Quantization of a theory of 2D dilaton gravity

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<ul><li><p>Physics Letters B 289 (1992) 278-282 North-Holland PHYSICS LETTERS B </p><p>Quantization of a theory of 2D dilaton gravity </p><p>S.P. de Alwis 1 Department of Physics, University of Colorado, Box 390, Boulder, CO 80309, USA </p><p>Received 3 June 1992 </p><p>We discuss the quantization of the 2D gravity theory of Callan, Giddings, Harvey, and Strominger (CGHS), following the procedure of David, and of Distler and Kawai. We find that the physics depends crucially on whether the number of matter fields is greater than or less than 24. In the latter case the singularity pointed out by several authors is absent but the physical interpre- tation is unclear. In the former case (the one studied by CGHS) the quantum theory which gives CGHS in the linear dilaton semi-classical limit, is different from that which gives CGHS in the extreme Liouville regime. </p><p>Recently Callan, Giddings, Harvey, and Stromin- ger [1 ] (CGHS) , discussed a model for two dimen- sional (di laton) gravity coupled to matter. They showed that classically the theory has solutions cor- responding to collapsing matter forming a black hole. This solution is in fact a linear dilaton fiat metric one, patched together with Witten's [ 2 ] 2D black hole so- lution, along the infall line of a shock wave of 2D massless matter. In order to incorporate the quantum effects (in lowest order) CGHS included the contri- bution of the conformal anomaly coming from the conformally non-invariant measure in the matter sector path integral. </p><p>In this paper we examine the consistency of this procedure. It is argued that one way of carrying out the quantization of the theory is to follow the proce- dure of David, and of Distler and Kawai [ 3 ] ~1. Then we rediscover the singularity pointed out in refs. [ 5,6 ] when the number N of matter fields is greater than 24, and furthermore we find that the quantum theory which leads to the CGHS action in the semi-classical l inear dilaton region is different from the one which gives the CGHS action in the extreme Liouville re- gion. For N&lt; 24 there is no field space singularity but </p><p>I E-mail address: dealwis@gopika.colorado.edu. ~' Similar methods have been used in ref. [4]. However, these </p><p>works do not discuss the particular conclusions for the CGHS theory which is our main focus here. I wish to thank Dr. Chamseddine for bringing these references to my attention after an earlier version of this paper had been circulated. </p><p>it seems to lead to an unphysical theory with a nega- tive flux of black hole radiation. The classical CGHS action ~2 is </p><p>S=~--~f d2a x / / -~ (e-2~ [R+4(V~)2-422 ] </p><p>where ~ is the dilaton andf i are N (unitary) matter fields ~3. The corresponding quantum field theory is defined by </p><p>y [dg]g[d]g[df]g exp( iS[g, ~,./] ) . (2) Z= [ Vol. Diff. ] </p><p>The measures in the above path integral are de- rived from the metrics, </p><p>II 8gll ~ = d2tr x / / -~ g'~Yg~ ( 8gaa 8gra + 8gc,~ 8gpa) , </p><p>r </p><p>To evaluate the path integral one needs to gauge- </p><p>~2 We use MTW [ 7 ] conventions. ~3 This lagrangian comes from the low energy limit of string </p><p>theory. </p><p>278 0370-2693/92/$ 05.00 1992 Elsevier Science Publishers B.V. All rights reserved. </p></li><li><p>Volume 289, number 3,4 PHYSICS LETTERS B 10 September 1992 </p><p>fix it. We choose the conformal gauge g= e 2p ~, where is a fiducial metric. Then the path integral becomes </p><p>Z= ~ ( [dp] [d0] )~e2p </p><p> exp [iS(0,p) ] Af(e2p ~)Avp(eZP ~), (4) </p><p>where S(0, p) is the pure gravitation-dilaton part of ( 1 ), the last factor is the Fadeev-Popov ghost deter- minant, and </p><p>Af(e2p~) = ~ [dJq~e:pexp[iS(f) ] , (5) </p><p>S( f ) being the matter action. The measures in (4), (5), are again given by (3) </p><p>except that we must put g= e zp ~. In particular we have (up to a constant) </p><p>I18pI12= J" d2ax /~Sp 2 . (6) </p><p>From the well known transformation properties [ 8 ] of the matter and ghost determinants, </p><p>Ai(e 2p R)Avp (e 2p R) </p><p>=ay(R )Ave( ~ , exp [ i( ~ ( N - 26 )St (p, R ) </p><p>where </p><p>1 SL(p, ~) = ~-~ I d2a V/-~ [ (~p)2+/~p]. (8) </p><p>The quantum theory is then given by </p><p>Z= f ( [dp] [dO] )eZo~[df]o~( [db] [dc] )e </p><p> exp [ iS(p, O, f ~) + iS(b, c, ~) ] . ( 9 ) </p><p>S(p, ,fR) _lnf d2ax/-S-~(e-2~[4('O)z-490.