Download wormholes and supersymmetry

c
WORMHOLES AND SUPERSYMMETRY
JAMES
R.
ANGLIN
Physics Department
McGill University, Montreal
October 1990
A thesis submitted ta the Faculty
of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
(
Master of Science.
( c) J. R. Anglin 1990
(
TABLE OF CONTENTS
Page
(
c
1. Introduction . . . . . .
3
II. The Bosonic Worrnhole .
8
III. The Superwormhole .
19
IV. Conclusion. . . ..
32
Appendix: Review of supersyrnmetry, -space, and -gravity .
35
References .
40
Bibliography .
42
1
,i
a
ABSTRACT: Following work by Coleman and Lee using ordinary gravity, we
obtain a low-energy effective Lagrangian for a supergravity system, in which superwormholes are represented by local operators in superspace. Supersymmetry is not
broken by these new interactions.
RÉSUMÉ: En suivant un argument de Coleman et Lee pour un système avec
la gravité ordina.ire, nous trouvons un Lagrangien effectif à basse energic pour un
système avec la supergravité. Les effets des "superworml1oles" sont représentées par
des opérateurs locaux dans la super-espace. La supersymmetrie n'est pas brisée spontanément par ces nouvelles interactions.
Note to the reader:
This thesis is written using the "Victorian wc" throughout. We do not know
where we acquired this affectation, but it secms to he common in scientific papers,
even those with only one author. Perhaps it is intcnded t.o include the readers, making
them feel more involved in the discussion. Perhaps it givcs ncrvolls authors a pathetic
illusion of safety in numhers. Or perhaps it is only strict accuracy, rccognizing that
even physicists are composed of many parts and particles.
From ail my quarks, to all of yours ...
2
__
~~:.~
. ......
~
~~
. ...
~'i".
. ._ ...
1··
r-
.... ";
.,~
. .. ......
--r-"'>I· ,
..
~._.~-
"'_.. _"'.1<' ..",_ •.• ;,- - _ ........ ?l- ··-7·...
,1J·~
·u\,"·
-.;;~·1~"'I-'1'2i"'~~_·
... •· .....
·~~-
···r.,,· ...
~-
..... _. __ ....---
·M~"
w··
~~._.-
1. INTRODUCTION
The idea that small-scale geometry might be more than meets the eyc seems
to pre-date General Relativityl.
Wheeler originated the modern hypothesis t.hat
quantum mechanics and gravit y tpgether imply that space has non-trivial, fluctuating
topology on the small scale 2 . The key point in his argument was that the Einstein
Lagrangian allows drastic m~trical changes on small scales to have small action and
so make s gnificant contributions to transition amplitudes. The idea is intriguing,
in that by giving space a rich texture as weIl as large-scale ~eomctry, it promises to
endow with explanatory properties the space-time background that our theories all
have to have anyway. Economy i<;: an elegance.
The problem with quantum topological fluctuations is that th<:' causality embed-
(,
ded in a Lorentzian geometry forbids them on pain of singularities 3 . Thus Wheeler's
idea for a spacetime froth sc<:'med just that, until Hawking's daim t.hat quantum
gravit y should use Euclidean geometry4. Hawking's arguments for imaginary time
are based on the fact that his event horizon thermodynamics, obtained now by many
independent approaches, appears very elegantly in a Euclidean gravit y formalism
where time is analytically continued to imaginary values. Since many calculations
that are simply impossible in Lorentzian gravity are rather easy in a Eucliclean theory (because functions tend to clecay insiead of propagating), imaginary time has been
a boon to quantum gravit y physicists clespite its uncertain l'ole in quantum gravit y
physics.
A less extreme position than I-Iawking's that still lets one use Euclidean space is
3
(
u_
1------------------1
that in whirh irnaginary time in gravit y is compared to imaginary time in conventional
second-quantized field t.heory or quantum mcchanÎcs. In such theorics, quasi-classical
approximations for "tullneling" phenomena can be donc by \Vick rotating t
---+
lt and
cxpanding pat.h integrals around Euclidean spaC(' saddlC'point.s known as ln8tantons.
Since instantons arc by dcfinitioll fidd configuratio'ls that arC' localizl'd in (ElIclidcan)
space, they arC' amcnable to varions conv('nif'nt non-int.cract.ing approximations. 5 In
convclltional thcories such as Yang-Mills, thesc tcchl11qucs ar(' irr('proachablc mathematical tricks; in gravit y, they are
1101. 50
irreploachablc, bccausc time is a dynamical
variable whose propf'rties arc radically changed by analytic cont.inuation.
In this eontext, adual instanton solutions to Euclidean-spacc Einstein equations
are of int.erest. Glddings and Stromingel' round the first one, in a system with an
axionic matter field 6. It is the simplest non-trivIal topology, namcly a w01'mhole.
Wormhole configurations are geometrics in which two asymptot.ically fiat spaces are
connected by a narrow 14throat" region. Ot.hers have RincC' appearcd 7,8, including
one found by Coleman and Lee, in which a conscrvcd global charge is manifestly the
thing that makes the worrnhole, by keeping the throat from collapsing into nothing9 .
(IIawI..:ng has also u5rd wormhole gcometrics that, arc not instantons, in that they do
not saUsfy ail the.' C.'<]uations of motion. 10)
As the calulablc harbingcrs of quantum gravit y, worrnho!c effeds have been invoked to producc the "wild and wonderful" phenomena of fundarnental quant.um
incoherence ll and the vanishing of t.he cosmological constant 12. There has been great
controversy ovcr thesc notions, with disagreemcnts on fundamcntals and details alike
leading to dramat,ieally diffcrent results. 13 ,14,15 The state of the theory of wormholes
4
is itself rather like a worrnhole geometry. On opposite sicles are the two asymptotically
elegant formalisrns of Hawking's Euclidean gravit y, and c1assically causal relativitYj
they cornmunicate throllgh a channel cqually inaccessible to wcll-grounded theory
and to experimcnt.
In order to have sorne hope of finding sorne facts in a rnorass of speculation,
a good thing to do is to find the low-energy effccts of wormholcs, in an attempt to
make contact with observable physics. There is widespr<'ad agrecmf'nt that wormholes
should be so srnall t.hat. wc never sec them explicitly, but they ought
1,0
give rise
to effective local operators in a low-energy approximation. Even if su ch operators
turn out not to be observable at currently accessible seales, local field thcory is weIl
ullderstood (in cornparison with quantum gravit y), and the bcttcr we manage to fit
wormholes into a familiar framework, the better is our chance of appreciating their
(
real significance, great or small.
Another interesting but speculative idea is supergravity, which an extension of the
Coleman-Mandula theorem (still in fiat space )16 strongly suggests to be the "wildest"
quantum theory reasonable in the curved space rcgime below the Planck scale. There
is no experimental evidenn> whatever for it, but neithcr is thcre damning evidence
against it. It is rnainly of interest b('cause any analysis that works with supergravity
is invoking what is prcsumably the rnost general symmetry money can buy, and we
need powcrful symmetries to hope to explain things like the hierarchy problem and
the observed particle spcctrum.
