- Email: [email protected]
Quantization of the string inspired dilaton gravity and the Birkhoff theorem
9 April 1998
Physics Letters B 424 Ž1998. 265–270
Quantization of the string inspired dilaton gravity and the Birkhoff theorem Marco Cavaglia` a , Vittorio de Alfaro a
b,1
, Alexandre T. Filippov
c
Max-Planck-Institut fur ¨ GraÕitationsphysik, Albert-Einstein-Institut, Schlaatzweg 1, D-14473 Potsdam, Germany b Dipartimento di Fisica Teorica dell’UniÕersita` di Torino, Via Giuria 1, I-10125 Torino, Italy INFN, Sezione di Torino, Italy c Joint Institute for Nuclear Research, R-141980 Dubna, Moscow Region, Russia Received 9 May 1997; revised 20 February 1998 Editor: L. Alvarez-Gaume´
Abstract We show that, for the pure dilaton CGHS model, the Gupta-Bleuler quantization is viable because the constraints can be linearized due to positivity conditions present in the model Žand not in the general string case.. There is no ambiguity nor anomalies in the quantization. The expectation values of the metric and dilaton fields obey the classical requirements, thus exhibiting at the quantum level the Birkhoff theorem. q 1998 Published by Elsevier Science B.V. PACS: 04.60.-m; 04.60.Kz; 03.70.q k Keywords: Quantum gravity; Two-dimensional models; Canonical quantization; Field theory
Gravitational models in 1 q 1 dimensions may hopefully clarify some difficult problems of quantum black holes and, more generally, of quantum gravity in higher dimensions w1–6x. Most of the models studied in some detail in the literature are dilatongravity ŽDG. models coupled to gauge and scalar fields Žfor a very compact overview and references see w4x.. In general, these models are not integrable and their solution cannot be found even classically. As a consequence, not much is known about their quantization: only the so-called CGHS model w1x has been studied in detail w2,3,5x. Two-dimensional pure DG is classically inte-
1
E-mail: [email protected].
grable and both the metric and dilaton fields may be expressed in terms of one free ŽD’Alembert. field and of one invariant parameter which is a local integral of motion independent of the coordinates Ž‘‘mass’’ operator.. Using the gauge in which this free field is one of the coordinates, any solution depends actually on one coordinate. These properties constitute a generalization of the classical Birkhoff theorem for spherically symmetric gravity in any dimension Žwe thus call, somewhat improperly, ‘‘ static’’ these solutions.. The drastic reduction of DG to a finite-dimensional dynamical system signals that DG is actually a topological theory. Unlike the theories coupled to scalars, the pure DG model can be quantized by first reducing it to a finite dimensional dynamical system
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 2 2 6 - 3
266
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
with a single constraint. ŽThis can be regarded as a particular gauge fixing.. The resulting Hilbert space is spanned by the eigenvectors of the Žgauge invariant. mass operator. This quantization approach, that uses first the Birkhoff theorem and then the quantization algorithm, has been introduced in Ref. w7x in the case of the 3 q 1 dimensional Schwarzschild black hole; the procedure can be generalized to any pure two dimensional DG model. A further approach to the quantization of the pure DG model does not use the topological nature of the models nor the Birkhoff theorem. Since the classical solution can be written as a function of a single free field, one may try and find a canonical transformation of DG to constrained free fields. This transformation has been implemented in the CGHS case w3,5x, and recently an interesting new proposal in this direction has been advanced w8x. ŽThe main results of these investigations are clearly summarized in the report w9x.. However, up to now an explicit canonical transformation has not been constructed for the general case. Furthermore, the investigation of the CGHS model revealed two main obstacles to quantization: first, the quantum canonical transformations to free fields are highly nontrivial and difficult to construct; second, the quantization is obstructed by anomalies in the commutator of the constraints, and the theory needs to be modified. Due to the presence of anomalies, different quantum theories corresponding to the same classical model can be found. In particular, in the Schrodinger representation there exists a quantiza¨ tion scheme in which the anomaly in the CGHS model is cancelled w3x. However, this scheme is based on the use of negative energy states for the string fields instead of the standard string quantization that introduces negative norm states, and the result obtained has no evident connection to the quantum version of the reduced theory via the Birkhoff theorem. For these reasons we return to the subject and consider a further quantization scheme for the pure dilaton CGHS model, that directly proves the Birkhoff reduction at the quantum level and does not produce any anomalies. The quantization is carried out by use of the standard Gupta-Bleuler ŽGB. method. This is possible because the model has actually more information in it than the canonically transformed version described by a couple of string
