The canonical formulation of supergravity

Ncldcar I'h)sic,, BI .>-">~1978) 138-154 X,,r'h-lh>ll.ind Publishing C~mlpany

THE CANONICAL FORMULATION OF SUPERGRAVITY °

,%lartm Pl L A T i ** Joscp/l tlt'ttrv Lahoratorics. Prim'ct,m ('#tir~'rsit.l'. Prbtcct<~n. .N'J. 08540 Received I I July 1(t77 (Rc~i~cd l O O c t o b e r 1977}

The I[,linilhmi,in for the Ihcory of ,~upcrTravily i'~ ¢onipulc'd with l i l t frc'¢doln of arbitrciry [ o r ¢ l l t / rolalitHl~ inainlaincd lhrt~tight)tll. "l hc rL.'a,,Oll for lhc particular c'ouplill 7 tl%'d in thl., Iheory ;.llld lilt." c'oilnt'clion bt,'l~.~,t't,'il Stll~c'rs) inin~.'lry alld dcrivali~e motlpliilg arc tnad¢ pclrticularly clt,'ar hi lilt" c'alculali~m.

I. lntmduelion Recently, illllch aliClilioli has bc'cil given ttl the Sllpc'rsyiilillelric theory of gravily lhal resulls fioiu a ceitain ("llon-ininiinal") coupling ~lf Ihe spin-~ 14arila-Schwillgt;r field to il;e spin-2 gravil:ilional field II .21. This lht.'ory has the usual invariance Io c't~t~rcliiltiil." lr',lliStoi inalioliS alld local I,tnelllz rolalitqis. :iiid :in addilional g;itlg0 °l iii~,ariancc under lhe Inixing of lhe Splli-.; field wiih Ihc nlclric, The invarianee piopcrl its of till)" Ihcory tire iiitlsl clearly sl utlicd in ils I [anlilloni:in I'oiinulal ion, alld we delive Ihe I I:ilnillonian for supcr l,,iavily here. The I laiuihonian for supcrgravily has also been indcpl.'ndenlly o.unpuled I))' I:radkin and Vasilk'v 11 ~1, anti, in lhe liine gtillTc', by I~:~cr, I~a)', alid Siclh: Isl. "Io ililillducc' (ili ti ftlliiilitir COliiCxi) SOlii¢ of the lClllliliology ill the ¢aliOiiica[ Irealnicnl ol sysll.'lns ~'ilh gaiig¢ invaliancos, we fhsl review the Ilaillillonian tklrliltilalion I\lr Ihe Maxwell aild liloC:i fields. There are also siinilarilies bolwoen the Maxwell ik'ld couplecl Itl lhe gl:ivitaliona[ field and Stlpergravity, ]'lie forinal i.lifference bt,,l~,~,eeiisupergiavily wilh ils sil called "'ntln-iiiiniinal" coupling :illd lhc non-super s) inni¢liic iheoiy wilh nlininial coupling ix lhe stiin¢ :is the i.Iifferonce between tile Maxwell aild llltlca fiohls. Tile l lanlillonitin trcallncnl of the Rarila-.~chwingor field in t'lal space-lime is then given. The need for the Majiualla field I~ be anliclunnluling leads to inodificalions of lhe Ilainillonian t\lrinalisni. The properly tll'beilig first order in time derivalines lhal f~rinion [A.igrangitins havt.' auloinalicaliy leads Io ,i eonsliaini rclalin 7 Ih¢ •

i Viork suppt,ried in part by NSI' (lrallt No. IqlY-76-1"12(162. ** NSI I'rt.,-d,Jcior:il l'dlo~. 138

M. Pilati / Sttpc~, ra,'ity

139

field and its conjugate momentum. From this the basic anticommutation rules o f the fields are derived. Next the tlamiltonian for the full theory o f supergravity is found. As one expects. there are a total o f fourteen constraints representing the invariances of the theory: four from the invariance under changes o f coordinates, six due to the invariance under local Lorentz rotations, and four (one spinor equation) from the supersymmetry transformations are explicitly given. It has been shown elsewhere [ 11 ] that the anti-cormuutator of the supersymmetry generator with itself gives the generators of hypcrsurface deformations. This is what one would expect from the usual statement that the anticotnmutator o f two supersymmetry transtbrmatiotls is a translation. Let t,s explain some of our notation. Greek indices always range over 0, I. 2 . 3 and Lttin indices over I, 2, 3. Indices referring to the coordinate basis (world) arc left free. those with respect to a local orthonornlal basis (vierbein) are written in round b r a c k e t s ( ), and spinor indices arc written in square brackcts [ 1-Symmctri" zation is denoted as usual, e.g. for world indices (/at,), for vicrbein indiccs ((/.0(r)). etc. Our metric has signature ( - + + +). Tile representation of the Dirac matrices is chosen such that the matrices are real :,nd have the property (3,(°))-r = _ 7 ( 0 ) (7(i))T = 7(O Define 75 = i7(°)7(I)7t2)7 (3) with 75 = I. and tru ' ' = ~(Txt7 ~' - ..¢,,-/n). The Levi-Civita density has the property ¢(1123 = I. Deriv~,tives in tk~ur-space in the full connection (including spin connection) are dcnotcd by a senti-colon, tile derivative on a threestir face in the foil thrce-cot]ncction (also including spin cqt~nneclioll) is denoted by a stroke h and the derivative on the three surface in the ('hristoffcl ctmnecti(m isdcnotcd by a double stroke II.

