IC/79/136

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

SUPERGRAVITY

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

J. Hiederle

1979 MIRAMARE-TRIESTE

IC/7? ' l i ' -

I n t e r n a t i o n a l Atomic Energy Agency

United Uations Educat ional S c i e n t i f i c and Cul tura l Organizat ion

IIITERI.'ATIONAL CZHTRE ?n0R THEORETICAL PHYSICS

SUPERGRAVITY

J. Hiederle ""

Internat ional Centre for Theoret ical Fhys i c s , T r i e s t e , I t a l y .

ABSTRACT

The present s t a t u s and some recent developments of supergravity are

br ie f ly reviewed.

MIRAMARE - TRIESTE

September 1979

* Invited ta lk at the Workshop on Theoretical Problems in Elementary P a r t i c l e

Physics and Quantum Fie ld Theory, Serpukhov, July 1979.

* • Permanent address: Ins t i tu te of Physics , Czechoslovak Academy of Sciences ,

l8o UO Prague 8, Czechoslovakia.

1. INTRODUCTION

Modern physics i s b u i l t on three foundat ion-s tones: specia l

theory of r e l a t i v i t y , quantum mechanics and general theory of

r e l a t i v i t y . The un i f i ca t ion of these basic theor ies has always been

one of the pr inc ipa l aims of t heo re t i c a l physics . There have been

a number of notable successes in the un i f ica t ion of specia l theory

of r e l a t i v i t y and quantum mechanics in the frame wori of quantum

f ie ld theory. Let us r e c a l l tha t predic t ion of quantum e l e c t r o -

dynamics agree with experiments within a few par ts in 100 b i l l i o n !

For ins tance , the l a t e s t t h e o r e t i c a l and experimental values for

the e lec t ron anomalous magnetic moment are f l ] :

theor .a e ~ (

a exper . = (

On the other hand, successful attempts to unify the g rav i t a t i on

theory and the quantum f i e ld theory lay undiscovered for long. The

turning point appeared in 1975 when new tools were discovered which

hopefully not only bring together theory of grav i ty with the pr inciples

of quantum mechanics but a lso c l a r i fy the connection of the g r a v i t a t -

ion i n t e r ac t i on with the other fundamental i n t e r a c t i o n s . Moreover,

there are even hopes tha t the g r a v i t a t i o n , electromagnetic , weak and

strong in t e rac t ions are jus t four aspects of one more fundamental

interaction.

So far as the energy scales are concerned such a full unification

may go through a number of intermediate scales - through the unificat-

ion scale of the electromagnetic and weak interactions around 100 GeV

(i .e . within 10~ - 10 cm) and the unification scale of theIt G

electroweak and s t rong in t e r ac t ions as low as 10 - 10 GeV ( i . e .18 —?0

10 - 10 cm for the Pati-Salam model) or much higher than

-2-

101Q

GeV ( ±. 10 cm) to the unification of all interactions at

GeV (i.e. 10 cm). In this connection let us only note that

in 1961 one expects to reach the energy ICO GeV (c.ir..).

'Aliat are the main tools which allow us to attack the problem of

the unification of the basic interactions? First, it is the gauge

principal and second, it is the structure of Lie auperalgebras.

1.1 The gauge principle

.Roughly speaking, the gauge principle consists of the following

steps:

/ i / I t begins with the theory of matter f i e lds y> (x) invar iant with

respect to global transformations from a siaiple connected Lie

group G , i . e . the corresponding Lagrangian s a t i s f i e s

whenever the fields cotransform under the continuous unitary

representation g t—> T(g) of group Q as

(l.if) y>(x).—> f'U) = T(g) y>(x) .

It is clear that such a theory will not, in general, be

invariant with respect to local transformation g(x) since

/ii/In order to restore the invariance of the theory with respect

to local transformations g(x) , it is sufficient to introduce

the covariant derivative

(1.3) (x))

by '2 = ar.G of the Tau^e potent ia l A^ (x ) , i . e . a vector f i e ld

over spaceuime with values in the Lie algebra of group G on

which group G can therefore act by conjugation

(1.4) A^ (x).-^Ar'(x) = g U H ^ x ) g"1(x) + \ g(x) % g ^ t x ) .

