2 2

Note that since {θ, π} = 1 the expressions (3.32) are just the classi-

cal analogs of the corresponding quantum quantities. From (3.29),

(3.27d) and (3.27a) we have

1 1

{A, A— } θπ + AA— {θ, π}

HScl =

2 2

12 1 2

= p + W ’ iW θπ (3.33)

2 2

Here the potential VScl = 1 W 2 ’ iW θπ matches with the one

2

obtained from the classical limit of the SUSYQM Lagrangian given

in the previous chapter. To see this we rewrite (2.127) up to a total

derivative as

12 1 2 i ™ ™

x ’ U ’ ψ + ψU + ψ+ψ ’ ψ+ψ

LScl = ™

2 2 2

12 i ™ ™

ψ+ψ ’ ψ+ψ ’ V

= x+

™ (3.34)

2 2

where

1

V = U 2 + ψ + ψU (3.35)

2

and an overhead dot stands for a time-derivative. Now from (2.127)

the canonical momenta π for ψ is iψ + which when substituted in

V gives V = 1 U 2 + iU ψπ. We thus recover VScl if we identify

2

W = ’U and note that ψ plays the role θ.

To complete our discussion on the classical supersymmetric La-

grangian we write down the equations of motion which follow from

(3.34) and (3.35). Writing LScl as

1 1 i ™ ™

LScl = x2 ’ U 2 (x) +

™ ψ ψ ’ ψψ ’ U (x)ψψ (3.36)

2 2 2

we see that the equations of motion are [see also (2.128)]

™

ψ = ’iU ψ

™

ψ = iU ψ

x = ’(U U ) ’ (U )ψψ

¨ (3.37)

© 2001 by Chapman & Hall/CRC

Setting ψ0 = ψ(0) and ψ 0 = ψ(0) and looking for a solution of x(t)

of the type x(t) = x(t) + c(t)ψ 0 ψ0 , the solutions for ψ(t), ψ(t) and

c(t) turn out to be

t

ψ(t) = ψ0 exp ’ U (x(„ ))d„

0

t

ψ(t) = ψ 0 exp i U (x(„ ))d„

0

™ ™ t

x(t) x(0) » ’ U (x(„ ))

c(t) = c(0) + d„ (3.38)

™ µ ’ 1 U 2 (x(„ ))

2

x(0) 0 2

In the expression for c(t), » and µ enter as arbitrary constants of inte-

gration but are linked to the conservation of energy E = µ + »ψ 0 ψ0 .

On the other hand x(t) may be interpreted as the quasi-classical

contribution to x(t). A quasi-classical solution has the feature that

it describes fully the classical dynamics of the bosonic along with

fermionic degrees of freedom [13,14].

In conclusion let us note that spin is a purely quantum mechan-

ical concept having no classical analogy. Thus we cannot think of

constructing a classical wave packet having a spin 1 angular momen-

2

tum. Pseudo-classical mechanics is somewhat in between classical

mechanics and quantum mechanics in that even in the limit h ’ 0 ¯

we can persist with Grassmann variables. Historically, the role of

an anti-commuting variable in relation to the quantal action was ex-

plained by Schwinger [15]. Later, Matthews and Salam [16] tackled

the problem of evaluating functional integrals over anticommuting

functions. Berezin and Marinov [17] also developed the Grassmann

variant of the Hamiltonian mechanics and in this way presented a

generalization of classical mechanics.

3.4 References

[1] H. Goldstein, Classical Mechanics, Addison-Wesley, MA, 1950.

[2] E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A

Modern Perspective, John Wiley & Sons, New York, 1974.

[3] M.G. Calkin, Lagrangian and Hamiltonian Mechanics, World

Scienti¬c, Singapore, 1996.

© 2001 by Chapman & Hall/CRC

[4] N.D. Sengupta, News Bull. Cal. Math. Soc., 10, 12, 1987.

[5] R. Casalbuoni, Nuovo Cim, A33, 115, 389, 1976.

[6] J.L. Martin, Proc. Roy. Soc. A251, 536, 1959.

[7] L. Brink, S. Deser, B. Zumino, P. di Vecchia, and P. Howe,

Phys. Lett., 64B, 435, 1976.

[8] A. Barducci, R. Casalbuoni, and L. Lusanna, Nucl. Phys.,

B124, 93, 521, 1977.

[9] R. Marnelius, Acta Phys. Pol., B13, 669, 1982.

[10] P.G.O. Freund, Introduction to Supersymmetry, Cambridge Uni-

versity Press, Cambridge, 1986.

[11] J. Barcelos - Neto, A. Das, and W. Scherer, Phys. Rev., D18,

269, 1987.

[12] S.N. Biswas and S.K. Soni Pramana, J. Phys., 27, 117, 1986.

[13] G. Junker and S. Matthiesen, J. Phys. A: Math. Gen., 27,

L751, 1994.

[14] G. Junker and S. Matthiesen, J. Phys. A: Math. Gen., 28,

1467, 1995.

[15] J. Schwinger, Phil. Mag., 49, 1171, 1953.

[16] P.T. Matthews and A. Salam, Nuovo. Cim., 2, 120, 1955.

[17] F.A. Berezin and M.S. Marinov, Ann. Phys., 104, 336, 1977.