In this thesis we will combine these two interesting possibilities, thus bringing
wormholes further into the arena of conventional theoretical physics. We will con-
5
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1
I~--------------
....
struct the effective Lagrangian for a supersymmetric multiplet of n'attcr fields couplf'd
to supergravity, ignoring loops, but including the trec-levf'1 dfefts of sllperworrnhole
instantons. W(' will follow and cxtf'nd a paper by Col('man and L('('9. Our results will
support the common assumption thal stlpersymm<,try should supprC'ss t.he effective
scalar self-couplings that wormholes genel'ically induCf'lO.
In the remaining part of this introduction,
W('
will dcscribe the basic features of the
argument wc will follow. ln Scctior. II w(' Will <tpply
tllIS
argunH'nt to a simplf' system
of a scalar field roup!ed to 8instein gravity. Tllls will follow part of RefefC'nn'
closely. 1'hell in Section III wc
Will
[9] very
tackIr :mpcrgravity, and obtaill an original l'csult
exactly like that of Section II, hut in superspace. Sediu!1 IV is a hrier conclusion.
An appendix is attarhed giving a whirlwind Sllnlmary of supcrsymmpt,ry, Sllperspace,
and supel'gl'avity.
Our goal is to determine the effect at experimentally acr('ssi ble 8ral(,8 of \VOl mholes
that arc very much smaller than those srales, yet sufliciently largC'1' than the Planck
scale for us to use gcneral relativity (and ultimately supcrgravity) in our action. 1'here
is pIenty of room - about 170rders of magnitude -- betwcen thosc seales to fit in
our wormhoIes. Our approach will take advantagc of this faet to insNt yet another
seale (hcreaftcr "ro") th:-tt is to be much largcr than the wormholC' scalp and yet al80
rnuch sm aller than tlw experimental. Wc will th('JJ constluct a field ('Onfiguration in
Euclidean four-spaœ in which a 3-ball of radius
7'0
is eut. out. of a flat. harkground
space and replaccd with a worrnhole, matching our fields al, the hOllndélry. Wc will
perform a saddle-point approximation 1.0 the path intcgral using t.his "cut and patch"
configuration, and wc will find that t.he lcading cont.rihutions to t.he act.ion rorning
6
from the wormhole insrrtion are boundary terms on the sphere at ra. Since ra is smaU
(
on the laboratory scale, these terms may effectively be taken as point-likc terms "at
the wormholc."
There are t.hree steps in this procedure that nced justifying. The first is the use of
imaginary time to givc Euclidean four-spacc instead of L\)rentzian spacetime. Thcre
is no r~al justification for t.his procedure, bl!t in these days when therc is no king in
Israel, every one do('s what secms right in his or her own eyes. Thelc ie;; no compelling
argument against imaginary time in gravlt.)', and il. works \\,('ll in fiat spac(' tUJlncling
calculations for Yang-Mills theorif's. Admittedl)', time is in no scns(' a dynamical
variable in sneh tlwories, but theorists l,avc to carn a living somehow.
The second step is the use of the abrupt "cut and patch ,. configurat.ion. As wc will
see, it is not rcally very abrupt, because ail our fields will approélch t.lwir background
(
values quite rapidly as wc rnove out from the wormhok to the eut-off
7'0'
The third great leap is identifying the thrcc-spher(' of radins ra as the point "at
the wormhole." This seems reasonahlc insofar as ro can easily he very small indeed,
and we will not try to Justify it any further hcre. There are depths to the idea,
however, that may weIl f('ward exploration.
In the end we must confess that in the wormholc geometry of theorics about
wormholes, wc will sit far out in th(' ësymrtotieally fiat spacc 011 Hawking's si de of the
controvcrsy. We idealizc any fundamcntal conccptual fiaws in Euclidean path integrals
or instanton approximations as point sources of anguish and regret. Hopcfully the
loss of coherence will not be too great.
7
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II. THE BOSONIC \VORMHOI E
To warm up for the supergravity case, we will dctermine the ('ffective action
incorporating worrnholrs for the simpk'r system of a massless, complex scalar field
couplcd to Einstein glavity. In Euclidcan
~pé1(·C. we
assume the Lagrangian
Lo = -Jij(U~R - \1 ,1 4>\1 /1 4>*)
(1)
- JgM;, R
where cP
= jc °1V2, MI'
1
+ ~Jg(\1,d\,JI j + j2\1 ,I
'OL
0\1 J
is the Planck mass t and li is t1}(' Hirci scalar.
Our initial task will he
(0
find a wor:llhok. If wc
four-space integral of t.!w above Lagrangian
\V('
~illlply
look for ext.rema of the
will dis«)\,p\ that nOI1<> of th<>m can be
worrnholC'sU (lIlllesf> \ve allow th<, 0 fi('ld to })(' imagmary). 'l'hi" fol'(,('s
ilS
to go back
1,0
t.he fnndamental, Ilamlltonian form of the path integral, from which the' Lagrangian
form is dcriv{'d.
Suppressing the' gravitational sector to fit, tll<' formula on tht' page, the fundamental form of the path integral is
whcrc
7r
and Q arc the mOP1cnta conjugate
Hamiltonian dCI1Slty. WC' have set Il cqllal
1,0
f
1,0 OIJ('
and O. 1'<>sppctivcly, and 1i is the
and Wick rot.ated 1 -+ -ir.
\~ 1
and
\l1 F arc wavc functionah.. fol' the init.ial and findl q1lant.llm states, which wc assume
to be standard
Q). IIcreaftcr
~-i-particl(' ~t,at('s
(and Ilf'nce
e'ig(,lI~tafco~
of th(' total chargc' op('rat.or
we will omit \II's frolll our fOJ'mulae, but th('y arr undC'rstood to he
lO-ïkg ~ 10 17 TcV. wherC' r; is N(~wtoll's constant, and the
converfiion to TeV a~sulTlcs 1/ = C = 1.
Afp
== (167l'C,')-~
= !.G ><
8
(
implicitly present. Sincc 1i is quadratic in the momcnta, the mornentum sect or of the
path intf'gral can \)(' cOllyerted il1to a Gaussian forl11. Tllls 1I1tegral can he perforrncd
to yicld a constant (of an arbitrarily larg(' kind) that i~ ahsod)('c! in th(' Ilorrnalization
of the whole integraJ. \Vhat 15 left is the usual pat hint egral in\'oh'ing only t.JlC' fields
and the Lagrangian. Nothillg forces us to do this Gau5sian integral, ho\V('v('r; in this
case, we will choose not to do it for some modes of the 0 fi('ld and thelr conjugate
momenta.
We will look fol' a saddle-point having central (thrce-splwrical) symnH'try, and so
to evaluate the fUJlctional integral we ,l'ill male the radial ('o-orclinat(' r discrde, and
expand ail fi('lds al. each value of
l'
in series of orthonormal angular modes. Thus the
osC'ctor of OUI' path intC'gral will look like
II JIl dakl71,(r = n~r)
00
c
n=l
Cdr
Jd n A(O=Lum aklt1l(n~r)rUm(n)),
3
(3)
klm
where we understand that ~r -+ O. The
f
and Yll/J intc$!;rat :f)ns and the momentum
sector are of similar form, (A == 11. - i1r\1 rf - iQ\1 1'0, and is introduced only for
brevity.)