In (1.3) T( )

representa t ion g»~»T(g). Thus making the s u b s t i t u t i o n

( X ) ) is Lhe representa t ive of A (x) in the

the invariance of under g(x) i s maintained.

/iii/ln order to interpret A (x) as a physical field, one has

to add its Icinetic term to Je* in a gauge invariant manner,

e.g. (by means of the inner product (...) in the group apace)

in the form

(1.6) - i (F (x),t,'f

Here, the gauge field (curvature) F ^ x ) is defined by

(1.7) F..., = 0. A.. - -?.. A. + £ r k . A..7

- 3 -

One may conclude that the gauge principle: "replace all deri-

vatives ^, in J^o by covariant derivatives D,, and add a

kinetic term - j ( F ^ , F^w )", represents a systematicfcrocedure

for constructing theories invariant under local transformations.

It is typical of this procedure that it introduces not only gauge

potentials A^ but alao their interactions with themselves (via the

kinetic term) and with the matter field as well (via D^ ).

Remarks:

1/ As is well known, the theory of electromagnetic or electroweak

or strong interactions can be constructed within the framework

- i t -

of gauge theories with gauge group G equal to U(l), SU(2) $ U(l)

and SU , £3), respectively. The corresponding gauge fields are

the fields mediating the interactions.

2/ As pointed out firat by Utiyama [2] in 1956 and then by a number

of physicists (see [3] )| gravitation may be looked upon as a

gauge theory too. However, diverse answers can be found to the

questions: What is the gauge group of gravitation?

What are the gauge potentials?

What is the form of the corresponding Lagrangian?

What about the metric tensor g^v ?

For example, Utigama [2] identified the gauge group with the

Lorentz group and the gauge potentials with the coefficients

of the Riemannian connection of spacetime. Kibble [A] criticized

Utiyama and took the gauge group to be the Poincare' group and

consequently derived field equations of gravitation with spin

and torsion in the Einstein-Cartan theory. Yang [5] proposed

a Lagrangian analogous to (1.6), i.e. quadratic in Riemannian

curvature tensor R^ „ . According to Fairchild [6] this

Lagrangian does not have a correct Newtonian limit. On the other

hand, Hehl, Ne'eman, Nitsch and von der Heyde 11] claimed that

theory of gravitation based on a Lagrangian quadratic in both

curvature and torsion has a correct Newtonian limit and al3o

an Einstein licit yielding the Schwarz3cnild solution. Recently,

Trautman (3j proposed the ^auge field to be a linear connection

(or a connection closely related to a linear connection), the

C«uge groMp to be G L (4, £ } or the affine group (depending

on tne bundle) and that the metric tensor g ±.u plays the

role of a Higgs field. Ivanov and mrayself (. 0] in the spirit of

[9] suggested as the most natural gauge theory of gravitation the

following choice: the gauge group to be the Poincare group (P which

ha3, however, to be spontaneously broken down to the Lorentz group

S0(3,l) (the fibres can be identified with the coset space ^VsO(3,D)

and which is realized non-linearly. The gauge potentials are the

vierbein eC and the gauge field the Riemannian curvature

tensor H&j-*1' • The metric tensor g^v is a form and the

Lagrangian is taken linear in the curvature.

Summarizing we have seen that all fundamental interactions

can be constructed in the framework of gauge theories.

3/ The gauge theory will be a physical theory if it i3 renormalizable.

4/ One must use the Higgs-Kibble mechanism for a spontaneous symmetry

breaking whenever some gauge fields are massive.

1.2 Lie superalgebra3

What are the difficulties when one tries to unify various inter-

actions? First, one has to find a common origin of theories, i.e.

a common gauge group in which the corresponding f;auge groups of

unified interactions are non-trivially embedded. However, the comnior;

gauge group does not exist if (roughly speaking) the gauge groups

are Lie groups and one of them of spacetime transformations (no-po

theorems; for details see e.g. flOJ ). Such a situation appears if

one unifies the gravitation interaction with the other interactions.

