Physics Letters B 304 ( 1993 ) 70-76 North-Holland PHYSICS LETTERS B

2D dilaton gravity in a unitary gauge

Aleksanda r Mikov i6 Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK

Received 19 November 1992; revised manuscript received 13 February 1993

Reduced phase space formulation of2D dilaton gravity coupled to matter is studied in an extrinsic time gauge. The correspond- ing hamiltonian can be promoted into a hermitian operator acting in the physical Hilbert space, implying a unitary evolution for the system. Consequences for the black hole physics are discussed. In particular, this way of defining the quantum theory rules out the Hawking scenario for the endpoint of the black hole evaporation process.

1. Introduction

In a pioneering work [ 1 ], Callan, Giddings, Harvey and Strominger (CGHS) have proposed a theory of 2D dilaton gravity coupled to matter as a toy model for studying the formation, evaporation and back-re- action o f black holes. The attractive features o f the model are that it is classically exactly solvable, it pos- sesses black hole solutions and it is a renormalizable field theory. The last feature raises a possibility that the corresponding quantum theory may be tractable, and hence allow for the investigation o f the elusive issues associated with the endpoint of the black hole evaporation [2 ]. As shown by a series of authors [ 3 ], the solutions of the one-loop matter corrected equa- tions of motion are not free from singularities, in contrast to the initial expectation by CGHS. Hawk- ing has even argued [ 4 ] that the solutions of any semi- classical approximation scheme will be singular, sug- gesting that the possible stabilization of the black hole by the quantum effects could be achieved only if the gravitational field is quantized together with the matter fields.

Non-perturbative quantization of the gravitational field in four spacetime dimensions is still an un- solved problem. However, in 2D, significant simpli- fications occur, most notably the number o f physical degrees o f freedom of the gravitational field is finite. In addition, the CGHS model is a renormalizable field

E-mail address: UMAPT66@VAXA.CC.IC.AC.UK.

theory. However, the non-perturbative analysis is still a complicated problem. Instead of using the path- integral techniques, one could try using the canonical quantization methods, which were developed in the context of 4D quantum gravity (for a review and ref- erences see ref. [ 5 ] ). In ref. [ 6 ] the canonical anal- ysis of the model has been performed, and the Dirac type quantization has been investigated. It was shown that a set of non-canonical variables can be found, forming an SL(2, E) current algebra, such that the constraints become quadratic in the new variables. For a compact spatial manifold (i.e. circle) and piecewise continuous field configurations, Fourier modes can be defined, and the physical Hilbert space can be obtained form a cohomology of a Virasoro al- gebra. Although exactly solvable, the configuration space of this model does not contain singular solu- tions which can be associated with black holes. As suggested in ref. [ 6 ], a Schr6dinger type equation would be more appropriate for quantization of a more general configuration space, which naturally leads one to employ the extrinsic time variable approach [ 7 ].

In this paper we discuss the reduced phase space formulation of the CGHS theory in an extrinsic time gauge. Our gauge fixing conditions contain only the canonical variables, in contrast to the usual gauge fix- ings, where the Lagrange multipliers are involved, like the conformal gauge. Since we are dealing with a re- parametrization invariant system, a consistent ca- nonical gauge fixing must contain the definition o f a time variable [18 ]. We construct a time variable

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Volume 304, number 1,2 PHYSICS LETTERS B 22 April 1993

T(x, t), and in the gauge T(x, t)=t solve the con- straints in terms of the independent canonical vari- ables. We obtain an explicit expression for the ham- iltonian of the system in this gauge. That hamiltonian can be promoted into a hermitian operator, acting on the physical Fock space, implying a unitary evolu- tion. Hence in this theory there are no anomalies as- sociated with a non-unitary evolution, like transi- tions from pure into mixed states, a pathology expected at the endpoint of the black hole evapora- tion process [ 2 ]. However, it is still difficult to ex- plicitly see what happens during the gravitational collapse in this theory. This problem together with some other caveats is discussed at the end of the paper.

2. Canonical formulation

The CGHS action [ 1 ] is given by

S= - ~ ~ d2x x / t ~ (e-2~[ R d-4(VtTI) )2-t-,~ 2 ] M

where M is a 2D manifold, gu~ is a metric on M, • is a scalar field (dilaton), 2 is a constant and R is the 2D curvature scalar. ~ are massless scalar fields, minimally coupled to gravity. We will label the time coordinate x°=t and the space coordinate xl=x, while the corresponding derivatives will be denoted as" and ', respectively.