Our next stcp will be to split up the four-space integral in (3) into a large outel
region and an inner bail of radius
1'0,
where 1'0 is very much sm aller than any observable
lcngth scale. W(' will illtegrate away the mOllwnta just as usual in the large, outer
region, to obtain an ordinary, Lagrallgian pat h integral whose action stops short at
r = ra. VVe will also expand the gravitational sector of this integral around the
saddlepoint of flat spac(\ and suppress tlw frce gravitons to conccntrate on tllC' scalar
field. Terms
111
which fluctuations of the metric (ê,bout any saddlcpoint, not jllst fiat
9
(
1
space) couple to matter fields are suppresscd by inverse powers of the Planck mass.
We will c:all this portion of the path intcgral
BACKGROUND
=
J
VIVO e -
1ro r
00
3
3
drd nCo(J,O).
(4)
The other part will hr callcd "\VORMIIOLE", so thaL
(Fie -
JdrIll!) = BACKGROUND
X
WORMHOLE.
For this part of the path integral, we will integrate out aB the momenta as usual
exc;,~pt
for the momentum conjllgatc ta a
= aooo,
the angular mode of 0 that is
the constant mode. This particnlar momcntum is t.he totcll charge' Q. WC will now
suppress Crom our formlliac aIl the highcr angular modes, in orde'r to sr arch for a
stationary point of the path integral that Îs ccnt.rally symmctric. This is of course
consistent, bec alise aU snch modes enter the path i ntcgral quadratically, and sa a
centl'ally symmctl'ic extremum is stationary against ::tngular variations. The "strippcd
where wc have assufficd that on the
1'0
scale the background 0
i8
very dose tü constant.
We can now do the a intcgration, to obtain a D-function which forces Q tü he constant
for aIl r
<
To. This ma'<es the
Q int'~gration trivial, lcaving
lis with a path intcgral
whose centrally symmctric sector is simply
WORMHOLE ~
é'~'QO(,o)
.
J
VYfwVfe
-471'2 rro dl'
J
R+
(_ M p-lii,fi,
2
1\7
\Ir
,/ f
+l
2
2) ,
J'g~' ft
(6 )
10
(
where we have included the metric sector for the purpose of finding a wormhole
saddlepoint.
Q is now merely a constant parameter within r = ra, the (conserved) charge
flowing into the wormhole. The constant angular mode a(r < ra) has di'3n.ppeared as
a degree of freedom, which is all right, since it was never a physical degree of freedom
at aU, but a cyclic variable.
We have replaced the familiar Lagrangian with a function known to classical
mechanics as the Routhian, formed by turning
and adding a surface term. It should he emphasized that this procedure is perfectly
legitimate despite the fact that it is not what is usually done with a path integral.
(
It is also worth noting at this point that this treatment is actually equivalent to an
-:,pproach in which one allows the cyclic mode of () to he imaginary,9,17 on the grounds
that the fields in the path integral are ultimately operators on quantum states, and
may have real or imaginary eigenvalues depending on the kind of states hetween
which one is computing the tunneling amplitude. (For tunneling hetween states of
real Lorentzian :-harge we require imaginary Euclidean charge in the Euclidean path
integral.) The argument we are ff)llowing rather sweeps the issue of quantum states
un der the rug, but it is of intercst as an alternative derivation based on the path
integral itself. It is perhaps also easier to digest for any who filld that declaring a
matter
fi~ld
to be imaginary is even more &uspicious than having Wicked time.
We will now find a wormhole saddle-point for our part path integral WORM11
(
1
HOLE, and so justify its prophetie name. We let our Eudidean metric be
where dn~ is the line element on a round unit three-sphere.
Our field equations therefore become
Q2
3
V'r(gr V rf) =
----r-f3
gr
(7)
-4r6 M;G~ = 12M;r\1- g2) =
~: -
(gr 3 '\lrf)2
(8)
-4r6M p2aO
0= M2pr 4( l-g 2 -2grV'rg ) = (gr 3M
Vr f)2
+ Q2
f2'
(9)
Differentiating (8) and applying (7) yields
(r
V l' 4(1 - g2))
= 0,
which irnplies 1 _ g2 =
~: .
(ID)
(This metric is exactly the same as that found by Giddings and Strominger, 6 with a
,~
.H-
!
different matter field.)
Evidently r is restricted to the range L < r < ro. The apparent singularity in
the metric at r = L may be removed by changing to co-ordinates y = arecos(L 2 /r 2),
i
which rang(~ from y = ~ at r =
00,
decreasing with r through y = 0 at r = L, to
y = -~ at another asymptotically fiat infinity on the other si de of what is clearly a
1
1t
wormhole. Since the volume of a three-sphere of constant r deereases from infinity
= ~ to a minimum of 27r 2L3 at y ::-: 0 and then increases to infinity again at
y = - ~, we ean see that L is the radius of the wormhole. Sinee the r co-ordinates
at y
l
t
~
t
match asymptotically onto flat-space polar co-ordinates we will retain them, but
!
f,
the complete wormhole is covered by two identical co-ordinate patches with r ± both
1
ranging from L to ro.
~
12
te.
,...
;.
c
Putting our wormhole metric (10) into (8) yields
This can be integrated by re-arranging it as
ta reveal
for any real constant C. Defining
x
=6M;L (arccos( 2") +C)
r
= 6M 2L2
p
(
L2
2
=
('I-r +C 2
6M 2L2 (
p
-L2
r;
L 4)
+0(-)
r+
2
-'2 + c + Lr:
'Ir
on one side of the throat
+ O(rL_) 4)
on the other side.
we therefore have
(11)
At this point we can recall (9) and note that it is satisfied by the solutions in (10)
and (11). The remaining components of the Einstein equations aU vanish triviaUy.
We have now to choose boundary conditions to fix our constants of integration L
and C by matching
f
to the two BACKGROUND values
we will neglect all powers of
J±
at r ± == ra· In doing so
.g.,
sinee this quantity can easily be of order 10- 30 for L
ra
near the Planck length. We will, however, keep terms up ta second order in the ratio
of the boundary values of f to the Planck mass, beeause wc will need them to Clbtain
13
(
non-trivial results in Section III.
(12)
where in the second line we drop terms of or der ~. We therefore fix C by requiring
ro
(13)
We also have
which with (13) implies
(14)
(For small Q, (14) may be pushing L too close to the Planck scale for comfort,
but we will continue undaunted anyway. The only alternative is to take up string
theory.)
We can now evaluate the WORMHOLE sector of the path integral with tree-Ievel
accuracy by ignoring fluctuations around the wormhole saddlepoint. We will calI
this contribution Ro, because it is Ilot an action but the integral of a Routhian, and
14
(
,
"
(15)
where the limits of integration
+ and
- stand for T± = ro. We have neglected
gravitational boundary terms that are of order (L/ro)4.