This principle difficulty was overcome in 1973 by using a new

algebraic structure - the Lie superalgebras (or the Z-,-/*raded Lie

•algebras), the gen^mtors of which satisfy '.;omiTiUtaio;-;L; or anti-

commutators ,

Remarks:

1/ The Lie superalgebras were fully classified by Kac fll] and by

Kaplansky [127 Cfor further details see e.g. [l3j ).

2/ The basic properties of representation theory of Lie superalgebraa

are known fllj but various techniques for the construction of

concrete representations are in progress (see e.g. a discussion

in fl3j )•

3/ The Lie superalgebras have been used in mathematics aince 1955.

In physics, these algebras were introduced by Gol'fand and

Likhtman in 1971, by Volkov and Akulov in 1973 and, in parti-

cular, by Weas and Zumino in 1974. However, the origin of the

Lie superalgebraa can be found in the works by Lipkin, Stawraki,

Flato, Hillion, Schwinger and others (for references and more

details see e.g. (13J )•

The most interesting Lie superalgebras are those, which contain

the Lie algebras of spacetime transformations (such aa the Poincare"

algebra or the cor.formal algebra or the de Sitter algebra etc.) aa

their subalgebras. They are often called supersymmetry algebras.

They were classified by Haag, Lopuszanski, Sohnius £14] and by Nahm

[l5j {for some critical remarks see also [8] ).

As an illustration let us consider the simple supersymmetry

algebra consisting of the Poincare' algebra (generated by iA and

P ) and the venerator Q

(1.8) 1%*, ftf*] - <'-%,?

(1.9) [P^ , P, 1 = C ,

d . io ) (.\>t, , pf ] = i 7

fulfilling the relations:

-3- *• ' fa ^"f ' 'Jff ^<r ~ '

(1.1X)

(1.12)

and

(1.13) 1 /

where Q, = Q ft , V = ? L • dW " " ' " = ° . 1 i 2 . 3 , and

diag( 'Jfti, ) = (-1) + l f +1, +1). From relation (1.11) one sees

that generators Q^ have a spinorial character.

The simple supersymmetry algebra has remarkable properties even

considered as a global transformations algebra.

/ i / It may be realized in the space of fields. Then because of a

spinorial character of generator Q and the connection between

spin and s ta t i s t i cs , generators Q transform fermions to boaons

and vice versa so that the aupersymmetry is the f i rs t example of

a genuine relat ivis t ic spin containing symmetry.

/ i i / The structure of an irreducible representation of the simple

supersymmetry algebra when reduced to i t s Poincare' subalgebra

can easily be found by applying the spinorial generator Q^

to a particle state \ p iTl ... y • Here, pA is the particle

momentum and J i ts spin or helicity. Then from £q. (1.12)

and (1.13) one aee3 that generators Qa , Q form a

Clifford algebra of creation and annihilation operators so that

the Fork space technique can easily be applied. Finally, one

obtains the following structure:

1 / Let us note that by using anticommuting npinorial paraiiir tr> •"•-•a -y , Qe,, the antieoiunutat ion relation (1.13) can be ».:-;U.enas

(1 .14 ) {m, J - •£, ^ , - • •> © (m, J" , i < 2 , . .

© (m, J + ^ , - , . . . )

p rov ided m i- 0 , J i- 0 and p a r i t y

( ± i for J integer,

) + 1 for J" half-integer,

(1.15) (m, 0, i^,...) ® (m, 0, -i/£,...) + (m, \f\ ,.-•)

provided m / 0, J - 0 and

(1.16) (0, A ,... ) « (0, A + ±, ... )

provided m = 0 and helicity is equal to X .

Summarizing all particles in the irreducible representation of

the ainple superaymmetry algebra have the same mass and are

grouped into spin pairs ( J" , J" + |) _pr__{ JrJ_ J - | ) .

/iii/The simple supersymmetry algebra can be realized in terms of

(local, renormalizable) interacting field theory which ia

invariant under global simple supersymmetry transformations.

