Fermionic coordinates and supersymmetry in quantum mechanics

Fermionic coordinates and supersymmetry in quantum mechanics

Nuclear Physics @ North-Holland B196 (1982) 509-531 Publishing Company FERMIONIC COORDINATES IN QUANTUM P. SALOMONSON AND SUPERSYMMETRY MECHANICS ...

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Nuclear Physics @ North-Holland

B196 (1982) 509-531 Publishing Company

FERMIONIC

COORDINATES IN QUANTUM P. SALOMONSON

AND SUPERSYMMETRY MECHANICS

and J.W.

VAN

HOLTEN

CERN, Geneva, Switzerland Received

25 September

1981

We describe the quantum mechanics of particles with fermionic degrees of freedom, both in the Schradinger wave function and Feynman path integral formalism. In particular we derive an exact expression for fermionic path integrals in (0+ 1) dimensions. Under suitable circumstances the theories we consider can exhibit supersymmetry. As an application we analyze in detail how the spontaneous breaking of supersymmetry, occurring in certain models recently discussed by Witten, can be derived as an instanton effect in the path integral formalism.

1. Introduction In this paper we will concern ourselves with the quantum mechanics of particles which have both bosonic and fermionic degrees of freedom. To keep the mathematics simple, we concentrate on particles described by one bosonic and one fermionic coordinate, but this is by no means essential and the formalism can without difficulty be extended to any desired number of degrees of freedom. The quantum mechanical hamiltonian operator for such a particle, with unit mass, reads fi=&Y2+

v(q^)+$iw(q*)[&,

j,].

In this expression (3, 4) are the usual bosonic momentum while f,,, are two real fermionic operators describing Equivalently one may start from the c-number lagrangian

(1.1) and coordinate the fermionic

operators variables.

(1.2) where

the components of which are to be interpreted as anticommuting c-numbers. The gi denote the usual Pauli matrices. Besides being of intrinsic mathematical interest, these theories can be regarded as a restriction of quantum field theories to (0 + 1) space-time dimensions. As such 509

510

P. Salomonson,

J. W. van Holten / Supersymmetry

they serve as a useful mathematical

laboratory

for the development

of field theoreti-

cal methods. The purpose of the present investigation is to develop some methods which are relevant to theories with fermionic degrees of freedom, in particular supersymmetric theories. We apply them to give a detailed analysis of the nonperturbative mechanism for supersymmetry breaking recently found by Witten [l]. This paper has been organized give an expose of two formalisms

as follows. In the first part (sects. 2 and 3) we that can be used to describe fermionic degrees

of freedom in quantum mechanics, to wit canonical quantization and path integrals. In particular we show how to evaluate fermionic functional integrals in (0 + 1) dimensions exactly. Some related work can be found in ref. [2]. In the second part (sects. 4 to 7) we study a particular class of potentials, which are distinguished by the property of supersymmetry. Supersymmetries in quantum mechanics are generated by conserved fermionic charges 6,, which satisfy the anticommutation relation <6+, 6-j

= 2A.

(1.3)

By constructing solutions to the Schrbdinger equation we show that for certain supersymmetric potentials, supersymmetry is broken spontaneously by an exponentially small ground-state energy. Using the methods developed in the first sections we demonstrate that this breaking of supersymmetry can be interpreted in the path integral formalism as an instanton effect, i.e., as the result of the existence of solutions to the equations of motion with euclidean time. Some details of our calculation have been collected in an appendix.

2. The matrix representation

of the fermions

In canonical quantum mechanics the observables fi,G, $1,2 and the hamiltonian (1 .l) have to be interpreted as operators. The postulated (anti-) commutation relations for these operators are:

[@,41=-i, Actually

it is more convenient

tion operators

$&, defined $* = J$j,

(2.1)

b4L &I = &P ’

to work in terms of fermionic

creation

and annihila-

by l

ilj,, ,

G+, J-1 = 1 3

qc:=o.

(2.2)

The algebra (2.2) resembles more closely the bosonic commutator algebra (2.1), because $* anticommute with themselves. The bosonic operators can, in the coordinate representation, be taken to be q*=x,

p^= -i(a/ax)

,

(2.3)

the states of the system being described by wave functions P(x). The fermionic anticommutation relations (2.1) or (2.2) may conveniently be represented by the

P. Salomonson,

anticommuting

c-number

511

J. W. LKUIHolten / Supersymmetry

operators:

G+= dlX,

$-=L

(2.4)

acting on wave functions*

or one may represent

them by finite dimensional

matrices

(2.6) whence

the wave functions

become

two-component

W(x)=

41(x)

objects:

(42(x))

(2.7)

.