2. Ila,uiltonian formt,latio,I of the Maxwell and i'roca fields 1 3 - 5 ] We review here tile Ilamilh~nian l'~_)rnmlali~n I'c~rthe massless and tnassive vector

fields. In ll:,t space-tinle the l.agrangian for the Maxwell field is '~

=

_ !4/ ", ' h i ' "/,'n,'

(2.1)

with 1'"m, = a ~ , A , , - ~,.4~, .

(2.2)

Tile basic field is A~ and its conjugate m o m e n t u m is 6£ rr~ . . . . . . . . 6,-l~u. 0

/.,~o.

(2.3)

The variables A~ and n"~ satist~' canonical COlmnutation relations. From (2.3) there is the constraint n° = 0.

(2.4)

140

31. Pffati / Supergravity

The Hamiltonian is H= nUAu.o - £, I L-

= a-,s-

i~rsa. I

r

- irr n r -

(2.5)

~ , r A o + ~rr °

where ~, is a Lagrange multiplier. Requiring that rr°o = 0 gives the constraint 7 - rr",. = 0 .

(2.6)

It is immediately true that 5' = O, and, since [n °, 7] = 0, the two constraints are first class. The constraint 7 generates the standard gauge transformation 6 a i ( x ) = f d x ' [Ai(x), ATG(.,")] = - A . / ( x ) .

(2.7)

For the Proca field £ . . . . 4"u,'-t-r" F~u,'+_ m A U A u ,

(2.8)

n u = ! ''u° ,

(2.9)

n° = 0 ;

(2.10)

so the momentum and primary constraint are the same as for the Maxwell field. The I l a m i l t o n i a n is II

=

t l"

a.,,¢.,

Irs

+

~nrnr

-

n~.rAo

-

m A U A u + ~n °

(2.1 I)

and/r ° = 0 implies the f u r t h e r constraint r 7 = n,r + 2mA ° = 0 .

(2.12)

It is no longer true that [7, nO] = 0 so tile constraints are by definition second class. They cannot generate a gauge transl'ornmtion since a transfornlation generated by one of them would change the other, i.e. it would transform us out of the theory that we started with. In general, first class constraints correspond to gauge invariances of the Lagrangian and second class constraints do not. For purposes of comparison with the supergravity I lamiltonian we will give the Ilamiltonian for the Maxwell field in curved space-time [15]. The Lagrangian is £ = 1(_(4)g)1/2((4)

R _ 2I1." " u v "i..p,,~, ,

(2.1 3)

and the l lamiltonian is I!

=

N[~g-tlZ(niiTrii + Nq-g,,,

I~.z) - :~

_

Io11217_ + ~.,',

I I/2[.rsFrs ~g

, " # l q + ~nq",vl - A o n ' . , + ~n ° •

+ ~g I -1/2~TrlT rlI

(2.14)

14 l

M. Pilati / Supe~'rat'ity

(See the appendLx for definitions of N, N i, n/i, etc. Torsion plays no role in electromagnetism.) Because it is useful below, we will make clear where the N i n i F d term comes from. Denote by [ the spatial index in the four-basis (see appendix) while i is a spatial index with respect to the three-surface. The indices are related by I? =

I" i ,

V i = vi _ N i V o ,

N i = I/;-

IV

(2.15)

1 "t

and so the expression I";~f"rs (which occurs in the process o1" the canonical decomposititm) is given by N g l l Z l.';~l.'.s = Ngtl?-t.'rsb'cv _ 3vsl~llz

l,'r° l;r.~

(:.t6)

= A~,nl a i,.~Sl..s _ 2N.~rd-t.r.~ .

3. Rarita-Schwi,lger fidd in flat space The l.agrangi:m for a (Majoram,) Rarila-Schwinger field in Minkowski space is [6J

(3.1)

.C = -- ~i d\w"' ~a y s y u a ,, 0,, •

The fidd ¢,~, must be anticomnlt, ling or (3.1) is a total divergence (the s:une is true of the I)irac l.:lgrangi,m in the Majm:ma c:,s¢), l.agr;mgians with anticommuting variables have been studied 171. Their only properly Ihal concerns us here is Ihat tile I'oisson bracket of two anlicommuti,lg Fields is symmetric to Ihe interchange of the two fields (i.e. the Poisson bracket algebra is graded ). (This gives the Poisson brackets the same algebraic properties as quantum mechanical anticommutators and makes it in many ways more satisl':,ctury to have "classical" fermion ileitis a,lticommuting even wheu they are not bl:,jorana.) We follow the convention that all derivalives with respect to anticommuting fiekls act to the right. The cammically conjL, gate fields for this kagrangian arc ~a. and a'x; satisfying the I'oisson bracket relations X

,

[~,q~l(.V), 7TI~sI(X )] = The conjugate UlOlnellttun 71o

I. XpO~sTF, .. ~. = ;~e VXrSI,,,

X

,

I%~l(.v ), ~.l.l(.~.-)l

7T~'

,X

.

= 6v~l,~ll~SlS(X,x').