For example, one may consider a system consisting of a spinor

field A (x) and a vector field A ., (x) with the Lagrangian

fie;(1.17)

where F.^ = ^ A v - /3V k^ . It is relativistically

invariant but also invariant under the supersymmetry trans-

formations

(1.18)[5 Q, A^J = ix

J

-9-

since these transformations change Lagrangian (1.17) by a total

divergence. Consequently (via the standard Noether's procedure)

the spinorial current

r o' h V at -i

(1.19) J = ^ $ f # <*

is conserved. Let us note, however, that applying (1.18) twice

one must introduce a new pseudoscalar field in order to close

the algebra. The added field is auxiliary because it can be

eliminated by using its equation «fmotion. Thus, for the first

time, a new phenomenon happens in the representation space of the

symmetry - the number of fields which are needed to close the

algebra of mass shell is greater than the number of fields which

realize the symmetry on particle states. Let us also note that

recently auxiliary fields have been successfully applied to

derive the so-called tensor calculus which leads to a remarkable

simplification of the structure of the theory considered.

Once one has a field theory invariant with respect to global

supersymmetries (like that described by (1.17)) one can construct

a theory invariant with respect to local supersymmetries (by applying

the gauge principle). One obtains the theory called supergravity.

Since the local superaymmetry algebra always contains the local

Poincare" algebra, in the supergravity the gravitation field emerges

as a gauge field. However, one always gets yet another field - the

gauge field with respect to pure local supersymmetry transformations.

It is a spinorial Rarita-Schwinger field with helicity _+ 3/2.

2' The other possibility, i.e. a spinorial field with helicity + 5/2,

is usually rejected for technical difficulties to construct a

consistent higher spin quantum field theory (see e.g. &7]).The

situation might be changed due to the works fl8j,fl97«

-10-

Since i t accompanies the gravitation f i s ld , i t s quantum is called

gravi t ino.

2, SIMPLE SUPEEGRAVIIX

In 1-J76 two formulations of tut supergravity based on the

local simple supersyasietry appeared: The formulation by Deser and

Zuir.ino [20] and the formulation by Ereedaan, Nieuwenhuizen and

Ferrara [21]•

In the&imple supergravi ty , the vierbein e£ , the connections

t^'c" and the Sarita-Schwinger f i e ld ip? appeared to be gauge f i e ld

under transformations generated by ^ , ii „ and Qw r e s p e c t i -

vely . The Lagrangian in the Deser-Zumino formulation is of the form

where <¥£ ia the usual Lagrangian of the Einstein g rav i t a t ion

theory

( 2 . 2 ) # p = •

and =^^s is the Lagrangian of the Earita-Schwinger field in a

curved spacetime

•RS(2 .3 )

Here , R =

(2 .4 ) **„£" = 3, w^" - '

e =

(2.5)

and covariant derivative D^ is given by

(2.6)

Lagrangian (2.1) is invariant with respect to local tranafonnations

generated by I,: ^ , ? and Q . The transformation properties

of the fields under the local supe rsynur.e try traridf ormations (generate;

oy QA ) are given by

(2.7)

r

B fa-where

(2.8)

/ - y - \J •

Applying this transformation twice one must introduce new scalar,

pseudoscalar and axial-vector fields to make the algebra closed.

These fields are auxiliary fields of the simple supergravity analogous

to the scalar field of the supersymmetric Yang-xills theory (1.17)

discussed above.By taking variations with respect to yj/f ,

one can easily write down the equations of motion

and

(2.9)

R

C'

= i

which are not independent (due to the local aupersymmetry invariance)

and in which torsion tensor C.g- is given by

and the Einstein tensor by

- 1 1 - - 1 2 -

From the point of view of quantum f ie ld theory the simple

/ i / It is the f i rs t consistent field theory a&fe$ri*iii& 1jha&

An «uSai*i'ta'*-Sehyraj!i5Ee!irp fdpldJ: ^ imflerafctaragnwdite atutUoBati £S»4d

/ i i i / There

ai wort snoJJ P I E *•••:

Theyew6uldiB©ollvon*qheJjI

gra*aJLy liBleesIthcsir eoSffttfieart* ¥aftinh.iB9qq&

t o

OI JO

However, many exten3i§HSj:(5fit!fi'iosiiii|ieI

They are based on supersymmetry algebras,,(^Ls^gif^gd jn,^4-jj

containing either the Poincare algebra or"tTie"de^srE"ter"il"gebira~or ""*"

the conformai algebra.