Here we will explore the last possibility (2.6), (2.7). In sect. 3 we will discuss the anticommuting c-number representation (2.4), (2.5), and show the equivalence of the two formalisms. In the matrix formulation, the hamiltonian (1.1) becomes a 2 x 2 matrix as well:

A=

(-&$+V(x))l-tW(x)cT,,

where V~ is the usual diagonal

(2.8)

Pauli matrix:

(2.9) The stationary equation:

states of the system are the normalizable

solutions

tip==?& with energy E. The interpretation operator

of the two components

of the Schrodinger

(2.10)

of the wave function

i= $[$+, c-1 = $(T3

is now clear. The

(2.11)

commutes with the hamiltonian and is diagonal in this representation**. From the algebra (2.2), or from (2.11), it follows that its eigenvalues are f= ZIZ$.Clearly the states 41,2 are eigenstates of the hamiltonian corresponding to fermionic quantum number f = f 1, respectively. Hence the state space of the system is defined by all This representation presupposes the wave function to depend only on the single variable L not on both [ and its complex conjugate I*. For details, see ref. 121. l * Alternatively one may introduce the operator I$+$_ = $[$+, $-I+ 5, with eigenvalues 0, 1, the fermion number. l

512

P. Salomonson,

the normalizable

solutions

J. W. van Holten J Supersymmetry

of (2.10) and the individual

states

are characterized

the energy E and the fermionic quantum number f. An alternative approach to describe the state space and dynamics is by the path integral [2,3]. function and may be defined

The path integral by

&(qt Clearly, number

with the hamiltonian f is conserved. Hence

describes

1q’t’) = (qf 1e-iir(r-r’)jq’f) (2.8), Kf

is diagonal,

by

of the system

the evolution

. i.e., the fermionic

of the wave

(2.12) quantum

(2.13) where (2.14) The functional integral is taken over all trajectories from 4’ to 4 between the times t’ and t. Knowing the path integral (2.13), it is sufficient to specify the initial wave function Pr(q’, t’) to obtain all possible information about the system at any later time f, by (2.15) with [cf. (2.7)]:

~*1/2(S,t) = 41,&L t) .

3. The anticommuting

c-number

representation

In this section we construct the abstract space of eigenstates of the conjugate operators Get, having anticommuting c-number eigenvalues. We will then construct wave functions (2.5) on which the fermion operators are represented by (2.4), and show the equivalence with the matrix representation (2.6). Finally, we will define a fermionic path integral, which is shown to reproduce the result (2.13), (2.14). We start by defining the state loo-), which is the normalized zero-eigenstate of 4 and 6-L i(oo-)=O, The state 100 +) is defined

$-loo->=o.

(3.1)

by ]00+) = $+]OO-)

)

(3.2)

which leads to the relations $+]00+)=0,

&loo+)=

loo-).

(3.3)

P. Salomonson, J. W. van Holten / Supersymmetry

Obviously,

the f sign indicates

f Because

$1 = I,&, one has

the eigenvalue

= (*001 .

(3.4)

eigenvalues of $, 4, p, 9, . . . for the c-number eigenvalues of $+. Consistency requires:

Finally we introduce the notation and cy, 0, . . . for the anticommuting A 1 ff*r, = -$+a, We are now able to construct

*$ of the states for the observable

(rOO&

= 0 )

(r001$*

513

a loo*) = *100+x.

the eigenstates

(3.5)

of $, I,_ as follows:

lqcv-)=e~iq8-“j+100_),

(3.6)

which satisfy

l&T-)=aIqa-).

4lsQ-)=4lP-), The p^ and $+ eigenstates

are obtained

lqp+)=

by Fourier

-

I

(3.7)

transformation:

da eapIqcx-), (3.8) dq e’4PIqa =t) .

One easily proves

P*lP*)=PlPf)~ From this and the continuity

~+1(4,P)P+)=Pl(4,P)~+).

of the spectrum,

(3.9)

we find

(*~*414’pf)=e”PG(q-q’), (rLY*qlq’p*)=r6(a-B)6(q-q’), (3.10) (fa*plqp*)=e”’

eeiqp,

(*cu*plqP*)=rs(cy-P)e-‘qP. Here the anticommuting

S-function

JdP S(P-aIf Finally

the following

by

=f(a) ,

completeness -

is defined

relations

[email protected])=-s((Y-p).

(3.11)

hold:

da dqIqcrf)(ra*ql=

I, (3.12)

-

da$$m*)(h-z*p~=

1.

514

P. Salomonson, J. W. van Holten / Supersymmetry

The wave function can now be defined as usual. We denote the state of the system at time t by It). Then the wave function is obtained by expanding this state with respect to the coordinate basis:

It)= Inverting

this equation

-J da

dq K(qat)lqa

we get P_(qat) = (+qa*lt)

By Fourier

(3.13)

-) .

transformation

.

one finds the conjugate

P+([email protected]) = -I

(3.14)

wave function:

da ep”‘P_(qat)

= (- qp*lt) .