(3.2)

is given by 13.3)

M. Pilati / Supergmvity

142 w h i c h leads to the constraints rr°=O , ~ki = 7"gi - - ~ i c m n O i - ~ m " / 5 " / n

=

0

.

(3.4)

i , )l is The bracket [Xi~d(.~'). Xl~l(X

c(j I[~l

" ~ ['~,l(-v) •

~i~l](.X.')]= --Olall~l ~

ij

~o"

(3.5/

st) these constraints are second class (it" they were first class, i.e. gauge generators, an arbitrary initial 0i could be gauge-transformed into any other ff~). These constraints result from (3.1) being first order in time derivatives, and they say that the field is "self-conjugate". The second class constraints do not represent atwthing fundamental in the theory and we would like to eliminate them. For the usual Poisson bracket [t", Xi] 4- 0 for a general ft,nctiotlal I'" o f the canonical variables: so we cannot eliminate the constraints Xi = 0 if we continue to use this bracket. Instead we use the l)irac bracket [3-51 [ I" which has the property [/", Xi I " = 0 for all I". The Dirac brackct is tlefincd to be

It, el" = I/", (;I - 1/.. ,~[,',t I t),,,,,i,,u,i

['i'll, (;I,

(3.6)

where/),.,~ is tile inverse o l ' C ii defined in (3.5), i.e.

( ,iiI"11~1/)i,. 131DI = 5,.i 51,~ID i .

(3,7)

It is found to be

th,,,t,,It,I

= ~('~,,,'YJ)l,,u,I

•

(3.s)

The basic Dirac bracket relations of the field ~i are derived from (3.6) [~il,~l(X), ~iDl(X')] °=

Diil.iDih(x,x ')

= - }(')'i Yi)I-II~I 5(x, x').

(33))

It is tiffs expression that gives the anticommutation relations of the field upon quantization. The constraints Xi = 0 can now be used before taking (Dirac) brackets, I m,'lOi-y, i.e. we use the expressions rri aqd ~e ~ ' , , r- . s ~' , vn ---- ~ltl'~O0 Itli interchange:tbly ~o,,,,,, ~jl" =

(1~,,,

~O.

Now compute the il:mfiltoniaq. (Notice that [rr°(x'). ~)o(x)]" = [,'r°(x'). ~0(x)J = 5(x, x').) l)efine H = 4;,,," - £,

(3.10)

3L Pl"lati / Supergravit.v

143

(the order of the ( a n t i c o m m u t i n g ) fermion field and its conjugate m o m e n t u m is d e t e r m i n e d by our convention for taking derivatives with respect to a n t i c o n m m t i n g variables). The tlamiltonian for the Rarita-Schwinger field is then [8] I.

tl = - ~ t e

i/kT

,

""

(3.11)

t,/iYs7o~i ~k + ie'/k ~oYsTiO/~k •

Requiring 7:o = 0 gives -

Oi(~,,,

(3.12)

7°o '''i) = 0 •

tim constraints aT° = 0 and 0 = 0 are first class, with 0 = 0 generating a Maxwell-like gauge transformation. J d x ' [ ~ i ( . v ) . A ( x ' ) 0 ( x ' ) l " = - ~A.i(x). and rr° = 0 implying that ~o is a Lagr;.mge multiplier. After the completion of this work. the author learned that the Ulanliltonian for tile l;',arita-Schwinger field in flat space has been independently c o m p u t e d by Senjanovich [I q].

4. Tile ilamiltonian d e c o m l m s i t i o n for st,pergravity

We COml)ute here the Ilamiltonian l\)r supergravily. Tile I'recdom to make arbitrary local [.orentz rotations is maintained throught)tfl Isl. Mr,oh of the m~t:,titm used in this section is explained in the appemlix. The l.agr,'mgian for supergr,'tvity is .£= ~(-('lik')l/2 (4)R

I. K/.~vp-i ~tc" V,'.TSTu/),,~.

(4.1)

wilh /),, = ~, + ~m.(,,)(#) ~/")(~)

(4.2)

¢o.(,0(~) = ~' [~,'~(,,)(Ov~ " , y(~) - 3.rG,(~ )) + ( ::(.)c(d)toTjQ(o)l " . . . . ,,~,") -

(c,

"" ~)1

+ C;,(,,)(~), V

(4.3) p

,

Cu(.)(~) = c(-)e(iO ( u , t , , and C'~,,~ is the contmtim~ tensor. The absence h, (4.2) of c(mnectiun terms appropriate to a covariant derivative of a space-time vector is the "non-minimal" coupling of s u p e r g r a v i t y . The Lagrangian can be written as (see appendix) £ = ~,\i~ "112 [ R + ( K ( q ) q ' P i + - P li ....

-

2r(i/)) K ( i i ) - K ( K

' P , I - ~te

....

2r) - K I , / I K l i / I

~,~ Y S Y , I ; , , ~ / ,

,

+ 2rl,ilKliil

(4..';)

3L Pilati ] Supergravity

| 44

with rii = 2SiLl. Pi = 2Si) "'/, qi = 4Si',. ".l. We take our basic variables to be cu(~), vu,' and Su,, x with conjugates p~(,~). ~ru, and puvX respectively, satisfying canonical commutation relations. The constraints

p~,~) = O,

(4.6)

rr° = 0 ,

(4.7)

~i = lri _ gll2~l.Tloii

= 0

(4.8)

,

pu,,x = 0,

(4.9)

are obtained immediately (we have used tile identity ei/kTs3, i = 2igll2TloJt'). Tile nlomentunl conjugate to c~ ~O is i p(,~) = _gl/2 [(/~-(ik)_

i . __ 1")] 1.(ik)) Ck(c~) _ t,(~x)[K

-- 5,l i ~,' l l 2 A X u IP~/k')'STp(O'.(c~) i.

