Two dimensional quantum gravity coupled to matter

Physics Letters B 294 ( 1992 ) 310-316 North-Holland

PHYSICS LETTERS B

Two dimensional quantum gravity coupled to matter R B Mann Department of Physlcs, Umverstty of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Received 5 July 1992, revised manuscript received 18 August 1992

A classical two dimensional theory of gravity which has a number of interesting features (including a newtoman limit, black holes and gravitational collapse) is quantlzed using conformal field theoretic techniques The critical dimension depends upon Newton's constant, permitting models with d=4 The constraint algebra and scaling properties of the model are computed

1. Introduction It has become clear m recent years that two d i m e n sional q u a n t u m gravity has much to teach us, both m terms o f obtaining a p r o p e r u n d e r s t a n d i n g o f two dimensional systems and as a theoretical laboratory for the (3 + 1 ) d i m e n s i o n a l case In the latter context it is h o p e d that the reduction m the n u m b e r o f degrees o f freedom will yield a better understanding o f short&stance problems, topology change, singularities and the cosmological constant p r o b l e m In the former context the p r o b l e m o f confronting two d i m e n s i o n a l q u a n t u m gravity was a v o i d e d by considering massless matter systems Classically, general coordinate m v a r l a n c e and Weyl mvarlance m a y be used to gauge fix all three degrees o f freedom o f the metric [ 1 ], u p o n quantization, Weyl m v a r i a n c e no longer holds except for particular values o f the central charge o f the m a t t e r (1 e the critical d i m e n s i o n s ) By restricting attention to these special cases the p r o b l e m o f quantlzing the two d i m e n s i o n a l gravitational field itself was a v o i d e d This restriction was lifted when it was shown in hght cone gauge that such systems can be quantlzed in the presence o f the Llouvllle m o d e o f the metric for a genus zero surface [ 2 ] Shortly afterward it was shown how to obtain this result in conformal gauge for all genera [ 3 ] However, the central charge in such models is restricted to be less than or equal to one, and it is not clear how to recover the known critical cases A key problematic element in the above p r o g r a m is the choice o f classical action for the gravitational 3 l0

field The Einstein tensor vamshes m two d i m e n sions for all metrics and the Einstein-Halbert action simply yields the Euler n u m b e r o f the m a n i f o l d A number o f suggestions have been made to address this p r o b l e m [ 4 - 1 0 ] These typically involve taking the classical action to be trivial (either zero or the Erastein-Halbert action) and then gauge-fixing the symmetries o f the metric to obtain a q u a n t u m action, or alternatively making use o f topological interactions [ 9 ], a variant o f which [ 11 ] permits constant curvature solutions classically In all cases the m a t t e r gravity Interaction is classically trivial, and quant u m - m e c h a m c a l l y quite limited However in the last few years it has been shown that classical gravity in two spacetame dimensions need not be so trivial [ 12 ] An interesting relativistic theory o f gravitation m this context m a y be f o r m e d by setting the Riccl scalar R equal to T = TuU, the trace o f the conserved stress-energy tensor, R=8nGT

,

(1)

where G is N e w t o n ' s constant m two spacetlme dimensions In spite o f its slmphclty, this theory has a n u m b e r o f remarkable classical and semi-classical features, including a well-defined n e w t o m a n limit [ 12 ], black holes [ 13,14 ], a post-newtonlan expansion, gravitational waves, F R W cosmologies, gravitational collapse [ 15 ] a n d black hole radiation [ 1618 ] These features suggest that it is potentially a very useful tool in the study o f q u a n t u m gravity since its classical features are so similar to those o f (3 + 1 ) &menslonal general relativity, one might hope that Its

0370-2693/92/$ 05 00 © 1992 Elsevier Science Pubhshers B V All rights reserved

V o l u m e 294, n u m b e r 3,4

PHYSICS LETTERS B

quantlzatlon would bear a similar resemblance to (3 + 1 ) dimensional q u a n t u m gravity Indeed, it has been shown the system ( 1 ) along with the conservation of stress-energy V~TU~=0,