(3.15)

On wave functions (3.14), $* are represented by (2.4), as follows directly from the definition (3.13). On wave functions P+, the rbles of $+ in (2.4) are interchanged. We now prove the equivalence of the representations (2.4) and (2.6) of the fermion operators. The most general operator constructed from p*,4, I$* has the form n=A+~+$++&_+&+$_,

(3.16)

where a,. . . , 2 may still depend on BIG. Using and (2.6), (2.7), respectively, it follows immediately

the representations that

(2.4),

(2.5)

fl!P(x,Kg= (~++)a~+~+a*+~(S_a~+da*)) (3.17) ii

;’ (

(A + C),,

= 2

)

(

+S+(P* >

Lp,+dcp*

.

Comparing these expressions, one concludes that u1,2 may be identified with 41,2. It now remains to describe the dynamics of the system in terms of the anticommuting c-number representation. specified by the hamiltonian the kernel

The time evolution of the state It) is, of course, through the operator emitit. From this we can calculate

K(qat Iq’a’t’) = (+qcu*l epiti(r-r’)jqrar-), (3.18) ?+_(qat) = This kernel can be evaluated correct definition, we divide (t - t’)/(N + l), and define

dcr’ dq’ K(qatlq’a’t’)K(q’a’t’)

,

by a path integral as in eq. (2.13). the time interval into N + 1 steps

77”=(y),

To obtain of length

the E=

qN+l=ff,

(3.19) 40=

4’7

qN+l=q.

P. Salomonson,

Using the completeness

relations

515

J. W. van Ho&en / Supersymmetry

(3.12) 2N + 1 times, we find:

Nfl

x ,lJ, (+qn77ZlpnSn+N-pnC eeiEAlqn-lw-). In the limit of small E, and with the pn integrations

performed

(3.20)

this leads to

where

(3.22)

This is the discretized

form of the lagrangian

limit E + 0 these equations

provide

(1.2) with $ = v%( i(r_:))

the definition

. In the

of the path integral

To end this section, we prove the equivalence of (3.23) with (2.13). As was proved above, the components u1,2 of the wave function (3.14), as defined by (2.5), correspond

to f = *$, respectively.

part of i times the action

Now the fermionic

reads [cf. (3.22)] 1 - $E W, -dl

-1 -$eWl * .

+hW&

+

(7)l

’ ’ ’ qN+d

(3.24) Using the Grassmann

integration

rules

dn.l=O,

dq.n=l,

(3.25)

516

P. Salomonson,

J. W. van Holten / Supersymmetry

and eq. (3.18), one obtains:

!f_(q&)

= lim E’O

1

d&+1 dn0 dqo _ J2 7rie

(3.26)

with .Zf(q) as in (2.14). This is the desired

result,

4. Supersymmetry We now turn to a specific class of potentials, metry. To that effect we take [l]

V(q) = h-m2

7

namely

those that exhibit

WI) = v”(q)

supersym-

(4.1)

3

where u(q) is an arbitrary function of q, which may be called the superpotential, and ~1’and U” are its first and second derivatives. It is then easy to verify that fi = k6+,

(4.2)

6-1,

with 6, Moreover, generators [6,,

ql=

= (p*+ iv’(q*))$+ .

(4.3)

6, commutes with fi, hence these charges are conserved. of supersymmetry transformations between 4 and 6:

-ii*,

[&,fi]=

W(~>lj*,

{6*,

qQ=fi*iv’(ij>,

a,

They

are

h=o. (4.4)

In two-component

notation,

the hamiltonian

and supercharges

read:

J. W. van Holten / Supersymmetry

P. Salomonson,

517

and

6+_(0 +(&-v’(x))1, 10

i

0

(4.6)

Because degenerate.

the 6’s commute Namely,

with the hamiltonian,

if !P =

is an eigenstate

all eigenstates

of fi

are doubly

of fi, then

is an eigenstate of the same energy. There is, however, one possible exception, namely if the energy is zero, in which case the state may be annihilated by 6, and &. Since in a supersymmetric theory the hamiltonian is positive definite, such a state is necessarily the ground state of the system. It follows from the equation d*P=O, that such a supersymmetric

ground ?P=

(4.7)

state must have the form

‘I ( 0 )

or

P=

0 ( 42 )

)

(4.8)

where ~$r,~= c eTUCx).

(4.9)

As was observed in ref. [ 11, one of these solutions is acceptable only if v(x) becomes infinite at both x + *CD with the same sign, as in the case of the harmonic oscillator, with v(x) = $JX*. If this condition is not satisfied, neither of the solutions (4.8), (4.9) is normalizable, and they cannot represent the ground state of the system. In this case we necessarily have 6, ?P # 0 and supersymmetry must be broken. This happens for example if (see figs. 1 and 2) u(x)=;AX3-~2X,

(4.10)

which is the superpotential for the symmetric double-well anharmonic oscillator. In the following sections we will analyze the mechanism of supersymmetry breaking in these theories in detail. Although most of our derivations do not depend on the

518

P. Salomonson,

J. W. van Holren / Supersymmetry

Fig. 1. The qualitative form of the superpotential. Au = v(q_) - ~(9,).

specific choice of potential, (4.10) in mind.