~p] </p><p>N </p><p>-xfTp.~Tp- E V f i~f ' 1 </p><p>e2(p-)), (10) +/~ (e-20- top) -4~ 2 </p><p>where ic= ~ (26 - N). For ~-- )/the Minkowski met- ric, this reduces to the CGHS action with conformal anomaly term ~4 </p><p>There is, however, something strange about the path integral (9). The measures for matter and ghost are defined relative to the fiducial metric ~ while the p and 0 measures are still defined in terms of the orig- inal metric g= e 2p ~. In particular this means that the p measure is not translationaUy invariant, and there- fore that for example the (Dyson-Schwinger) quan- tum equation of motion gets modified from the equa- tion derived from (10). In order to formulate the quantum theory in a manner which yields a system- atic semi-classical (or 1 IN) expansion it is necessary to rewrite all measures in terms of the fiducial metric ~. Thus we need to do what David, and Distler and Kawai [ 3 ] did for conformal field theory coupled to 2D gravity. </p><p>Assume (as in ref. [3] ) that the jacobian which arises in transforming to the measures defined in terms of ~ is of the form e iJ where J is a local renor- malizable action in p and 0. Putting XU= (O, P) we may write </p><p>Z= f [dXU]~[dj]e( [db] [dc] )~, </p><p>exp[iI(X,~)+iS(f,~)+iS(b,c,~)], (11) </p><p>where </p><p>I[X, ~1 = - ~ x/r-~ [ lgabG.u u OaX u ObX" </p><p>+ l~rP(X) + T(X) ] . (12) </p><p>In the above Gu~, q~ and Tare functions of Xwhich are to be determined and the measure [dX u ] is de- </p><p>In the above equation S(b, c, ~,) is the ghost action and ~4 Eq. (23) of ref. [ 1 ] except that the ghost contribution is ig- nored there. </p><p>279 </p></li><li><p>Volume 289, number 3,4 PHYSICS LETTERS B 10 September 1992 </p><p>rived from the natural metric on the space ]lSXull2= f d2a x /~ Gu~ ~XU ~X~. </p><p>The only a priori restriction on the functions G, O, and T comes from the fact that Z must be indepen- dent of the fiducial metric ~, i.e., the theory defined by the action I+Sy+Sbx with the standard transla- tionally invariant measures is a conformal field the- ory with zero central charge. So we must satisfy the B-function equations </p><p>Bu,,=&amp;,,+2V~O~O-OuTO,,T+..., </p><p>fl~ = - :~ +4G *'~ OuO OvO-4 V20 </p><p>+ ~ [ (N+2) -251 +G u~ OuTO,,T-2T2+..., </p><p>fiT= -2 V~T+4G u" OuOO~T-4T+ .... (13) </p><p>In the above Y/is the curvature of the metric G. These conditions are not sufficient to determine the functions uniquely, but clearly they are necessary. If no further restrictions are imposed, they define a class of quantum 2D dilaton-graviton theories. The anal- ysis of CGHS and others [1,5,6 ] will be valid pro- vided that the functions G, O, and T, defined by (10) satisfy (13) at least in the semi-classical regime. Be- cause of what happens in the corresponding case studied in ref. [3] we will make x [see (10) ] a pa- rameter to be determined by (13). Comparing (12) with ( 10 ) we have </p><p>G~=-8e -z~, G~p=4e -2~, Gpp=2x, (14) </p><p>O=-e-2~+xp, T=-422e 2~p-~. (15) </p><p>It is easy to see that the curvature ~ = 0. So we may transform to a field space coordinate system which is euclidean (or Minkowski). The transformation </p><p>p=x -t e-2~+y (16) </p><p>gives for the metric in field space </p><p>ds2= -8 e-2(dO2-d0 dp) + 2x dp z </p><p>= _ 8 e_40 ( 1 +xe 2) d02+ 2~ dy 2 . ( 17 ) K </p><p>In the latter form we see (for x&lt; 0) the singularity pointed out in refs. [ 5,6 ]. Now let us introduce a field space coordinate </p><p>x= f e -zp ( 1 +/ e2') 1/2 . (18) d </p><p>Note that if x&lt; 0 x is real only in the "linear dila- ton" region x e 2~ &lt; 1. It is also convenient to intro- duce two more coordinates, </p><p>x_-2 U- x, Y_- ~/IKI </p><p>Then we have </p><p>ds2=- 8 dx2+2~ dy2= T- dX2__dY, (19) K </p><p>where the upper or lower signs are to be taken de- pending on whether x is positive or negative respec- tively ~s. In the Liouville region e-2U I xl &lt; 1, we de- fine the coordinate </p><p>.(=2V/2~dOe-(l+e-xC~)'/2dO, (20) </p><p>we get </p><p>ds2- - -d.