The wave functional of a super-clock

Nuclear Physics B27411986) 60-70 North- Holland. Amsterdam

THE WAVE FUNCTIONAL OF A SUPER-('LOCK S. ELITZUR"

CERN. Gene~'a. S~'tt-erlaml

A, FORGE

Racuh Institute of Phy~,'.~. Jcrttsalt'ot. l~rat'l

E, RABINOVICI"

('ERX. (h'nt'~'a. Swtt:¢rh~ml Received 17 January' 1'486

Wc discuss a model with h~.-al suf,crsymnlctrical features in two tliH~cnsitms. S~l~,in[~. the c~mstraint equations, a .',ingle 'a'zmve I'unctior~al for a .,,upcr-ch,ck ix foum.I. It c,dlibit~, m~ corrclalions between geometry and the I'ermion ilunlbcr.

I. IIitr~xluction It has been conceptually useful to study gauge theories in the hamiltonian formulation. Abelian and non-abelian gauge theories have bccn analyzed in tile el() = 0 gauge. As this is not a complete gauge fixing, the hamiltonian conmlutes with tile generators of the residual gauge transformations. These in turn are imposed tt) vanish when acting on the physical states. The eigenstates of the hamiltonian are

thus functionals depending only on singlets under time-independent gauge transfornmtions. Gravitational theories are treated by analogy [1] in the synchronotis gaugc, that is the gauge in which gO, = 0; g(X)= 1. In this cl,se the hanfiltonian itself is part of the residual gauge symmetries. These consist, in addition, of the generators of spatial reparametrization transformations. The wave functionals of the system (the Universe) are thus to depend only on spatial reparametrization invariant quantities and must be annihilated by the hamiltonian. This latter property is more difficult to characterize, it is instead labelled by calling this condition the Wheeler-DeWitt (WD) equation. The WD equation was solved on a subset of possible geometries •Pcrrnanent addrc.,,s: Racah Inr.titutc of Physics. Jert, salcm. Israel. This work ix supported in part h,, the fund for basic research of the Israeli Academy of Sciences and the A,ucrican-lsracli Ill-National Science Foundation.

0550-3213/H6/$03,50 " Elsevier .~icnce Publisher.,, BV. ( North-I Iolland Physics Publishing Division)

S. Eht=ur et aL / Super-ch~'l~

6l

[1.2] and in perturbation theory [3] in various dimensions of space-time. Its cosmological implications were studied as well [1-3], In this paper we wish to study the wave functional in the presence of fermions. To this end one needs first to identify the set of spatial reparametrization invariant quantities, involving fermions, on which the wave functional may depend. Next. by solving the WD equation, one would obtain in genera[ information about the correlation between the geometric content of the wave functional and its fermionic structure, in particular, we wish to study a model in which the fermions are coupled in a locally supersymmetric manner. The simplest such system is supergravity in two dimensions. By a simple counting of degrees of freedom, gravity in two dimensions is an over-constrained system. The metric tensor g,,, consists of three degrees of freedom. Two are removed by working in the synchronous gauge, quantum mechanically two more should be removed by imposing, as mentioned earlier, the WD equation and spatial reparametrization invariance. Banks and Susskind [4,5]. appealing to Polyakov's [61 work in the conformal gauge with zero internal degrees of freedom, have suggested that due to the conformal anomaly in the ghost sector, a modified system with a total of one quantum degree of freedom per point in space time may emerge. A calculation of the effective action due to the ghost in the synchronous gauge has not been perftwmed. Adding tire two-dimensional gravitino. X~, does not e,alarge tile number of degrees of freedom as the gravitino being a gauge particle is over-constrained as well. in the gauge X'~'~= goJ = 0, g~M~= 1, one gets a rnodcl with a bosonic field gll and a fermionic one X". With the same nmuher of fields we construct a lagrangi:m which is the supersymmetric extcnsioq of the one suggested by Banks and Susskind [41. The system will have all the residual symmetries including timc-indepetadcnt supersymmetry. On top of being a solution of the W[) equation and spatial rcparametrization invariant, it must be annihilated by the local supersymmctric generator. We have not been able to show that the supersymmetrical extension of the lagrangian of ref. [4] is indeed 2d supergravity. We thus view it rather as a toy mt~lel, cont,'fining fermions and some supersymmetrical elements, for the study of the interrelations between fermions and geometry. It has been suggested by Witten [7] that the bothersome question of handling several solutions of the WD equation may be resolved in supergravity in which only a first-order equation must be solved. In the model we study, this indeed turns out to be the case, supersymmetry necessitates a negative cosmological constant which leads, already in the WD equation, to only one normalizable solution. This solution is picked also by the first-order equation. The paper is organized as follows: in sect. 2 we describe the supersymmetrical system studied. In sect. 3, canonical quantization and the various quantum constraints are discussed. In sect. 4, the constraints are solved and the wave functional is obtained.