the reader

Fig. 2. The classical

may find it convenient

bosonic

potential.

to keep the example

5. Solution of the Schriidinger equation In this section we will develop an iterative scheme to find the approximate ground-state solutions to the Schriidinger equation (2.10) for the supersymmetric potentials (4.1). We will then use these solutions to calculate the parameters which measure the breaking of supersymmetry such as the ground-state energy. As we have seen above, solutions for non-zero E come in pairs of the form (4.8), with bl.* related by supersymmetry. Indeed one finds the following relations between energy eigenstates with fermionic quantum number f i:

Since E is supposedly small (the ground.-state energy), we may devise an iterative approximation scheme to solve (5.1) by taking a trial wave function for ~$2,substitute this into the first equation (5.1) and integrate it to obtain an approximation for 41. This can then be used as an ansatz in the second equation (5.1) to find an improved solution for 42, etc. The procedure can be shown to converge for well-behaved potentials, with a judicious choice of initial trial function: In the case where U’(X) is odd [as in (4.10)], one can even short cut eq. (5.1) by noting that ~$i(-x) = &(x), since they satisfy the same eigenvalue equation. Hence

($+ d(x))4,(x) = JGh(-x) . Taking

x = 0 this yields an expression

(5.2)

for the energy:

J/2E = v’(0) +4;

(O)/&(O)

,

(5.3)

519

P. Salomonson, J. W. oan Holten J Supersymmetry

where the prime denotes differentiation with respect some approximations to q!~,,*and use these to estimate

to x. We will now calculate E and some matrix elements

of $*-, 6, to check the self-consistency of our approximations. Suppose, as in (4.10), that the superpotential has a maximum, at x-, and a minimum, at x+. We take the trial wave functions .

C#$: =6(x-x*) After one iteration,

(5.4)

we obtain +(r)(x) =10(x 1 N

-x_)

eeuCX) (5.5)

~~“(x)=~S(x+-x)e”~x~, where N is a normalization

factor.

C+:*‘(X)= $

The next approximation emvtX)I,“,,,-.

leads to

x_) e-2”(X’) dx’ , (5.6)

mint-x,x+) 4:“(x)

2

-&

eucx)

1

e

2U(X’)

dx’

.

-cc

However, for x’ > x_ the exponential e-2”(X’) will peak sharply around x+ and may be approximated by a S-function &(x+-x’); similarly ezvCX’)is approximately cS(x_ -x’) for x’< x+. With these approximations, eqs. (5.6) reduce to (5.5); hence to this level of precision (5.5) is a self-consistent solution. The normalization constant c can easily be calculated to be j,“_ e-2”(X) dx. Then a straightforward calculation of the normalization in (5.6) leads to N’2 =

m

,

(5.7)

)

(S XWe use (5.6) and (5.7) to calculate

3

e-2“(X) dx

the energy

expectation

value and find (5.8)

Remarkably, almost using (5.6) for d1,2:

the same expression

is obtained

E =;(~“e-2u(X~)~2_

for odd potentials

from (5.3),

(5.9)

Assuming the exponential e-2U(X) to be small between x_ and 0, which is correct to the same approximations underlying (5.8), the difference is negligible and the integral in both cases may be replaced by gaussians around x+, leading to the value fiu”(x+) E=----e

2lr

-2Av/h

(5.10)

520

P. Salomonson,

We have reinstated is supposed

J. W. van Holten / Supersymmetry

zt, to make clear the order of approximation

to hold, as well as its non-perturbative

nature.

Au = U(L)-u(x+)

to which this result

In (5.10) we have defined (5.11)

.

Although (5.10) gives direct evidence for the breaking of supersymmetry, a more practical measure for this phenomenon, in particular in field theories, is given by the quantity (5.12) This quantity have replaced

corresponds to the expectation value of an auxiliary field, which we by its equation of motion right from the start. Using that 6+w=o,

and inserting

a complete

set of states,

(5.12) becomes

w =i(K d+x)k, 6-w >

0

X=

() 42

(5.13)



because 6, commutes with l$ and hence the intermediate state must have the * same energy as P’. Using the matrix representations of 6+, I+_ and the wave functions (5.6), we obtain

(5.14)

where Ax = x+ - x_. Consequently (F)= Moreover,

v”(x) -Te

-26

(5.15)

“Ax.

by taking E = (!P, A*)

= $(P, 6+x)(*,

we re-obtain the result (5.10). In sect. 6 we will investigate origin of the results (5.10), (5.14). 6. Supersymmetry

(5.16)

d-P),

breaking in the instanton

the non-perturbative

picture

In this section we will show that the results (5.14) are obtained in the path integral formulation of the theory by calculating the matrix elements in an instanton

P. Salomonson, .I. W. van Holten / Supersymmetry

background*.

