obeying (4.42) and (4.43) and F, F + are, respec-

tively, the q-deformed fermionic annihilation and creation operators

subjected to

F F + + qF + F = q ’M (4.52)

In contrast to the usual fermionic operators whose properties are

summarized in (2.12) - (2.14), the deformed operators F and F +

do not obey the nilpotency conditions: (F )n = 0,F(+ )n = 0 for

n > 1 and q ∈ (0, 1). This means that any number of q-fermions

can be present in a given state. For a study of the properties of

q-superoscillators in SUSYQM see [55].

To evaluate δ we need to look into the plausible ground states

of H d (note that H d forms a part of Hs ). As such we have to search

q

for those states which are annihilated by the deformed operator A.

In the following we shall analyze Ferdholm index δ for those

situations when the elements in dim ker A as well as dim ker A+ are

countably in¬nite and so the index, when evaluated naively, may not

be a well-de¬ned quantity. Indeed such a possibility occurs when the

Fock space of the underlying physical system is deformed and the

deformation parameter is assumed complex with modules unity (for

preservation of hermiticity)

2iπ

q = e± k+1 , |q| = 1, k > 1 (4.53)

However we shall argue that since both the kernels (coresponding

to A and A+ ) turn out to be countably in¬nite modulo (k + 1), the

question of building up an in¬nite sequence of eigenkets (on repeated

application to the ground state by the creation operator) is ruled out

and the deformed system has to choose its ground state along with

the spectrum over some suitable ¬nite dimensional Fock space. This

has the consequence of transfering δ from the ill-de¬ned (∞ ’ ∞) to

the zero-value.

For the deformed oscillator the roles of A and A+ are

A|k >= [k]|k ’ 1 > (4.54a)

A+ |k >= [k + 1]|k + 1 > (4.54b)

© 2001 by Chapman & Hall/CRC

where

1

(A+ )k |0 >

|k >= (4.55)

[k]!

We see that the bracket [k + 1] = 0 whenever q assumes values (4.53)

for which ¦(k) = 0 (We do not consider irrational values of θ in the

present context). It follows that for these values of q the Fock space

gets split into ¬nite-dimensional sub-spaces Tk . One can thus think

of the kernel A to consist of a countably in¬nite number of elements

starting from |0 > with the subsequent zero-mades placed at (k + 1)-

distance from each other. Similar reasoning also holds for the kernel

of A+ . We can write

ker A = {|0 >, |k + 1 >, |2k + 2 >, . . .} (4.56a)

ker A+ = {|k >, |2k + 1 >, . . .} (4.56b)

Note that since ker A+ is nonempty, the process of creating higher

states by repeated application of A+ , on some chosen vacuum be-

longing to a particular Tk , has to terminate. By simple counting

(which we illustrate in the more general parabose case below) δ cor-

responding to (4.56) takes the value zero.

Fujikawa, Kwek, and Oh [60] have shown that for values of q

corresponding to (4.53) the singular situation discussed above allows

for a hermitean phase operator as well as a nonhermitean one. They

have argued that since rational values of θ are densely distributed

over θ ∈ R, the notion of continuous deformation for the index can-

not be formally de¬ned which means, in consequence, that singular

points associated with a rational θ are to be encountered almost ev-

erywhere. These authors have also shown how to avoid the problem

of negative norms for q = e2πiθ .

4.7 Parabosons

The particle operators c and c+ of a parabose oscillator obey the

trilinear commutation relation [61,62]

[c, H] = c (4.57)

where H is the Hamiltonian

1

cc+ + c+ c

H= (4.58)

2

© 2001 by Chapman & Hall/CRC

The spectrum of states of parabosons of order p can be deduced

by de¬ning a shifted number operator

p

N =H’ (4.59)

2

and postulating the existence of a unique vacuum which is subject

to

c|0 > = 0 (4.60)

N |0 > = 0 (4.61)

While (4.59) implies that the commutation relations

[N, c] = ’c, [N, c+ ] = c+ (4.62)

hold so that c and c+ may be interpreted, respectively, as the annihi-

lation and creation operator for the parabose states, the conditions

(4.60) and (4.61) prescribe an additional restriction on |0 > from

(4.59)

c c+ |0 >= p|0 > (4.63)

Using (4.63) one can not only construct the one-particle state

c+

|1 >= √p |0 >, p = 0 but also recursively the -particle state.

n

However, one has to distinguish between the even and odd nature

of the states as far as the action of c and c+ on them is concerned

√

c|2n > = 2n|2n ’ 1 > (4.64)

c|2n + 1 > = 2n + p|2n > (4.65)

c+ |2n > = 2n + p|2n + 1 > (4.66)

√

c+ |2n + 1 > = 2n + 2|2n + 2 > (4.67)

It is obvious from the relations (4.64) - (4.67) that these reduce to

the harmonic oscillator case when p = 1. It is also worth pointing

out that for zero-order parabosons the above properties allow for the

possibilty of nonunique ground states [61]. Indeed one readily ¬nds

from (4.64) - (4.67) that in addition to c|0 >= 0, the state |0 >

also obeys the relation c+ |0 >= 0. Also c|1 >= 0. However, for a

physical system, though the level |1 > is of higher energy, owing to

the condition c|1 >= 0 the operator c can not connect |1 > to level

© 2001 by Chapman & Hall/CRC

|0 >. As such |0 > plays the role of a spectator leaving |1 > as the

logical choice of the ground state.