fields. In the first place, there exists a gauge invariant local mass that cannot be expressed purely in terms of the string fields. Moreover, further information comes from the canonical transformation between the original fields and the string ones, under the form of positivity conditions for certain functions. This allows an essential step: the linearization of the constraints. There is no anomaly in the algebra of the linearized constraints Žthe algebra of the quadratic constraints does not concern us any more.. This quantization of the string-like fields is similar to the ‘‘Schrodinger’’ quantization that uses positive ¨ norms and negative energy states w3x. The only gauge invariant operators are the mass and its conjugate momentum Žconjugate operators were discussed in Refs. w10,11x. and the ground state of the string must be labeled by the eigenvalue of the mass operator Žas discussed in a different context in Refs. w2,11x.. Our quantization procedure explicitly shows how the reduction to quantum mechanics is achieved by the GB quantization of the field theory. The quantized field theory has the same content as scalar-longitudinal electrodynamics and is pure gauge. The set of GB states corresponds to reparametrization of the coordinates, expectation values of the metric and dilaton fields agree with the corresponding classical quantities and the ‘‘Birkhoff reduced’’ quantum theory is recovered. Our starting point is the two-dimensional action S s d 2 x y g Ž w R y lr2 . ,
H '
Ž 1.
where w is the dilaton field. The two-dimensional metric gmn can be written w3x gmn s e f
ž
a 2-b 2 b
b . y1
/
Ž 2.
Here a and b play the role of the lapse function and of the shift vector respectively; f Ž x 0 , x 1 . represents the dynamical gravitational degree of freedom. The coordinates x 0 , x 1 are both defined on R. Eq. Ž1. can be cast in the canonical form S s d 2 x f˙p f q wp ˙ w yaHyb P ,
H
Ž 3.
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
where the super-Hamiltonian and super-momentum are respectively H s p f pw q f Xw X y 2 w XX q l e fr2 s 0 , P s 2p fX y pw w X y p f f X s 0 .
Ž 4.
ŽWe neglect boundary terms as they are irrelevant for the following discussion. See e.g. Ref. w5x and references therein.. We may also define a functional M of the canonical variables which is conserved under time and space translations Ž M˙ s M X s 0. w4,11,12x: yf
M s lwr2 q e
ž
p f2 y w X 2
/.
Ž 5.
The Birkhoff reduction may be stated as follows. Setting a s 1 and b s 0 and introducing the coordinates u s Ž x 0 q x 1 .r2, Õ s Ž x 0 y x 1 .r2, the two-dimensional line element corresponding to the metric tensor Ž2. becomes ds 2 s 4 r Ž u,Õ . dudÕ Ž r s e f .. A metric is called static iff it can be cast in the form w4x
r Ž u,Õ . s h Ž C . aX Ž u . bX Ž Õ . , C ' aŽ u . q b Ž Õ . ,
Ž 6.
where a and b are arbitrary functions. If, in addition, w depends only on C , the solution will be called static. In the pure DG case the general solution can be written
r ' e f s F X Ž C . EuCEÕC , wsFŽ c . ,
Eu EÕC s 0 .
Ž 7.
So any solution of the generic DG model is static. In particular, for the CGHS case we find F ŽC . s C0 eylC r2 q 2 Mrl. Here C0 is an integration constant; the second integration constant was expressed in terms of the constant M by using Ž4.. M appears thus as a zero mode of the field w . Now, let us use the canonical transformation w3x
l A 0 s 2 eyf r2 Ž p f cosh S y w X sinh S . , p 0 s yl e f r2 cosh S y l AX1 , l A1 s 2 e
yf r2 f r2
w Ž x 0 ,c .. To make the transformation invertible, one has to supplement the new field variables Aa ,pa by a pair of conjugate variables, e.g. w Ž x 0 ,c s `. and 2 S Ž`. Ž w 0 will commute with all the field variables if we choose c s `.. Using Ž8., Ž9. the two constraints become 1 Hs
2l
l
p apa q
AX aAXa s 0 ,
2
P s yp aAXa s 0 ,
Ž 10 .
where the indices are lowered and raised by the two-dimensional Minkowski metric h s Ž1,y 1.. The functional M is
l M s M0 q
x1
H 4 b
dxX1 Ž ´ a bpa Ab q l A aAXa . ,
Ž 11 .
where ´ ab is the two-dimensional Levi-Civita tensor. The value of b is irrelevant and M0 must be a constant since M is independent of t. On the equations of motion we have M s M0 . Let us introduce the operators C a and D a C a s p a y l´ abAXb , D a s p a q l´ abAXb .