Following the analysis in ref. [ 61, we perform the field redefinitions [ 9 ]

¢ = l e -z~ , gu,=4¢e-~gu~, (2.2)

so that the action becomes

s=-½ M

× ( (~¢)2+/~q~+ ~;t2 e~+ i~1 (VOi)2) " (2.3)

The canonical formulation requires that the 2D man- ifold M has a topology ofZ ×~, where Z is the spatial

manifold and R is the real line corresponding to the time direction. E can be either a circle or a real line. The compact spatial topology is relevant for cosmo- logical solutions and string theory, while the non- compact spatial topology is relevant for 2D black holes.

After introducing the canonical momenta, (2.3) takes the form [ 6 ]

Y Go-nGl) S= ~ dt dx(pg+ n~+ ni~i - - ~

(2.4)

where we have omitted the tildes, g=gl 1, X a n d n are the laps and the shift vector and

Go(x) = - 2g2p2- 2gpzt + 1 ( (b') 2+ ~2 2g e 0

1 g' N - ~ g ¢ ' + O " + ½ ~=1 ~ [~ '~+(0;)21 '

N

G,(x)=lrO'-2p'g-pg'+ Z rtt(~. (2.5) i = 1

The constraints Go and GI form a closed Poisson bracket algebra,

{Go(x), Go(Y)} = - ~ ' ( x - y ) [a l (x) + GI (Y) ] ,

{GI(x), Go(y)}= -6 ' (x-y)[Go(x)+Go(y)] ,

{GI(x), G~(y)}= -6 ' ( x -y ) [GI (x )+G, (y ) ] , (2.6)

where the fundamental Poisson brackets are defined a s

{p(x), g(y)} = 6 ( x - y ) ,

{n(x), 0(Y)} = 6 ( x - y ) . (2.7)

G1 generates the spatial diffeomorphisms, while Go generates the time translations of Z, in full analogy with the 4D gravity case. Note that the algebra (2.6) is isomorphic to two commuting copies of the 1 D dif- feomorphism algebra, which can be seen by redefin- ing the constraints as

T+ =½(Go_+G1). (2.8)

Since we are dealing with a reparametrization in- variant system, the hamiltonian vanishes on the con- straint surface (i.e. it is proportional to the con- straints). Therefore the dynamics is determined by the constraints only. Since Go and Gi are indepen-

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dent, there will be (2 +N) - 2 = N local physical de- grees of freedom. When N= 0, there is only a finite number of global physical degrees of freedom (zero modes ofg and ~), and one is dealing with some kind of a topological field theory. When N # 0, these global degrees of freedom will be present, together with the local ones.

The variables (g, p, ~, It) are not convenient for quantization, since Go is a non-polynomial function of these variables. First we perform a canonical transformation in order to get rid of the e ~ term in Go:

g = e - # ~ , p=e~/~, q~=ff, n=~+/3~. (2.9)

The constraints now become

Go(x)=-4g2pZ-2gpTr+ ( f ) ' )Z+Ag- 2 ~ ~'+0"

N +t Y,

i = 1

N

G,(x)=tcO'-Zp 'g-pg '+ ~. ~'¢);, (2.10) ~=~

where we have dropped the tildes and A = ]2 z. Now it is convenient to define the SL(2, ~) variables in- troduced in ref. [ 6 ],

j + = x / /2T_+ A_._A__ 2g 2x/~ '

jO = gp + ~ ;z- ~ ,

1 J - = - ~ g , (2,11)

and a U ( 1 ) current

1 2~') p o = ~ ( ~ - l g '

The (J~, PD) variables satisfy an SL(2, ~ ) ® U ( 1 ) current algebra

{Ja(x), jb(y) } = f a6jC(x)6(x_ y ) __ l q , b 6 , ( x _ y ) ,

{PD (X), PD(Y) } = -- O'(x--y) , (2.12)

where fab~=2e"bdtla~ with q+- = q - + =2, qoo= _ 1, and {J, PD} = 0. Instead of using the canonical vari-

ables (n/, ¢i), we introduce the left/right moving currents

1 I Pi = ~ ( ~ i + ~ ) , /~i = ~ ( ~ i - q ~ ) , (2.13)

satisfying

{P~(x), Pj(y)} = - 6 u 6 ' ( x - y ) ,

{P~(x), ~ . (y ) }=6o f i ' ( x -y ) , (2.14)

and {P, P}=0. Now one can show that the energy- momentum tensor associated to the algebra (2.12) via the Sugawara construction