From the point of view of the Fredholm index de¬ned for the

parabox system as

δp ≡ dim ker c ’ dim ker c+ (4.68)

(where obviously dim ker ± corresponds to the dimension of the space

spanned by the linearly independent zero-modes of the operator ±),

the elements of both dim ker c and dim ker c+ for the p = 0 case are

nonempty and ¬nite

dim ker c = {|0 >, |1 >} (4.69)

dim ker c+ = {|0 >} (4.70)

Taking the di¬erence the index δp turns out to be 1. (Acutally for the

harmonic oscillator as well as for the normal parabosons, |0 > is the

true and genuine vacuum so that the unity value of the index holds

trivially). It is also interresting to observe that even the physical

interpretation o¬ered above, which distinguishes |1 > as the natural

choice for the ground state of p = 0 parabosons, also leads to δp = 1.

We now turn to the case of deformed parabose oscillator of order

[p].

4.8 Deformed Parabose States and Index

Condition

Let us carry out deformation of the parabose oscillators by replacing

the eigenvalues in (4.64) - (4.67) by their q-brackets [62,63].

B|2n > = [2n]|2n ’ 1 > (4.71)

B|2n + 1 > = [2n + p]|2n > (4.72)

B + |2n > = [2n + p]|2n + 1 > (4.73)

B + |2n + 1 > = [2n + 2]|2n + 2 > (4.74)

As in the case of deformed harmonic oscillators here too we can

connnect the operators (B, B + ) to the bosonic ones (b, b+ ) through

B = ¦p (N )b, B + = b+ ¦+ (N ) (4.75)

p

© 2001 by Chapman & Hall/CRC

where

[N + p]

¦p (N ) = , N even (4.76)

N +1

[N + 1]

= , N odd (4.77)

N +1

Accordingly we get for the Hamiltonian of the q-deformed para-

bosons the expressions

1

pd

H2n = {[2N + p] + [2N ]} (4.78)

2

1

pd

H2n+1 = {[2N + p] + [2N + 2]} (4.79)

2

where the su¬xes indicate that H pd is to operate on these states.

The transformations (4.75) mean that we are adopting a de-

formed quantum condition of the form

BB + ’ q » B + B = f (q, N, [p]) (4.80)

where

» = p or 2 ’ p

f (q, N, [p]) = q ’2N [p] or q ’2N ’p [2 ’ p] (4.81)

depending on whether (4.80) operates on the ket |2n > or |2n + 1 >.

We may remark that when p = 1, the q-deformed relation (4.80)

reduces to (4.42) which is as it should be. Further hermiticity is

preserved in (4.80) for all real q and also for complex values if q is

con¬ned to the unit circle |q| = 1.

To examine the index condition for the q-parabose system con-

trolled by the Hamiltonians (4.78) and (4.79) we note that B and

B + can be expanded as

∞

B= [2k + p]|2k >< 2k + 1|

k=0

∞

+ [2k]|2k ’ 1 >< 2k| (4.82)

k=0

© 2001 by Chapman & Hall/CRC

∞

+

B = [2k + p]|2k + 1 >< 2k|

k=0

∞

+ [2k + 2]|2k + 2 >< 2k + 1| (4.83)

k=0

Taking q = exp(2πiθ) and setting θ = 2k+1 , k > 1 , which implies

1

2

[2k + 1] = 0, it is clear (we consider rational values of θ only) from

the ¬rst term in the right hand side of (4.82) that unless p = 1 the

coe¬cient of |2k >< 2k + 1| remains nonvanishing except for cases

when p assumes certain speci¬c values constrained by the relation

p = 1 + 2m + m , where k = m/m , m and m are integers and

m = 0. However, barring these values of (and of course p = 1),

p

there is no other suitable choice of q for which this coe¬cient can

be made zero. Similarly, for B + . We therefore conclude that, for

such a scenario, ker B = {|0 >}, ker B + = empty implying that the

condition δ = 1 holds.

1

On the other hand if θ = 2k it results in the possibility of a

singular situation with dim ker B = ∞ and dim ker B + = ∞

ker B = {|0 >, |2k >, |4k >, . . .} (4.84a)

ker B + = {|2k ’ 1 >, |4k ’ 1 >, . . .} (4.84b)

We would like to stress that the above kernels depict a case simi-

lar to the truncated oscillator problem where the available degrees of

freedom are ¬nite [25]. In the present setup the degrees of freedom

are those from the chosen vacuum state to the one just before the

next member in the kernel.