S. EItt'ur et aL / Super-chwl~

62

2. A locally supersymmetric model for Imo-dimensional graviD' Two-dimensional supergravity involves four zweibein components e,~ and four Rarita-Schwinger fields X~,'. in the supersynchronous gauge, X~ = 0 thus, off-shell, there is one gravity variable e and two fermionic ones, a single scalar auxiliary field is needed to complete a superfield E. E=

(2.t)

e + O x + ~ 0 + OOF.

We assign to E the transformation properties of a density in superspace. It is more convenient to define the superfield (2.2)

~b = 2 E t / :

in terms of which a supersymmetric extension. expressed by:

A~.

A , = f d:x d 0 d t J ,, (d,Pa,

of the system in ref. [4] is

(2.3)

- .,p:).

where by convention c~ >1 0. 0 tire left derivatives and d.d are _ a

ii d=

= -- - iO-- . i~O 3t

a

(2.4)

~ - iO-- . 30 i~t

The action A, is indeed still invariant under some rcsh.lual gauge transforn+ation.,,. These can be expressed in supcrspace Z ~= (t, x. O, 0 ) = ( Z °. Z L. W'. Z i ) by Z ~ -. Z 4 +,~ 4(7. ).

(2.5)

Time-independent local supcrsymmctry transformations follow from:

C=i(~O+~'~),

~'

=0.

C=~.

~ = ~.

(2.6)

where ~', ~: tire time independent. Under these transfor,uations, the superficld q, is changed by 6q~: (2.7)

$tP = i( ~Q + Q ~ ' ) @ .

where Q and Q are given by: a

;i

Q = 0 0t - i t - ~ '

a

Q = -°~t

+ i/~.

(2.8)

The action A is also invariant t, nder spatial reparametrizations for which: ~"= O.

sct = h ( x ) .

~6=~i =0.

(2.9)

S. Ehtzur et al. / Super-ch~'k

63

E transforms under both symmetries as a density v, hich. for (2.9) reduces to 0 a E = -~--~x( h E ) .

(2.10)

These invariances emulate some aspects of two-dimensional supergravity and in any case by observing the wave functional of this system, one may study the properties of the wave functional in the presence of fermions. Expressing the superfield q~ as:

(2.it)

¢, = q + O~ + ~,o + o+s,

the lagrangian reads, in component notation, after eliminating the auxiliary field as 1 "~ ..~'= ~_q" - 5a'q" + i ~ + a~l'~. I

~

•

(2.t2)

In the absence of fermions and recalling that by (2.2). q = 2e t'-'. the bosonic sector reproduces the lagrangian of Banks and Susskind [4 I. The terms !a"q'- and a,~4, are the supersymmetric pair corresponding [81 in higher dimensions to a gravitino mass term and a cosmological constant which is forced to be negative by supersynunetry. The term ~O" is tile conjectural dynamical content of pure gravity in two dinlensions and ~ that of its supersymmetric Rarita-Schwinger partner. The bosonic and fermionic sectors are coupled only by tile quantum constraints of spatial reparamctrization invariance and tile fermionic (]auss law which will be discussed in tile next section. In conlponcnt notation, the SUSY tra,asformations (2.7) and (2.8) are expressed a s:

aq=+~,+~,+,

a,l,=+:(-i,?+aq),

Q=~k(LI-iaq),

,~Zv=+(i,'l+,q).