-

+ ~igl/;A,~.k~,{XTSTuck(o)o

tt(t~)O ~i ) ~l' -- ligllZ"13#aiPf,k'YS'fUOl('~)~P

li~)p

_

(4.10)

,

,Id

whcre e xu''° =: N,k,I/ZA }qa''p and/t(t O - II~a { (,~). [t iS useful to define the tensor density h-t

,

k(h)

= g~/.,

m

i(k.(J,,,,)

l,m(~,

_ r(J,,,,))

. ,~:'k'"'"~^

r}l

+ ~igll'*(A ~ k P ~ T,~.Tu o I,. ~t, ÷ (k '" m))

gt/'. [(K(t,m) _ r(k,,,)) .. gk,,(K

__ r ) l + B ~'''' •

(4.1 1)

l:mm (4.10) comes am~ther constraint [9],

j(-)(t~) -. pkt,,)C~) _ pk(~)c(,) _ rtt, ot,)(~k

= 0.

(4.1 2)

As in the flat space case, (4.8) are a set tff second class constraints and we eliminate them here. The only i)irac brackets that differ ft'onl the original canonical I~oisson brackets are

tO,,, I,, I (-"). ll"~l(x),

¢',,t,Jl(x')l" = ~."-~/"(~'t

~,,(x

)1

~t,, "r., T(0))l.ll~l ,S(x. x'),

(4.13)

= - O,i~,~ h'"l D,.,, = ~.I-Os 3 ' 0 0 " / , ,

oxi'7(O)~(X,

X'),

( 4 . I 4)

M. Pilati / Supc~rarity [pi(c,)(x), pi(~)(x')]" = _[pi(~)

145

X"] O,,,,, [X n, pJ(~)]

= ~.~112 fsT(a)Oir..fL 0 i~7 (~3)~r~(X, X') .

(4.[ 5)

From now on rrk and g i / , 5~i71oik will be used interchangeably. If, as in ref. [8], one uses ¢,(,~) as basic fields (with proper density factors absorbed into them) rather than 0u then. instead of (4.14) and (4.15). one would have [pi(~), 0(~)1" = 0 and [pi(~)p/(d)] • = 0. but the Ilamiltuniatl that is obtained below wot,ld be more complicated in terms o f those fields. The ftmction j(~,)(,3)generates Lorentz transl\mllations in the Dirac bracket. The tlamiltonian is

II = ~;k(,O p ~('~) + 0,o rr ° - - £

+ g-I(Biil]i]

_ ~112) __ 2g-I(Hiirrtl - {Btr) - I ~ + "i(i/)T(ij) --T 2

+ KliilKliil

- 2TI#IKIiil + qipi - 2pili + 2pipi]

+ N }ig t/2A'~aiP ~a 757, Vt ~o + A' ak(trto Lk~t) + .~ieau''p ~aTsTu (7]v,,¢;~'" ~j, ,

(4.16)

w h e r e / i s as in sect. 2. The three-dimellsional ;malog of the derivative D~,, defined in eq. (4.2). is Vk = 3~, + ~k(,~)(~s)o ('~)(~s) ,

(4.1 7)

where ~/, (,~)(~s) is the three-dilnensional spin ctmnection defined by [!0,1 g~i(,,)(~)

I

k

k

k

"1- II(.,t)ll(c~)C,(ij)~k

e~ v)

i

- e3/,. ('i03)) + e(,O('(g)(al('k(~))

= ~. {('(g)(~i('/,(.c.~)

--

• s t 0 . (o~ +, ~)} + ~is, e(,~)e(,

7I

(,!"t) ' + n(p)~i/t(,~) (4.18)

The full three-dimensional covarianl derivative h which i~clt,des the cormecti~m ~?,i(,~)(~), has the property

ek(,~)li = 0 .

(4.19)

The derivative V~ in (4.16) is the derivative ~7~ without the torsion term CisteS(,~)e~s).

146

M. Ihlati / Supereravit.v

At this point we require that [>u,,,t = 0. Using (4.16), this implies [.5_

Su,.x - a vu %', if,. = 0 .

(4.20)

The constraints (4.20) and pu+,a = 0 are a second class s e t so eliminate them t'rotn the theory. The new Dirac bracket is such that i

[/,~,,x, 1:1.~,, = 0.