That is, supersymmetry

breaking

is, from this point of view, the result

of tunnelling between the classical vacua of the theory. The instantons are solutions of the euclidean-time equations a’;(r - v”(q)v’(q)

521

+&I”‘(~)~=~~~

= 0

of motion:

, (6.1)

(&

where 8, = -i(a/at).

-

(+2z1”(LI)M

Requiring

=

0,

the solutions

to have finite action one finds:

4 = *v’(sc) , (6.2) &=O. From now on a dot will indicate differentiation with respect to r, a prime, differentiation with respect to q. The subscript c will denote evaluation at the classical fields (6.2). The qualitative features of the solutions are sketched in figs. 3 and 4. They interpolate between positions q*, and iC is negligibly small outside a relatively narrow interval around ~~(7;). We now expand the fields around these classical solutions:

q(7) = 9c(T)+v%(7), The euclidean

action

becomes

SE=S,E-fi

I

dr[tB(a,~vE)(a,rvP)4_77(a,+u”)~

-$G(v~u,” +~v~“‘v)$“-Jszv~477f+o(h)],

(6.4)

with SF =; We use this expression

dr (4; + 0;‘) = f dr(icv: = Au,. I I for SE to calculate the matrix elements

Fig. 3. The qualitative features of the instanton solutions.

l

Fig. 4. The qualitative

(6.5) of $+, 6,

behaviour

This result was noted in ref. [l]. To the best of our knowledge no derivation For the non-supersymmetric double-well potential with fermions a calculation energy splitting by instanton methods was carried out in ref. [4].

with the

of (jJr).

has been published. of the ground-state

522

P. Salomonson, J. W. van Hoiten / Supersymmetry

path integral the problem normalizable

(3.23). However, before we can do this, we first have to deal with of the zero modes of the bilinear terms in (6.4). The existence of zero modes is easily established; they read 9 = 50U:(T) , if d===zI:.,

77= w:(T),

(6.6)

or

5=

&d (7) ,

ifd==-v:,

with

They arise from time-translation of instantons, and supersymmetry transformations on them, respectively. Note that a supersymmetry transformation gives rise to only one fermionic zero mode, the other one being identically equal to zero. The existence of zero modes gives rise to non-gaussian behaviour of the functional integral. This problem is dealt with, as usual, by introducing a collective coordinate T() replacing the bosonic zero mode [5]. The collective coordinate, to be called the “instanton time” is defined by writing

= qc(7-

7”) + Ji C’ h7 (7 - 7”) ,

(6.7)

n

where {(uL/N), 4”) form a complete orthonormal set of eigenfunctions of the differential operator (a,*v1)(&rvE), and Cl, denotes a sum over the non-zero modes q,(7) only. Clearly, TV is defined by the requirement co=;

In general,

the functional

Iq(T)U:(T-Ti))dT=o.

integration

measure

(6.8)

for q is

(6.9) Trading

(to, &) for (T,,, c,,) we get a jacobian

factor

(6.10)

523

P. Salomonson, J. W. uan Holten / Supersymmetry

where D4 denotes

the functional

integration

that the normalization

for the shifted

quantum

field (6.11)

=q(7)-qd-TO).

Jh+-70) We remark

measure

factor N is given by (6.12)

The fermionic zero mode one-instanton contribution integration rule

cannot be removed by a similar procedure. In fact, the to the path integral vanishes, due to the Grassmann

I dqo.l=o.

(6.13)

In order to see this better, we note first that the fermionic field ~(5) can be expanded in terms of the same eigenfunctions of (8, f ~:)(a, T vi) as q(T), and we write (6.14) for the case dC = v:; similarly for 5 if dC= -vL. On the other hand, the fermionic field l(v) must be expanded in terms of eigenfunctions of the reversed operator (a, r ~:)(a, * up), which does not possess a normalizable zero mode:

(or similarly

for q), with (& 7 ul)qn (7) = w,z, (7) > (6.16) (& f ul )zn (7) = -w&I, (7) >

n#O.

Hence (6.16) does not establish a complete l-l correspondence between the amplitudes v,,, [,,. Since the action contains TJ and 5 always in the bilinear combination ~5, there is no way to obtain a term from which the zero mode nO(lO) can be factored out, thus leaving a non-vanishing functional integral over the fermionic fields orthogonal to the zero mode. However, the same argument no longer holds if we calculate matrix elements of fermionic operators, because they may provide the necessary extra factor of n(l) which reduces the effective functional integral over 5, 7 to one over the non-zero modes only. This is precisely what happens if we calculate the matrix elements of $+, 6, in the background of the classical solution gC = -us, as we will discuss now. We actually want to calculate the quantities (+Oq+, e-iA(TMJ+

e-‘A’T+‘)/q_O_)

) (6.17)

(+Oq+l ep’A’Tp”(~ + iu’(Lj))$+ e-rA’T+t)lq_O-), in the limit

T + -ion. Using the methods

of sect. 3 it is straightforward

to show

524

P. Salomonson, J. W. van Holten / Supersymmetry

that this is equivalent

to calculating

the functional

J J

Dq Dn Dl l(~) e-(l’*)SF

integrals

, (6.18)

Dq Dn DC i(4 + ~‘)4’(r) eP(l’h)S” .

