Nuclear Physics @ NorthHolland
B196 (1982) 509531 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 cnumber lagrangian
(1.1) and coordinate the fermionic
operators variables.
(1.2) where
the components of which are to be interpreted as anticommuting cnumbers. 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) spacetime 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+, 6j
= 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 groundstate 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+, J1 = 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
cnumber
511
J. W. LKUIHolten / Supersymmetry
operators:
G+= dlX,
$=L
(2.4)
acting on wave functions*
or one may represent
them by finite dimensional
matrices
(2.6) whence
the wave functions
become
twocomponent
W(x)=
41(x)
objects:
(42(x))
(2.7)
.
Here we will explore the last possibility (2.6), (2.7). In sect. 3 we will discuss the anticommuting cnumber representation (2.4), (2.5), and show the equivalence of the two formalisms. In the matrix formulation, the hamiltonian (1.1) becomes a 2 x 2 matrix as well:
A=
(&$+V(x))ltW(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= $[$+, c1 = $(T3
is now clear. The
(2.11)
commutes with the hamiltonian and is diagonal in this representation**. From the algebra (2.2), or from (2.11), it follows that its eigenvalues are f= ZIZ$.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 1eiir(rr’)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
cnumber
representation
In this section we construct the abstract space of eigenstates of the conjugate operators Get, having anticommuting cnumber 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 zeroeigenstate of 4 and 6L 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 cnumber 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(qq’), (rLY*qlq’p*)=r6(aB)6(qq’), (3.10) (fa*plqp*)=e”’
eeiqp,
(*cu*plqP*)=rs(cyP)e‘qP. Here the anticommuting
Sfunction
JdP S(PaIf Finally
the following
by
=f(a) ,
completeness 
is defined
relations
[email protected])=s((Yp).
(3.11)
hold:
da dqIqcrf)(ra*ql=
I, (3.12)

da$$m*)(hz*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 cnumber 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(rr’)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+NpnC eeiEAlqnlw). 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 $eWl * .
+hW&
+
(7)l
’ ’ ’ qN+d
(3.24) Using the Grassmann
integration
rules
dn.l=O,
dq.n=l,
(3.25)
516
P. Salomonson,
J. W. van Holten / Supersymmetry
and eq. (3.18), one obtains:
!f_(q&)
= lim E’O
1
d&+1 dn0 dqo _ J2 7rie
(3.26)
with .Zf(q) as in (2.14). This is the desired
result,
4. Supersymmetry We now turn to a specific class of potentials, metry. To that effect we take [l]
V(q) = hm2
7
namely
those that exhibit
WI) = v”(q)
supersym
(4.1)
3
where u(q) is an arbitrary function of q, which may be called the superpotential, and ~1’and U” are its first and second derivatives. It is then easy to verify that fi = k6+,
(4.2)
61,
with 6, Moreover, generators [6,,
ql=
= (p*+ iv’(q*))$+ .
(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 twocomponent
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 doublewell 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 groundstate 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 groundstate energy. As we have seen above, solutions for nonzero 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 ~$2,substitute this into the first equation (5.1) and integrate it to obtain an approximation for 41. This can then be used as an ansatz in the second equation (5.1) to find an improved solution for 42, etc. The procedure can be shown to converge for wellbehaved potentials, with a judicious choice of initial trial function: In the case where U’(X) is odd [as in (4.10)], one can even short cut eq. (5.1) by noting that ~$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 $*, 6, to check the selfconsistency of our approximations. Suppose, as in (4.10), that the superpotential has a maximum, at x, and a minimum, at x+. We take the trial wave functions .