N

~ga=2qabJajb (jo),+½p~ t , + PD+½ Z i=l

(2.15)

satisfies 5 e = G~. Therefore the constraints become

J + ( x ) - # = 0 , b°(x) = 0 , (2.16)

where iz=A/2x/~. Now it is convenient to introduce three new vari-

ables fl(x), ~,(x) and PL(X) [ 6 ] such that

J+ =/~,

j o = _ B e - ½Pc,

J - = fly2+ yPL -- ½y', (2.17)

where

{fl(X), 7(y)} = - -6 (x - -y ) ,

{PL (X), PL(Y) } = 6' ( x - - y ) , (2.18 )

with the other Poisson brackets being zero. Then the expressions (2.17 ) satisfy the SL(2, R) current alge- bra (2.12 ), and represent the classical analogue of the Wakimoto transformation [10]. The 5 e constraint then becomes

N 5 e = f f y _ ½ p 2 + l , l 2 l , 2 ~PL +~PD+~PD+½ ~ Pi

i=l

= 0 . (2.19)

If we define B ( x ) = f l ( x ) - 2 and F(x)=?(x ) , then the J+ constraint implies that B=0, and conse- quently we can omit the canonical pair (B, F) from the theory. Therefore we are left with PL, PD and P~ variables, obeying only one constraint:

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Volume 304, number 1,2 PHYSICS LETTERS B 22 April 1993

N

25¢=-P2+P[+P2 +Pb+ ~ P ~ = 0 . (2.20) i = 1

The form of the Poisson brackets of PL and PD al- low us to introduce a canonical pair (P(x), T(x)) such that

1 1 PL=-~(P-T ' ) , PD=-~(P+T'). (2.21)

Note that the definition (2.21 ) implies that the zero- mode parts of PL and PD are equal. When N = 0, this is true on the constraint surface, but away from the constraint surface these zero modes are independent. Therefore we are going to modify eq. (2.21 ) by intro- ducing an independent zero-mode momentum p such that

1 1 P L = ~ ( P - T ' ) , P D = p + ~ ( P + T ' ) .

(2.22)

Then the 5 econstraint becomes

N

25e= (p+v/2T ') (p+x/~P) +x/~P'+ Z p2 i = 1

= 0 . (2.23)

Now one can easily solve eq. (2.23) for T o r P, and therefore put 5 p into a form which is linear in one of the momenta, a step which is crucial for formulating a Schr~Sdinger type equation [ 7 ]. Although in this way one preserves the manifest diffeomorphism covari- ante, the corresponding multifingered time Schr6- dinger equation is difficult to solve. Instead, we fix the time reparametrization invariance by choosing the gauge

T(x, t)=t. (2.24)

Then from eq. (2.23) we get

P P(x ) = - , / 5

x

_ ~e-VX dyepy • p2(y). (2.25) x/ z i = 1

Hence the independent canonical variables are (hi(x), (hi(x)) together with the x-independent vari- ables (p, q). The (p, q) variables are the global rem- nants of the graviton-dilaton sector, and they repre-

sent the physical degrees of freedom of that sector. The hamiltonian for the independent canonical vari- ables can be deduced from the f dx P~r part of the ac- tion to be

cp

+ ~ dxe-VX dye~Y pE(y), - - o ~ i = I

(2.26)

where c is a constant. In the compact case c is propor- tional to the volume of Y., and can be set to I. In the non-compact case, the value of c can be determined from the requirement of the asymptotic flatness of the black hole solution, whose ADM mass is asymp- totically conserved energy [ 11 ], and therefore M=H=cp.

The formulas (2.25) and (2.26) simplify if we use the Fourier modes of Pi,

Pi(x)=~nidkeik-~Odk, (2.27) - o o

and analogously for P~. In particular one gets for the hamiltonian (2.26)

H= - ~ + ~ dk a~_koli~ , (2.28)

which is almost like a free-field hamiltonian, except for the non-polynomial dependence on the momen- tum p.

It is a non-trivial task to deduce directly from our approach what is the range of p, but the identification o fp with the ADM mass M (up to a proportionality constant) implies that p ~ + . Also, when compared to the results of the Dirac analysis on a circle [ 6 ], p can be identified with the energy of a free relativistic 2D particle, whose range is positive. There is also a disconnected piece, corresponding to the negative energies, or negative M solutions. Since M < 0 solu- tions have naked singularities, we will quantize only the M > 0 sector.