The functional integrals include an integration over the instanton time TV.However, the integrands (6.18) depend only on the difference r - T,+ Hence we may equivalently take r. to be fixed and integrate over --7 instead. For convenience one may now define the origin of time at TV. The calculation of the first integral (6.18) is easy. We get, to lowest order in fi,

J

@a

X

exp

-Au,/h

-=

eJ%

d~‘(t4(a,,-v~)(a,,+v~)q_77(a,,+v,”)~)

-al

J

cc

det’ (c+& + 0,“) det’(-&+u,“)(&+u,“)

__,drvb(T)

1’ l/2

(6.19)

Here the prime on the determinants indicates removal of the zero mode. It is shown in the appendix that the ratio of the determinants is 20,” (q+). Hence we find (+Oq+l$+lq-0-)=

r-

yePAyc’“dq,.

(6.20)

This is identical to the result (5.14). To obtain (6.19), note that there is a factor 4% coming from the integration over the fermionic zero mode; this cancels against the same quantity in the denominator of (6.10). Next we turn to the second integral (6.18). Because dC+ v: = 0, it vanishes at the tree level. Hence we will have to go to the one-loop level (O(h)) to find a non-zero contribution. Using (6.4) and (6.10) we obtain ieZVe-Au=‘”

J dr J d10

J DQDv

DL(-I+$&

J zI~(T')Lj(T')dT')

(

x 50+$+~(~))((&+U~(~))q(7)+#iU~(T)q(i)~) x [ 1 -;Jh

-A

J dr’

(&“u: +fz$“‘“u:)(r’)~(r’)3

J dr’Vr(r’)q(r’)rj

(T~)(~I~v:(T')+~(T'))]

e-cl"')s~,

(6.21)

P. Salomonson, J. W. van Holten / Supersymmetry

525

where SF is the quadratic part of the euclidean action. The term of order J\/h vanishes because it is odd in the integration variable. As a result we only get terms of order h, and we can rewrite

+ (a, +

+f(~)j

(6.21), performing

u: (T))~(T) ($ U:(T) j

dT’

ZJ~(T')U~(~')~(T')~(T'))]

An explicit

T') =

calculation

(6.22)

e-cl'wsf.

DLjq(~)ij(~‘) e-“‘h’SH

I

over lob:

dr’ ~W)&T’)

These terms are represented by the diagrams bosonic non-zero modes is [cf. (6.16)]: K(T,

the integral

of fig. 5. Now the propagator

for the

(6.23)

=c’ -$n(+nb). I <,Tl=o n wn

gives*

It satisfies the relations: K(T,

7’)

=

7) ,

K(T’,

(-a, + U:)(& + U:)K(T, (a7+vP(T))(a,,+v,“(T’))K(T,

7') = s(T.- 7’) -

lJ;(T)O:(i’)

N2



(6.25)

T’)=8(T-T’) .

I--\

[email protected]:::‘;

+.._,I

y---o

,,-‘ I

I

&B

y----y

Fig. 5. The Feynman diagrams contributing to the matrix element (+Oq+ld+lq_O-). I denotes fermion source produced by the instanton, and J the boson source resulting from the jacobian.

l

Without

loss of generality

we choose

u(q+) + u(q_) = 0.

the

526

P. Salomonson, J. W. van Holten / Supersymmetry

Similarly,

the fermionic

propagator

S(T, T’)=

I

is

Dn Dfq(r)f(r’)

ePsF”=z’

= -(a, + v:(T))K(T,

sz.(r)q,(r’) ”

7’).

(6.26)

The explicit form is s(T,

We now insert

7’)

=

v:(r’)

-

v:: (7) [

v,(r)

$E(T-T’)+-

Au,

1

(6.27)



(6.23) and (6.26) into (6.22), obtaining -&L(T)v~(T)K(T,

+j+

V;(T)

1

dr’

-V:(T)

dr’

j

V~(T’)d,~(&+V:(T))~(T,

dr’ (vrv,”

+&L(T)

+

dr’

V:

T)

7’)

+$v~v~)(T’)(&

(T’)(&

?J! (T’)v:(T’)(&

+

v:(T))K(T,

+

V:

(T))K(T,

T’)(&+

+

V:

(T))K(T,

~‘)(a,,+

112 1

det’((+3&+vF)

T’)K(T’,

V:

(T’))K(T’,

21: (T’))K(T,

7’)

7’)

T’))

(6.28)

.

det’(-&+v,“)(a,+v,“)

One can now perform a partial integration in the fourth term, and use (6.25) to find a cancellation of this term against the first and third one. This leaves the second and the last term of (6.28) only. Using (6.24) and (6.25), the complete result reduces to:

-

[

dr’

V:.(T’)V:(T’)(:-:B(T-T’)(

vc(T;;;c(T’)

-

vc(;v”(T”))

.

