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Physics Letters B 315 (1993) 267-276 North-Holland PHYSICS LETTERS B

Symplectic structure of 2D dilaton gravity

A. M i k o v i d a a n d M. N a v a r r o b

" Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK b Departamento de Fisica Tedrica y del Cosmos, Facultad de Ciencias, Universidad de Granada,

Campus de Fuentenueva, E-18002 Granada, Spain

Received 21 June 1993 Editor: R. Gatto

We analyze the symplectic structure of 2D dilaton gravity by evaluating the symplectic form on the space of classical solutions. The case when the spatial manifold is compact is studied in depth. For the free theory we find that the reduced space is a cotangent bundle and we determine the Hilbert space of the quantum theory. In the non-compact case the symplectic form is not well defined due to an unresolved ambiguity in the choice of the boundary terms.

I. Introduction

The study of 2D di la ton gravity models has a t t racted a lot of interest because they can serve as toy models for unders tanding the physics of black holes (for a review and references see ref. [ 1 ] ). A model proposed by Callan, Giddings, Harvey and St rominger ( C G H S ) [ 2 ] has been an object of lot of studies, because of its nice properties, of which its classical solvabil i ty is crucial for our approach. As argued by many authors (see ref. [ 1 ] ), semiclassical quant iza t ion schemes of the CGHS model are not sufficient in order to unders tand the quantum fate of the corresponding 2D black hole. Non-per turba t ive quant izat ion schemes were proposed by many authors (see ref. [ 1 ] ). In the canonical quant iza t ion approaches (see ref. [ 3 ] ), the knowledge of the phase space o f the theory is crucial.

In the s tandard canonical approach, de te rmina t ion o f the true ( reduced) phase space becomes a non-tr ivial task because of the presence o f the constraints . In the covar iant phase space approach [ 4 ], the pa ramete r space of non-equivalent solut ions of the equat ions of mot ion is defined to be the true phase space of the theory. Since the classical solut ions o f the CGHS model are known, one can study the symplectic structure of the space of the solutions in order to f ind the reduced phase space o f the theory and also f ind suitable variables for quant izat ion.

This is usually achieved by evaluat ing the symplect ic form on the space o f the solutions, whose defini t ion was given in ref. [4] . This method was employed to study certain 2D gravity models, like the Jack iw-Tei te lbo im and induced gravity models [ 5 ]. In this paper we apply the techniques developed in ref. [ 5 ] to the case o f the CGHS model.

2. General solution

The classical act ion o f the C G H S model can be writ ten as

~" Work partially supported by the Comisi6n Interministerial de Ciencia y Tecnolog/a.

Elsevier Science Publishers B.V. 267

Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993

SCG~s=l ~ d 2 x x f ~ { e - 2 0 [ R + 4 ( V c b ) 2 + 4 ) 2 ] _ ½ (Vf)2}, (1) .//

where R is the scalar curvature corresponding to the 2D metric gu,, V is the corresponding covariant derivative, while 0 and f , i= 1, ..., N, are scalar fields. ,J# is the 2D manifold with the topology of Z'xR. When X=D, the solution of the equations of motion can be interpreted as a 2D black hole. When X= S 1 (circle), then it is unclear whether a black hole interpretation is valid, although some authors tried to argue its relevance in the limit of a "large" circle [3,6].

In 2D one can always chose the conformal gauge for the metric

d s 2 = - e 2 p d x + d x - , (2)

o r

1 ( 0 e~") (3, gu, = - e2p ,

where x + = t + x, x - = t - x. The equations of motion are then given by [ 2 ]

T++ = e - 2 0 [ O + (2p)O+ ( 2 0 ) - 0 L ( 2 0 ) 1 + 1 O + f O + f = 0 , ( 4 )

T__ =e-2~[O_ (2p)O_ (20) -02- (20) l + 1 O - f O - f = 0 , (5)

T+_ = e - 2 ° [ 0 + 0 (20) - 0 + (2q~)0_ (20)--/'(2e2P] = 0 , (6)

-20+0_ (20) +~+ (2400_ (20) +0+0_ (2p) q-.~2e2p=0, (7)

o + 0 _ f = 0 . (8)

Eqs. ( 6 ) and ( 7 ) are equivalent to

0+ 8_ ( e -2¢) + ,~2e(Rp- 2¢) ~--- 0 , ( 9 )

0+0_ (2p- 20) =0. (10)

Eq. (10) implies that

e (2p-2~) = 0 + p 0 _ m , ( 1 1 )

w i t h p = p ( x + ), r n = m ( x - ). Eq. (9) then becomes

0+0_ (e -gO) +1.20+p0_ r n = 0 , (12)

and can be integrated as

e-2¢= -a2pm+a+b, (13)

where a = a ( x + ) and b = b ( x - ). Thus we can write

O+pO m e2p=.

