Symplectic structure of 2D dilaton gravity

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<ul><li><p>Physics Letters B 315 (1993) 267-276 North-Holland PHYSICS LETTERS B </p><p>Symplectic structure of 2D dilaton gravity </p><p>A. M ikov id a and M. Navar ro 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, </p><p>Campus de Fuentenueva, E-18002 Granada, Spain </p><p>Received 21 June 1993 Editor: R. Gatto </p><p>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. </p><p>I. Introduction </p><p>The study of 2D di laton gravity models has attracted a lot of interest because they can serve as toy models for understanding the physics of black holes (for a review and references see ref. [ 1 ] ). A model proposed by Callan, Giddings, Harvey and Strominger (CGHS) [ 2 ] has been an object of lot of studies, because of its nice proper- ties, of which its classical solvabil ity is crucial for our approach. As argued by many authors (see ref. [ 1 ] ), semiclassical quantization schemes of the CGHS model are not sufficient in order to understand the quantum fate of the corresponding 2D black hole. Non-perturbative quantization schemes were proposed by many au- thors (see ref. [ 1 ] ). In the canonical quantization approaches (see ref. [ 3 ] ), the knowledge of the phase space of the theory is crucial. </p><p>In the standard canonical approach, determination of the true (reduced) phase space becomes a non-trivial task because of the presence of the constraints. In the covariant phase space approach [ 4 ], the parameter space of non-equivalent solutions of the equations of motion is defined to be the true phase space of the theory. Since the classical solutions of the CGHS model are known, one can study the symplectic structure of the space of the solutions in order to find the reduced phase space of the theory and also find suitable variables for quantization. </p><p>This is usually achieved by evaluating the symplectic form on the space of the solutions, whose definition was given in ref. [4]. This method was employed to study certain 2D gravity models, like the Jackiw-Teitelboim and induced gravity models [ 5 ]. In this paper we apply the techniques developed in ref. [ 5 ] to the case of the CGHS model. </p><p>2. General solution </p><p>The classical action of the CGHS model can be written as </p><p>~" Work partially supported by the Comisi6n Interministerial de Ciencia y Tecnolog/a. </p><p>Elsevier Science Publishers B.V. 267 </p></li><li><p>Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993 </p><p>SCG~s=l ~ d2xxf~{e-20[R+4(Vcb)2+4)2] _ (Vf)2}, (1) .// </p><p>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]. </p><p>In 2D one can always chose the conformal gauge for the metric </p><p>ds 2= -e2pdx+dx - , (2) </p><p>or </p><p>1(0 e~") (3, gu, = - e2p , </p><p>where x + = t + x, x - = t - x. The equations of motion are then given by [ 2 ] </p><p>T++ =e-20[O+ (2p)O+ (20) -0 L (20) 1 + 1O+fO+f =0 , (4) </p><p>T__ =e-2~[O_ (2p)O_ (20) -02- (20) l + 1O- fO- f=0, (5) </p><p>T+_ =e-2 [0+0 (20) -0+ (2q~)0_ (20)--/'(2e2P] =0, (6) </p><p>-20+0_ (20) +~+ (2400_ (20) +0+0_ (2p) q-.~2e2p=0, (7) </p><p>o+0_f=0. (8) </p><p>Eqs. (6 ) and (7 ) are equivalent to </p><p>0+ 8_ ( e -2) + ,~2e(Rp- 2) ~--- 0 , ( 9 ) </p><p>0+0_ (2p- 20) =0. (10) </p><p>Eq. (10) implies that </p><p>e (2p-2~) =0+p0_ m , ( 1 1 ) </p><p>withp=p(x + ), rn=m(x- ). Eq. (9) then becomes </p><p>0+0_ (e -gO) +1.20+p0_ rn=0, (12) </p><p>and can be integrated as </p><p>e-2= -a2pm+a+b, (13) </p><p>where a=a(x + ) and b=b(x - ). Thus we can write </p><p>O+pO m e2p=. _22pm+a+ b . (14) </p><p>The constraint equations (4) and (5) can be now expressed in terms of the functions p, m, a, b as </p><p>o=r++=@a- j </p><p>0=r =(02-b ) __ -ff---mO_b +0_fO_ f . (16) 268 </p></li><li><p>Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993 </p><p>The general solution of eqs. ( 15 ) and (16) is given by x+ y+ </p><p>a=ao+ap-~ dy+O+p dz + ~ O+fO+ , (17) x- F - </p><p>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]. </p><p>Therefore the general solution of the system ofeqs. (4 ) - (8 ) can be written as x + v + x - y - </p><p>M-22pm- ~dv+O+Pldz+(~-~ f) f f (~'~O__fO__f), (19) e-2= T " ~O+fO+ -- dy-O_m dz- </p><p>e2P= (O+pO rn)e 2~, (20) </p><p>f=f+(x+)+f T(x -) , (21) </p><p>where M/2 = ao + bo. In the case when there is no matter one gets </p><p>M 0+p0 m e-2~=--22pm+ -,, e2P= - (22,23) z -22pm+M/2 " </p><p>The space of the solutions ( 19 )-(21 ) is invariant under the conformal transformations </p><p>x+=x+(y+), x -=x- (y - ) , (24) </p><p>0x + 0x- (25) O(x+'x-)=(Y+'Y-)' P(X+'X-):fi(Y+'Y-)+1n~y + Oy-" </p><p>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]. </p><p>3. Symplectic form </p><p>Given a field theory with fields T" and an action S[ T], one can obtain the corresponding symplectic form as </p><p>~o= f dau( -S j z ) , (26) X </p><p>where Z is a spatial hypersurface, while j** is the symplectic current, j ~ can be obtained from the variation of the action </p><p>5s= j '0~.+ j" ~is 5T ~ 5 T ~ </p><p>(27) </p><p>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=const, hypersurface. With this choice of Y. we have </p><p>269 </p></li><li><p>Volume 315, number 3,4 </p><p>o9= _I dx ( -S j ) , t=to </p><p>where jo = j + + j-. </p><p>PHYSICS LETTERS B 7 October 1993 </p><p>(28) </p><p>In order to calculate the symplectic form of the theory ( 1 ) we can start from the lagrangian in the conformal gauge. We get </p><p>N </p><p>~=e-2O[0+8_ (2p) - 0+ (20)0_ (2q~) +j.2e2p] + 8+fS_fi. (29) i </p><p>Variation with respect to ~ and p gives </p><p>8S 8S 8 = 80+Vsp </p><p>+8+ (e-2%_ 8(2p)-e-2O_(20)8(20)+~ ~i O_fSf ) </p><p>+8_ ( -8+ (e-2)8(2p)-e-28+ (2q})8(20)+ ~ 8+fSf). (30) </p><p>Thus the symplectic current is </p><p>j+ =e-2%_8(2p) -e-28_ (2~)8(2~) + ~ 8_fSf, (31) i </p><p>j - = - 3+ (e-2O)8(2p) - e-2~0+ (2~)8(2~) + ~ O+fSf. (32) </p><p>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: </p><p>09= .f dxo9 , (33) t~lO </p><p>where </p><p>o9= (0+ -0_ ) [8( -22prn+a)8 In 0_ m+ 8( -2Zpm)8 lnp+ 8a 8 In 0+p- 8a 8 In 0+ a+ 8b 8 In 8_ b] </p><p>+l[-saS(o-!+-aS+fS+f )-Sbs(ol~_b S_fS_f ) +(SfSO f +Sf80+f)l. (34) </p><p>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 </p><p>- 8a \~+p 8+ aJ - 8b 8 \OT_ m O_ bJ </p><p>=- (0+-O_)[Sa 8 ( ln0+p- ln0+a) ]+80+a 8( ln0+p-lnO+a) </p><p>-(O+-~_)[8b8(lnO rn-lnO_b)]+80_ b 8(ln8 m- ln0_b) , we find a simpler expression for the symplectic form </p><p>(35) </p><p>270 </p></li><li><p>Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993 </p><p>CO= [5( --22pro +a+ b)6 In O_ m+ 6( -22pm)5 In p]~ X2 t~ </p><p>+ j dx[50_b 51nO_m+SO+a 51nO+p+(6fSO_f+Sf60+f)]. (36) Xl </p><p>which we rewrite as </p><p>co= [5(a+b)51n 0 m+ 8( -2Zpm)~ In (p 0_ in) ]~ </p><p>+ ~ dx(50_ b SlnO_,n+ 80+a SlnO+p) + ~ ~ dx SfSf~ , (37) x I x I </p><p>where rl,2 are the endpoints of the x-interval, f= Of/Or and </p><p>IF(x) lx~ -F(x ix,- . 2)--F(xl). (38) </p><p>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 = - Be-z% (2p_ 2~) we get </p><p>co= - 6(a+b)6 In ~ + 8( -- )~2pm po_ rnj~, </p><p>+~dx(80_b61nO m+50+a61nO+P)+~dxgfSfj. A-I X l </p><p>However, we will use eq. (37) since it is simpler for calculations. </p><p>(39) </p><p>4. CGHS model on a circle </p><p>Until now the discussion of the CGHS model has been general, without specifying the topology of the mani- fold where it is defined. In this section we will choose the two-dimensional spacetime manifold to be of the form Sl~. </p><p>4.1. Pure dilaton gravity </p><p>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 </p><p>M 0+p 0_ in e -2=-22P m-I- ~ , e 2p= _2Zpm+M/2. (40) </p><p>In this case the symplectic form becomes </p><p>x+2n , </p><p>o-_- j (0+-a_ )w. (41) x </p><p>Since O+-a =04 we have </p><p>co= W(x+27r) - W(x), (42) where </p><p>271 </p></li><li><p>Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993 </p><p>W=8~)81nO_m+ 5( -22pm)81n (pO_m) . (43) </p><p>It is important to notice here that p and m do not have to be single-valued functions on the circle. They can have a nontrivial monodromy transformation </p><p>1 p(x++Z~z)=qp(x+) , m(x- -27~)=qm(X- ) , (44) </p><p>where q is an arbitrary positive 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 arbitrari 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 inserting (40) into (38 ) so that </p><p>co=-sMSlnq . (45) </p><p>We conclude that in the compact case without matter 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 momen- tum conjugate to the monodromy parameter q. It is tempting to identify M with the mass of the black hole. However, it is not clear whether in the compact case such an identif ication makes any sense. </p><p>Our task now is to determine the exact reduced phase space of the model. To do that we have to find out the exact reduced phase space of the model. To do that we have to find out the maximal set of functions p and m which, beside fulfilling the right monodromy transformation properties, define physical fields with the appro- priate sign. In the present case we need to find out the functions p, m, with the appropriate monodromy prop- erties such that both e 2p and e -2 are positive everywhere and everywhen. If we take q= e r with re[R, then any functions with the appropriate monodromy transformation properties can be written as </p><p>p= e(r/2~)X+u, m= -jse~r/2~):'-v, (46,47) </p><p>where u = u (x + ) and v = v(x - ) are periodic functions on the real line. With this choice for p and m, the physical fields take the form </p><p>M (1 -uve (~/2~)2t) , (48) e-2O= </p><p>e2p= (M/2)(1/22) e(r/2~r)2t[~+ U+ (r/2rOu] [0 v+ (r/2rOv ] (M/2 ) ( 1 -uv e C~/2~)2t) (49) </p><p>The periodicity of the functions u and v, together with the positivity of the exponential e (r/27rI2t and the fact that the coordinates x and t, or what is almost the same, x + and x - , can be varied independently of each other, makes it easy to realize that the posit ivity of both the metric and the exponential of the dilaton field imply that the parameter M and the product uv are restricted to be positive (we are taking 2&gt; 0). The positivity of the product uv restricts in turn the allowed values for r and t. It is clear form the expression (48) that the range of their allowed values does not cover the entire real line. For example, if we permit t to be positive then r is bounded by above, and r is bounded by below if t is negative. </p><p>To go on with the analysis let us consider the solutions above after an appropriate 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 </p><p>m ( 1 - -e (r/2~)at) , (50) e -2O= A </p><p>272 </p></li><li><p>Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993 </p><p>1 (r/2g)Ze (r/2~)2' e2p= 22 1 --e (r/2Jr)2t (51) </p><p>It is clear from (50), (51) that, for M&gt;O, e -2 and e 2v are positive if and only if r0 or r&gt;0 and t&lt; 0. The reduced phase space splits then into two disjoint pieces, one with r&gt; 0 and the other with r&lt; 0. </p><p>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 </p><p> o=aa ap. (52) </p><p>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 </p><p>T 'R+ wT*R_ . (53) </p><p>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 </p><p>~ = L2 ( t+ , ~h) (~)g2 (g_ , -~) , (54) </p><p>where the measure dh/h accounts for the restriction in sign of the parameter h. </p><p>4.2. Inclusion of the matter </p><p>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 </p><p>co= -8(a(x) +b(x) )8 In q x+2z~ x+2~ </p><p>+ f dy(aa_bSlna_m+6a+aflnO+p)+ ~ dySfi~j[:. x x </p><p>(55) </p><p>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). </p><p>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><p>~+p=e(r/2rOX+W+, 0_m=e (r/2")x w_, (56,57) </p><p>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. </p><p>We then have </p><p>o9= - 8(a(x) +b(x) )5 In q x+2n x+2n </p><p>r +~sr + ~ dy[aa+b(alnw_+a-~y-)+aa+aa(alnw+ ~y+)]+ ~ dyf fa f . (58) x x </p><p>273 </p></li><li><p>Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993 </p><p>Note now that if a + b is to be a single-valued function on the circle, then a and b can be expanded as </p><p>a(x+)=ao+s( t+x)+ ~ ane -i"X+ , b (x - )=bo+s( t -x )+ ~ bne - inx- . (59,60) n4-0 ns~0 </p><p>Replacing this expansion into (58) we find after some integrations by parts x+2n </p><p>e)=-8~-Sr+ dy[ (8O_bSlnw +8~+aSlnw+)+(SfSJ~)] , (61) x </p><p>where M is the zero mode of a + b, M = 2 (ao + bo ). The symplectic form (61 ) c...</p></li></ul>