(2)

may be understood as the D--, 1 limit of the ( D + 1 ) dimensional Einstein equations [19] The present paper is concerned with taking a first step towards quantlzatlon o f this theory The system ( 1 ), (2) may be derived from the action [ 17 ]

1

S= ~ G f d2x x / - g ( UgU" Oa~O~+!vR -- 8~G,~aM) ,

(3)

which in turn yields the field equations

1

~7==__gOU(x/--gg u" O . q / ) - R = O ,

=8nGTu,,,

(4)

(5)

plus the equations o f motion for the matter fields Ins e m o n of the trace of (5) into (4) yields (1), and conservation of the stress-energy tensor (eq ( 2 ) ) is guaranteed since the covarlant divergence of the lefthand side of (5) is identically zero when (4) holds This also follows from considering the invarlance of the action (3) under infinitesimal co-ordinate transformations, analogous to the ( 3 + 1 ) dimensional case The action (3) with Tu~=0 has been considered before [ 20,21 ] in the context of finding a local action which produces the Polyakov theory In such cases the ~,u-field is fully eliminated in terms o f the metric degree of freedom by imposing boundary conditions which break reparametrlzatlon lnvarlance, thereby inducing an anomaly into the Vlrasoro algebra In the present case, q/is treated as an auxlhary field, since the classical evolution of the gravity/matter system IS Independent of the evolution of V However the ~, field obeys the equation

and so depends upon the evolution of the gravity/ matter system ( 1 ), (2) Eqs (4) and ( 5 ) form a system of 4 equations with 2 1dentines, which is equal to the number of unknowns (the metric degree of freedom and ~u) Reorganized in the form ( 1 ), (6) (and ( 2 ) ) , one easily sees that upon setting Tu~=gu~l that they reduce to a proposal considered previously by Jacklw and Teltelbmm for two dimensional gravity [5,6], whose quantlzatlon was recently considered by Chamseddine [ 11 ] Such a restriction permits only constant curvature solutions to the field equations This could be avoided by including ~-dependence in the cal LM term in the action (3), such models have recently been considered by a number of authors since they arise in the context of non-critical string theory [22 ] From a relatlvlst's viewpoint, such models are variants of Brans-Dxcke theory [23 ], in which the interactions of the Brans-Dxcke scalar field ~u (or some field redefinition thereof) cannot be decoupled from the classical gravity/matter Interaction In contrast to this the theory based on ( 1 ) and (6) allows curvature and stress-energy to act upon one another in a manner which is very similar to the (3 + 1 ) dimensional situation Motivated by the above, the classical gravitational action is taken to be (3), and its quantum properties are considered herein The newtoman gravitational constant G will be seen to slgmficantly modify the relationship between the central charge and the conforreal dimensions o f the matter fields, permitting critical dimensions larger than 1

2. Constraint algebra It is instructive to consider the constraint algebra which follows from (3) Working in conformal gauge, w l t h g u ~ = e ~ r/u~ yields from (4) and (5) ¼ [ ~/,/2.~ (~/jt) 2 ] __ ~/tt ~ 1

~O~'+ ~1 t ~' ¢= T o o ,

½~,q/,+l~q~, ~ + ~1 ,0 - ~ , = T o t ,

(6)

(7) (8)

for the constraint equations and

[ q/2_{_(~t') 2] _~//+ ½~q/+ ½~i'~p,'= T,1,

½( ~ ugt ~ -- ½guv~ a~,a) + ½guvga#gt,,~,;a-- V u v =8zcG( Tu. - ½gu.T) ,

19 N o v e m b e r 1992

~,- ~ " + ¢ - - ¢ " = 0 ,

(9) (10) 311

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for the dynamical equations Here ~,'-O~u/Ox and gt=_ O~,lOt Subtracting (9) from ( 7 ) allows this latter set to be rewritten as

(~- O" = - ( Too - Tl l ) ,

(ll)

~,- u/"+ 0 - 0 " = 0

(12)

Following ref [20], it is straightforward to show that P~,= - (~+q~)

(13)

is the momentum conjugate to ~u and that the momenta conjugate to the metric is //00=0,

//°l=--½e-¢lff',

Hll=e-~v/,

(14)

where H " a ~ 8S/8g "p Clearly only P~, and H 1~are mdependent canonical variables in conformal gauge It is useful to make a canomcal transformation on these varmbles so that Z= ~ ' + 0 ,

Pz=P~,=-Z,

(15)

19 November 1992

?