_ 2 2 p m + a + b . (14)

The constraint equations (4) and (5) can be now expressed in terms of the functions p, m, a, b as

o=r++=@a- j

0=r =(02-b ) __ -ff---mO_b + ½ 0 _ f O _ f . (16)

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The general solution of eqs. ( 15 ) and (16) is given by x + y +

a=ao+ap-~ dy+O+p dz + ~ O+fO+ , (17)

x - F -

f d,, 0 mfd ( 0J0 0 where ao, bo, a, fl are arbitrary constants. One can get rid of the ol and fl terms by redefining p and m by constant shifts. The CGHS parametrization is obtained by defining 0+ p = e w+ and 0_ m=e w- [2].

Therefore the general solution o f the system ofeqs. ( 4 ) - ( 8 ) can be written as x + v + x - y -

M-22pm- ~dv+O+Pldz+(~-~ f) f f (~'~O__fO__f), (19) e - 2 ° = T " • ~O+fO+ -- dy-O_m dz-

e2P= ( O + p O rn)e 2~, (20)

f = f + ( x + ) + f T(x -) , (21)

where M/2 = ao + bo. In the case when there is no matter one gets

M 0+p0 m e-2~=--22pm+ - , , e2P= - (22,23) z -22pm+M/2 "

The space o f the solutions ( 19 ) - (21 ) is invariant under the conformal transformations

x+=x+(y+), x - = x - ( y - ) , (24)

0x + 0 x - (25) O(x+'x-)=¢(Y+'Y-) ' P(X+'X-):fi(Y+'Y-)+½1n~y + Oy-"

This residual symmetry can be fixed by specifying the functions p and m. This choice depends on the topology of~./¢, and for the time being we will leave p and m unspecified. In the non-compact case, the choice p = x + and in = x - corresponds to a 2D black hole, whose mass is given by the parameter M [2].

3. Symplectic form

Given a field theory with fields T" and an action S[ T] , one can obtain the corresponding symplectic form as

~o= f d a u ( - S j z ) , (26) X

where Z is a spatial hypersurface, while j** is the symplectic current, j ~ can be obtained from the variation of the action

5s= j '0~ .+ j" ~is 5T ~ 5 T ~

(27)

The symbol 5 in (26) is an infinite-dimensional generalization of the usual exterior derivative [ 4 ], and it co- incides with the usual field variation. The form ~o is conserved in time if the equations of motion are satisfied. We chose Z to be a t=cons t , hypersurface. With this choice of Y. we have

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Volume 315, number 3,4

o9= _I d x ( - S j ° ) , t = t o

where jo = j + + j-.

PHYSICS LETTERS B 7 October 1993

(28)

In order to calculate the symplectic form of the theory ( 1 ) we can start from the lagrangian in the conformal gauge. We get

N

~=e-2O[0+8_ (2p) - 0+ (20)0_ (2q~) +j.2e2p] + ½ • 8+fS_fi. (29) i

Variation with respect to ~ and p gives

8S 8S 8 = 80+Vsp

+8+ (e-2%_ 8(2p)-e-2°O_(20)8(20)+~ ~i O_fSf )

+8_ ( - 8 + (e-2°)8(2p)-e-2°8+ (2q})8(20)+ ½ ~ 8+fSf). (30)

Thus the symplectic current is

j+ =e-2%_8(2p) -e-2°8_ (2~)8(2~) + ½ ~ 8_fSf, (31) i

j - = - 3+ (e-2O)8(2p) - e-2~0+ (2~)8(2~) + ½ ~ O+fSf. (32)

After a long but straightforward computation it can be shown that the symplectic form takes the following form in terms of the phase space coordinates:

09= .f dxo9 ° , (33) t ~ l O

where

o9°= (0+ - 0 _ ) [8( -22prn+a)8 In 0_ m+ 8( -2Zpm)8 l np+ 8a 8 In 0 + p - 8a 8 In 0+ a+ 8b 8 In 8_ b]

+l[-saS(o-!+-aS+fS+f )-Sbs(ol~_b S_fS_f ) +½(SfSO f +Sf80+f)l. (34)

It can be seen from (34) that the only contribution of the chiral functions p, m is through boundary terms. By making use of the constraints ( 15 ), ( 16 ) and the equality

