2 2

— —

’ (ψ+ ) + W ψ+ ψ+ + W ψ+ (4.29)

—

where the dependence on (x, k) of ψ+ , ψ+ and W has been sup-

pressed. Simplifying the right hand side one ¬nds

1— — —

E(k)Ψ(x, x, k) = ’ ψ+ ψ+ + ψ+ (ψ+ ) + 2(ψ+ ) ψ+

4

d —

+2 (W ψ+ ψ+ )

dx

1d d — —

=’ (ψ+ ψ+ ) + 2W ψ+ ψ+

4 dx dx

1d d —

=’ + 2W (ψ+ ψ+ ) (4.30)

4 dx dx

© 2001 by Chapman & Hall/CRC

From (4.26) one thus has [6]

d 1d d

[K+ (x, x, β) ’ K’ (x, x, β)] = + 2W K+ (x, x, β)

dβ 4 dx dx

(4.31)

d∆β

The above identity greatly facilitates the computation of dβ . Indeed

inserting (4.31) in the right hand side of (4.23) we can project out

the β-dependence of ∆β in a manner

d∆β 1 d d

= dx + 2W K+ (x, x, β) (4.32)

dβ 4 dx dx

which can also be expressed, using (4.24), and (4.6a) as

d∆β 1 d +’

e’βEk

= dx 2Ek ψk ψk (4.33)

dβ 2 dx k =0

Let us consider now the solitomic example W (x) = tanh x cor-

responding to which the supersymmetric partner Hamiltonians are

1 d2 1

1 ’ 2 sec h2 x

H+ = ’ + (4.34a)

2

2 dx 2

1 d2 1

H’ = ’ + (4.34b)

2 dx2 2

If one employs periodic boundary conditions over the internal [’L, L]

the associated wave functions of H± turn out to be of the following

two types

1

ψ’ = N cos k1 x (4.35a)

N

1

ψ+ = {k1 sin k1 x + tanh x cos k1 x} (4.35b)

2

k1 +1

2

ψ’ = N sin k2 x (4.36a)

N

2

ψ+ = {’k2 cos k2 x + tanh x sin k2 x} (4.36b)

2

k2 +1

These wave functions are subjected to

k1 > 0 : k1 sin k1 L + tanh L cos k1 L = 0 (4.37)

k2 > 0 : sin k2 L = 0 (4.38)

© 2001 by Chapman & Hall/CRC

i i

which have been obtained from the considerations ψ± (L) = ψ± (’L),

i = 1, 2 and E(ki ) = 1 (1 + ki ), i = 1, 2. The energy constraints

2

2

i i

from the Schroedinger equations H+ ψ+ = E(ki )ψ± , i = 1 and 2. It

is worth mentioning that (4.35) and (4.36) are consistent with the

intertwining conditions (4.6).

d∆

We can now use the formula (4.33) to calculate dββ correspond-

ing to the wave functions (4.35) and (4.36). It is trivial to check that

d∆

as a result of the associated boundary conditions dββ = 0. [Note that

in addition to (4.35b) and (4.36b) H+ also possesses a normalizable

zero-energy state »sech2 x, » a constant, but it does not contribute

to the sum in (4.33)].

We now pass on the limit L ’ ∞ when one recovers the con-

1

tinuum states. Employing the usual normalization N ’ √π , the

d∆β

derivative is found to be

dβ

β 2

e’ 2 (k +1)

∞

d∆β 1

dk √

= [cos kx {k sin kx + tanh x cos kx}

dβ 2π k2 + 1

0

1 + k 2 + sin kx {’k cos kx + tanh x sin kx}

x=+∞

1 + k2

x=’∞

’β ∞

1e βk2

2

dk [tanh x]x=+∞

e’

= 2

x’∞

2 2π 0

’β

1e 2

√

= (4.39)

2 2πβ

which, clearly, is β-dependent.

The above discussions give us an idea on the behaviour of a reg-

ulated Witten index. Actually the evaluation of the index depends

a great deal on the choice of the method adopted and in ¬nding a

suitable regularization procedure. Also the behaviour of ∆β depends

much on the nature of the spectrum; a purely continuous one extend-

ing to zero may yield fractional values of ∆β . In this connection note

that if we use the representation (4.32) we run into the problem of

determining exactly the heat kernels. There is also the related issue

of the viability of the interchange of the k and x limits of integra-

tion. For more on the anomalous behaviour of the Witten index and

its judicious computation using the heat kernel techniques one may

consult [6].