(6.29)

Now we use the relations dr’

J J

dr’

V,“(T’)V;(T’)2

=

-:V;2,

V;(T’)V::(T’)V,(T’)

dr’ vr

(T’)v:(T’)

=

=

-vi

-V:V,+:V:z,

,

(6.30)

P. Salomonson, J. W. van Holten / Supersymmetry

and the property

V,(T) = -v,(-T),

terms of (6.29). Hence ih

Again

J

to get cancellation

d7~=

‘,n(q+) -Av,/h

-e

this is identical

between

we are left with the extremely

?i-

Au,

to the result

ih

527

the second

and third

simple result

J0:

(q+)c-Au,/h

(6.31)

7r

of the wave function

calculation

demonstrates that (to first approximation) supersymmetry instanton effect in the path integral formulation of the theory.

breaking

(5.14).

This

is a one-

7. Discussion After calculating the effect of tunnelling between two classical vacua by using a one-instanton background, a few remarks are in order. First, the amplitudes calculated by us do not receive any contributions from either no-instanton or anti-instanton configurations, because of the fermionic zero-mode problem discussed above. Multi-instanton configurations could contribute in principle, provided they have not more than one normalizable fermionic zero mode. However, their contribution is clearly smaller by factors e-A”c’h with respect to the result (6.31). Therefore, they have been disregarded. Finally we consider briefly the effects of instantons when the superpotential ZI(q) has more than two extrema qy, Y = 1,2, . . . , N. We assume that v(q) + ~o((-l)~-‘oo) when q + -CO(CO), and that the extrema are well separated, i.e., Ji;+’ lu’(q)ldq >> 1. Around each of the classical minima qy of the potential v’(q)* we can approximate the theory by a supersymmetric harmonic oscillator. Then there are N ground states which have energy e0 = 0. These states are described by the upper or lower component of the wave function, depending on whether v is odd or even. The corrections to this picture are due to higher order terms and quantum tunnelling effects. In the wave-function picture, we can describe the results as follows. If N is even, there are ;N zero-energy states both in the upper and lower component of the wave function, and supersymmetry dictates that they remain in pairs of one upper and one lower component state with the same energy, even when the energy is no longer exactly zero. If N is odd, there is one upper component state left over. Because of this, this state, described by the wave function e-“(‘), remains

an exact zero-energy

state.

Alternatively, the splitting of the N states with &o= 0 can be analyzed by field theoretic methods. The harmonic oscillator approximation corresponds to the tree approximation in Feynman diagram language. Since supersymmetry cannot be broken by radiative corrections, the N states remain zero-energy states to all orders in ordinary perturbation theory. The states will not split until instantons are taken into account. Instantons (anti-instantons) give non-zero matrix elements IV) denotes the perturbative vacuum (2v* 11&12vM24c?+12v~ I)), where centred around qY. The calculation of these matrix elements uses the form of the

528

P. Salomonson,

superpotential taken

between

over directly.

J. W. van Holten / Supersymmetry

two adjacent

With suitable

qy’s only,

and the result

choice of phases

of sect. 6 can be

for the perturbative

vacua, we

obtain (2y f 1]&2v)

= (2~]~+]2v

l

1) = C2VC2V*l e-“2~+“2~*1 ,

(7.1) u:(q”) “4 u, = uc(qY) . ( T )y It is easy to verify that the N perturbative vacua now form a representation of the supersymmetry algebra. As a consequence, the states come in pairs with the same energy, except for a single zero-energy state in the case of odd N. C = Y

We thank H. Osborn and R. Stora for useful discussions. One of us (P.S.) wishes to thank Kevin Cahill for stimulating his interest in the problem. Appendix CALCULATION

OF DETERMINANTS

In this appendix

we calculate

l/2 1

the ratio of determinants

det’ (cr& + 0:) det’(-&+u1)(&+v:)



qc=-UC,

,

in eq. (6.191, taking for convenience v(q) = -v(-q). The fermionic obtained by the methods of sect. 3. There it was shown that

(A.1) determinant

is

64.2) implying I

Dn DC exp (I”,ca,+w,rd~)=exp(fIT’Wdr), 7

I

for the evolution of states with fermionic quantum number f. The functional integral is taken over all n, 5 including possible zero modes. On the other hand, we defined in sect. 6 T’ [det’ (c+& + vl)] “’ = Dn Df exp &,+vE)cdr , (A.31 I ) (I 7 where Df runs over non-zero modes only. However, we can reinsert mode by reversing the derivation of (6.19) which leads to [det’ ((+& + vE)] l/2 =~lD~DCC(I)exp(l”$(d,+v~)idi). E

the zero

(A.4) i

P. Salomonson, J. W. van Holten / Supersymmetry

If we start out at time -T

with a zero-fermion

at time r and afterwards

a one-fermion

529

state, we have creation

of a fermion

T. Applying

state until the time

(A.2) we

obtain [det’ (a&

N =-exp d(T)

l/2

+ v:)]

