13

their

support

To

of

expansions by

arb i trari

1y ,

In

g e o m et r y

cl-aims we undergo form

equivalent their

physics. the

curvature metric

(the

geneTalized

equations

guage invariance

This

is

event

gauge

rather

from

the

However, ghosts

mass into m et h o u

( 3z;

from the

.

IOrm

the

the

since

metric

nAg

But, can

what

limit.

differentiaL due to

choose

a gauge

one expects

of

the

equations.

required

elimination

such

order

one

all

system

asymptotic

spinors.

tensor,

of

the

from

non-dynamical

theory.

the

gauge fixing

the

theory. by

the

procedure

This

is

ttf -meson

does not

confirmed dominance

by of

eliminate introducing

gravi tytt

.

superspace, (34)

second

saLisf y

is

theory

Although flat

are

metric

strange

transformations

variables

all

the

properties

space the

in

reveals

particle

elementary

flat

including

f ields

r.rhere some spinor

of

ab ove

the

covariant,

a coordinaIe

the

to

an entirely

in

algebraic

the

obtained

fields of

(f .f )

manifestly

metric)

Minkowski

all

To examine

not

I^le adopt

i*posed

introduction

wri ting

clarifies

equations for

of

Ban approaches

the

rather

wave equations

tensor.

that

The linear

the

form

This

task

although

that,

but

a brief

s upe rs pace .

in the

to

relation

we give

four

Chapter

Riemannian

condition,

covariant

any

obtaineci

form

special r.vhich \^/as not

of

the

metric,

superfield

the

very

considered

they

claim

loca1 its

supersymmeLry metric

tensor

does not satisfies

correspond an equation

to of

a the

r5 is

The invariance

gauge fields.

compensating

comes from At

this

s tage

We only r{ay of

on

In

the

(39) Ferrar&'--',

.3

spiniinvariant bv

under

think

to

abandon

of

the

idea

the

an

of

eight

of

work

by

Deser

lie

groups

1oca1

F reedmaa,

which

Nieulrenhui

( 4 0 )' , a n and Zumrno'

general

some

as a convenient

superspace

can be

a

manifold.

space-time

Chapter

we

six

where

supergravity,

a

zan

and

, . combined

r^Tas given

relativity

spin

rolhich

2

is

transformation

supersymmetry

we

give

of

the

Because #t

be

reaLi"zed

"annot

covariant

potentials

an

relate

gauge invariance

alternative

the

obtained

supersymmetry

by

linearly studying

1ocal

obtained

by

an rnUnu Inligner contraction(41).

of

Physics.

Chapter

to

1oca1

w€ get

gauge inter4al

cLA 0SP(+lrltazl

in

of

on space-time.

on space-timer

Ehe orthosymplectic

contained

lagrangian

algebr ^A

of

The research

formulation

combined

symmetry

Journal

the

rnethods.

trialIn

the

the

of

Again,

equations.

generalized

of

extension

Lagrangian

of

fields.

spinor

to

the

then

free

the

a se t

wor1d.

recen t

the

field

tr/e have

have

reaLLzLng

symmetry

the

physical

dirnensional

the

introducing

by

From

we derive

guage fields, trouble

achieved

three

of

is

whi"n-{

published

is

in

T7

2.L

G R A D E DL I E

ALGEBRAS

a unif ied

theory

f errnions

and bosons,

the

state

to

a ferrnion

1ocal

density

In

a t

I

state

can only

a f undamental

of

generator

symne try is

symmetry connectittg

a fermion,

itself

be written

terms

in

between a boson

and thus 1oca1

of

its

fermion

i

r].elCls, However,

the

symmetry

because

the

cannonical

algebra

generetors

trvo fermion commutator

ttanalogous

replaced

by we

direct

an

either

the

in

anti

define

a1low

t.he

a Lie

relations

calcul-ation

r,,ras overcomed

through

algebrastt

no\r

which

graded

the

of

for

for of

their

Lk,

group, E

odd

o,

the

introduction

as

would

spa"u(to1

spaces

abelian

integers

two

the

known

commutator

vector

vector

indexing

group

a

and

commutator

a

sum of

some gi.ren

Lie

of

(G.L.A.),

of

form

anticommutation

r,yi11 not

difficulty

algebras

the

cannot

algebra.