- 8a \ ~ + p 8+ aJ - 8b 8 \OT_ m O_ bJ

= - (0+-O_) [Sa 8 ( l n 0 + p - l n 0 + a ) ] + 8 0 + a 8( ln0+p- lnO+a)

-(O+-~_)[8b8(lnO rn-lnO_b)]+80_ b 8(ln8 m - l n 0 _ b ) ,

we find a simpler expression for the symplectic form

(35)

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CO= [5 ( --22pro +a+ b)6 In O_ m + 6( -22pm)5 In p ]~ X2 t ~

+ j dx[50_b 51nO_m+SO+a 51nO+p+½(6fSO_f+Sf60+f)]. (36) X l

which we rewrite as

co= [5(a+b)51n 0 m + 8( -2Zpm)~ In (p 0_ in) ]~

+ ~ dx(50_ b SlnO_,n+ 80+a SlnO+p) + ~ ~ dx SfSf~ , (37) x I x I

where rl,2 are the endpoints of the x-interval, f = Of/Or and

IF(x) lx~ -F(x ix,- . 2)--F(xl). (38)

Note that the expression (37) is not symmetric under the exchange of x + and x - although the space of the solutions is. A symmetric expression can be achieved by adding a total derivative of a two-form 0 ~ to oo. This does not change the symplectic form in the compact case. By taking (5 = - ½ Be-z% (2p_ 2~) we get

co= - 6(a+b)6 In ~ + 8( -- )~2pm p o _ rnj~,

+~dx(80_b61nO m+50+a61nO+P)+½~dxgfSfj. A-I X l

However, we will use eq. (37) since it is simpler for calculations.

(39)

4. CGHS model on a circle

Until now the discussion of the CGHS model has been general, without specifying the topology of the manifold where it is defined. In this section we will choose the two-dimensional spacetime manifold to be of the form Sl×~.

4.1. Pure dilaton gravity

Let us consider first the case without matter, i.e. f = 0, for all i. The solution for the metric and the dilaton field is then given by

M 0+p 0_ in e - 2 ° = - 2 2 P m-I- ~ , e 2p= _2Zpm+M/2. (40)

In this case the symplectic form becomes

x + 2 n ¢,

¢o-_- j ( 0 + - a _ ) w . (41) x

Since O + - a =04 we have

co= W ( x + 2 7 r ) - W(x), (42)

where

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W = 8 ~ ) 8 1 n O _ m + 5 ( - 2 2 p m ) 8 1 n ( p O _ m ) . (43)

It is impor tan t to notice here that p and m do not have to be single-valued functions on the circle. They can have a nontr ivial m o n o d r o m y t ransformat ion

1 p ( x + + Z ~ z ) = q p ( x + ) , m ( x - - 2 7 ~ ) = q m ( X - ) , (44)

where q is an arbi t rary posi t ive real number . Hence Wis not a single-valued two-form on the circle and therefore co is not zero. co should be independent of the arbi t rar i ly chosen point x, and hence it should be independent of the functions p and m. Indeed this is the case, which can be seen by insert ing (40) into (38 ) so that

c o = - s M S l n q . (45)

We conclude that in the compact case without mat ter the CGHS model has no local degrees of freedom. In fact it behaves like a mechanical system with one degree of freedom. The parameter M behaves like a momentum conjugate to the m o n o d r o m y paramete r q. It is t empt ing to identify M with the mass of the black hole. However , it is not clear whether in the compact case such an ident i f icat ion makes any sense.

Our task now is to de te rmine the exact reduced phase space of the model. To do that we have to f ind out the exact reduced phase space of the model. To do that we have to f ind out the maximal set of functions p and m which, beside fulfilling the right m o n o d r o m y t ransformat ion propert ies, define physical fields with the appro- priate sign. In the present case we need to f ind out the functions p, m, with the appropr ia te monodromy properties such that both e 2p and e -2° are posi t ive everywhere and everywhen. I f we take q = e r with re[R, then any functions with the appropr ia te m o n o d r o m y t ransformat ion propert ies can be writ ten as

p = e(r/2~)X+u, m= -jse~r/2~):'-v, (46,47)

where u = u (x + ) and v = v ( x - ) are per iodic funct ions on the real line. With this choice for p and m, the physical fields take the form

M (1 - u v e (~/2~)2t) , (48) e - 2 O =

e2p= ( M / 2 ) ( 1 / 2 2 ) e(r/2~r)2t[~+ U + (r/2rOu] [0 v+ (r/2rOv ] (M/2 ) ( 1 - u v e C~/2~)2t) (49)