L z~ = j d a e -1too (az± 2 -T-bx+) ,

(23)

Lg'_+ = i d i e

(24)

. . . .

(a~+ T-b.+) ,

where the periodic spaUal variable a t ( - n, n) The Polsson brackets of these variables are

{L;± , Lx% } =+l(n-m)Lx±

n .+- r ~

--

+4mn3Jn+mo,

(25)

= +_l(n--mj~Ln+ ~_+m -Y-4rein 3J.+ m 0,

(26)

{Lg±, L;"+ } all other brackets being zero Adding the constraints (19) and (20) yields respectively L~m_+= ( iV~o-T-T~'I ) ,

(27)

where T- um. IS the Fourier transform of the stress-energy tensor The quant~tles

and H=e~HI1=~

(16)

This guarantees that on a spatml shce

~ _ =-Lx'± - L ~

(28)

obey

,

(17)

{~q~n+_,~ _ } = l ( n _ m ) ~ q ~ + m

{~t(x),//(x') } = d ( x - x ' ) ,

(18)

for their Pmsson bracket algebra Each of the Z, gravity sectors yields a classical V1rasoro algebra with a non-zero central charge However these cancel out in the c o m b m e d z - g r a v l t y system, as is clear from (28), (29) Classical couphng to conformally mvarmnt matter will not affect this result, since the stress-energy tensor wdl in general have a Fourier decomposition m terms of operators which obey the Virasoro algebra (29) For example, a massless scalar field ~0will have stress-energy tensor components

{X(x), e x ( x ' ) } = 6 ( x - x ' )

are the canomcal Poasson brackets Under this transformation the constraint eqs (7) and (8) become respectively

a~+ - b z + - (a~_ + b ~ _ ) + a 2_ + b z_ - (a~+ - b o + ) =2Too,

(19)

-(a~+ -bz+ )+a~_ +bo_ +a~_ +b z_ - (a~+ -bo+ ) =2To,,

(20)

where

Too++_ToI = ½ ( p +q¢)2,

ax+ - ½ ( P z +_Z') ,

a~± =-½(P~ +~O') ,

(21)

and

bx± =-2a'x± ,

b~± =-2a'o±

(22)

On spatml sections topologically equivalent to S ~ these variables may be Fourier transformed so that 312

(29)

(30)

which upon Fourier decomposition will yield operators Lg'± whose Polsson brackets form a Virasoro algebra with vanishing central charge Other approaches to two dimensional gravity typically impose additional constraints to the system described by the action (3) which introduce an anomaly at the classical level For example, if the ~/degree

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19 November 1992

PHYSICS LETTERS B

of freedom is frozen out (~ e if in (10) there are no homogenous solutions permitted for V) then Z = 0 and consequently Px= 0 The action (3) then becomes the non-local Polyakov action [ 1 ] and the constraint algebra develops an anomaly (29) due to this non-locahty Conformally coupling to two dimensional matter and quantizing then permits one to cancel off this anomaly against the central charge of the matter [10] Alternatively, one could fix the two dimensional metric gu~=~u~ which would freeze out the degree of freedom In this case the action (3) becomes that of the LlOuvllle action [24] The field is Interpreted as the quantum-dynamical field whose exponential yields a conformal factor which provides the quantum corrections to ~ . 1 e the full two dimensional metric g.~ = e~'R~ [2,3 ] The approach taken here does not involve freezing out any of the degrees of freedom in the system described by the action (3) Classically the metric is locally gu.=e~ r/~. and the ~u field is an auxlhary field whose classical evolution has no effect on the gravity/matter system

P Z= J [ ~z]

~gO ~gb ~gc ~,V ~,cl)

X e - ' s [ v/O]- S ~ [ b cl --SM[~]

= f [~zl~O~b2~c~w~ >( e - s [ ~ O l - s ~ [ b , c ] - s M [ ~ l - ~ c [ e ~ ' l

,

(32)

where [ ~ r ] represent the integration over the Telchmuller parameters and

- Q ~ f i + # e "~ )