~2___ dY 2 . (21) </p><p>As before the upper or lower signs are to be taken de- pending on whether x is positive or negative. Note that X is real in the linear dilaton region and imagi- nary in the Liouville region while the converse is true for )?. From ( 15 ) and the above we also have the form of the dilaton in the new coordinates, </p><p>c19=xy= + x/ + 1c Y. (22) </p><p>Thus in these new coordinates we have a euclidean (Minkowski) metric linear dilaton theory in field space, and the first beta function equation (13) is satisfied if we ignore quadratic terms in T. From the second equation in (13) we then get </p><p>x= ~ (24-N) . (23) </p><p>Thus the metric signature on field space as well as the absence or presence of a singularity depends on whether N&lt; 24 or N&gt; 24. As is well known the linear dilaton euclidean (Minkowski) metric theory is an exact solution of the beta function equations, i.e., our solution (with T= 0) exactly satisfies the sufficiency criterion discussed in the sentence before ( 13 ). </p><p>Let us now discuss the tachyon T. We do not know </p><p>~5 One may consider the coordinates x, y (or X, Y) as the field space analog of Kruskal-Szekeres coordinates! Of course the physical interpretation in terms of the original physical coor- dinates is valid only outside the field space coordinate singularity. </p><p>280 </p></li><li><p>Volume 289, number 3,4 PHYSICS LETTERS B 10 September 1992 </p><p>how to incorporate the contribution of the tachyon exactly. All we can do is to work to linear order in T. Thus our discussion is valid only for 22 &gt; Ixl. Using the expansion of (18) and the expression for y (16), we find </p><p>= -422 e 2p-2 h(x e 20) , (25) where h(xe2O)=l+O(xe 2') and indeed can be written out exactly. </p><p>Now let us discuss the theory in the "Liouville re- gion" x&gt; e -2~. The appropriate coordinates are X, Y defined in (20). Solving the tachyon equation in these coordinates and imposing the boundary condition that the CGHS expression (15) is reproduced in the extreme Liouville regime x &gt;&gt; e-2~ &gt;&gt; 1 we find #6 </p><p>T=-22tc[cos(X)exp(N/U~ Y)-e'~+r], (26) </p><p>where a+=-xfZ~+~. By using the transformations (20) in the large I r l limit it is easily seen that Tgoes over to the expression given in (15). For x 24) which gives CGHS in the large N limit we have two wrong sign fields and the conformal symmetry is not sufficient to gauge them both away. However, since neither the graviton nor the dilaton are propagating modes, the above probably does not mean that the theory is non- unitary. </p><p>~7 The theory given by (27) is of the Liouville type and in fact can be solved. Also given that the Liouville theory is supposed to be a conformal theory at the quantum level it is likely that the same is true of (27) i.e., to all orders in 22 (provided that the functional integration range is extended to the whole real line even though the interpretation in terms of the original variables is valid only over the half line for X). These issues and their physical implications are currently under investigation. </p><p>281 </p></li><li><p>Volume 289, number 3,4 PHYSICS LETTERS B 10 September 1992 </p><p>Note added. While this work was being prepared for publication, a preprint by Strominger [ 9 ] was re- ceived, in which the N&lt; 24 case with what is effec- tively a modif ied G, to avoid the problem of a nega- tive flux of Hawking radiation, is discussed in some detail. This theory can in fact again be written in the form (27) with T given by (2 5 ) except that the relation between x, y and , p is modif ied from (16) and (18) top=x I (e-2~+4~) +y, d r= [ (e -2 ' -2 )2+ /c(e -2 . - 1 ) ]1/2 dO. </p><p>The author is grateful to Leon Takhtajan for a dis- cussion. This work is partial ly supported by Depart- ment of energy contract No. DE-FG02-91-ER-40672. </p><p>References </p><p>[ 1 ] C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D 45 (1992) R1005. </p><p>[2] E. Witten, Phys. Rev. D 44 (1991) 314. [3] F. David, Mod. Phys. Lett. A 3 (1988) 1651; </p><p>J. Distler and H. Kawai, Nucl. Phys. B 321 ( 1989 ) 509. [4] A. Chamseddine, Phys. Lett. B 256 ( 1991 ) 379; B 258 ( 1991 ) </p><p>97; Nucl. Phys. B 368 ( 1992 ) 98; A. Chamseddine and Th. Burwick, preprint hepth 9204002. </p><p>[ 5 ] J.G. Russo, L. Susskind and L. Thorlacius, Stanford preprint SU-ITP-92-4 ( 1992); L. Susskind and L. Thodacius, Stanford preprint SU-ITP- 92-12 (1992). </p><p>[6] T. Banks, A. Dabholkar, M.R. Doublas and M. O'Loughlin, Rutgers preprint RU-91-54. </p><p>[7] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). </p><p>[8] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [9]A. Strominger, Santa Barbara preprint UCSBTH-92-18 </p><p>(1992) [hepth@xxx/9205028]. </p><p>282 </p></li></ul>