We choose Q to be positive at this point without loss of generality, because if it
were negative, Q ± x would simply interchange and the two ends of the worrnhole be
switched.
(
Using
we can approximate as follows:
211'2Q ln (
(Q
= - 471'2Qln
(Q + x )2)
+ x)(Q - x) r+=ro
f+
(j+3M ) + 411'2Q ( )3Mp 1l'
and similarly, with
~ 211'2Q ln (3M;1I'22+ 2f~)
+ and
)2,
p 7l'
- interchanged, for the other end's term. We therefore
have
~ '" -ln (
471'2Q -
f+
f _ ) + ( f+
V311' Mp y!37r Mp
y'371' M p
15
(
) 2+(
f_ ) 2
y37r Pr/p
(16)
Thus the total path integral including the tree-Ievel effects of a single wormhole is,
taking subscripts
±
to mean evaluation at r ±
= ro,
BACKGROUND x e-'Roé i 1l"2QO+e- 4i1l"2QO=BACKGROUND x Aq 4>t ,p:q e-!Aq(f~+f:)
(17)
~BACKGROUND x Aq 4>t(1 - !AqJ.~) ,p:q(l - ~Aqf~) ,
If we assume that the threc-spheres r ± = ro can be taken as the "effective points"
x~ in Cartesian co-ordinates, then the effective path integral for a single wormhole is
It is quite evident that translations of x~ are zero modes of this system, and so
will be integrated over upon evaluating quadratic fluctuations in the saddlepoint
approximation. We introduce an unknown normalization constant
B;, which contains
the 1-loop determlnant for a wormhole of charge q, and which we assume has the
correct rnass dimension of 4 to make path integral dimensionless.
(Fle- JHdrlI) =
xB;
f VfVOe- J
4
d xlo
J
d4x+ Aq/24>q(l- Aq</J*4»
J
tx_ Aq/2</J*q(1 - Aq,p*<jJ).
A further improvement we can makc in our result is to indude configurations with
sorne number N of worrnholes just like the one we have becn studying. As long as
their ends are aH separated by at least ro from each other, we can simply add thcir
action contributions together. When we integratc over aU their possible locations
we are actually counting sorne arrangements in which thcy are doser than this, but
16
. ' ...."... -
(
>l'r,--··-~~
····v ...
U.~';:C~T
.•.• -i •
•
_~._-
-_."t' ... _.
~._
....... - .. _ .. _-
sinee ro is very smaU, this discrepancy can safely be neglected. Of course, the path
integral contribution for each N-worrnhole configuration must be divided by N! to
make up for the zero mode integration having counted separately the N! identical
configurations where the wormhole3 are rnerely permuted. Thus the effective path
We can add up aIl such contributions from N =0 to
00
to get an exponential. We
must also surn over an positive q to count aU different types of worrnholes. Since 0 is
an angular variable, its constant mode a takes values on a circle and the conjugate
mornentum q is quantized. Letting Q be a canonical rnornenturn operator again
instead of a constant, we have
[O(r, n), Q(r, n')]
= inÔ 3(n -
n'),
where the operators are evaluated on equal-r three-spheres of the skeletonization
given by (3). We can integrate over both ns to obtain [a, q]
eigenvalues q
= nn for integers n;
= in, which
makes the
q is like the momentum of a particle in a box of
length 271", because a is periodic.
Thus we finally obtain
(Fie - J H drlI) =
X
exp
J f'D9 eJ
'D
4
d
x.c O
L [j d'x+ Jd'x_ B~[Aq/2".q(l- Aq~'''')lx+ [AQ/2".,q(l- Aq""~)lx-l '
q>O
(18)
This result is essentially the same as that of Reference [9], except that we show
the effects of both ends of the worrnhole explicitly, and we go to higher order in
17
(
r
1
inverse Planck rnass. We have a bi-local expression. We can make it into a local
operator, however, by introclucing a Gaussiar. integral over complex dummy variables
0,
following Reference [18]. The last exponential in (18) is equal to
fd
fI
q>O
~
2
Qq
e-O';O'q e-Bq
f d x[O'qAQ/24>'Q(1-AqI/J-I/J)+0';Aq/2I/JQ(1-AqI/J-I/J)] .
4
7r
(19)
This gives us an effective Lagrangian which is local and real, but which includes sorne
new
QQ
pararneters that may be thought of as creation/annihilation operators for
baby universes earrying global charge q.
The wormhole throat may be regarded as the "history" of a small, closed "baby
uni verse" which pinches off from a large, fiat parent unÎverse. The other side of the
worrnhole may be another parent uni verse which the baby universe breaks into, or it
may be a distant region of the original parent uni verse wllich the baby universe rejoins.
One can even eut the wormhole at its narrowest part and have the baby universe
remain separate from any parent. The details of these geometries are unimportant in
the low-energy effective Lagrangian pieturc, wherc wc are concerncd with tunneling
between states defined in one large parent uni verse. We will thereforc consider (19)
to give a low energy effective potential which breaks the global U(1) symmetry of the
original Lagrangian, but whose coupling constant depends on baby uni verse dynamics.
Note that in the limit where M p -+
00,
which is implied by our decoupling of
gravit y in the BACKGROUND, A vanishes. Thus our result l'cally presupposes that
if wc take l'vIp to be large we must also allow
Qq
to be large
50
that B qAqj2 Qq remains
finite. Even in this limit, the A<jJ*<jJ factor in (19) becomes vanishingly small, but we
retain it because it will be important in Section III. This concludes our study of the
bosonic wormhole.
18
III. THE SUPERWORMHOLE
(
We will now attack wormholes and supersymmetry. To do this, we will take
the sirnplest supersymmetric matter multiplet, and couple it to supergravity. At
first we will simply study supersymmetric Lagrangians; when wc have built up the
necessary background of formalism, we will insert a wormhole into the discussion. For
a thorough treatment of supergravity, the rcader may consult such authors as Wess
and Bagger 19 or van Nieuwenhuizen 20. A brief summary of supersymmetry appears
in the Appendix. Before we bring in supergravity, let us examine a supersymmetric
model in fiat Lorentzian spacetime.
The simplest supersymmetric system is the massless "scalar multiplet" consisting
of a real scalar A, a real pseudo-scalar B, and a spinor X. The scalar and pseudo-
«
scalar are grouped togcther into a complex field 4J
= (A + iB)fv'2.
The simplest
Lagrangian involving these fields is
(20)
where
$ == '''(l'Sjl
for any vector Sil' We take the trace +2 metric. t
The Lagrangian (20) is invariant up to a divergence under the supersymmetry
t One rnay, for example, choose the spinor representation of the "'( matrices:
where k E {l, 2, 3}. We define the chiral projection operators PL =
PR == !(1- "'(5)'
19
(
Hl + "Y5) and
l
transformation
é<p = ..,fi'fPLX;
(22)
6x =
V2( PL 11 cP + PR YI <P*)f,
where fis a constant, infinitesimal, Grassman, Majorana spinor with mass dimension
-~. It is to be noted that these transformations only satisfy the proper supersymmetry commutaiion relations (see Appendix) if wc employ the equations of motion.