Ž 12 .
M˙ and M X are proportional to C a . The classical quantities H , P, r , w , M can be written as functions of C a , D a and Aa . They read 2 l H s C a Da ,
2 l P s ´a b D a C b ,
l2r s D a Da ,
Ž 13 . a
w s 2 Mrl y l Aa A r2, M s M0 y Ž lr4 .
x1
Hb
dxX1 ´ a bAa Cb .
Ž 14 .
The Poisson brackets of C a and D a are
Ž 8.
X
Ž p f sinh S y w cosh S . ,
p s l e sinh S q l AX0 , x1 where S Ž x 1 . s 12 Hy` dxX1pw Ž xX1 .. 1
267
C a Ž x 0 , x . , D b Ž x 0 , y . s 2 l´ a bE y d Ž x y y . .
Ž 15 . Ž 9.
The transformation Ž8., Ž9. is canonical for the field variables Žyw X , S , f,p f ; Aa ,pa .. Note that the inverse transformation only defines w up to a zero mode w 0 Ž x 0 . '
The constraints can be linearized, using a suitable redefinition of the Lagrange multipliers. The resulting linearized constraints are the functions C a. In this form of the theory the generators are linear in the string variables Aa .
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
268
It is useful to introduce the conjugate variables Ž P ",X X " . w5x 2'l P "s C 0 " C 1 , X D 0 y D1 , 2'l X " s "D
Ž 16 .
and define the constraints Hq and Hy that generate reparametrization in u and Õ; they can be written as functions of the constraints P " and of the space X derivative of the conjugate constraints, X " : 0
1
0
X s y2 Xy Py ,
s yhab
Ž 18 . X X"
A crucial point is that have definite signs, as a consequence of Eqs. Ž8., Ž9.. Then it is easy to control that the constraints H s 0 and P s 0 are satisfied iff C a s 0. Accordingly, the coefficients X X" can be reabsorbed in the Lagrange multipliers. We have L s X˙q Pqq X˙y Pyy rq Pqy ry Py, where the Lagrange multipliers have been defined as X X rqs 2 Xq ly, rys y2 Xy lq. Finally, since C a Ž x0 , y . r Ž x 0 , x . ,C s 4´ ably1 Db Ž x 0 , x . E y d Ž x y y . ,
H 2p d Ž k
2
. ´ Ž k 0 . e i kŽ xyy. .
Ž 22 .
Aa s
dk
yi v x 0 qi k x 1
Hy` 2'pv b Ž k . e a
q h.c. ,
Ž 23 .
where v s < k <, or
Hy' H y P s Ž D 0 y D 1 . Ž C 0 q C 1 . r2 l X s 2 Xq Pq .
d2k
The field expansion is `
Ž 17 .
Ž 19 .
the linear constraints, P " or Ca , generate the reparametrizations of the metric. As a consequence of the linearization, the theory can be quantized in the GB scheme. Before doing that, let us see the form that the classical solution takes in terms of the fields Aa . The equations of motion are
Ž 20 .
and the solution is Aa s Ua Ž u. q Va Ž Õ .. The linear constraints C a s 0 correspond to h abEa Ab s 0, ´ abEa Ab s 0, or U0 Ž u. s U1Ž u., V0 Ž Õ . s yV1Ž Õ .. One can prove that the above solution coincides with Ž7.. Substituting Aa into Ž13., Ž14. one obtains
r ' e f s 4 U0X Ž u . V0X Ž Õ . , w s 2 Mrl y 2 lU0 Ž u . V0 Ž Õ . .
Aa Ž x . , Ab Ž y .
1
Hq' H q P s Ž D q D . Ž C y C . r2 l
l A˙a s ha b p b , p˙ a s lh a bAXXb ,
The quantization is achieved by imposing the commutation relations Žfrom now on we set l s 1, there is no real loss of generality while the formulae become more elegant.
