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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


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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)




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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


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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


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|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



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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



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+
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.

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