This is a general feature of supersymmetric theories. A thcory with a symmet,ry relating bosons and fermions that does not satisfy the supcrsymmetry CRs is known by
Reference [16] to violate such basic principlcs as causality and locality, so satisfying
them is important.
This will pose a special problem for us, however, in that we plan ultimately to construct an effective Lagrangian in which wormhole terms are added to (20). We expect
the low-energy action to he supersymmetric, inasmuch as supergravity is a gauged
symmetry and thus generically unhroken hy wormholes. ll Since the equations of
motion will change in going to the effective theory, though, the supersymmetry transformations may differ from (22). Finding the proper supersymmetry transformations
could mean sorne awkward gucssing.
Fortunatcly there is an elegant way out of this difficulty. Wc can add to our
system an extra, non-physical complex field F whose sole purpose is to keep track of
the Lagrangian-dependence of the supersymmetry. The Lagrangian becomes
Co = -\1 JJc/>\1 J'<p* 20
~xYlx + p* F .
(23)
c
We change the transformation rules to
6ifJ = JëÙPLXi
6X = V2PdYlifJ + F)f+ Vi,PRCYlifJ* + F*)fi
6F
(24)
= V2tCYl PLX .
F is obviously not a physical degree of freedomi it can be rcmoved entirely from a
path integral bya Gaussian integration, and it does not propagate. Now, though, the
transformations (24) represent supersymmetry regardless of the equations of motion
for <p,
x,
and F. We need only find an effective Lagrangian involving <p,
x,
and F
that is invariant under (2·1) to have, and know we have, a supersymmetl'ic effective
Lagrangian.
Another convenient technique we can use is to construct supcrspacc. (Again, see
the Appendix for more ddails.) We define the scalar (or ch1ran supcrfield
(
whereTJ is a constant, Majorana, Grassman spinor of mass dimension -~.
This superfield is to be thought of as a function defincd on Sl1perspace, the union
of ordinary four-space and the Grassman four-space spanned by "1. Since aIl powers
of Majorana Grassman 4-spinors higher than the fourth vanish identically, the expansion in "1 of (2.5) completely defines 4>. Our supersymmetry transformation (24)
corresponds to
A property of cI> that will be very convenient in our superwormhole argument is that it
is equal to the resuIt of a finile supersymmetry transformation of parameter "1 acting
on <p.
21
(
We can defme ~* to he just the complex conjugate of~. We may then note that
t
Thus its ".,4 componer,t is four times the Lagrangian (23), up to a divergence. Using
the convention thdt Grassman integration oheys
== a, for a and b
ordinary numhers, we can therefore write
(26)
In fact, any function of
~
and
~*
provides a superspace Lagrangian wHrh is !'uper-
symmetric, although not necessarily one having canonica} kim'tic terrns.
We can now indicate why it was worth taking the trouble in Section II to extend
the 4>q result of Reference [9] to the next higher order in cP. The gencralization to
.'
superspace of the Coleman- Lee effective vertex would he 4>q. This vertex is itself a
scalar superfield, and by comparison with (25) we see that its
.,,4 component is just a
total derivative.
Wc will now start our analysis by coupling our supcrsymmctric matter to supergravity, which is the gauge theory of local supcrsymmctry. (Wc cannot combine
supersymmetry and gravit y wi1,hout snpcrgravi1,y; s('c 1,h(' Appendix for why this is
so.) If we restrict oursclves to the (0+,0-., ~) paI ticl<> content of our simple example
r
If
above, we find that the simplest locally supcrsymmctric Lagrangian is th(' special
.
,,'
"
l d11i(a."i + b)
ca~e !(
i'
1.
= q>*<I> of the non-linear sigma model discussed by Baggcr in Rcferen-:e [21].
~
,~
ï
(This case is simplest in that it is the unique Lagrangian that yiclds canonÎcal killetic
\\
22
,
.
~
.
!.
~,#"
~
.....
'1.,'It~-~J.".:,.~-
....
~
,,~......
"'I·· .... · , - , ·..·,·f'\.... · i - · o
..
~;"
..
. . . ,~
-
--.,.~.
terms.) Thus, in Lorent zian spacetirne, we take
(
J:,
=
2
Jl *
1
'dJ
1 ./.
IlVpU
-NlpygR - ..;9\1,,4>\1 </> - "2..;9X", X - 2'PJl IS IV \1 plf;(ff.
+ 16~~ (4)*\1 Jlt/> -
</J'V Jl4>*)( lf;vIP1f-'UfJlVPU - ygX/sIJlX.)
(27)
1 .;;;_ Jl v
+ 2Afp v gx./
1
/
-
*
\1 v(PL</>
+ 128Af~ X/5/U '( (4lP Jl /vlPpf 'lVPU -
+ PR 4> )lPJl
u
Jl
4..;9lP Il/5/u lP - \, gX/51 X),
where lPJl is the gravltl110, a Majorana ~pinor-vector that is the gauge field for local
supersymmetry, and 9 ::::: - det gJtV. F, as well as the auxiliary fields required by the
graviton-gravitino multiplet. have been eliminatcd by Gaussian integration.
This Lagrangian is invariant -
up to a total derivative -
undcr the following
supergravity transformations:
8t/>
= V2fPL'(;
(
V2(\1 Jl4>PL
-
+ \1,,4>* PRh"f. + 4~M~ (X(4>* PL -
1
y'2Af (f/Jl'I/Jv
p
q.PR)f.h5X
(28)
+ flvlPl');
1
= 2V2Mp \1 Jlc - 4J2M~ (\.( 4>* PL - 4>PR)f.h51f-'1l
+ 2~Afp (~0"1Iv(:\/51v\) + (</>*'\7 ,1 </> for
*
f
an infinitesimal Grassman Majürana spinor field. *
These transformation laws are more complicatcd than thüse obtained by just addin~
gravitino terms tü (24). This is because sorne field redefinitions have been done. 2
The original field, übeyed much simpler transformation rules, Lut the kinetic terms
in the Lagrangian were not of canonical fonn in those fields.
23
(
</>\1,,</>*)h5f.,
t
Our system is invariant under the global U(l) trah30rmation
(29)
for a an ordinary real number. This symmetry provides a generalization of the charge
that was
50
important in Section II, most easily expressed through a newly defined ()
field. Let
<P
=
X
We choose
f
~feif)
= é'Y5 0À.
to be real, but this stillleaves À as an unconstrained Majorana spinor.
We then have the conserved current
",
which appears in the re-written Lagrangian
2
C = -Mpy'gR -
1
, , 1 -m
2y'gV,,!V
f - 2V9 )..rv>"
1-
- 2~,,'5,"'Vp~O'f
""PO'
1
+ 2V9!2\1 pO"V"O + j 1L "V Jl O
+ 2~Mp JgX,Jl'"~/l 'V v !
+
(30)
1 2X'5/(1'À(4~Jl,"~p()l/'P(1' -4y'g"p1L I51 (1'"p" - J9X'sÎ''T)..).