Q=~(Ll+i,,q).

(2.13) (2.14)

Next we proceed to the quantization of the system.

3. Canonical quantization and the algebra of constraints Quantizing the lagrangian in component notation, one ascribes conjugate momenta P = O and ,'n'¢= - i~b to q, ~,

[q(x),p(y)l_=iS(x-y),

[~(x).~,(y)].=~(x-)')

(3.1)

in terms of which

Q(x)--+(x)(p(x)-iaq(x)),

~)(x)---~(x)(p(x)+ic~q(x)),

(3.2)

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S. Eht.zuret al. / Super-,h~ch

v, hich generates transformations (2.13) by:

8,,(,.) = ,f d,.

+ C)( ,'),( , ). ,(

u = (q. ff.,~).

)]

(3.3)

The measure in q space is d q / q due to spatial reparametrization invariance. The hermitian representation of (3.1) is thus:

p ( x ) = 7 qt'-(x)°--77qoq~xj

'."-'(x).

'~(~)

8ff(.~-)

(3.4)

The wave functional is in general q'(g,, q) and its inner product defined by:

(q'tl'i',_)= f D(Inq(x))Dq,(x)'t'~(~.q)(* 'l':(,l,.q)).

(3.5)

where the duality operation * acts on the form basis for the fermionic degrees of freedom. A zero form being the empty state, a one-form contains one fermion and. in the presence of some cut-off, an N-form is the full state. The du;,lity • operation tt, rns n forms into N - n forms. In form space the inner product is (fltl//.,)= //'ll.~(*fl,) = f f l , A ( . f l l ) , hi terms of (3.4). Q(x) and Q ( x ) arc hermitian conjugates. With the definition (3.5). q'(de,, q ) D ( I n q ) expresses the amplitude to find thc Universe in a specific m form and q(x) configuration. This wave function is subject to so,no constraints. The residual gauge symmetries :ire to annihilate the wave function. The first constraint is i,wariance under spatial rcparamctrization. The transformations (3.9) ,,re explicitly given by: 3u 3x

8h

,, -

=

(,/.

t,. ~,.

~)

(3.6)

Ox

under which 8(h~) 3'd'= - -

(3.7)

the hernfitian generator of this symmetry, ~r(x). is found to be:

i I aq(.~.) ) Op(.,-)1 '~(") ~(")= 2[ ,--G--.~ p(')- q(" -o.-7-! '-"(~(")-~(~) ,--G--.~)" -

"

"

(3.s) reprodvcing (3.6) through:

8,(.,:) =

ifd.l'h(.v)[,~(.v). ,(.~)1

.

(3.9)

S. Elitzur et al. / Super-cl~'l~

Under a finite reparametrization x - , x ' , density

ax t/2 q'(x') =[~x" q(x).

65

the superfield q~ transforms as a half

Ox t/,~ ' ( x ' ) = [ O-~'

,(x).

(3.10)

(x)/q(x) is thus a scalar, a fact which will presently become useful. Invariance of g'(q. 4') under *r(x) implies that: (3A1) It will be more convenient to define a functional {'(q. if) by: '/" = D[,n'~'.

(3.12)

Dl,~z=exp( ~f dx~5(O)lnq(x)).