[&,,,,,,

-

•

(4.2i)

' T

," 1,,<.,, = [a v,.,'ra

,,:',,, Vl

*

(4.__)

where I'" is an arbitrary functional or" the fields. The bracket on the right-hand side of (4.20) is that given in (4.13) and (4.14). The expression (4.16) intlst be further reduced. If we expand the tellsor density II d defined in (4.11) we find ijii = _/ifll2

[ ._

~ll(
+

,v<

,iflkTi ) t~g + "-- f,4.(t'ir"l [ + gj,'yi) t~ r N

2 + ~.(~.V~dik)

i N* ] ~ii _ , i f / - - - ~,< "/'~,.}

(4.23)

The coustrahits (4.6) and (4.7) imply that N, N k, and ~ , are lagrange ulultipliers. I:uitilcrnlole r<,,ff .. (il) - r 2 = g,-I (IJ i/ Ill~ ~- all I 2 ) ,ittcl , • ,. tlSlllg .. , ( 4 , _~0 ) , . S('l . . tile . . .,... I;xpz'cssltHI .if IlBiilld -- }B z) + riii)-r (i/) -. r z is quadr:.ltic ill I.aglange multipliers and it nnlst be c:ulcellcd frtnn the I I;uniltouian. The ICllns that jusl tltl this t2tlllle fli.lln I • )l,ut'p ale" ~.yS"lu(;,.t,,o't'*l~+i. I f : l UliUini:llly cut, pied [;igt;.,ngi:.in, i.e. tnle ill whicli 1) z, ~ is ,eplaccd b)' +p;x, in (4.1), is used this will I,I, II haPl+cn. As ill tile t?ase tlf the plroca field, l al~l;.llige Illtlllipliers (specifically ~u) will elilCr Ihe I I:.llniltoilian quadralicaliy and there will be IiO first class gencratt)r of supers)li,lnl2tly lr:_lnsfurni:liiOliS. Now hlok at tim telin N, if I/2(lliitrij

i - ~lJ,7) = -~0(--5"~[j~i17 ij}

+ N k - ~)k(~'~j~i

~il') ,

14.24)

I v,'hich rest,Its from the derivative coupling of tile theory. The expressiun - 5"71 ~r i "_ii , which contributes to the supersynunetry generator, is v,'hat makes the theory super{¢~) • , symmetric since [ek , Yi f~irriil ~ ~ and [~k, Ti ~inii[ ~ ,-r'/; i.e. I'ermion :rod bosou degrees of freedom are mixed. There are no such terms in the I lamilttmkm (2.14) for the Maxwell field; the gauge symmetry ~1" electromagnetism is purely internal, h is remarkable that twu theories that are so similar in Ilat space should difl~'r so fundamentally in curved space (although the reasoq is clearly the derivative coupling in supergravity). After reducing the expressioq (4.16), getting rid ~fl"terms qt, adratic in Ligralige

M. Pflati / Supcrgravity

147

1 7~u , multipliers, using Su,, a - ,3, 7~ ~',., etc., we find

t f = A':t(~ + ,V"h~',,, + 77~0 S r

(4.25)

with ,3c'~ = ~[g-I/"(,-r;/~r u - ~rr") - (112 R n

_ -g,:2

ieilk~i..lsT,gi~ k

+,'r+ e:/~;'>~ ¢'/1,

(4.26)

, j ,Jr,,' = [ g,,irrikLi k + rtk(Vk ~,,, -- V,n ~k) + ~I . T~,, ~i ~i nil"'

_

gt/2 f,n ~t oik Vi ~k + /gl 12g,,,i(~;,7, ~i)lg

+

~glla~,,,V~)s~sVkO~ ]

,

(4.27)

am.I

S'=ieilkTsTiVl~k

_ ~TiClirrij

--

I,,I/2_~ i ~.,..7.&. ~t{ ¢ j w 1 1 5,u/ ' (4.28)

It shovhl be mired that the derivative Vi contaius torsion, i.e. terms qvadratic itl tile fiekl ~,~,. As in the case o1" the I)i,ac I lamilt(mian [101, terms of tile form ;zi/ok. lh;.,I indicale tile deriva, live Cul,pling in tile Iheory. are m~t present in ,"K'I. This.seeming contradiction is reconciled by the uon-c:mouical COI1UllUt;.llit)U relations (4.14) aud (4.15). Certain oF the terms iu .'ff'[ and A'"'.'R';,, ;.ire ;.lualogot,s to terms in(2.14). tile most i,nporta,II being N'"n'~( g;, ¢,,,, .. gm ~'a) which is k,loWl! [5 1 to be characteristic of vector gauge ileitis. The prese,vatio,i of the co,lst,ai,lts (4.6) and (4.7) in lime imply the furlher cousir:finis ,~c', = 0 ,

~c,',, = 0 ,

s' = 0.

(4.2,0

Tile constraint S ' = 0 is identical to that obtaiued directly from the equation of motion

eoiik 7S.,lilk ! ~ = 0 .

(4.30)

l)t,e to tile complexity of the co,numtatitm relations (4.14) and (4.15), directly pr
148

?,t Pilati / Supergrarit.v

forms of those constraints here and from the previously derived c o m m u t a t i o n relations for the new constraints it will follow that (4.29) are first class. To do this. the expression Wo(~o(,3)J (a)td) nlust be c o m p u t e d . Using (4.3) it is found lhat k • ~o(~)(,~)J (a)('~) contains terms of tile form (ct,oe k (~) + nUhh(~))J(~)(~). The time derivatives are c o m p u t e d with the tlamiltonian • ' + N"t'~(',~ + 7. ! t = ,~'~1 ~o S , + M(~)(d)J(~)(~) .

where

(4.31

)

,I/(~0(~) is a L~grange multiplier. Tile result is I :Z f f j j i i ) + N ' " [g,ni Jg/ k (Oo(~)(¢)J (~)(~) = N( -"~Jlg . k - ~Vi71 + ~t Z, , , 7 ~ i r j r / _

~I -i ' ~ , , , t~j J u" -- 2 g - I / Z ( n k , ,

, ,jlk] -- ~trr gk,,,)

+ ~ 0 ( ~t 7 / ~ i J i / - 7 L (u k"r g , - ~ J -- 2M(,~)(~) J("~(~ + total derivative.