128Mp
Now to deal with wormholes in our supcrsymmetric theory, one must corne to
grips with the euclideanization of this action. White this is a trivial exercise for the
!
t
purely bosonic part of the action, it is somewhat morc subtle for the fermions because
,~
,t
1
,~
there are no Majorana spinors in four Euclidean dimensions. This prohiem has heen
resolved 23 ,24,25. The essentia.l point is to surrendcr the usuai definition of adjoint
r
~,
24
,
spinors and use instead Majorana conjugation23 . We leave this construction and the
details of the supersymmetry zero modes (see below) to elsewhere26 . We will be able
to obtain our result with minimal calculation, by relying on supersymmetry itself to
give us a supersymmetric generalization of the bosonic result that we already have.
It would be straightforward to define the Routhian analogous to that of Section
II. Since both the charge and the Lagrangian (30) reduce to those of Section II when
aIl fermion fields vanish, so the Routhian must also be identical in the absence of
fermions. AIso, sinee the fermions must be introduced in bilinears of spinor fields, it
is c1ear that a bosonic saddlepoint is stationary against fermionic variations. Con sequently we know that the bosonic wormhole solution of Section II is a saddlepoint of
the supergravity Routhian in Euc1idean space. Having determined that a wormhole
instanton exists for our supergravity system, we will eonstruct an effpctive Lagrangian
(.
incorporating a vertex representing the wormhole. We will do the same things we did
in Section II, namely patch a WORMHOLE saddlepoint eut off at a radius ra into
a fiat BACKGROUND. Eq. (28) describes our WORMHOLE sector; we should now
consider the BACKGROUND.
As in Section II, we want to expand around a fiat space saddlepoint and concentrate only on the matter sector, while disregarding the supergravity fields. We
cannot, however, simply follow Section II in taking as BACKGROUND Lagrangian
the full supergravity Lagrangian (27) in the limit where M p becomes infinite. That
limit yields the simple Lagrangian (20), but it requires disearding terms that will be
of interest. To be precise, if we take the limit M p
~ 00
we would find that the lowest
order U(l) symmetry-breaking effective wormhole vertex vanished. So instead we
25
(
1
return to the superspace matter action that appears in the construction of (27) and
for the present ignore the supergravity sector for simplicity. This leaves the globally
supersymmetric matter action
(31)
This is the special case !(
= <b*~ of the general sigma model obtained in Reference
[21]. Note that after the Grassman integration has been done and the auxiliary field
F eliminated, this action has non-renormalizable vertices and non-canonical kinetic
terms:
Cu
=e-~t:f ((1- 6A1
</J*~)(-'\lIl</>\1Il</>* p
!xYlX)
2
1 (2
</J*</»- Il (,.I.*V ,.1.)
24M2p - 6A12p X,5, X '+' Il'+'
il
_ (XX)22 - 3MJ,
48Mj
It is only in the limit that M p -+
~
+ 36M~)
(1-~)
6Mp
00
(32)
.
that one recovers (20). It is also only in this
limit that we can really say that supergravity is decoupled From the matter. Since
as we have explaiüed we do not wish to take this limit, we must accept that the
supergravity sector needs to be included in a full analysis of wormhole effeds, which
may modify our final result by the addition of sorne new interactions. For simplicity
we will nevertheless continue to ignore it here, and rclegate the full analysis to a
future investigation 26 . In the meantime, we may seem somewhat inconsistent in
discarding sorne terrns suppressed by inverse Planck masses, but keeping others. Our
justification will be that the terms we keep will be leading order symmetry breaking
terrns, whose presence is significant even when they are very small.
26
...
The fermion in (32) is related to that of (27) by a field redefinition involving the
scalar and the gravitino; the (fixed) metric is related to that of (27) by a Weyl rescaling
involving the scalar field. We emphasize, however, that while the field redefinitions
used to obtain (27) from coupling (31) to supergravity rescaled the metric and altered
):, they did not affect the scalar field
4>. This will be important, because as we
know from :'ection II, the bosonic wormhole saddlepoint of (28) will yield an effective
vertex that is a function of the scalar field. Thus we will be able to compute the
WORMHOLE contribution to the path integral using the redefined fields of (28), and
then interpret the result as a function of the BACKGROUND scalar field.
Supergravity being the gauge theory of supersymmetry (as weIl as diffeomorphisms), one must fix the gauge when evaluating a path integral.
A eonvenient
convention is fix local supersymmetry transformations by imposing ,ll'IjJJl
(
= 0, for
this makes the gravitino a pure spin-3j2 field just as we make gravitons and photons
transverse. As with diffeomorphisrns, however, there remains sorne residual supersymmetry after gauge-fixing. In the case of an asymptotically fiat space sueh as our
wormhole configuration, this residual symmetry does not vanish at infinity, but tends
to a rigid, global supersyrnmetry. Such a symmetry constitutes a zero mode of a
wormhole instanton: the path integral is invariant along the subspace of field configuration space spanned by the symmetry gcnerators. Just as the translational zero
modes of the bosonic worrnhole in Section II led to an integration over position of the
effective wormhole operator, so the translational and supersymmetric zero modes of
the supergravity wonnhole lead to an integration üver superspace "position" of the
superwormhole effective operator. Just as we had independent four-spaces of transla-
27
(
tional modes for each end of the wormhole in Section II, so additionally we have four
independent supersymmetry modes for each end of the superwormhole26 .
In the no-fermion limit, the superwormhole is identical to the bosonic wormhole
of Section II, and so yields an effective operator
plus a complex conjugate term for the other worrnhole end.
(Recall that A -
2/{37r 2 M'fi) .) This expression is iden!ically equal to the superfield operator
Because of the fad (noted ab ove ) that a superfield is equal to the value of the scalar
field after a fini te supersymmetry transformation of measure
7],
we can easily write
the effective operator for the superworrnholes related to the bosonic one by a global
......
supersyrnmetry transformation as
Since translation in Tl is a zero mode of the path integral, as wi th ordinary translation in xJ.l in Section II (and here) wc must intcgrate over it to obtain tree-Ievel
accuracy in our saddlepoint approximation. Thus the one-superwormhole end effective operator looks just like a superspacc action term
Bq Aq/2
J
aIxd4ry
~q+l-r>*,
where we have dropued the ~q term because it gives the total derivative D~q after
Grassman integratlon, and adjusted the power of A to give the correct mass dimension
,
28
(
of 2 for a superspace Lagrangian. (That is, we assume that the zero mode functional
integrations involve a normalization constant of the correct mass dimension for Bq
to be dimensionless.) Apart from the suppression of the leading order term, we have
merely translated our bosonic result into superspace in the most obvious way.
Adding the complex conjugate term for the other wormhole end -
with its own
four independent supersymmetry zero modes, summing over all N-superwormhole
configurations to obtain an exponential, summing over positive q, and introducing
dummy a parameters to obtain a local action aIl proceed exactly as in Section II. We
obtain
as the tree-level action incorporating the effects of superwormholes. Doing the Grassman integration, and defining
V(4),4>*)
= L:B Aq/2[ a;4>q+l4>* + a 4>*q+l</> 1
q
q
po
dV
DV
=d~
we obtain
Ce!! = D* D V( -4V7 p4>\1I'4>* - 2f1Jx +4F* F)
+X7s,PX(D* D2V V7 ,,1> - D*2 D V V,,</>*)
- 2( D*2 D V F >:PRX + D* D 2 V F* XPLX) + ~ D*2 D2 V (XX)2.