(3.13)

where Dlu I is defined by

Using this notation one has

,,(x),v =

(3.14)

where ,rT"(x) is obtained from ~r(x ) by inserting the more conventional ( i / i ) 6 / 6 q ( x ) = f,(x) instead of p(x) as given by (3.4). Solving ~'(x)~' = 0 ensures ~(x)q" = 0 as well. The second residual gauge invariancc is imposed by the WD equation

X'( x ) q'( q, ¢; ) = O,

(3.15)

..,'¢'(x) = _t,p2(x) + ~a-'q2(x) - ~ , [ ~ ( x ) . ~ ( x ) ] _

(3.16)

where

and ~ " ( x ) is semi-positive definite by: [Q(x ). Q ( y ) ] , = 28(x - y ) . ~ ' ( x ).

(3.17)

The third imposed gauge invariance is the fermionic Gauss law:

Q(x)q'(q,~)

=0,

O.(x)q'(q,~)

=0,

(3.18)

which are first-order equations containing the same information as (3.15). This model bears great similarity to Witten's supersymmetric quantum mechanics.

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S. Ehtzur et al. / Supt'r.(h~cl~

Defining Q in analogy to .k. one may consider the equations

Q(x)'~'(q,4')

=0.

Q ( x ) ' k ( q . ~) = 0.

(3.19)

We conclude this section by writing out the algebra obtained by the constraints:

[q/x~.Q(>,)l.=o.

[O(x).Ol>,)l+=o.

[Q(x). Q(.,.)], = 26(x-y)oU(x),

[ Q ( x ) . ~-(y)]. =

[~(x).

[~(-,).

~(.,.)1 _ = ,~;(.,- -y)~2(>).

[~(_,-). ~(y)]

~(.,)]

i6;(x-y)Q(y).

= ~:(., -.,).Y{

y).

(3.2o)

= '_,i,S:(x-.r)(~(x) + . ( y ) ) .

This algebra can be consistently realized over the physical states ~hich are simultaneously annihilated by all these operators. We now solve these constraints.

4. Solution for the WD st,pers.~mmelrie equations 4 1. SI'AI-IA[. REPARAMETRIZATION

INVARIANTS

in tile proposed model for two-dimensional gravity, the wave functional depends upon ol'~c par.,mctcr: the length of the Universe. The question which transcends the particular st, pergravity model we arc studying is: which new global invaria,u forms appc:tr in the presence of fcrmions': The functional space as defined by DcWitt consists of wave functions depending on spatial rcparamctrization invuriant quantities. We denote these by ~2,. In a general n o t , t t i o n , 0'~q(.~ -)

s,, = f d.,- ,;:(,,'*'(.,). 4'~*'(x) =

q~'(.~.) =

i).~

i'~'~'4' (x) fix ~

(4.1)

Uqdcr arbitrary variations 6q(x) and 84'(x), the corresponding variations of the f u i l c l i o n s arc:

8u(x--'----) = ~" 8t-~x) +~.~, '

u(x) =

(q(x), 4 ' ( x ) ) .

(4.2)

t

The variations of .(2, arc given by:

8u(x----~ = ,,,Y"-.,,(-1) ~

[ ,)u'""(x)

"

(4.3)

S. Ehtzur ¢t al. / Super-chM~

67

The constraint equations (3,11) and (3.19) are written as:

Oq( x ) o.~

,~ ~q( x )

0

,~ q( X ) Ox ,Sq( x ) +

o~(.,:)

o.~

g

o

,~¢(x)

q,(X)ox ,~,(.,~

g

) /

q,=o.

(4.4) •4,(x)

+,q(x)

'('

8~,(x) 8q(x)

7,=0.

(4.5)

aq(x) ~'=0.

(4.6)

)

We first proceed to solve the condition of spatial reparametrization invariance. (4.4). The spatial volume of the system is still a suitable ~2, variable.

p.,, = f dxq-'(x).

(4.7)

The addition of fcrmions allows us to construct new such invariants. In the zero form sector. $,~, is the only available invariant. In the one-form sector, an additional invariant. I2,. arises by integrating over the position of the single fcrmion

l-) t = f d x q ( x ) d/( .~.").