3.

g^seof "thef:

J

tfie*;'iri

tb vector1

ihL in

is the

SQ.f a) : is limited.

is

i n d i c a t e d i n T a b l e I . ^ J J V B T ^ I ? ; . - : « (fl)Oc- « , ' • rtj i ^ j j i ' r v i

J e n s v 9 c

B es (

? R 3 i o

Xs r t fL

- s , •

oMl1

n = 1

abiolq

-

'-O E i i i

SO{n) - extended supergravity

2

>ol-r

1

3

3

4

;:• U a

6

* ; > , - • C,B

o; wo

5,.

10

, &;

16

c

-A!

• - I

- i ^

28

5 '

S i i

' i i

OJ

;

r

0

bi

8

1 '

5b

70

Tab. I

of

field

^ ^ ^^IJa n d

/ii/ For n > 4, the theories have very complicated

structure (mainiy due to the presence of spinlesa fields).- O - T J - ; - ) I & baa S ' O T - 1 ' ; &i*i' ii> sq^-rv:; ^y:-;:.; - i ^ ~ ^ r : ' " ^^ -1 ' sri;

If turns out, however, that the next most important case

' ' is n = 8, since n = 5 and S'-'tfre expected to be subcases of

the S0(8) - extended supergravity and n - 7 ia equivalent

to the n = 8 case. ' ~

/iii/ The dase -n~'=' 8 is' extenalvefy1"'studied by'Creimer, Julia and

Scherk f26j (partial results were obtained in [27j). They

- 1 3 -

use a special reduction technique. They start with the

simple supergravity (i.e. n = 1) but in 11 dimensional space-

time, then they compsctify 7 spatial dimensions and reach

the SOC7) extended supergravity in 4 dimensions which should

be related to the 30(8) supergravity.

What are the advantages of the extended supergravities?

/i/ The extended supergravities unify fields with helicity 3/2,

1, 1/2, 0 and 2.

/ii/ Their S-matrix elements up to two-loop order are finite C2B].

On the other hand, the extended supergravities have also several

drawbacks.

/i/ First of all, they uae the internal symmetry S0(n) as a

global symmetry. If one considers S0(n) as a local symmetry

(which can be done due to a right number of fields in the

irreducible representations of the extended supergravity),

one gets a coamological term ~ 7 which is more than

100 orders of magnitude lqrger than the astronomical upper

limit. Let us remark in this connection that it is claimed

that by using the Higgs mechanism or the Hawking approach

one might be able to overcome this problem (see e.g. [29J).

/ii/ In any case, the local S0(8) aymsetry is too small to include

the present minimal gauge groups of the strong and electro-

weak interactions SJ(3) ® SU(2) O U(l) . Consequently,

several well established particles are missing {e.g. if one

takes 56 dimensional spin 1/2 multiple!, then it breaks up,

3/

according to Gell-Mann [20], into the following SU(3) multi-

ple ts (electric charges) of Dirac spinors

3(2/3) 9 3(-1/3) <» 3(-1/3) © 3(2/3) » 6(1/3) « 8(0) <B 1(-1) ®

© lt (0) « lt (0)

and thus can accommodate the u,d,s,c quarks, a fifth quark

flavour (colour sextet), a neutral octet, one charged lepton

(e.g. electron) and two neutrinos, so that leptons /< and r

are not present. Analogously, from the structure of spin 1

multiplet one may see that W bosons are absent.

There are several suggestions how to overcome these difficulties.

One may construct field theory based on several representations with

spin 2 of the extended 3upergravity. One may also use the S0(9) or

S0(10) extended supergravity (then necessarily spin 5/2 particles

and several gravitons appear in the theory). One may also say that

the fields of supergravity do not correspond exactly to known

elementary particles at least at present energy. Finally, one may

try to use other approaches of the supergravity based e.g. on

superconformal or super de Sitter symmetry or on contractions of

0Sp(4,N) symmetry or on geometrical formulations.

4. CONFORKAL SUFEHGHAVITi

For n ^ 4 it is not 9 constant but a s-alar fieldDOt°nt i a l .