The the

generators

Graded

be

=

which group

r,-@ L,KK

where

k

is

/o\

to

be

an element

we will of

Lie

simply

generaLors"', L

of

two

take

as

elements

v-z Define

on

this

vector

space a bilinear

map

t ,J I LxL"-'DL satisfying

the

f ollowittg

conditions:

I t. , Ltl c Lu*d ,xl btlk{E,l l*,vJ.rI x ,r't,ilJ - [r x,z1,aJ+ (t)ou[y,l.l,x]J

(2'L'L)

1B

where

y,

X,

are

z

Lt

of

The elements

graded

The

(L,

numbers

complex

is

operation

a

fermions)

or

-1. over

defined

grade
field

the

of

above bilinear

the

algebra. of

basis

the

G.L.A.

The

L.

by

def ined

are

constants

structure

L

a homogeneous

be

A

or

respectively.

(bosons

odd

and endowed with

ca11ed

Z^

Let

+1

is space

vector

or

even

(-1)k

to whether

according

be

will

Lg and L,

Lk,

of

elements

Z, , CnZeC, lzr,z*7=C^"u The generaLors type

a = 1,

{X^,

are

{ZO} ,..

.2)

form:

t,

Xr,J=

No te tt

that

the

{*"}

is

Cefined

it

pr eserves The

7 of

=IeV'

V V

n into

n4

t he

{ Q' c- , - o = 1 ,

the

..'

following

(2.r-.3)

v

Aq

g. nerate

us ua1,

bose

n o P

Udu

algeb ratt

Li e

as

B

La{

Qpl=

A homomorphism

Le t

lt

Qn,

un de rlyittg

type

or

A g

L ab

t X*, 9n] =

r

fermion

decomposed into

can be

Even

sets!

n c v

1

Xq

two

into

and odd or

M},

(2.f

Equatioo

divided

(2 .L.2)

the

it

which

is

ca11ed

the

G.L.A.. graded

one

where

algebra

a Lie

of

from

.

is

Lie

algebra

additionally

to

another

required

that

grad ient.

G.L .A. be

i tself

of a

is

End(If) g raded s uch

constructed

vector that

XV'

space. CVrr*k

as Let form

follows: the

linear

EndU(V).

maps

tt}

50

Therefore

;< T tlart

{(f'd'fr')1

= -f;iflo^*

+-

( ilt + iy't'+''/t)

+(i d+*) r ( i / + v ^ ) z

(f /+n

),

f,.a

K r b b1 t ,e

rvhich

(24)

a a- n- d1 L a ti e r

One easily the

Greents

1

^

S.

by

passes

that

obtained Q3)

^a C hang

to

the

by

Gural-nik

and

' . momentum space

expressions

of

funcEions

( , 1 f,1^ t f ' ) =

C*

with

agrees

( 3.3 .L2)

1ap{$,n'f,)

32 + t^z a result

I

3(rt-Xr) .

i-,e4f . ( l-

G++ {4F ,dt1?')=

h

( t i r,"i { {; 4 z - + €'{ e') { y r f + } , ^ )(,/ , f + n) r \ ffil(qfi,*,1t,)

vxl(r* 4rif {sdtu-{7r f er)4Ptzt , (3 .- $'r *I2, !A,-lr) ,) , 1a13,a,f

At,: ( ttg),rx'g'!)=-fi; 4f (t lf 4rf Flsot,-lA, fol fire,J ( Y'lr- l,fr) ( sf),satp,])) ,

+ - +-

("p ,o'f')

r

-

flt G'+f (.tf , c,f') =

Since

a factor

gives

rise

binations

to like

€

-

VO = ln(0,

6++

(oft , a/f') ,

Gt;

(4f ,,1,/1,)a

O) is

applied

4-momentum conservation e

f

t rO and 0 y50) .