3. Quant iza t iun

Given the reduced phase space and the corre-

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sponding hamiltonian, one can define the quantum theory by the Schrrdinger equation

i0t ~r't----/t~ T't , (3.1)

where/~ is an operator corresponding to the classical expression (2.28). ~Ubelongs to a Hilbert space con- structed from the canonical algebra of the basic vari- ables (p, q, hi(x), q~i(x) ), which are now promoted into hermitian operators. As in the classical case, it is convenient to use the Pi and P~ operators, satisfying

[P;(x), Pj(y) ] = - i,~' (x--y),~u,

[/~(X), ej(y)]=it~'(x-y)t~ij, (3.2)

and [P~,/~] =0, while forp and q we will take

[p,q]=ip, (3.3)

since psR +. Given the relations (3.2) there is an im- mediate problem of how to order the P's in expres- sion (2.28). However, given the simple form of H in terms of the a modes, and the fact that they resemble particle creation and annihilation operators, we can define a quantum theory based on the Hilbert space

~ * = 9¢~ (p) ® ~r(ot) ® ~ ( & ) , (3.4)

where Y~(p) is the Hilbert space associated with the (p, q) algebra, while if(or) and ~-(fi) are the Fock spaces built on the vacuum

a ~ 1 0 ) = ~~ct_kl0)=0, k>_-0. (3.5)

One can now introduce the standard field-theory cre- ation and annihilation operators as

a j ( k ) = x ~ k a ~ , k > 0 ,

1 a A k ) = - - , ~ k < 0 (3.6) k ,

Therefore ot k corresponds to the right-moving (k> 0) quant, while ~k correspond to the left-moving (k< 0) quant.

Given the Hilbert space Y:*, the hamiltonian H can be promoted into a hermitian operator

/ t = ~ + ~ - p o C P 1 i d k l k l a t ( k ) a , ( k ) . (3.7)

Note the absence of the left-moving modes in expres- sion (3.7). This is the consequence of the fact that

the b ° constraint does not depend on the P~ variables. This asymmetry arises from our choice of variables and the gauge-fixing procedure. In (2.11 ) we set J+ ~ T and subsequently Sex T+. Then we solve the J+ constraint by setting fl=/2 while the ~ constraint is solved by choosing the gauge (2.24), which is a choice of the time variable and therefore the 5 ~ con- straint is transformed into a Schr/Sdinger equation. Hence in the gauge (2.24) the T+ constraint gener- ates the time translations, while T_ generates the spatial diffeomorphisms and consequently P~ are fro- zen (integrals of motion). Clearly our choice of vari- ables and the gauge is convenient for describing a one- sided collapse, i.e., when initially one has only a right- moving matter.

4. Concluding remarks

The hamiltonian of our theory is a hermitian op- erator in the physical Hilbert space, and therefore any time evolution will be unitary. This implies in partic- ular that there will not be any transitions from pure into mixed states, which rule out the Hawking sce- nario [ 2 ] for the endpoint of the black hole evapo- ration process. However, in order to see what really happens during the gravitational collapse one has to carefully study the black hole solutions in this theory. The spatial metric g(x) can be written as

4-oo

~ 2 ~ ) 2 ( x ) - - ~ ) ( X ) (p+ 1 f dkLk eikx g(x) -~ ~ \ ~ J p+ik} ~ oc~

- v/2y' (x) , (4.1)

where

+oo

L k = dq k-q~'q . (4.2) - o o

Classical equations of motion imply

k p={H,p}=0, ~k={H, Olk}=-~pOlk, (4.3)

SO that from eq. (4. l ) one can find the spatial metric at any time. Note the similar structure of the expres- sion (4.1) and the corresponding expression of CGHS [ 1 ], where ourp is analogous to their M (ADM mass of the black hole), our arbitrary gauge function y (x)

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is analogous to their we (x) gauge functions, and the dependence on the scalar fields is similar. The differ- ences come from the fact that we are working in some type of the Polyakov light-cone gauge [ 12 ], while CGHS are in the conformal gauge.

When N = 0, then the black hole solution is equiv- alent to a choice ofT(x) such that

1---6 ~'2 (x) - P-x/r27'(x)

eZ.~

1 - (M/2)e - ~ " (4.4)

Solutions of this equation exist; however, we could not find an explicit expression. Such an expression will give the relation between the parameters p and M, and it will constitute an independent check of M=cp.

Since H,~p in the N = 0 case, one has an eternal black hole. Clearly in order to get some interesting effects, N must be different from zero. Then g(x) is given by expression (4.1), which becomes an opera- torial expression in the quantum theory. A normal ordering ambiguity in the Lk operators then appears. The standard normal ordering prescription causes a c-number anomaly in the commutator [R(x), ~(y) ], and we believe that this is a technical problem which could be resolved by an appropriate modification. Note that in the Dirac approach a c-number anomaly appears in the diffeomorphism algebra [ 6 ]. It would be interesting to see whether this anomaly is in any way equivalent to the metric anomaly in our ap- proach. A more difficult problem is the construction of a hermitian operator associated with the scalar curvature R. This operator is important since it will give a measure of a singularity. R is certain to be a non-polynomial function of the p and the a (k) ' s , which will be the main source of difficulties in con- structing the/~ operator.