The conformal supergravity is built on the Lie superalgebra

generated by the generators of the conformal algebra, generators

of the internal symmetry algebra U(n) (for n = 4 also SU(4)),

by two sets of apinorial generators Q1 and S1 , i - l , . . . , n which

transform according to vector representations of U(n)^nd by generators.

i- rom the point of view of gravity alone, the confon..3. nuper-

y i:-> closely reliteci to V/eyl a gravitation theory. The present

. #• • * • !

status of the conformal supergravity can be summarized as follows:

/i/ For n = 1 the theory is known /3lJ -

/ii/ For n > 1 only partial results are available because the

corresponding Lagrangians are pretty complicated (they contain

terms with higher order derivatives like ^ i D ^ , f ^ )•

/iii/ The internal symmetry U(n) for n > 5 is big enough to include

SU(3) c o l o u r <S SU(2) ® U(l) gauge groups. However, as argued

by de Witt and Ferrara [32] , thi3 perhaps does not mean any-

thing since the supergravity theory beyond n = 4 probably

does not exiat.

/iv/ It is stated that the conformal supergravity (with the

Lagrangian quadratic in R ) analogously to the Yang-Mills

theory might be renormalizable.

FDHJiULATIoNS OF 3UPi,:-:GRAVIIY

/v/ {asThe conformal supergravity is asymptotically free in

3hown by Fradkin).

/vi/ On the other hand, since X ~~ R2 the conformal aupergravity

does not agree with the macroscopic predictions of general

relativity. However, there is hope that by applying the Higga

mechanism the Lagrangian changes and a new term linear in R

appears which will reproduce the usual predictions of general

relativity,

/vii/ Since the Legrangian consists of higher derivative terms,

negative metric ghosts appear in conformal supergravity. It

is not completely clear how to remedy this difficulty (see,

however, £33j)«

Up to now, the supergravity theories have been discussed in

the framework of quantum field theory. One may ask whether one cannot

construct the aupergravity theories by applying geometrical methods.

-17-

The answer in yea and as 3 matter of fact the geometrical

approaches seem to be natural, systematic and quite comprehensive.

One can make steps as Einstein did but on fibre bundlesa the base3

of which are rather superspaces or L- and R-chiral auperapaces than

spacetime and the structure groups of which are Lie supergroups.

Such constructions use various notions which were intuitevely

introduced by 3alam, Strathdee, Okulov, Volkov, Zumino etc. (see /34])•

It is advantageous that we have at our disposal now the precise

definitions of the notions due to berezin, Leites, Kac, Kostant,

Hochschild and others [35j.

There are at least four geometrical approaches treated in

the literature. Their primary objects as vtell as the number of their

components are summarized in Table II.

Primaryobjects

Number ofcomponents

Extensionto n >1

Approach

Zumino,Wess poj

supervier-bein E(x,9)

super-connection

S - 1024

r - 1792

not known

Ge 11-Mann, BrinkSchwarz, Ramond;Me Dowall,Mansouri ;Ne enan, Regge D7J

E(x,6)

iridep.112

n = 2,3

Arnowitt,Wath DbJ

supermetri*

gMW(x,e)

1024

all n

Ogievetsky,Sokatchev

axialsuperfield

64

all n

TAB. II,

-18-

T.n geometrical approaches (except the Cgievetsky, Soiatchev

one) there are technical difficulties to eliminate a large number of

unwanted high spin components. Usually it, is JOIUS by imposing variousconstraints by hinJ. OP.J- -. \<r\ily a serious problem of their physicalcor.siutenees arises. ...any pnyaioists are, tnerefore, trying to

construct a "tensor calculus" v.riich hopefully will simplify not only

the construction of Lagra.-.gians but also of invariants vhich can

play a role of counter terms and thua make a systematic discussion

of renorcalizability possible.

6 . CONCLUSIONS

/ i / The s.upergravity is a supersyametric extension of the Einstein

(Einstein-Cartan or V/eyl) theory of gravity in which gravitat-

ion is unified with very restrictive combinations of lower

spin particles in one irreducible representation of the

supersymmetry algebra. For example, the gravitation field is

always accompanied by the spin 3/2 Harita-Schwinger field.