C#$: =6(xx*) 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_) e2”(X’) dx’ , (5.6)
mintx,x+) 4:“(x)
2
&
eucx)
1
e
2U(X’)
dx’
.
cc
However, for x’ > x_ the exponential e2”(X’) will peak sharply around x+ and may be approximated by a Sfunction &(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 selfconsistent solution. The normalization constant c can easily be calculated to be j,“_ e2”(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
e2“(X) dx
the energy
expectation
value and find (5.8)
Remarkably, almost using (5.6) for d1,2:
the same expression
is obtained
E =;(~“e2u(X~)~2_
for odd potentials
from (5.3),
(5.9)
Assuming the exponential e2U(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 nonperturbative
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, 6w >
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 reobtain the result (5.10). In sect. 6 we will investigate origin of the results (5.10), (5.14). 6. Supersymmetry
(5.16)
dP),
breaking in the instanton
the nonperturbative
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 euclideantime 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,Efi
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 nonsupersymmetric doublewell 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 groundstate
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 timetranslation 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 nongaussian 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 nonzero modes q,(7) only. Clearly, TV is defined by the requirement co=;
In general,
the functional
Iq(T)U:(TTi))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)qdTO).
Jh+70) We remark
measure
factor N is given by (6.12)
The fermionic zero mode oneinstanton 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 ll 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 nonvanishing 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 nonzero modes only. This is precisely what happens if we calculate the matrix elements of $+, 6, in the background of the classical solution gC = us, as we will discuss now. We actually want to calculate the quantities (+Oq+, eiA(TMJ+
e‘A’T+‘)/q_O_)
) (6.17)
(+Oq+l ep’A’Tp”(~ + iu’(Lj))$+ erA’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$+lq0)=
r
yePAyc’“dq,.
(6.20)
This is identical to the result (5.14). To obtain (6.19), note that there is a factor 4% coming from the integration over the fermionic zero mode; this cancels against the same quantity in the denominator of (6.10). Next we turn to the second integral (6.18). Because dC+ v: = 0, it vanishes at the tree level. Hence we will have to go to the oneloop level (O(h)) to find a nonzero contribution. Using (6.4) and (6.10) we obtain ieZVeAu=‘”
J dr J d10
J DQDv
DL(I+$&
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'))]
ecl"')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)
ecl'wsf.
DLjq(~)ij(~‘) e“‘h’SH
I
over lob:
dr’ ~W)&T’)
These terms are represented by the diagrams bosonic nonzero 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(TT’) .
I\
[email protected]:::‘;
+.._,I
yo
,,‘ I
I
&B
yy
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)
$E(TT’)+
Au,
1
(6.27)
’
(6.23) and (6.26) into (6.22), obtaining &L(T)v~(T)K(T,
+j+
V;(T)
1
dr’
V:(T)
dr’
j
V~(T’)d,~(&+V:(T))~(T,
dr’ (vrv,”
+&L(T)
+
dr’
V:
T)
7’)
+$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(TT’)(
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+)cAu,/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 oneinstanton background, a few remarks are in order. First, the amplitudes calculated by us do not receive any contributions from either noinstanton or antiinstanton configurations, because of the fermionic zeromode problem discussed above. Multiinstanton configurations could contribute in principle, provided they have not more than one normalizable fermionic zero mode. However, their contribution is clearly smaller by factors eA”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 wavefunction picture, we can describe the results as follows. If N is even, there are ;N zeroenergy 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 zeroenergy
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 zeroenergy states to all orders in ordinary perturbation theory. The states will not split until instantons are taken into account. Instantons (antiinstantons) give nonzero 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 zeroenergy 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 nonzero 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”$(d,+v~)idi). E
the zero
(A.4) i
P. Salomonson, J. W. van Holten / Supersymmetry
If we start out at time T
with a zerofermion
at time r and afterwards
a onefermion
529
state, we have creation
of a fermion
T. Applying
state until the time
(A.2) we
obtain [det’ (a&
N =exp d(T)
l/2
+ v:)]
I
(
i
=$)a*
ii+)
(A.5)
c
Using N =
v: dr'+fJrTv:
JT
=
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 vacuumtovacuum 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+PX2A
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. ZinnJustin (NorthHolland) [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)