(4.8)

in tile two-form sector there exists a whole class of new invariants depending on the invariant distance between the two fermions. This now generalizes to any n-form. It is convenient to present these variables in terms of their "Taylor expansion (which coincides with an expansion in covariant derivatives). In a given n-form sector, a general term in this expansion is:

s),~"'J = fdx "" (q"'(

,-t (, q2(x) &r ]

I q(x) ]

B

(4.9)

m~ is the rank of the derivative appearing in the j t h term of the product which consists of n terms. { m~} is the set of these integers. Recalling that ~/(x)/q(x) is a scalar. ~[/t",,,,l(q~(x)) must also be a scalar, that is it may depend only on 12o. We denote:

' j

,.,~ ,_,

q:(x) 0x/

/

(4.10)

b8

S. Elttzur et al. / Super-chn'k

where {m} = (m t, m . . . . . . m,,) with 0 ~< ni t < m. --- < m,, because of Fermi statistics. I2.t) and ~i coincide with those defined in (4.7) and (4.8). ~(.") is given for example by:

~;">=

( q-'(x) '

fdxq(x)ff(x)

ax

~

"

m

= 1.3.5

(4.11)

....

The calculations simplify by defining

O(x) = q(x),

~(x)-

,~(x) q(x ) "

(4.12)

from which it follows:

~l

8

7,(x)

8q(x---~ = &7(x)

8

~7(x) 8~,(x)

(4.13)

8 / 8 ~ ( x ) and 8/8~,(x) commute, thus (4.5) and (4.6) become:

)

7(x) ,s,7(.,-----S+,,,7(.,) ~,=o.

'('

87,]x) 8,7(x)

~

=o.

t4,4,

Ill forlu ilot;ition, the general r e p a r a m e t r i z a t i o n i n v a r i a n t wave f u n c t i o n is:

,b='i4)(.LJ,,)+L),~',(.f.),))#- ~ L~(_,"')~l")(L)o)+(scctors>_.3). (4.15) ,)i <~.h.l

It renlains to select between all these wave functions the supersymmetric one. 4 2.

T111:.

SU PERSYMMETRIC SOI.UTION

We first study explicitly zero-, one- and two-fornls and then present a general solution, w(x) and ,Jf'(x) can be diagonalized separately in each form sector. (i) Zero-form sector. (~(x)~'. = 0 reads, using (4.2) and (4.3): O(x)~/(x) ~ +

)

~ci~'o = 0 ,

(4.16)

automatically annihilates the empty state. On the contrary. ~,(x) will turn the zert)-form into a one-form, on top of that ~(x)~e(x) is x-dependent, thus one must dem:,nd that the x-independent parentheses vanish. From which follows: ~'o(12o) = coexp( - ~a,(2()).

(4.17)

S. Elit:ur et aL / Super-ch~'k

69

For the full state Q and Q would change roles, it is the convention of having a >/0 that has chosen the normalizable zero-form solution over the non-normalizable full state. (ii) One-form sector. Q(x)~'t(I2o)f? ~ = 0. reads:

=,~(x)~,(x) = 0). (4.18) However. for a ~: 0. Q cannot annihilate ~t(I2o) = clexp( - ~cK9o) thus ~'t(P.o) = 0. (iii) Two-formsector. ~I',=~.,. oad'~'~')~I22 ~ o) 12~,"'~. with: ~ ~"~

I

s~T"= fdx#'-(.~)~,(.~) #-'(x)

ax) (5(x)),

m

=

1.3.5 ....

(4.11)

Annihilation by Q(x) requires: d ~/'.~" ~

,~(.~)/,(x) ,,, d

dft,,

/

1

+ ~,,,/,~"' ~,'"' + ~ ~',~"~~ , ""' "

,,,>~

"

SO(x)

2Ct(x)

]

--0.