/ i i / It is crucial for supergravity to find out gravitinos - the

quanta of the Barita-Schv.inger field in experiments. It is

not an easy task since the properties of gravitinos are not

well established (e.g. the mass of gravitinos is predicted

in the range lCf19 - 1O19 GeV) and probably yield subtle

effects,

/ i i i / In any ca3e, the supergravity ia the first consistent quantum

field theory for spin 3/2 field interacting with the spin 2

gravitation field and the spin 1, 1/2 and 0 fields.

/ iv/ Two technical triumphs were remarked in the last 2 years -

off-shell component approach to simple supergravity (via

auxiliary fields) and superspace formulation of the simple

supergravity.-19-

/v/ The status of divergences may be summarized as 1'ollows:

The conforj.al supergravity is believed to be renoriLalizable

In simple ind extended supergravities (m contradistinction

to the tinstein gravitation theory interacting with matter)

the S-matrix elements in one- and two-loop orders are finite.

However, starting with three-loop order the invariants which

can play a role of counter terms Jo not vanish in the simple

aupergravity. For extended supergravity there is no techniqut1

available to calculate these invariants for higher loop order,

/vi/ The supergravity can be considered as a possible unification

scheme of gravitation, electromagnetic, v.eak and strong

interactions, tor a realistic scheme it will be necessary to

solve the problem of renormalizability and quantum field

theories for higher spin particles, in realizing that, it is

perhaps worthwhile to Steep in mind the Einstein idea;

"Conviction is a good main spring, but a bad regulator".

ACKNOWLEDGMENTS

The author would like to thank Professor Abdus Salsun, the

International Atomic Energy Agency and UNESCO for hospitality at

the International Centre for Theoretical Physics, Trieste.

-20-

fl] S.D.Drell, SLAC preprint, FUB-2222, 1978.

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CH:(HEMT 1CT!J I'HSPIIINTS AND INTERNAL KSi'ORTi';

Ic /79 /20 J . T . MacMULLSN and M.D. SCADRON: Low-enert-y photo- and electro-productionfor physical pions - 1: Ward Iden t i t y and ch i r a l breaking s t r u c t u r e .

IC/79/21 J . T . MacMULLEV and M.B. SCftDRONi Low-enerpy photo- and e lec t roproduct ionfor physical nions - I I : Photoproduction pbonomenolopy and a t t r a c t i o n ofthe quark masn,

Ic/79/22 5.K. PALs Oarm fragmentation function from neutr ino da ta ,INT.SEP.*

IC/79/23 M.D. SRINIVAS: Collapse pos tu la te for observable3 with continuous spec ta .

IC/7^/24 HIAZUDDIN and FAYYAZUDDINs Oluon co r r ec t ions to non- leptonic hyperondecays,

IC/79/25 H.P, MITA.L and U, NARAIN: Die lec t ronic recombination in sodium i a o -INT.REP.* e l e c t r o n i c sequence.

IC/79/26 MUBARAK AHMAD and M, 5HAPI JALLU: Gel'fand and Tsa t l in technique andINT.REP.* heavy quarks.

IC/79/27 Workshop on d r i f t waves in high temperature plasmas - 1-5 SeptemberINT.REP.* 1978 (Reports and summaries).

IC/79/28 J .G. ESTEVE and A.P. PACHECO: Renormalisation group approach to th«INT.REP.* phase diagram of two-dimensional Heisenberp spin systems.

IC/79/29 E. MAHDATI-HEZAVEH: Production of l i ^ h t leptons in a r b i t r a r y beamsINT,REP.* via many-vector boson exchange.

IC/79/30 M. PAXKINELLO and M.P. T05I: Analytic so lu t ion ofthe mean sphe r i ca lINT.REP.* approximation for a multioomponent plasma.

IC/79/31 H. AKQAY: The leading order behaviour of the two-photon s c a t t e r i n gIHT.BEP.* araplitudea in QCD.

IC/79/32 U.S. CRAIOIE and H.P. JOKES: On the i n t e r f ace between BID»11-P_ non-per tu rba t ive and larfie-anp;le pe r tu rba t ive physics in QCD and t eaparton model.