(33)

IS the Llouvllle action with arbitrary coefficients and gu~=e"~u~ Note that R [ e " ° ~ ] = e - ~ ( R - a V 2 ~ ) where/~ and ~,2 are respectively the curvature scalar and laplaclan of the metric ~ Sr~ is the action for the ghost fields b and c The approach is then to determine the parameters ot and Q from the requirement that the conformal anomaly vanish and that e "~ is a conformal tensor of weight (1, 1) The action STOT=~+S[~', g]+S~h + S~ may be rewritten as

STOT=S~ + SM 1

3. Quantum corrections

-t- Or2 " Following the remarks of the previous section, quantlzatlon of two dimensional gravity coupled to matter may be carried out by considering a functional integral over the field configurations of the metric, ~, a and matter fields This may be done using the path integral Z=

f ~~g ~

~

e -S[q/gl--SM[~I

(31)

where S = S[ ~, g] + SM [ ~ ] is the euchdeanlzatlon of the action (3), with SM being the matter part of the action and • representing the matter fields The volume of the dlffeomorphlsm group, Vcc, has been factored out Making the same scaling assumption as in refs [ 3,11 ] about the functional measure yields

1

- ~ [ ½~'" a~(~,+a0) a.(~,+ o~0)-(~,+ o~0)~] + ( # + 8ztA )e"~],

(34)

where A is the (possibly zero) classical cosmological constant Upon rescaling the fields V and ~ so that ~=

/

o~2 1+ ~ - ~ ,

1 ~ = ~ 7 ~ (~,+ o~0),

(35)

(34) becomes

313

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19 November 1992

since 0£ must be real Setting yields

1 +0£2/2G

(o

(36)

The coefficients in front o f the ~ t e r m s are due to the sign o f the kinetic energy term in ( 3 ) This m a y be dealt with by rescahng ~U~l~ so that the functional integral converges The action xn (36) m a y now be analyzed using conformal field theoretic techniques The fields q~, ~u have propagators

<~(z)~(w)>=-ln(z-w)=<~(z)t#(w)),

(37)

and the contrIbutmn o f the m a t t e r a n d ghost fields xs as usual Hence ~t is straightforward to c o m p u t e the total central charge

CTOT = 1 + 3 (Q-0£/G)Z 1 +0£2/2G + 1 --6/G--26+CM ----CM- - 2 4 + 3 G Q 2 - - 2 ( 1 +0£Q) O+10£2 ,

(38)

using the o p e r a t o r product expansion o f the stressenergy tensor which Is associated with (36) The a n o m a l o u s d i m e n s m n o f the ( r e n o r m a h z e d ) cosmologlcal constant t e r m m (36) is A=

1

using (40)

0£2= ( 12--CM)G-- 6 _+x/(CMG-- 6) [ (CM - 2 4 ) G - 6 1 CM -- 1 2 + 6 G (41)

+ (#+8~zA) exp x/1 +0£2/2 G +S~+SM

CTox= 0 and

0£

for 0£ m terms o f CM AS 0£ IS real, 0£2 must be positive, imposing constramts on CM and G There are three distinct cases to consider (1) CM< 1 2 - 6 G In this case the negatwe sign for the square root must be chosen in (41 ) and so 0£2= x/(CM G - 6 ) [(CM -- 2 4 ) G - - 6 ] + (CM -- 1 2 ) G + 6 12--6G--CM IS the only possible solution Posltlvaty o f CM l m p h e s G<2 (11) CM> 1 2 - - 6 G The d l s c r l m m a n t in this case is not positive if 6 / G < CM< 24 + 6/G It IS straightforw a r d to show that if CM lS larger than 24 + 6/G then 0£2 1Snot p o s m v e , yielding CM< 6/G There is no upper limit on G, but posltlVlty o f 0£2 l m p h e s G > 1 regardless o f which sign o f the square root is taken In the G ~ limit this is the famahar restriction 1
0£2= 2 -

which i m p h e s 2 > G > 1 when CM IS posItwe Finally, one can compute the scaling dependence o f the correlation functions Inserting the factor