(4.32)

. "(")(~) to (4.31) we find Subt ratting and adding ~t ~o(-)(~)." t l = N,'/f~ + N'",'ff,,, + ~ o S - ~too(~,)(~).l (~')(~J)

(4.33)

with I {..~llk I ~('l = 3('~ + i , . . ~ , k -- , i ~ i T J --

~jjij)

I [g- t / 2 ( ¢ h r i i - irr i 2 ) - gll'. R -

• qk--

V/~ k

--r g l / 2 ( ~ i T i ( a j ) l i + ikg I/2 ~ST[ 6'~S"fL ~ t + I g l / 2 T~iTJ(vi~l.,[k fig t -+ 2PgL. k + ~¢i7t d/iplil = 0 ,

(4.34)

, I r, .kj 1 -" I-"' ~ ( m = ~f,n + ~ L~,,, ;J , k + ~ ~ m 7 i (aiJ t/ _ a f l i T , , , ffi J ' t

- 2g I/2(rrk,. -- 5 n g k , , , ) J tk ] -

= [-g,,,in

I

ik

I t ~" Ilk + g , , d O k P lig + i V , , T i t ~ i W! + rrg(Vk d;,.

-- g l / 2 ~ , n T t o i k V j ~ k

--

V , . t~k)

I ~, I . , I / 2 ."E. + ~J,,, 7 i ~ i l )tl -- ,~.t~ ~,,, 71 ~d/'~J"[k~k '

+ ~ f i T m t~jP ii - 2g-112(~rk,,, -- ~rrgk,,,)J lh ] = 0 ,

(4.35)

S = S ' + I ' Y j ~ i j i j - ~7±1 ~k j lk = 2gl/2710/kvj~k

I_112.. I i - ( ~ ) ' "~.', + 5,g ti -- ~P(#)7

I~/i ~ i T i ~ j = 0

(4.36)

M. Pilati / Supcrgravio, pi/= pi(COeic,),

pk~ =_ _n(COp~e,).

149 (4.37)

The tlamiltonian (4.33) is the starting point tbr the derivation of commutation relations given in ref. [1 I ]: so the constraints given explicitly in (4.34), (4.35), (4.36), and (4.12) satisfy the commutation relations given there. Notice that it is the supersymmetry generator given in (4.36) and not (4.28) that gives the standard [1,2] supersymmetry transformation for the vierbein field (i.e. 6c'k(,~) ~ 7(~)~k). [laving obtained (4.33) the tlamiltonian decomposition is complete. To summarize the steps involved in finding the tlamiltonian (4.33): starting from the Lagrangian the momenta conjugate to the basic fields are found, immediately leading to ant, mber of constraints (among them the generator of Lorentz transformations (4.12)). Also included in this initial set of constraints, are a set of second class constraints characteristic of systems with fermions. Eliminating these constraints from the theory leads to modified commutation relations. The llamiltonian is defined in the standard way, and, eliminating time derivatives of fields in favor of momcnla, expression (4.16) is arrived at. Requiring that one of the initial constraints is preserved in time leads to a further constraint (4.20) relating the torsion tensor and the Rarita-Schwinger field. We use this constraint to eliminate the torsion tensor from the theory. On the st,rface expression (4.16) contains terms that are quadratic in l~lgrange multipliers, but these miraculot,sly cancel, leaving a theory with only first class constraints. In addition to this cancellation, the supersymmctry of the theory results from the derivative coupling of the theory (i.e. the term B i~ in the expression (4.11) for the gravitational momenta rdi). The t lamiltonian (4.25) does not contain arbitrary rotations. In an effort to include these with a geometrically n~eaningful l=lgr,mge multiplier (wo(.)(~)) the final form of the I lamiltoni:m (4.33) is arrived at. ! would like to thank ('. Teitelboim for many helpful discussions.

Appendix

Canonical decomposition o f I.'instein Lagrangian with torsion [ 12/ To establish notation, and because it has never been published, we outline here the canonical formulation of pure gravity with torsion [I 21. The torsion tensor is deliqed by [13,16J

2( I m ,

,

(A.I)

with I'X,, given by

bla: v =

hal

"h u,,,

(A.2)

M. Pilati / Supergrarit)'

150

where the bu are basis vectors. The c o n n e c t i o n FX,, can also be expressed as

FL' = {~v'} - C~:£x

(A.3)

where C,,~ x is the c o n t o r t i o n tensor C~,, ~' = S,,. x - SuX,, + SX~,, ,

(A.4)

and {l~u} are the Christofl'el symbols. The I lamiltonian formalism gives file evohttion of initial data from one arbitrary space-like hypersurface to a n o t h e r [4.141: so we focus on a space-like hypersurface. Take the basis b,, to be a coordinate basis with b i tangent to the surface. The comp o n e n t s o f the metric I~)g m, m this basis are [14,15] (4)Co ° = N ) N i _ N2 . .