© 2001 by Chapman & Hall/CRC

4.5 Index Condition

We now analyze the Fredholm index of the annihilation operator b

de¬ned by [23,24]

δ ≡ dim ker b ’ dim ker b+ (4.40)

where b and b+ are, respectively, the annihilation and creation oper-

ators of the oscillator algebra given by (2.6) and (2.7). In (4.40) dim

ker corresponds to the dimension of the space spanned by the linearly

independent zero-modes of the relevant operator. Since b|0 >= 0 it

is obvious that dim ker b = {|0 >} while dim ker b+ is empty. Thus

δ = 1.

Apart from (4.40) we also have [24]

dim ker b+ b ’ dim ker bb+ = 1 (4.41a)

where dim ker b+ b represents the number of normalizable eigenkets

|ψn > obeying b+ b|ψn >= 0. To avoid a singular point in the index

relation it is useful to restrict dim ker b+ b < ∞. A deformed quantum

condition often leads to the existence of multiple vacua when δ may

become ill-de¬ned.

It is interesting to observe that for a truncated oscillator one

¬nds in place of (4.41a)

dim ker b+ bs ’ dim ker bs b+ = 0 (4.41b)

s s

where bs and b+ are the truncated vertions [25] of b and b+ de¬ned in

s

an (s + 1)-dimensional Fock space. To establish (4.41b) we transform

the eigenvalue equations

b+ bs φn = e2 φn (4.42a)

s n

to the form

bs b+ χn = e2 χn (4.42b)

s n

by setting χn = e1 bs φn with en = 0. We thus ¬nd the normal-

n

izability of the eigenfunctions φn and χn to go together. For a

¬nite-dimensional matrix representation we also have T r(b+ bs ) =

s

+ ). In this way we are led to (4.41b).

T r(bs bs

When applied to SUSYQM δ can be related to the Witten in-

dex which counts the di¬erence between the number of bosonic end

© 2001 by Chapman & Hall/CRC

fermionic zero-energy states. This becomes transparent if we focus

on (4.41a). Replacing the bosonic operators (b, b+ ) by (A, A+ ) ac-

cording to (2.34) in terms of the superpotential W (x), the left hand

side of (4.41a) just expresses the di¬erence between the number of

bosonic and fermionic zero eigenvalues. We thus have a correspon-

dence with (4.8).

In the following we study [26] the Fredholm index condition

(4.40) for the annihilation operator of the deformed harmonic os-

cillator and q-parabose systems in a generalized sense to show how

multiple vacua may arise. We also look into the singular aspect of δ

and point out some remedial measures to have an ambiguous inter-

pretation of δ.

4.6 q-deformation and Index Condition

Interest in quantum deformation seems to have started after the work

of Kuryshkin [27-30] who considered a q-deformation in the form

AA+ ’ qA+ A = 1 for a pair of mutually adjoint operators A and

A+ to study interactions among various particles. Later Janussis et

al. [31], Biedenharn [32], Macfarlane [33], Sun and Fu [34] and sev-

eral others [35-48] made a thorough analysis of deformed structures

with a view to inquiring into plausible modi¬cations of conventional

quantum mechanical laws. Recent interest in quantum deformation

comes from its link [49,50] with anyons and Chern-Simmons theo-

ries. The ideas of q-deformation has also been intensely pursued to

develop enveloping algebras [51] quasi-Coherent states [52], rational

conformal ¬eld theories [53] and geometries possessing non commu-

tative features [54].

Let us consider the following standard description of a q-deformed

harmonic oscillator

AA+ ’ qA+ A = q ’N , q ∈ (’1, 1) (4.42)

with A and A+ obeying

[N, A] = ’A, [N, A+ ] = A+ (4.43)

As the deformation parameter q ’ 1, A ’ b and we recover the

familiar bosonic condition bb+ ’ b+ b = 1 for the normal harmonic

oscillator.

© 2001 by Chapman & Hall/CRC

The operators (A, A+ ) may be related to the bosonic annihila-

tion and creation operators b and b+ by writing

A = ¦(N )b, A+ = b+ ¦(N ) (4.44)

where N is the number operator b+ b and ¦ any function of it.

Exploiting the eigenvalue equation ¦(NB )|n >== ¦(n)|n >, the

representations (4.44) lead to the recurrence relation

(n + 1)¦2 (n) ’ qn¦2 (n ’ 1) = q ’n (4.45)

The above equation has the solution

1 q n ’ q ’n

¦(n) = (4.46)

n + 1 q ’ q ’1

which implies that ¦(N ) is given by

[N + 1]

¦(N ) = (4.47)

N +1

with

q x ’ q ’x

[x] = (4.48)

q ’ q ’1

From (4.44) and (4.48) the Hamiltonian for the q-deformed har-

monic oscillator can be expressed as

ω

Hd = A, A+

2

ω

= {[N + 1] + [N ]} (4.49)

2

Note that the commutator [A, A+ } reads

A, A+ = [N + 1] ’ [N ] (4.50)