2x/l +0£2/2G ,_-

×(x/Ql+2/2~2G+ 41+~2/2G)

(42)

(39)

In the G - * ~ h m l t these results yield what one would obtain from the Polyakov approach [2,3 ], except that CM--'CM+ 1 due to the presence o f the a d d l t m n a l scalar q7 Setting A = 1 so that e ~ IS a conformal tensor o f weight ( 1, 1 ) yields

Into the functional integral (32) implies that

0£2+ Q 0 £ + 2 = 0 ,

~=--(h-1),

exactly as in ref [ 3 ] This yields the constraint 314

(40)

Qz> 8

/zI~I eC~zti~')A=A~-l( tl~I e~'~')A= ,

(43)

with

QG-0£ 0£G

(44)

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where A ls the area o f the surface a n d h the n u m b e r of handles

4. Discussion T h e m o d e l p r e s e n t e d here is that o f a two d i m e n sional q u a n t u m theory o f gravity which has a n o n trtvlal classical ltmtt As such it has a n u m b e r o f interesting features w h m h m e r t t f u r t h e r study U n l i k e v i r t u a l l y all other two d i m e n s i o n a l theories o f gravity, xt has a well-defined n e w t o n t a n h m t t a n d tts classical features closely resemble those o f (3 + 1 ) dim e n s i o n a l general relativity [ 19 ] T h e restriction o n the central charge n o w d e p e n d s o n the d i m e n s i o n l e s s N e w t o n c o n s t a n t G, a n d so for small e n o u g h G o n e c a n a v o i d the c o n s t r a i n t s t m p o s e d b y the critical dim e n s i o n s p r e s e n t m other m o d e l s ~1 [3,10] F o r example, o n e could use this m o d e l as the basis o f a n o n critical string theory in four d i m e n s i o n s p r o v i d e d G< 4

T h e restriction to c o u p l i n g to c o n f o r m a l l y m v a r l a n t m a t t e r m a y be lifted b y e x t e n d m g the m a t t e r act i o n so that SM=SM+

y" rn, C,,

where m, are ( d l m e n s m n a l ) c o n s t a n t s a n d (9, are operators o f c o n f o r m a l d i m e n s i o n A, U p o n c o u p l i n g to gravity, this t e r m is m o d i f i e d to (46)

l

in ( 3 4 ) E x p a n d i n g in the p a r a m e t e r s m,, o n e m u s t as before require that each t e r m in ( 4 6 ) be a ( 1, 1 ) o p e r a t o r a n d so ( 4 0 ) b e c o m e s

oz,

( Q-o~IG + 06 ) +oe2/2G x/1 + 0 ~ 2 / 2 G

1 x/1 + o e 2 / 2 G !"x/1 +d,=l

,

~),

(48)

p r o v i d e d at least o n e o f the c o n s t a n t s (m~ = m, say, with c o r r e s p o n d i n g o p e r a t o r o f d i m e n s i o n d ) ts n o n zero It is o f course possible to e x t e n d these results to i n c l u d e negative values o f G, although such m o d e l s have antigravity, they have b e e n o f s o m e field-theoretic interest [13,25] F r o m ( 3 4 ) it is clear that in this case a critical value for G exists (Gent = 2o~2) bey o n d which the m o d e l is n o longer well-defined At this p o i n t the k m e t l c t e r m for q~ v a n i s h e s a n d the m o d e l b e c o m e s singular, suggesting a phase transit i o n to a n o t h e r m o d e l

Acknowledgement T h i s work was s u p p o r t e d by the N a t u r a l Sciences a n d E n g i n e e r i n g Research C o u n c i l o f C a n a d a

References

(45)

l

SM =SM + Y, m, e"'°¢,

Q2>8(1-A)(1-

19 November 1992

(47)

which forms a c o n s t r a i n t o n the e x p o n e n t s a , I f / t + 8m4 is set to zero in ( 3 4 ) , t h e n the restriction that Q2 > 8 n o longer holds, b u t i n s t e a d is m o d i f i e d to ¢~ Although the model m ref [ 11 ] has no restnctmn on the critical dxmensmn, it ~s also hmlted to constant curvature solutions

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