The per iodic i ty of the funct ions u and v, together with the posi t ivi ty of the exponent ial e (r/27rI2t and the fact that the coordinates x and t, or what is a lmost the same, x + and x - , can be var ied independent ly of each other, makes it easy to realize that the posi t iv i ty of both the metr ic and the exponent ial of the di laton field imply that the pa ramete r M and the product uv are restr icted to be posi t ive (we are taking 2> 0). The posi t ivi ty of the produc t uv restricts in turn the allowed values for r and t. It is clear form the expression (48) that the range of their al lowed values does not cover the entire real line. For example, if we permit t to be posit ive then r is bounded by above, and r is bounded by below if t is negative.

To go on with the analysis let us consider the solutions above after an appropr ia te gauge choice has been made. A choice of gauge that fulfills all the requirements above for u and v is u = 1 = v. In this gauge, the physical fields take the form

m ( 1 - - e (r/2~)at) , ( 5 0 ) e - 2 O = A

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1 (r/2g)Ze (r/2~)2' e2p= 22 1 --e (r/2Jr)2t (51)

It is clear from (50), (51) that, for M>O, e -2° and e 2v are positive if and only if r < 0 and t>0 or r > 0 and t< 0. The reduced phase space splits then into two disjoint pieces, one with r> 0 and the other with r< 0.

By making a canonical change of variables, it is possible to rewrite the symplectic form in a way that makes clear the cotangent bundle nature of the reduced phase space. Let us introduce new coordinates p, s defined by M/2=e p, h=re p. In these coordinates the symplectic form reads

o=aa ap. (52)

Since the range ofp covers the entire real line, it is clear that the reduced phase space for the free theory and compact spacelike sections is given by

T ' R + wT*R_ . (53)

The cotangent bundle nature of the reduced phase space makes it possible to determine the Hilbert space of the quantum theory. General principles of the quantum theory indicate that the Hilbert space .;g is given by the square integrable functions on the configuration space. We have then

~ = L2 ( t + , ~h) (~)g2 ( g _ , - ~ ) , (54)

where the measure dh/h accounts for the restriction in sign of the parameter h.

4.2. Inclusion of the matter

In the case when matter fields are present the symplectic form is given by eq. (61 ) with p and m obeying the monodromy transformations (44), while a + b and f + + f - are periodic functions. By proceeding in the same fashion as in the pure dilaton gravity case we arrive at

co= -8 (a (x ) +b(x) )8 In q

x+2z~ x+2~ + f dy(aa_bSlna_m+6a+aflnO+p)+½ ~ dySfi~j[:.

x x

(55)

Once again o9 should be independent ofx. This can be checked by taking the total derivative with respect to x. The explicit dependence on x of the boundary term compensates with the nonperiodicity of the integrand in (55).

Let us now make a change of variables in order to separate the monodromy part of the functions from the parts which are periodic:

~+p=e(r/2rOX+W+, 0 _ m = e (r/2")x w_, (56,57)

with q= e" in order that p and m fulfil the monodromy transformation properties (44). Note that (57 ), (56) is a change of variables that does not imply any gauge fixing.

We then have

o9= - 8(a(x) +b(x) )5 In q

x+2n x + 2 n

r + ~ s r + ~ dy[aa+b(a lnw_+a-~y - )+aa+aa(a lnw+ ~ y + ) ] + ½ ~ d y f f a f . (58) x x

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Note now that if a + b is to be a single-valued function on the circle, then a and b can be expanded as

a ( x + ) = a o + s ( t + x ) + ~ ane -i"X+ , b ( x - ) = b o + s ( t - x ) + ~ bne - inx- . (59,60) n 4 - 0 n s ~ 0

Replacing this expansion into (58) we find after some integrations by parts x + 2 n

e ) = - 8 ~ - S r + d y [ ( 8 O _ b S l n w + 8 ~ + a S l n w + ) + ½ ( S f S J ~ ) ] , (61) x

where M is the zero mode of a + b, M = 2 (ao + bo ). The symplectic form (61 ) can be brought to an even simpler form by making the change of variables given

by

~+a=ciw+, ~_b=/Tw . (62,63)

In these variables the symplectic form is written as x + 2 n

~ o = - 8 ~ - 8 r + dy[ (g f fSw + S d S w + ) + ~ ( S f 6 f ~ ) ] . (64) x

The constraints as well take in these variables an especially simple form since they are given by