We can finally eliminate F by using its non-propagating field equation,
D*D 2V
"2 XPLX D*DV .
1
F=
29
(
.-
•
This gives the ultimate result
Ce!! =D* D V
(-4\7 1'<p'VJJ~* - 2XWX)
+Xfs"I,JX(D* D2 V \7,,<P - D*2 D V \71'~*)
+ !.2 (D*2 D
2 V _
2
ID* D V1
D*DV
2
)
(34)
(ï )2
XX·
Since this action was written in superspace, it is manifestly supersymmetric; using
the F equation of motion above we can write its supersymmetry in ternis of physical
fields only as
(35)
To see the low-energy behaviour of this system, we let the matter fields be small in
comparison with the Planck seale. We will look for the lowest order U (1) symmetrybreaking term. To find it, we will need to keep wormhole tenns having co-efficients as
small as M;3. The justification for keeping these terms but suppressing others of similar dimension that do not carry U ( 1) charge is that violation of charge conservation
is a drastic effect significant even if highly suppressed. We therefore approximate
CeJ/
~-
\1 /J~\7I'~* 3
+L
~x~x
Bq(q + 1)Aqj2 ((a:~q
+ aq~*q)( -4\7 p~'\ll'~* -
2X'X)
(36)
q=l
+ X"l5~JI.)(\7/J(Q:~q -aq~*q))
J2B
*
*)( -)2 O( 1 )
+ 3V3Mft7l'
(al ~ + al~ XX + Mi .
1
To inter~ ret this Lagrangian we need to make its kinetic terms canonical, by
30
defining
3
cp
=4> +4 L BqAq/2 o ; 4>q+l
q=l
tP =[1 +4
3
L Bq(q + I)AQ/2(a;4>QPL + aq4>*Qp )]X.
R
Q=l
This field redefinition does not affect the S-matrix in perturbative expansion in
0"5
because it is invertible to any given order in a's. Eq. (36) then becomes, to linear
order in a's, and keeping only lowest dimension terms,
(37)
Thus we ot~ain a four-fermion vertex coupled to baby uni verse creation/annihilation
operators which carry off or inject one unit of scalar field charge. This is what one
would expect an instanton with four fermionic zero modes to reduce to in the lowenergy limit, because those zero modes will make all Feynman diagrams without four
(
external fermionic Hnes vanish in the presence of a wormhole. Note that while the
new vertex of (37) does break the scalar field's U(l) symmetry, it does not break
supersymmetry. As we know from (35) above, the full theory is supersymmetric
when we include aIl of the higher order terms.
31
-------""-----------------
IV. CONCLUSION
1
We have found that, to tree level accuracy, the effect of wormholes in a supergravity system is to add a new operator which is a local operator in superspace, and is in
fact a simple generalization to superspace of a purely hosonic expression. The leading bosonic term found hy Coleman and Lee9 is cancelled by supersymmetry. This
is an example of supersymmetric suppression of worrnhole-induced scalar vertices as
suggested by Hawking 10 . There was no new dynamics in our derivation. Our answer
follows from the bosonic result hy transformation properties alone.
How much one should believe in this answer? Right now, the only voice that can
he heard on this issue is that of aesthetics. Insofar as our analysis has been smooth
and "inevitable," it is another piece on the pile of evidence that wormholes should be
part of nature because they are so nice. Insofar as we have had to beg sorne serious
questions, we remain open to the criticism that wormholes are only nice because we
make them that way.
The effective operator that wc have found cannot be use"" to break supel'symmetry
spontaneously. The potential term in (36) - the term left when aIl fields are set to
be constant - has a minimum value of zero; this means that supersymmetry remains
unbroken. (See the Appendix for an explanation of why this is so.)
Our result could be improved by computing the one-Ioop tcrm found by doing the
Gaussian integral of the wormhole action to second order in fluctuations about the
saddlepoint. Unless we believe with IIawking that quantum gravity is fundamentally
Euclidean, going any further than this would he to exceec1 the limits of validity of the
32
tunneling approximation that put us into Euclidean space in the first place, and
50
would probably be self-defeating.
One could also improve the dilute gas approximation by considering the classical
interactions between worrnholes. Sinee on shell the scalar field falls of as 1/r3 near
one of our wormhole points, and sinee the scalar field action is just like that for an
electromagnetic potential, we can see that the classical interactions turn our dilute
gas into a Coulomb gas in four-space. The Coulomb gas partition function can thus
he used to improve our accuracy.27 Note, however, that this improvement is only
the same improvement made when calculating quantum corrections to any classical
Lagrangian. As a tool for ohtaining Feynman rules, our Lagrangian needs no Coulomb
gas corrections.
Thus our result is a tree-Ievel result in two senses. We have neglccted fluctuation
(
effects in the quantum gravity saddlepoint approximation, and we have neglected
loops between effective wormhole vertices. Improvements could be made in both
areas, in tedious but straightforward ways.
A still harder task would be to examine non-centrally-symmetric wormholes. In
our superwormhole, based as it is on the centrally symmetric wormhole presented in
Section II, the fermion never carries any of the U(l) charge supporting the wormhole.
This possibility would be a new feature of non-spherical wormholes.
If what one most wants out of superwormholes is a method for breaking supersymrnetry spontaneously, however, it is unlikely that any amount of accuracy will satisfy
the requirement. This is because superwormholes, even non-spherical ones, have four
supersymmetry zero modes. Consequently the effective operators are generically "D-
33
(
terms" -
'1 4 terms like our result. D-terms are kinetic terms; they do not give rise to
non-zero vacuum energies. To break Ilupersymmetry, an "F-term" is needed. F-terms
are
.,,2
terms; to obtain one from a superwormhole we would need somehow to kill
off two zero modes. In a string motivated model, Park, Srednicki and Strominger are
able to do just this. 8 It would be interesting to determine what general features of
their modellead to this result.
34
-
.
_
-
-
_
.
_
-
-
-
-
_
.
_
-
-
~
-
____
-
__
-
w._ _ _ _ _
~
___.,,_r,,._..
~
.
_
.
h
_
~
Q
I
I
_
~
_
"
"-'-__
..• - ..
_.
APPENDIX: REVIEW OF
(
SUPERSYMMETRY, -SPACE, AND -GRAVITY
The following very brief introduction to supersymmetry presents knowledge gained
largely from a course taught by C.P. Burgess at McGill in 1989. Reference [19] provides a concise introduction to supersymmetry and supergravity in the eit::gûl1t superspace approach. It uses Weyl spinors. Reference [20] is an exhaustive review of
supergravity, presenting many formalisms.
Supersymmetry is a symmetry relating bosons and fermions. SuSy (for short) is
therefore generated by fermionic operators, which obey anti-eommutation relations.
Grassrnan algebra is therefore inevitable, sinee Grassman variables are numbers defined to anti-eornmute with each other and with any fermionic field.