(4)gO0 .

(4)gOk = (4)gko = Nk ,

(4)gOk = ~ " ,

(A.6)

(4)gi/

NIAq (4)g i/ = g i / _ .___N~,

(A.7)

= go '

I Ar2 ' N/~

(A.5)

where N is the lapse and Ni are the shift functions o f Arm)witt. I)eser, and Misner 1151 , ;lilt[ gt/ is the metric intrinsic to the hypersurface. ]'llenornlal n It) tile surlace is d e f i n e d by n • n = --I ,

(A.8)

n ' b, = 0 ,

(A.9)

and its c o m p o n e n t s

in the basis bs, are

n~ = ( - N , O . O , O ) .

( A . I O)

The extrinsic curvature tensor Ka/,, a nleasure o f tile change in tile normal as you

move across the surface, is defined by n;~ = --Kt,.dx a .

(A.I I)

We find Kat, =

Ar(4)po " t,a

-'"

(A.I 2)

!

=

~f~( -g,#,.o + Natlt, + Nhila) - Ca/,l,

(A.13)

.~L Pilati / S,pcr~mvity

15 [

where {I denotes a derivative in the Christoffe[ connection derived from the threemetric gii. The index J. denotes A L= _Al_~ _n~Au ,

tA.14)

and is related to quantities defined in the coordinate basis by

I

.4L = ~ ( . 4 o --,V'"A,,,) = - N A ° . From

(A.15}

( A. 13) one immediately gets

I

K(at, j = 2,\~,(-gah,O + NalIh + Nhlla) + T(ah),

(A.16)

K l,~, I = St,a t ,

( A. 17)

r,,h ~ 2 S . t h •

(A.I 8)

wit h

The iagrangian for ordinary gravity is

t, = ~ fd'~v(-v'k,)'/:

~'W~,

(,x.z,~)

where (4)g = det gu,,,

(_(4)g)1/2 = N g l l 2 ,

( 4 ) R = g U V ( l u. v .., ~. - l.t a., ~.. L., + l ~ l ~' u v "~

_

g = dot gii ,

.i"'~ ~v.I't~u , I ~ .

(A.20)

We wish to decompose the t,agrangian in terms o f quantities intrinsic to the hypersur(ace [4,14,151. Using the (;auss-Codazzi equatio0. [16] (4)1?'", H, = R ' " # k -- ( K / i K k " ' - KhiKi ''')

(A.21)

with R'ni/k the Riemann tensor co,nputed from gii t)nc linds " t ' h a + K2 _ "~ _ (4) R tl3~ (4)R = R - Kab~'.

The relations

(A.22)

[41

tt~3;'r tt'~;l3 = K a h K t'a ,

n~:~

•

- n,~;'d,

=

ttl3;13 = - K ,

RS~tJ'r n~ + 2 S ~ f bt~;a

(A.23) (A.24)

152

M. Hlati / Supergravity

imply R LHL~3 = (tt 7t1~3;~t -- tZ/3tZ7;.r)3j -- Kab K ba + K 2 - 2SaLPna;p •

(A.25)

T h e Lagrangian density is now

£ = ~ N g I/z [R + Kab K ~a - K 2 - 2(n'rn~:. t - n~n~':,t): ~ + 4S'~l°n,~wl. (A.26)

Define the vector density va = Ngl/Z(n~na:~

- nanZ':Q •

(A.27)

Tile Lagrangian has the form £ . ~ N g t.l Z [ R + K. a t ~ K o. a

. K z + 4 S., ~ ± p n , ~ . o l

a°~ + .~v ~ S,,,~o .

(A.28)

Ignoring tile total divergence and after some ft, rtller algebra the Lagrangian becomes .C = ~IVKI/Z(R + K ( a b ) K (at')

-

"~'- - t u n ~ ' " --k'("h) KI,#,I Kl'lt'l

+ 2rl,,hlK lat'l - K 2 + 2 K r - p a q a + 2pala - 2 p a p a ) .

(A.2q)

where I is the covariant derivative in tile full c o n n e c t i o n on the three-surface and qa - 4S,, Lt ,

(A.30)

Pa :_T3,t' .Oat , h

(A.31 )

.

I!q. (A.29) was first derived by Alvarez [I 2], and we t, se his notatio,i. Now that the Lagrangian has beeq d e c o m p o s e d in terms of quavtities defiqed on the hypersurface, we will obtain the tlamiltonian. The basic canonical variables will be taken to be the vierbein e ("1, its conjugate pU(, O [9,101, and the torsion .S'u~,x with conjugate puvX. (Given a coordinate basis b u and an ortht)normal basis b(, O the vierbein is defined by b u = e~a)b(a), b(,~) = eU(~,)bu. F r o m tiffs definition it ftfllows that ..u _ . let) _ 6 ( , ~ ) e p ( ~ l ) - ~(~)(~) and e u e,,(,~) - gu,'-) The Lagrangian contains no time derivatives o f torsion; so we immediately get the constraint 1" t ' x = 0 .