© 2001 by Chapman & Hall/CRC

CHAPTER 4

SUSY Breaking, Witten

Index, and Index

Condition

4.1 SUSY Breaking

As already noted in Chapter 2, while the Hamiltonian of the har-

monic oscillator is invariant under the interchange of the lowering

and raising operators, the vacuum, which is de¬ned to be the lowest

state, is not. On the other hand, when we speak of SUSY being

an exact or an unbroken symmetry both the supersymmetric Hamil-

tonian Hs as well as its lowest state remain invariant with repsect

to the interchange of the supercharge operators Q and Q+ . This is

due to the cancellation (corresponding to ω = ωB = ωF ) between

the bosonic and fermionic contributions to the ground state energy

thus admitting of a zero-energy lowest state for the supersymmetric

Hamiltonian.

Let us now study the case of SUSY being broken spontaneously

[1,2]. We know from (2.47) that

E0 = < 0|Hs |0 >

= |Q1 |0 > |2 > 0 (4.1)

where we do not assume a negative norm ghost state contributing.

© 2001 by Chapman & Hall/CRC

So Q1 |0 > = 0 means existence of degenerate vacua related by the

supercharge operator. This can be made more explicit by assuming

speci¬cally

Q|0 >= »|0 > = 0 (4.2)

where Q is de¬ned according to (2.31). Since Q anti-commutes with

Hs we have

Hs Q|0 > = QHs |0 >

= Qµ|0 >

= »µ|0 > (4.3)

where µ is the ground-state eigenvalue of Hs . Also from (4.2) we can

write

Hs Q|0 >= »Hs |0 > (4.4)

(4.3) and (4.4) thus point to

Hs |0 > = µ|0 > (4.5)

showing |0 > and |0 > to be degenerate states. The condition E0 >

0 is as necessary as well as su¬cient for SUSY to be spontaneously

broken.

The previous steps can also be formulated in terms of the con-

straints on the functional forms of the superpotential. For unbro-

ken SUSY we found from (2.56a) and (2.56b) that normalizability

of the ground-state wave function requires W (x) to di¬er in signs

at x ’ ±∞. This is of course the same as saying that W (x) pos-

sesses an odd number of zeros in (’∞, ∞). However if W (x) is an

even function of x, there cannot be any normalizable zero-energy

wave function and we are led to degenerate ground-states having

a nonzero energy value. Such a situation corresponds to sponta-

neous supersymmetric breaking of SUSY. For example, if we take

W (x) = 1 gx2 , g a coupling constant, the ground-state wave function

2

x

behaves as exp (± 1 x0 gx2 dx) which obviously blows up either at

2

plus or minus in¬nity.

Thus spontaneous breaking of SUSY is concerned with E > 0

with pairing of all energy levels. It is easy to see from (2.59b) and

(2.59c) that the following interrelationships among the eigenfunctions

© 2001 by Chapman & Hall/CRC

of H+ and H’ are implied

d n n

W+ ψ+ = 2En ψ’ (4.6a)

dx

d n n

W’ ψ’ = 2En ψ+ (4.6b)

dx

where n = 1, 2, . . . and the real-valued superpotential W (x) is as-

sumed to be continuously di¬erentiable. The set (4.6) brings out the

roles of H+ and H’ namely

1 d2 n n

’ + V + ψ + = En ψ + (4.7a)

2

2 dx

1 d2 n n

’ + V ’ ψ ’ = En ψ ’ (4.7b)

2

2 dx

where n = 1, 2, . . . and (V+ , V’ ) are given by (2.29).

4.2 Witten Index

To inquire into the nature of a system as to whether it is supersym-

metric or spontaneously breaks SUSY, it is necessary to look for its

zero-energy states. Consider the so-called Witten index [3-5] which

is de¬ned to count the di¬erence between the number of bosonic and

fermionic zero-energy states

(E=0) (E=0)

∆ ≡ nB ’ nF (4.8)

This is logical since for energies which are strictly positive there is

a pairing between the energy levels corresponding to the bosonic

and fermionic states. Thus ∆ = 0 immediately signals SUSY to be

unbroken as there does exist a state with E = 0.

For the spontaneously broken SUSY case note that the nonvan-

ishing of classical potential energy implies the vacuum energy to be

strictly positive in the classical approximation. A suitable example

is V (x) = 1 (x2 + c)2 ≥ 1 c2 > 0 (for c > 0) and SUSY is sponta-

2 2

neously broken. On the other hand, if the vacuum energy is vanish-

ing at the classical level, then SUSY prevails in perturbation theory

and can be broken only through nonperturbative e¬ects. However

© 2001 by Chapman & Hall/CRC

just from the vanishing of the quantity ∆ it is not evident whether

(E=0) (E=0)

SUSY is spontaneouly broken [nB = 0 = nF ] or unbroken

(E=0) (E=0)

[nB = nF = 0].

It is worth pointing out that as long as the basic supersymmetric

algebra holds the various parameters, such as the mass or coupling

constants, it can undergo changes leading to deformation in the po-

tential. Such variations of parameters will, of course, also cause

the energy of the states to change. But because of boson-fermion

pairing in the supersymmetric theory the states must move (corre-

sponding to their ascending or descending) in pairs. In other words