at

each vertex,

(Vg allnihilates

(3.3.13)

this eom-

62

Rna= + | rr)od+,d Tou*ro,,rtc- (-t)*df,aB,Dc e+oJ - 1't)od + Lc +LJ , + tt)b !u, ,rt?'l. o.3. 1o) A ot fi 13 "

t,+AfT,,LA,zcrcA + {,,or,tB-. 4^,3,cc F (-t)q+c A"crfrB] rshere v/e are

raising

for

.ito

example, We find

and lowering

hag

it

nuo

indices

linear

form

of

more convenient

the

the

metric

tAB,

= hcD. to

Tna=R.,rrB -+ilnaR The

with

(4.3.11)

curvature

solve

the

equation

=o

(4.3.12)

is

P.-= L, .LAt bt)oA ne,cA * 1.t)"Aru, *'3, T hu s

Tas=+2t (T*uflr",'u +Anc,,aBlaer.1(,,)nr'/.r1^u) trrf

Dc- /,t)'1", o "J] --Tna{u1'ArD, Equation

(4.3.13)

can

be

decomposed

into

a

set

of

three

(4.3. r:) equations:

Tr, = Lrtf Gvc t€r+ A*c,cv - /.r,,ec-ht)"',,u:^l *Tru(

lnf AcDtDc-p)? /.ccrDD)J=" (4.3.14a)

kn ' Lr)[- Aor,'^ + lr^e,to - {*4,c'-r4'Xri*]:o(4.3 .14b)

Trnrr= (lur,rf * Arr,t - lng,.t (u'i*r, t)[ on)

*|,rfl ( t,,;rA*a, D'* {-t)"/,r'rDoil= o (+.s. 14c)

74

For

simplicity

v/e lvill

alwavs

Vr\f nFl The

metri

The

inverse

c

take

n t) s d' t-eyt K ,

(4 .6 ,6)

agrees

wi th

metric

defined

by

f-

Fe) '

?o^'s Tru (t+

= tg (ulf y ?ooo %!t'I + TaP

that

(4.6 .7)

ob tained

9u,X"n=[rt t

by I^Ioo(3/r). S:

'i

lK

(4.6.8)

K$o The inverse

d.oes not

exist

in

case K = O and

the

the

metric

is

singular. The correspondirrg rth

1vf s

O

p0{

f /-\

=

*lthit

''

rl

connec ti ons

r*+ Jvd z

)

iq

lpF = lu(/3

af f ine

/Y '

7K

\tt

'\

7a

Lhv

)

are: ^. ?-

(YrY^8)a ,n Ei< -t
Jtt

(4 .6 ,91

* # [ trr4o(trrr^ uh (trdhFrfhn,il - (fteh( {f)!J, #t t{ro)n(Yr);

lE "qF The Ai cci tensor

is

computed

using

equation

(4.2.L2)

r we get

n 3'"i Ksrl..r = -fu'!ou

airsyu.4 = .5r 'nf , o 4 f i Equation

s

f fa

501.'* ,

(4,5.10)

r y q" l- q - f ,) K *-L SoaF

(4.6.10)

can

be

?oBJ

summar yzed

in

the

form:

=* FoFB #{,lao + YTo,rf;#'l

(4.0.11)

Suppose

that

tt

rt +; L6 (x,a) 3drt {r'(x,a) ,A

wnere

t_s a

A

smal1

expan sion

t a {r"'=

p arametre,

.-rt Jr'

,1 g

(5.L

L2)

then

A

4ft!' (5.4.13)

OA

h8' and

there

is

no

= [;Fil*Iui

difference

The rron-linear

between

quantities

upper

and

lcwer

into

the

linearized"

pass

= i ({nt);,i !,![,i#lA{gI?' s'),u'i&

indices,

ones,

.

A;f -1'({n{)ap[i[1i*]

+ i({hc)ai(f;*f,'*[;i{,!)#

(s.4.r4)

]

)ri] ?&r,-{*; ( &,^t## [o;(Y"e Sloa +Af-t.D{n$;^t;f gefrrh

(s.4.1s)

f ;! #- fr;*[#'gflslJ +i (y'c)ai(8{;* (i+ a$o;t = (,)tx+r) F )ai 1€r c Fr! Ard Ai

L

?^sr Li

The firs is the

not

f,,,** *,\( &Jf,a3 N t*rtfi;')J t

difficulty

satisfied

to

zeroth

in

rhis

order,

model the

is

that

d.iscrepancy

equarion being

(5 .4.16) (5.4.11) due to

te rm F

P-= ^l^L+i ( /")n'buflf - q (;d)a, ArF ' (€''t*)Yi

*$t However,

this

ean be

easily

Dr/t

b_

removed

by

Lt'cr)1I l1 J addirg

the

(5 ,4

1 ?\

term

f

jur l dF5 (5 .4. 1g )