(4.19)

Tim last summation starts from m = 3 because (~SI2[u/8#(.~'))-'--0. Explicit x dependence in (4.19) appears in the factor ~(x)~,(x) and in (IH2~"'~/8~(.~:)) x ( l / ~ ( x ) ) ( m >t 3). The latter are linearly independent as they contain a different number of derivatives, this requires ~'~'J---0 for all m t> 3. The reduced equation (4.19) is (d'~'~t~/dI2,,)+ i~'~t~ ,= 0; the soh, tion of this equation. ~ ' , = c : e x p ( - [,,[2,,)$2~,t' cannot be annihil:tted by Q(x), thus a[so ' ~ , m 0. (it') General solutions. This line of reasoning extends to all n-forms. The equations Q(x)q" is in general

a,b ,7(x)¢(x) ~ + ;,/, + ,,~_, E -

1 2O(x)

8~,'", 8~(x)

7,/, a~,;',

0.

(4.20)

(m

Tile n = 0 term is pulled out of the sum and the n = 1 term, which has already been shown to vanish, is omitted. Again, all the explicit x dependence in parentheses is in the linearly independent terms ~[2*,,"'~/O(x)80(x). These must be made to vanish separately by setting ~ - - 0

(except for n -- 2, m = I because (812["/8~/(x)) -- 0).

(4.21)

The resulting reduced (4.20) was just solved several times, confirming that the

70

S. Ehtzur et ul. / Super.~h~t~

solution is: q,

4,,, =

-

(4.22)

The wave function is frozen into the empty state. Due to fermion number conse~'ation and the absence of other fermions, it was clear that a solution would reside in some n-form, in the zero-form sector there exists no quantity to correlate with ~ , . Only for c~ = 0 there would be identical plane wave normalizable volume independent solutions in every sector.

5. Concluding remarks By working in the fermionic occupation number basis, we were able to give a general characterization of spatial reparametrization invariant states containing fermions. For c~ :~ 0, supersymmetry did indeed pick a single solution to the WD equation, the single solution appearing in the case of negative cosmological constant. This Universe does not gravitationally collapse but it is mainly concentrated at a volume of the scale of the cosmological constant. Generically, this implies that no long-range physics survives. Correlations could have formed in principle between bosonic and fcrmionic occupation numbers conspiring to maintain zero energy. however this was ruled out. the clock has zero-fcrmion number. The fermionic clock is frozen. It is not clear to us whether this is just a property of the simple model. Indeed the gravitino in general could bc an integral part of a super-clock or it could well behave as another piece of matter coupled to the clock. In the case studied, the "gravitino" behaves as the fixed fcrmionic label of the clock. We wish to thank Tom Banks for a useful discussion.

References [lj B.S. l)cWitt. Phys. Roy. 160 (1967) 11.t: C.W. Mi.sncr. K.S. Thorn and J.A. Wheeler. (;ravitatic, n (l:rccman. San I:rancb, co, 1973) [2] J.II. I{artlc and S.W. Ila',,,,'king. Phys. Roy. I)28 (1983) 2460: S.W. tlav.'king, Nucl. Phys. 11239 (1984) 257 131 T. Banks. W. I:ischlcr and L. Sus.,,kind, Nucl. Phy.s. 11262 (1485) 154: C. I [ill. I:crmilab-Pub g5/37-T ( 1q85) 14] T. Banks and L Susskind, Int. J. Thcor. Phys. 23 lit/N-l) ,175 151 S.C. Tcitclboim and R. Jackiw, tn t:est.-,,chrift for 11. DeWitt's 60th birthday, cd. S.C. ]'citclboim, Austin prcprint (1483) It'l A M . Polyakov. Phys. Lctt. 1031| (19811 207 171 See T. Banks, Nucl. Phys. 11244 (19~51 332 [8] S. l}cscr and 11. Zunfino, Plays. Roy. l.ctt. 38 (1477) 1433