T + + = O + c i - ~ w++~ ~ O + f S + f , T_ = 0_b'-~-~ w_+-~ E S _ f O _ f . (65,66) i i

The functions w + and w- are pure gauge, and we could set them to be constant, so that x + 2 z t

~ o = - s M s r + ½ ~ dyS~Sf . (67) x

The analysis of the reduced phase space is more complicated than in the pure gravity case, but the fact that the symplectic form decomposes as in eq. (67) can simplify the analysis. As in the pure case, the reduced phase space should be determined from the analysis of the solutions. Given the difficulties of that approach, we can say that the reduced phase space is a direct product of a subspace of T* (~ ) with the phase space for the free scalar fields.

5. The noncompaet case

In this section the spacetime manifold will be taken as ~ × ~. The spacelike sections will be now noncompact and with boundary. This fact introduces important differences with respect to the previous case, when the manifold did not have a boundary.

The general solution in the pure dilaton gravity case takes the form

M O+pO_rn e - 2 0 = - - 2 2 p m + ~-, e2p= (68)

z - 2 2 p m + M / 2 "

The symplectic form is now

+ c o ) ~ = dx~'c 8~-81nO-m+8(-22prn)81n(p~-rn)z " (69) - c ~

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o) must be such that it is independent o f t and of the hypersurface of integration, i.e., it must be given only in terms of constants o f motion and their conjugate momenta. If one makes the standard gauge choice p= x +, re=x- , the symplectic form (69) vanishes. This is contrary to our expectation that the system has only one degree of freedom, i .e.M. The reason why one gets a zero result is that there is no other parameter to play the role of conjugate momentum to M. In the compact case that parameter was the monodromy parameter q, which was introduced from the observation that p and m do not have to be single-valued functions on the circle. In the non-compact case we could try to introduce an analogous parameter q by means of the choice

1 p=qx +, m = - x - . (70)

q

However, this election cannot give a meaningful result for oJ since the scaling

p(x+)--,qp(x +), m ( x - ) - ~ l - re (x - ) (71) q

preserves the form of the solution (68). We would get a meaningful result for (o

M ~o=8~- 81n q , (72)

provided we evaluate the form only at one end of the real line. However, even if we somehow justify the evalu- ation of the symplectic form at one end, this prescription does not work in the case when matter is present, since then eq. (61 ) becomes

1 f 8 f 8fi. (73)

Introduction o f a wall, a device employed in the thermodynamics of 2D black holes [ 7 ], may provide some ideas how to proceed. Also one could try to exploit the ambiguity of the symplectic current under addition o f a divergenceless one-form, which in the non-compact case changes the symplcctic form by a surface term. Which- ever way one chooses to proceed, the final expression should be gauge independent and independent of the spatial surface.

On the other hand, knowledge of the symplectic form is not necessary in order to perform the quantization on the space of the solutions. It is only a device which simplifies the analysis of the structure of the phase space.

After completing our work, we received a preprint [6 ] where the standard canonical formalism is applied to the space of the solutions in the compact case with matter. Instead of working with the symplectic form, the authors o f ref. [ 6 ] work with the Poisson brackets. They find a canonical transformation which maps the constraints into a quadratic form similar to our eqs. (65), (66). However, they do not address the problem of the reduced phase space.

Acknowledgement

M.N. is grateful to the MEC for a FPU grant. M.N. acknowledges the Imperial College, where this paper has been written, for its hospitality. M.N. would like to thank J. Navarro-Salas and C. Talavera for valuable comments.

References

[ ! ] J.A. Harvey and A. Strominger, Quantum aspects of black holes, Enrico Fermi Institute preprint EFI-92-41, hep-th/9209055 (1992). [ 2 ] C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D 45 ( 1992 ) R 1005.

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[3] A. Mikovic, Phys. Lett. B 291 (1992) 19. [4 ] C. Crnkovi6 and E. Witten, Three hundred years of gravitation, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge,

1987) p. 676; C. Crnkovid, Class. Quant. Grav. 5 (1988 ) 259.

[ 5 ] J. Navarro-Salas, M. Navarro and V. Aldaya, Nucl. Phys. B 403 ( 1993 ) 291. [ 6 ] S. Hirano, Y. Kazama and Y. Satoh, Exact operator quantization of a model of 2D dilaton gravity, University of Tokyo preprint UT-

Komaba 93-3 (1993). [ 7 ] G. Gibbons and M. Perry, The physics of 2D stringy spacetimes, University of Cambridge preprint (1992).

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