IC/79/33 J . LOHENC, J . PRZYSTAWA and A.P. CRACKNELL: A comment on the ohainINT,REP.* subduotion c r i t e r i o n .

IC/79/34 M.A. SAMA2IE and D. STOSEY: Supersymmatric quant iza t ion of l i n e a r i z e deuperpravity.

IC/79/36 J . TARSKI: Remarks on a conformal-invariant theory of gravity.INT.REP."

IC/79/37 1. TOTfF: Additive quark model with six flavours.INT.REP."

Ic/79/39 A.O. BAHUT: Hadronie nmltipleta in terms of absolutely stableINT.REP.* par t ic les : An already-unified theory.

IC/79/40 A.O, BAHUT: Stable par t ic les ag building blocks of matter,

IC/79/41 A. TAOLIACOZZO and E, TO3ATTI: Effects of spin-orbit coupling onINT.REP.* charge and spin density waves.

Ic/79/42

Ic/79/43

Ic/79/441ST.HEP."

IC/79/45

TC/79/46

IC/79/47INT.REP.*

IC/79/46INT.HEP.*

IC/79/51

IC/79/52

IC/79/53

IC/79/56INT.REP.*

IC/79/59INT .HEP.*IC/79/61INT .REP .*

IC/79/63INT.REP.*

IC/79/66INT.REF.*

IC/79/71INT.HEP.*IC/79/77INT .HEP.*

IC/79/801ST.HEP.*

C.C, OIU.iAHDl, C. OMEUO and T . VfUBEU: '.Jiuintmn v u r a n s c l u n a i p a l laws I'cu-s e q u e n t i a l decay M o c e s n e s ,

G.C. (IHIUAIiDl, C1. OMUliO, A. I(!M]H1 ami T. WKliKH; ^rii-il 1 t in .e huhavjovjr 01quantum non—decay p r o b a b i l i t y and Z e n o ' s p' iratlox in qTjari tum m a n g a n i c - .

RIAZUDDITI and FAYTAZlfDDTM: tjnark mass r . i t i o n ih,e t.o H o l d s t o n ep s a u d o g c a l a r mesons p a i r .

FAYYAZUDDIN and RIA2UDD1N: A moiiel baaed on r a u p e syramatry p r o u p0 = G^ T [3U(3) x 3U(3)]c .

W.S. CRAIOIE and ABDUS SA1.AM: On the effect of scalar partons at shortdistances in unified theories wath spontaneously broken colour symmetry.

A.O, BA.RUT: Infinite-component wave equations describe r e l a t i v i s t i ccomposite systems.

T.N. SSIERHYs Comment on the question of hierarchies.

A.O. BARUT: Magnetic resonances between massive and maasless spin—Jparticles with mapnetic moments.

V. ELIAS: Gaup-e invariance and fermion mass dimensions.

RIAZUDDIN and FAYYAZUDDIN: CI non-leptonic decays as a tes t forcorrections to non-leptonic hyperon decays.

I . ROBASCH1K, 0. TROCER and E. WIECZOVEKi LiRht-cone expansion of matrixelements of current commutators.

V.V. M0LOTK0V and I .T. TOBOROVi Frame dependence of world lines fordireotly interacting olasaical r e l a t i v i s t i o par t i c les ,

G. SENATORE, M. PARHINELLO and H.P. TOSIi Optical absorption of dilutesolutions of metals in molten s a l t s .

H. YUSSOUF: A possible realization of Einstein 's causal theory under-lying quantum mechanics.

H.P. KOTKATA, E.A, MAHMOUD and M.K. EL-MOUSLYs Kinetics of oryetalgrowth in amorphous solid and supercooled liquid Te3d20 ueinff I1TA andd.C. conductivity measurements,

FANO LI ZHI and R. RUFPINI: On the dopplars S5433.

l.H. EL-SHIRAFY:of spiral plate .

First and second fundamental boundary value problems

On electrodynamics with internalA.O. BARUT, I . HABUFFO and C. VITI3LL0:fermionic exci ta t ions .

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