111

We requi re transformations

fP sf

the

invari

ance o f

(6,4. 16),

(6.4.L7)

?:!

lLt^'J. €)(A Ab Kr,

B ')r . f

we sha11

study

supersymmetry poirr""rE

invari

In

this

q

tln

only

=

A Uhu

the

Bb

under

e(\ + ft h v .d nUr" ) E O .

the

(6.5.9)

of

Lagrangian

will

(6.s.s)

under be obviouslv

ant.

case

equat,ion

(6.4.11)

reads:

e{ a \ r . ) f'D,.v i r a rac)e 'r it€'t ,Y ''C )a. z - ('l ry"{#

f {1 a6 A ,{' ^u = tJ

lD,.,:s

the

(6.4.j.B):

and

invariance

since

,r:ugrangian

the

(6.5.10) (6.5. i i)

(6.s.Lz>

+(fr,.b)oRr-oob. a

He re

Dh.s ?p+ + 8:o-on. Note

that

(d.s.13)

sinee

(?")nF =* (r"l: , (r")f =u*co.,$j^if.thb)f I

rhen as n-Dtro Equati on

WxaD;*,

)

( 6 .5 .9 )

can b e wri t ten

as :

f [ fl'"nb nr,"b + t f *J (i r*b)nJ

D^l go.

+[f"J The

lasE

term

in

(6.5.L4)

is

of

the

t6's'14]

form:

=f r&sLf * q,+tWrlq"il; 5{""^}ou* G Lf^

A Y77,

*[ yi; -( re- Dr?,,J r r{r^:Fi* Lglfo VW " AVp

(6'5rs)

_ 116 _

2*t'^af - g|}*Y^nf +?5Fay6yil_etE lnrl -?z*f

trap.

cof + ar"t ctf : o

(A.5)

, + ?)FasrttrJ-?^fp(oay]-\ryt

; cqy

+ r(rr ! c\y -. /*F r, { rl *(,), x"rr.trrl

*%f ( art-t-r - ;..ttf %?y+zrn*{yj lf f f f.!4 *tu),/ccrs {rl)=o,o.,, rrr 2^ ?oFut fyl + ?\), fi^t rFrl - ?rF^,/,)(?t.,z"F".tte/J - D^, Cfry * 2r*r*4 cf , * 7u Xpa, - To*( ara* rrr "/r, -

r ltyJ D,/.,)"F*.rro

* ? X*"4cpy-O"t Cprr /.,)b", cpr)= ?"tr"-otfrj rO2 Dntrascpr! + OrXagg

o (A.s)

!py1

-- ?A NJ^AEafiyJ il D*d gt^f y +D*,t c?r= o (a.e)

aefdFryrl + pa, C w + D*f tf6r; *, Dntttny; + 3i xf or.t y{J .n 7h xa)rf yrJ* /-r),D*, tnf ,l "{r r- Ql lu ( 3h}"Fh v { y{J +? y* fr frtJ-. Df f c rs ^ br)*p'F**rysJ * (, )c r)r, ( w) = o

(a.ro)

r_1B

'{h,

q

rr P\" C */*, ft * lr)c

Y"'rJ

1 A. 5

')

q A, 6 ' )

-- *1r fo"p

r j ^/-,rY Y" r 'N{")" 7dP fr1' C Fr* Dfrfil Fn*c$/3 e {n)'?r,

rrr f4'I p

rv

1A.9

{rd,f (y{1 + Q)'Tn,, Fc"crrt?

f4'

')

1A.8')

F*, r fYl

f|L

.r,

1 A. 7

')

1A,10

')

EA.11')

?'" N*rnJ p1L { H*un * /" )'

6A.12')

*fl* N*"(fr qA.13')

(-/)c ?,f xrt yJ + X1 * tr:L f /t{ v

rql f *NL ,*

Dn, 'F (a )"'luu D o

cJ

1A.14')

1 A. 1 5

Dru

qA.16')

+ {n}* TraD"J, fi'|?^ CD*F ObviouslY

obtainecl

in

these

section

4'4'

equations

are

the

')

sane

tYPe

as

those

L20

Q't*f

- ?r?r%or+

ktix)f 4*frqy+-L?*Feo;lpJ + $/, il/ Y*u, + ,* ?p*n7, tyf)f '#H- EYrN * ?)d^^r,tpJ + {rri{ -;:' ,Y&r^+dL Ferrtrrt (B'5) /*yr =-$t { n7r1'o W . F Srrt**p

?n f1lfayJ s ?iFxar prJ s

,k

(iil)-.

