then from (4.84a) the plausible states over which the system can run

are those from |2k > to |4k ’ 1 >. These are only ¬nite in number.

The point is that once a system takes a particular k-value (including

|0 >) for its ground state, all other states in the kernel and the

accompanying higher states (which can be created from them) are

rendered isolated in the sense that these are disjoint from the ones

constructed by starting from a di¬erent vacuum. It may be noted

that corresponding to the choice |2k > made for the vacuum, the

lowest k-value for the ground state in ker B + can be |2k > since the

state |2k ’1 > in (4.84b) is lower to |2k > and so acts like a spectator

© 2001 by Chapman & Hall/CRC

state as in the p = 0 case discussed earlier. It follows that not only

for the kernels (4.84) but also for (4.56) the index vanishes.

4.9 Witten™s Index and Higher-Derivative

SUSY

In Witten™s model of SUSYQM the operators A and A+ are assumed

to have ¬rst-derivative representations. However one can look for ex-

tensions of SUSYQM by resorting to higher-derivative versions of A

and A+ . Apart from being mathematically interesting, these models

of higher derivative SUSY (HSUSY) o¬er the scope of connections

to nontrivial quantum mechanical systems as have been found out

recently [64-78].

First of all we note that the factorization of Hs carried out in

(2.36) is also consistent with the following behaviour of H± vis-a-vis

the operators A and A+

H+ A+ = A+ H’

AH+ = H’ A (4.85)

These interesting relations also speak of the double degeneracy of

+

the spectrum with the ground state ψ0 , associated with the H+

component, being nondegenerate.

As a ¬rst step towards building HSUSY, we assume that the

underlying interwinning operators A and A+ are given by second-

order di¬erential representations

1 d2 d

A= + 2p(x) + g(x)

2 dx2 dx

1 d2 d

+

A = ’ 2p(x) + g(x) ’ 2p (x) (4.86)

2 dx2 dx

where p(x) and g(x) are arbitrary but real functions of x. The above

forms of A and A+ generate, as we shall presently see, the mini-

mal version of HSUSY namely the second-derivative supersymmetric

(SSUSY) scheme.

In terms of A and A+ we associate the corresponding super-

charges q and q + as

0 A 0 0

, q+ =

q= (4.87)

A+

0 0 0

© 2001 by Chapman & Hall/CRC

Pursuing the analogy with the de¬nition of Hs in SUSYQM, here

too, we can write down a quantity K de¬ned by

q, q +

K=

2

q + q+

= (4.88)

However K, unlike Hs , is a fourth-order di¬erential operator. The

passage from SUSY to SSUSY is thus a transition from Hs ’ K.

One of the purposes in this section is to show that in higher-derivative

models there are problems in using Witten index to characterize

spontaneous SUSY breaking [66,68].

Let us suppose the existence of an h-operator as a diagonal 2 — 2

matrix operator that commutes with q and q +

h’ 0

h= (4.89)

0 h+

[h, q] = 0 = h, q + (4.90)

It gives rise to the following interwining relations in terms of A and

A+

h+ A + = A + h’

Ah+ = h’ A (4.91)

These are similar in form to (4.85).

We now exploit the above relations to obtain a constraint equa-

tion between the functions p(x) and g(x). Indeed if we de¬ne h+ and

h’ in terms of the potential v+ and v’ as

1 d2

h± = ’ + v± (x) (4.92)

2 dx2

and substitute (4.86) in (4.91), we obtain

2

p 1 p µ

2

g = p + 2p ’ + + (4.93)

32p2

4p 2 2p

where the dashes denote derivatives with respect to x and µ is an

arbitrary real constant. Also v± (x) are given by

2

p 1 p µ

2

v+ = ’2p + 2p + ’ ’ +» (4.94a)

32p2

4p 2 2p

© 2001 by Chapman & Hall/CRC

2

p 1 p µ

2

v’ = 2p + 2p + ’ ’ +» (4.94b)

32p2

4p 2 2p

where » is an arbitrary real constant.