(A.32)

The conjugate to the vierbein is given by p ° ( , 0 = 0 , a constraint

(A.33)

,eL Pilati / Supergravity

15 3

and pi(,~) = _ g , / 2 [(K(it,) _ r(ik)) ek(,~) _ e i ~ ) ( K _ r)]

(A.34)

from which the constraint [9,10] J(~)(~) - p i(°') e[~) - p i(~) e[°') = 0

(A.35)

follows immediately. The symmetric tensor density 7Tij = I 1 . . i ( c 0 .,j + pj(aJ~,~tO ) 2 ~t ~ ~ (~)

(A.36)

has canonical commutation relations with gi/. The llamiltonian is I I = l~i(~)('i

(~x) -

,F.

(A.37)

= N,'JC c + N i s e i ,

with 3('1 = I z g l l Z ( t d / r'"r i i _

-- jI n 2 ) - g l l Z R

2gll:l~ala + 2 g l l 2 p a p a

+ gllZ(r(i/)r(ii)

__ ,'9 - g I / 2 "r[a/,J,'~r , h a t + ,k,I

- 7 .2 ) + g ' l Z q " p a 12Sat, l ,~'ahl

,

(A.38)

and 3Ci = - g h n ~ ' q l L j •

(A.30)

l!xpresscd ill terms of ti,e vicrbcins tile lapse and shift are Ni = t,o(~oe[ a) ,

U = N / N / - eo(,~)e(o c') ,

and the eo(c0 are Lagrange multipliers. o Reqt,iring the conservation of the constraint p(,~) = 0 in time (i.e. the additional constraints 3C~ = 0 ,

J() = 0 ,

(A.40)

[:,~,,),//I 0

= o)

(A.41)

are obtained. "['he preservation of the constraint 1'uvx = 0 requires the additional constraint [12] Su,,x = 0

(A.42)

?,L Pilati / Supergravity

154

and the torsion can be completely eliminated from the theory. The constraint J('~)('~) = 0 generates rotations and it is automatically true that [./('~(~), tlJ = 0 [gJ. The final tlamiltonian for the theory is

H = ,\'~± + ,,\'/,Tfi + M(,~)(~)J (c')('~) .

(A.43)

where 3I(~)(~) is a Lagrange multiplier. The following formulas were found to be particularly helpful in the course of calculating supergravity's tlamiltonian: ,0

_

1

(A.44)

('o(c~) = A'n(cO + A'mcm(cO ,

(A.45)

eiik'tsTi = 2 i T t o ik ,

(A.46)

n('t)cm(.~) = 0 .

(A.47)

Re ferellees J I] D.Z. Irc~:dman, I'. van Nieuwenhuitcr~ and S. l'crrara, l'hys. Rev. I)13 (1976) 3214; D.Z. l'rt.'¢dman and P. van Nicuw~:nhuizcn. l'hys. R~:v. I)14 (1976) 912, 121 S. l)eser and II. Zumin,~, Phys. I,etl. 6211 (1976) 335. [31 P.A,M. l)irac, I.ecturcs i~n quantunl mecharfics (l|elfur (;radtultc Sch~ol of Science, Yeshiva University, 1964). [41 A. I [;.In~.on, T. I(egge and ('. "l'¢il¢lb~inl, ('~nstrained 1lalnilt~niarl systents ( Accadcmia NaziLmale dci I.in~ei, I",~mlc, 1976). 15 j ('. T~.'ilvlb~)inl, 1lIe I lamilt~nivll str~aeture ~1" sPace-time, Ph. 17. thesis, Princet~ll (1973), tmpublisht:d; (7 "l'¢itt:lb~im, I{~:la tivity, fields, strings and gravity, Caracas, 1975. ed. (7 Aragonu' (Universidad Sim6n Ih~ffvar. 1975). [6] W. I,~arita and J. Schwinger, I'h~,s. I~,¢v. 60 (1941) 61. [71 P,. ('asalbu~mi, Nu~jw) Cim. 33A (1976) 115. [Sj S. I)user, J.II. Kay and K.S. Stt,'llt:, Brandeis Uni',ersity preprint (1977). j9j S. l)est:r and ('J. Isham, l'hvs. Ruv. 1714 (1976) 25(15. 110] j.l,i. Nels(jn and ('. "['¢itelboinl. Phys. Left. 69B (1977) 81. [ I I J ('. 'l'eitelb~)iln, Phys. l/,¢v. Left. 38 (1977) 1106. J 12 J O. Alvarcz, lille initial value prt~bleln and tile llamiltonian ~1" tile I':instem'.('artan-SciamaKihble th~.'c~ry,Scnit>r Thesis, I'rinceton University (1974), unpublished. [I 31 I.W. Ilehl, I'. wm der Ilcyde, (;.1). Kerlick and J.M. Nester, Rev. M~d. Phys. 48 (1976) 393. I 14 ('.W. ~,1isner, K.S. "l'hc~rne and J .A. Wheeler. Gravitati~)n ( Freeman, San I:rancisc(~, 1973). [ 15 R. Arn~w'itt, S. I)eser and ('.W. Misner, (;ravitati~n: an introducti~n to current research, ed. L. Witten (Wiley, N¢~, Y~)rk, 1962). J16 J.A. Sch~uten, I/,icci calculus (Springer, Berlin, 1954). [17 C'. "leitelboim. Phys. Ixtt. 69B (1977) 240. J 18] I'~.S. Iradkin and M.A. Vasiliev, l.cbcdev Institute pruprint (1977). [19J (;. Scnjanovich, Phys. I/,ev. 1716 (1977) 3[17.