#r_4

* S- ( ,F/r' H.r + ( {f ); f #Fptrarl '* trn7 rfrn {{r)f + b o,i rtr r ldfi *1, Xer{,{,i6rl tr auP"fr+ f (,ilr'%6r** y*Fr/ Y"Fr{tyJ*Xrhryyt * *'# FrorasrJ ({f)r' *+ xrrt'{q$yl"=-Ee '(B'6) 7K t<

fuwnr'

III

Af 9e.Fof txpJ+?'P}ufi^Ar4FJ * ?Thyr{ pJ*e-.}ofrrruf + { l + fl* (en?Afrut ?, ?tfro^ W {}n?^fivt + }^ fro^o bL&^,*e^Vuflff ) ep # + L ( (i x)*' ?*%rn + (;.{)*rLu*rp =u;(H}arr;vtrFJ t 3*F'f r yFJ( irf)u'n ouF*t,ryfjJ(,'{f)o'}

* *lrq*? Xv rc {dtri o k7" ?-K

Nhrc Topt

* ?*lrYf Trr( i yfA'* ?o?utr€TrapJ * -+ c",s E^r # f $';*,,Fafi', (8.7)

122

+ p--f f i ilr' ?pFrr rapl + {;4r'ou fv,lczfrj {KL

+ ?* xur c r rryi1ir4,f+ ?,xprETFrl(itril^r - L ti/)*T X^, c rFrl -? ?2?,Ft, cprJ(;yrS; 3 3v D, t t zaf v +f ?,/Dhr s rd4t, h { b " -* xrrc*pr1 .= (8.10) h?1^7u ffn^,caftJ I

f # ( Q'X*d rf?rtr- ?* DfXf* t fry{3 +2n Df LaFril

Y*rt +?'fl*dtr-?ff*rr4{l*?*{TfrotlCy,

" *b"Zf

+Lf

( Fx r { y{J + (,'H)ftx/-x {r yJJ 4K L ?* ,g)fT + {i rh' Xu rr dri J+ }* X*r { r yfJ( i y f)rr\ Srlr r rd yr/i ilJ{;

+ fi # teprr+{ tr1rt t?/*D*T "hDsr *'

h

#"'D*,r

taprf =-LX*^qyyf

I

(B.r1)

( 2f Dpf tevrI -?r DartFvtr ) -

Aq;Nf rrd.;IJ k[, - ztilr* xo,rc/ttfJ eDnr tqrt1t;{)r1-,zDF,{;Nr)rrtops

+ Da,ftiril{'rrrrtf - (/'{/rt tL*f rrtfJ 3

+ E 'i lffrrrd .; - ? rtr*/r,B/I ) *r "ho" xxVrrprJ zt{ + fr- Qz *,of tfrt * #- *q,ry{/J (8.12) -'/'!

l,(

' -^/

it c

{{"

L23

#i {}opf Drf +?, uf bn/ - et' D,.u--e*?,Dr,) * j 1r

(?*gf ForfwJ+?u afirrcyi.'-e,ro,cyrj-?fr,Fr, \-

"s

'oli

*F?-- L- ( (iyl*u 3n d.*r tf ril + tflf*rlu NArrf ji, T ,rr)

-d

fr*?,NrrcfruJ({f):

?#

* h ?olu Drt,,*f y{ = *

h, {a'Dhu- },^}r'fo)ffrsi"# -';' ?Yh*r$i3

Do,tofrr, I

(8.,3)

fa*ar F*rrrll

--, Bf Xrf ra gfJ+?/_y.ffrn{ fi) Cff

+' ,ff,-.,. * f;*4r'x{ :fr-tl ac{sf J +2*t}*r trytf(ir\; + /$)f D*, talrf- F*Df,ffrln'*rrfl=*too*Lfr{{.,o,

#

(HFrtdf,lJnx,tFrf t7t)cry+*

-tfu -

,*r.ncrryJcsy

{r;v)r' ,n, tas i? * (/ilr, Dor fptrrJ #

brr fqr ttr + i? h ,,0*t,tyfiY -';- *c --

'

qt

*{;

DoFtruf T '

(g'15)