SuSy is a quite specifie fermionic transformation. A vt'ry powerful theorem (Reference [16]) states that, under sorne very mild assurnptions - such as causality, locality
etc. - the most general symrnetry group of a field theory in fiat space is
(Poincaré) x (SuSy) )( (internaI symmetries),
where (Poincaré) x (SuSy) has the following (anti- )eomrnutation structure, using
[A,B]±
= AB± BA.
[JIlV, J~pl- = -i9JJpJv>.
[JIlV 1 P~]-
[PI" Pv ]-
= i(gll>'Pv -
+ permutations
[JJJv, Q]-
~
= 4bll ' ,v]-Q
[PJJ , Q]_ = 0
gv>.PJJ)
=0
[Q, Ql+
= -2,11 Pli
Here JIlV are the Lorentz generators, Pp generate translations, and
35
(
(38)
Q are the spinor
------------------
generators of SuSy. (Actually, the Poincaré group is a sub-group of this larger group
usually called the SuSy group, but SuSy is also taken to reCer only to thf' Qs.)
SuSy is evidently more like a
spacet~me
symmetry than it is like an internaI sym-
metry. The fact that two Qs can anti-commute to make a translation is remarkahle,
to say the least.
We can highlight this {act and represent SuSy in a way that gains elegance at
the priee of abstraction hy constructing superspace. Superspace is the union of ordinary four-space and a four-space of Grassman variables
1],
where the
1]'S
form a
Majorana spinor just as :cil form a vector (in fiat space). The superspace co-ordinate
transformation
(39)
represents SuSy.
Fields defined on superspace are called superfields. They can he expanded in a
series of powers of 1] up to the fourth, with the series co-efficients being ordinary fields
on four-space. The general superfield turns out to he a reducible representation of
SuSy. We can constrain the co-efficient fields in three independent ways to ohtain
"scalar superfields" and "vector superfields." (A scalar superfield and its comp;~x
conjugate are distinct SuSy representations; they are also callcd left and right chiral
superfields.)
For example, a complex scalar (ordinary) field, a Majorana spinor,
and a complex auxiliary field are sufficient independent co-efficients to form a scalar
superfield.
The
,,4 components of the product <I>*<I> of any scalar and conjugate scalar super-
field happen to transform into four-spaee divergences under SuSy. Such components
36
(
- known as "D-terms" - are therefore suitable candidates for supersymmetric Lagrangians. Since Grassman integration makes
J d4'17 pick out just such components
of a superfield, we can write supersymmetric actions as integrals over superspace of
Lagrangian superfields. 4>*<1> turns out to give a kinetic term.
The '17 2 components of either a scalar or a conjugate scalar superfieid likewise
transform into 4-divergences, and so these "F-terms" are also suitable for Lagrangian
purposes, although they do not work as ncatly in superspace integrals because they are
"picked out" of the superfields by only two Grassman integrations. It can be shown
that any function of scalar superfields (but not of mixed scalars and scalar conjugates!)
is a scalar superfield. (That is, the co-efficient fields of the product obey the proper
restrictions for a tlcalar superfield.) Consequcntly
J d'l.'17f(4)) and Jd2'17f(<l>*) are also
Lagrangian candidates, and these pl'ovide potentiai terms ("superpotentials").
(
The auxiliary fields that appear as components in superfields are those fields that
turn out to have non-propagating field equations when we Grassman-integrate these
Lagral'lgian superfields into ordinary Lagrangians. They can therefore be eliminated
and replaced by combinations of other fields, and do not represent physicai degrees
of freedom. There are always a few of them around. This is implies that, once we get
back to ordinary space, the exact form of the symmetry transformations on the fields
of a supersymmetric system depends on the Lagrangian of the system. This fact is
a feature of SuSy itself, and not a defect
III
the superspace approach. In fact the
chief virtue of the superspace approach is that we never have to guess what the SuSy
transformation looks like, or whether it exists as a symmetry. The chief drawback of
the superspace formalism is that we only know these things in terms of superfields
37
(
J
which contain auxiliary fields, and eliminating these is usually tedious.
For spontaneous breaking of supersymmCl.ry, we need a vacuum which is not
annihilated by a
Q. The anti-commutator in (38) implies, however, that
L IIQ.IO)11 2 + IIQ710)1I 2 = (OI([Q~, Qd+ + [Q;, Q2]+)10) = 4(0IEI0),
i=1,2
where Qi are the eomponents of the spinor Q and E is the energy operator pO. This
means that SuSy is spontaneous!y brokcn if and only if the vacuum energy is not
zero. Accordingly, if SuSy remains unbroken down to a low energy seale, it would
solve the eosmological constant problem.
Unbroken SuSy also turns out ta imply that bosons and fermions always come in
prurs of equal mass, because [Q, p 21_ = O. Since this is manifestly not the case, SuSy
must either be spontaneously broken at sorne scale, or eise be broken in the strict,
Î<
r
F
,~
't
i
'J-
I
old-fashioned sense of not really working at an. The bad news implieit in the Iast
paragraph is that breaking supersyrnmetry without inducing a cosmological constant
is difficult. It is, however, possible to break supergravity and yet maintain A = 022 .
!
field. The name "supergravity" is justified hecause, sinee [Q, Q]+ '" P, gauging P
1
•
forces Q to be gauged, and vice versa. Supergravity ean be expressed as a geometrie
~
theory of eurved superspacc. Sincc the rigid SuSy transformation (39) is not quite a
,~
rnere translation in superspace, the sllpergravity Wlcuum is not just fiat superspace;
~
r
(
~
:"
,
f
,,
<,"
Supergravity is local supersymmetry. The gravitino, a spin 3/2 field, is the gauge
it has no curvature, but it does have torsion. A further complication that arises is
that general relativity in superspace only reduccs to supergravity after one imposes
sever al constraints on the various fields involved. There is no natural way of deriving
~
38
----------------------------------------
(
these constraints from superspace geometry alone; one has to know the gauge theory of
local supersymmetry and choose those constraints that obtain it. Thus the superspace
formalism betrays itself in the supergravity case as an often convenient construction,
but not a fundamental one.
(
39
(
REFERENCES
1
1. W.K. Clifford, in Lectures and Essays, ed. C. Stephen and F. Pollock (MacMil-
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40
.'
c
14. W.G. Unruh, Phys. Rev. D40, 1063 (1989).
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~O.
(:
P. van Nieuwenhuizen, "Supergravity," Physics Reports 68, 189-398 (1981).
21. J. Bagger, Nu cl. Phys. B211, 303 (1983).
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41
r----------------,------------------.- - -
1
BIBLIOGRAPHY
Hawking, S.W. and W. Israel eds. General Relativity, an Einstein Centenary
Survey (Cambridge University Press; New York, 1979)
Strominger, A. "Baby Universes," in A. Jevicki and C-1. Tan eds. Particles,
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-
WeBB, JuliuB and Bagger, Jonathan.
Supersymmetry and Supergravity. (Princeton
Univer::;ity Press; Princeton NJ,1983)
42
l