An important issue which has to be analyzed is the Hawking effect. A natural way to do this in our model is to construct a state I ~Uo ) such that

~(X) I~o > =greg(X) I V,t/0 > , (4.5)

where gr~8(x) is a non-singular metric. Then evolve I ~Uo > in time by the evolution operator e -i~/t. At some

time tA an apparent horizon will form in the effective metric < ~Uo I e~atR e-iml ~Uo >. Then a reduced density matrix/~ can be introduced, by tracing out the states which are beyond the horizon [ 2,13 ]. How to define these states is not clear at the moment, but when this problem is resolved then one could in principle an- swer the questions about the thermal nature of~, i.e., when

1 e _ p 9 ' ( 4 . 6 )

and what are the non-perturbative corrections to the Hawking temperature

fl= ~ + .... (4.7)

Moreover, by analyzing the effective scalar curvature

Reff( x, t) = ( ~Uo I e~t/~ (x)e- iml ~Uo >,

one should be able to say what happens with the sin- gularity. Ideally, Refr(x, t) should stay a regular func- tion for any t.

We should emphasize that in our quantization scheme the topology of the space-time stays fixed. One could argue that this is the main reason why the theory is unitary, and no violations of quantum me- chanics occur. This may well be the case and we should point out that in the context of 2D gravity in- troduction of the topology change is equivalent to in- troducing the interactions in the corresponding string field theory. As a result, the string field 7 ~ will not satisfy the linear equation (3.1), but instead eq. (3.1) will be modified by ~v2 and higher order terms. This will directly violate the quantum mechanics. For- mally, one can invoke the third quantization in order to get around this problem; however, it is not clear how to define such a theory. Matrix models approach offers a definition [ 14 ], and there are indications that the Wheeler-DeWitt equation is not satisfied [ 15 ].

Clearly a lot of work remains to be done in order to answer all these questions. The main difficulty at the moment are the explicit calculations. However, we believe that the reduced phase-space quantization approach can answer, at least qualitatively, some of the issues raised in our discussion. Study of the the- ory in other gauges should also be beneficial, since then the questions of the diffeomorphism invariance

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cou ld be ana lyzed and the results in d i f ferent gauges

compared .

References

[ 1 ] C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D 45 (1992) R1005.

[2] S.W. Hawking, Phys. Rev. D 14 (1976) 2460. [3 ] T. Banks, A. Dabholkar, M.R. Douglas and M. O'Loughlin,

Phys. Rev. D 45 (1992) 3607; J.G. Russo, L. Sussldnd and L. Thorlacius, Phys. Lett. B 292 (1992) 13.

[4] S.W. Hawking, Phys. Rev. Lett. 61 (1992) 406. [5] A. Ashtekar, Nonperturbative canonical gravity (World

Scientific, Singapore, 1991 ); C. Isham, Conceptual and geometrical problems in quantum gravity, Imperial preprint Imperial/TP/90-91 / 14 ( 1991 ).

[6] A. Mikovi6, Phys. Lett. B 291 (1992) 19.

[ 7 ] K. Kuchar, Quantum gravity 2: a second Oxford Symp., eds. C.J. Isham, R. Penrose and D.W. Sciama (Clarendon, Oxford, 1981 ).

[8] N. Manojlovi6 and A. Mikovi6, Nucl. Phys. B 382 (1992) 148.

[9] J.G. Russo and A.A. Tseytlin, Nucl. Phys. B 382 (1992) 259.

[ 10] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605. [ 11 ] E. Witten, Phys. Rev. D 44 ( 1991 ) 314. [ 12 ] A. Polyakov, Mod. Phys. Lett. A 2 ( 1987 ) 893. [ 13 ] R.M. Wald, Commun. Math. Phys. 45 ( 1975 ) 9. [ 14] E. Br6zin, C. Itzykson, G. Parisi and J.B. Zuber, Commun.

Math. Phys. 59 (1978) 35; M.R. Douglas and S.H. Shenker, Nucl. Phys. B 335 (1990) 635; E. Br6zin and V. Kazakov, Phys. Lett. B 236 (1990) 144; D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127.

[ 15 ] A. Cooper, L. Susskind and L. Thorlacius, The classical limit of quantum gravity isn't, SLAC preprint SLAC-PUB-5413 (1991).

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