I

(

-i

=$)a*

ii+)

(A.5)

c

Using N =

v: dr'+fJrTv:

J-T

=

JAM,, we finally have [det ((+& + VP)]

112

Au,

t/2 _

-

v:(T)d(-T)

64.6)

*

1

We now turn to the bosonic contribution, det’ (a, - ~:)(a, + up). Its calculation will follow refs. [3, 61. As was proved there, the determinant of the operator (a, - ~,“)(a, + v,“) in the space of functions vanishing at the boundaries (-T, T), is given (up to a normalization factor B) by D(T), where D(T) is the solution of the homogeneous equation (a, - vl)(& with the boundary

vL(*T)+O.

form of the solution

1.

L4.8)

is

we may write approximately,

T

as follows.

In the limit

T + 03,

= 0 )

= c e-+,

w=vc”(q+).

(A.lO)

comes

Using

contribution to the integral (A.lO) leads to

J

(A.9)

for large )r] (SC+ 4+):

v:(r) Clearly, the main the approximation

$2.

for T = T we proceed

the integral Because

(a, + v:)v:

dr

_TvLcTj2=7

Hence

(A.7)

b(-T)=

= 0,

D(T) = v6(7)v5(-T)I_: To calculate

= 0 )

conditions D(-T)

The explicit

+ vE)D(r)

2

J

T 2w7

o e

1 dT=ze

from the end points.

2wT .

(A.ll)

we obtain det (a, - vl)(&

We now divide

+ ve) = B/w.

this by the lowest eigenvalue

(A.12)

of (8, - vE)(& + v:), which is the one

530

P. Salomonson, J. W. oan Holien / Supersymmetry

that becomes

zero in the limit T *CO. It is shown below that this eigenvalue

ho is

given by Ao=

4wc2

_

(A.13)

Au, e2‘“= ’ As a result

Au, e2wT

det’ (--a, + vE)(& + vi) = B 4.

(A.14)

w

In order to calculate the normalization B, we repeat the above procedure vacuum-to-vacuum amplitude, which. we require to be unity. This implies exp( Combining

-$[_Ludr)(det(-&+o)(d,+w))“‘=\/~=l,

(A.1.5)

l/2 1=

(A.6), (A.14) and (A.15) gives, in the approximation det’ ((T& + vi)

the same result is obtained

det’ (o&

(A.ld)

J2v,” (4+) .

det’ (-&+ve)(&+vz) Remarkably,

for the

(A.16)

by defining

+ ~8) = (det (8, + vP)(& - vl) det’ (a, - ~,“)(a, + v:))“~

,

(A.17)

and calculating the determinants as indicated for the bosonic determinant above. However, we have not been able to give a rigorous justification of this procedure. It remains to prove (A.13). The proof goes as follows [6]. The homogeneous equation (a,-v,“)(a,+v:)X=O has two linearly

independent

solutions,

which can be taken to be

Xi(T) = v:(r)

2 (A.18)

x2(7)= Their wronskian

v:(T)

7 dr’ I__Tm*

is xLt2-,t1x2=

1*

(A.19)

As a result, the function +*(T)=ffXl+PX2-A

T [x1(7)x2(7’) - x2(~)~1(7’M I -T

(7’) d+

is an eigenfunction of (a,- ~:)(a, + u,“) with eigenvalue A. We now look for a solution with A + 0 as T + 03, and 4A(T) = +bA(-T) = 0. These requirements

P. Salomonson, J. W. van Holten / Supersymmetry

immediately

531

yield

a=o, (A.20) A = x2(~) (x1(~) while p becomes

an overall

j_Lxl(r)2

dr -x2(T)

normalization

I_:xl(r)xz(r)

constant.

dr) -’ ,

We have neglected

of order h2. The integrals are evaluated in a way analogous to (A.ll), requires some care to sort out the leading contributions. We find T

J

J

2wT

T

~2(7)~ dr = ?-.-2wc2

_

-T

X2(T)

Substitution

of this into (A.20)

=c

WC ’

quantities although

it

Av, e40T x1(7)x2(7)

dT

=

~

T

402c4



(A.21) xl(T)

=

c

emwT.

leads to the result (A.13).

References [l] E. Witten, Princeton preprint (1981), Nucl. Phys. B 188 (1981) 513 [2] L. Faddeev, in Methods in field theory, Les Houches (1975), ed. R. Balian and J. Zinn-Justin (North-Holland) [3] R.P. Feynman, Rev. Mod. Phys. 20 (1948) 267; I.M. Gel’fand and A.M. Yaglom, J. Math. Phys. 1 (1960) 48 [4] E. Gildener and A. Patrascioiu, Phys. Rev. D16 (1977) 1802 [5] A.M. Polyakov, Nucl. Phys. B120 (1977) 429 [6] S. Coleman, in The whys of subnuclear physics, Erice (1977), ed. A. Zichichi (Plenum Press)