To proceed further let us consider now the possibility of factor-

izing the operators A and A+ . We write

1

A = bc (4.95)

2

where b and c are given by the form

d

b= + U+ (x)

dx

d

c= + U’ (x) (4.96)

dx

From the ¬rst equation of (4.86) we can deduce a connection of the

functions U± with p(x) and g(x) :

4p = U+ + U’

2g = U+ U’ + U’ (4.97)

Turning to the quasi-Hamiltonian K we note that for simplic-

ity we can assume it to be a polynomial in h (in the present case,

second order in h only). This leads to the picture of the so-called

“polynomial SUSY.” It must, however, be admitted that a physical

interpretation of K is far from clear. Classically, one encounters a

fourth-order di¬erential operator while dealing with the problem of

an oscillating elastic rod or in the case of a circular plate loaded

symmetrically [79,80]. Quantum mechanically the situation is less

obvious. However, HSUSSY, unlike the usual SUSQM, allows resid-

ual symmetries [64,65]. It may be remarked that coupled channel

problems and transport matrix potentials come under the applica-

tions of higher derivative schemes [69].

Expressing K as

K = h2 ’ 2»h + µ

= (h ’ »)2 + µ ’ »2 (4.98)

we note that

[K, h] = 0 (4.99)

© 2001 by Chapman & Hall/CRC

A particularly interesting case appears when K is expressed as a

perfect square

K = (h ’ »)2 (4.100)

with

√

»= µ, µ > 0 (4.101)

This means that K can be written as

(h’ ’ »)2 0

K= (4.102)

(h+ ’ »)2

0

But K is also given (4.88) which implies

K = qq + + q + q

AA+ 0

= (4.103)

+A

0 A

As such from (4.102) and (4.103) we are led to

AA+ = (h’ ’ »)2

A+ A = (h+ ’ »)2 (4.104)

To express the left hand side as a perfect square we see that a

constraint of the type

cc+ = b+ b (4.105)

can do the desired job. Indeed using (4.105) we ¬nd

(h’ ’ »)2 = AA+

1 ++

= bcc b

4

1++

= bb bb

4

1 +2

= bb (4.106)

2

Similarly,

1+ 2

2

(h+ ’ ») = cc (4.107)

2

In these words we can factorize h± in a rather simple way

1+

h’ = bb + »

2

1+

h+ = c c+» (4.108)

2

© 2001 by Chapman & Hall/CRC

We now utilize the constraint relation (4.105) by substituting in

it the representation (4.96) for b, c and their conjugates. We derive

in this way the result

2 2

U’ + U’ = U+ ’ U+ (4.109a)

When µ = 0 [which implies from (4.101) » = 0 as well] the functions

U± are explicitly given by

p

U+ = + 2p

2p

p

U’ = ’ + 2p (4.109b)

2p

where we have made use of (4.97) and (4.93).

We thus have from (4.108) and (4.105), a triplet of Hamiltoni-

ans ( 1 bb+ , 1 cc+ , 1 c+ c) of which the middle one playing superpartner

2 2 2

to the ¬rst and third components. More precisely we ¬nd the sit-

uation that H is being built up from two standard supersymmetric

Hamiltonians namely 1 (bb+ , cc+ ) and 1 (cc+ , c+ c).

2 2

To inquire into the role of the Witten index in the above scheme

we look for the zero-modes of the quasi-Hamiltonian K. These are

provided by the equations

AψB = 0, A+ ψF = 0 (4.110a)

It is a straightforward exercise to check that ψB and ψF are

x

√

ψB = cB p exp ’ pdt

x0

x

√

ψF = cF p exp pdt (4.110b)

x0

where cB and cF are constants.

Witten index ∆ of (4.8) is thus determined by the asymptotic

nature of p(x) that renders ψB and ψF normalizable. So it is clear

that ∆ ∈ (’1, 1) since the number of vacuum states can be [66]

NB,F = 0, 1.

The special case corresponding to

∞

|x| ’ ∞ : p(x) ’ 0, p(x)dx < ∞ (4.111)

’∞

© 2001 by Chapman & Hall/CRC

is of interest [68] since here zero modes exist corresponding to both

ψB and ψF . As a result a possible con¬guration can develop through

NB = NB = 1 implying ∆ = 0. The intriguing point is that the

vanishing of the Witten index does not imply absence of zero modes

and in consequence occurrence of spontaneous breaking of SUSY. On

the contrary we have a doubly degenerate zero modes of the operators

A and A+ .

Finally, we address to the more general possibility when the pa-

rameter µ is nonzero. For this, let us return to the constraint (4.93).

The sign of µ decides whether the algebra is reducible (µ < 0) or not

(µ > 0). In the reducible case we ¬nd for » = 0 and ν 2 = ’4µ the

following features