Ž 21 .
This solution coincides with Ž7. if C is defined by C0 eylC r2 s y2 lU0 Ž u.V0 Ž Õ .. The ‘‘staticity’’ of the classical solution is thus embodied in Eqs. Ž21..
`
Ua s
H0
Va s
H0
`
dk 2'p k dk 2'p k
aa Ž k . ey2 i k u q h.c. , ba Ž k . ey2 i k Õ q h.c. ,
Ž 24 .
and aa Ž k . s ba Žyk ., k ) 0. Consequently Ž k ) 0. the non-vanishing commutators are aa Ž k . ,ab† Ž kX . s ha b d Ž k y kX . , ba Ž k . ,bb† Ž kX . s ha b d Ž k y kX . .
Ž 25 .
X " . can be expressed The canonical quantities Ž P ",X linearly in function of q a, b Ž k . and pa, b Ž k . respectively, where q a s a 0 y a1 ,
qb s b 0 q b 1 ,
pa s a 0 q a1 ,
p b s b 0 y b1 .
Ž 26 .
Their non vanishing commutators are q a Ž k . , p†a Ž kX . s qb Ž k . , p†b Ž kX . s 2 d Ž k y kX . .
Ž 27 .
The classical quadratic constraints Ž10. – Ž17., Ž18. cannot be implemented operatorially, since as operator equations they are in contrast with the quantization rules Ž22., Ž25. and further they exhibit the usual bosonic string anomaly, c s 2 Žsee e.g. w3x.. However this does not concern us: in the present scheme the generators of reparametrizations are the C a and there is no anomaly for them because they commute. So we may carry out a different quantization scheme, quantizing the theory with the linear constraints Ž12.. Then the GB procedure can be carried out following the lines of QED w13x. The vacuum is chosen as
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
Žstring vacuum. aa <0: s 0, ba <0: s 0. This leads to negative norm states. Now we implement the constraints a` la GB by requiring that, for each oscillation mode, physical states be selected by q a
a
. qb† Ž kX1 . . . . qb† Ž kXn . <0: b
Ž 28 . satisfy the vacuum condition and have zero norm if n a / 0 or n b / 0. The general solution of the constraints is then
Ý
na
Hd kHd
nb X
k
na, nb
=Cn a n bŽ k 1 , . . . k n a ;kX1 , . . . kXn b . < n a ,n b 4 : .
Ž 29 . The Žnormal ordered. constraints H s 0, P s 0 hold for matrix elements. Using the Lagrangian and Ž28. the expectation value of r is ²C < : r Ž u,Õ . :
Ž 30 .
where dk
) y2 i k u q c.c. , 00 C10 e
Fs
H 2'p k C
Gs
H 2'p k C
dk
) y2 i k Õ q c.c. 00 C 01 e
Ž 31 .
The above result coincides with the classical relation Ž21.; we have of course F Ž u . G Ž Õ . s ²C < U0 Ž u . V0 Ž Õ .
Ž 32 .
Note that Eq. Ž30. is the essence of classical staticity Žcompare with the classical definition Ž6... Let us now consider the operator M, Eq. Ž11.. The integrand in Ž11. classically vanishes. In the quantum case, each term contains one of the operators q a, b or q†a, b . Adopting a normal ordering, the matrix elements of I between physical states vanish. This corresponds to the classical property. So, ²C 2 < M
269
eigenvalue of M0 . The only gauge invariant label of a state is m. This result is similar to the case of the Schwarzschild metric discussed in w7x, where staticity was imposed from the beginning, reducing the problem to quantum mechanics, and states were labeled by the eigenvalues of the mass operator. Finally, the expectation value of w reads ²C ;m <: w Ž u,Õ .:
Acknowledgements This investigation was supported in part by the grants RFBR97-01-01041 and INTAS 93-127-ext. One of us ŽM.C.. is supported by a Human Capital
270
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
and Mobility grant of the European Union, contract number ERBFMRX-CT96-0012.
References w1x C. Callan, S. Giddings, J. Harvey, A. Strominger, Phys. Rev. D 45 Ž1992. 1005; H. Verlinde, in: M. Sato, T. Nakamura ŽEds.., Sixth Marcel Grossmann Meeting on General Relativity, World Scientific, Singapore, 1992. w2x D. Cangemi, R. Jackiw, Phys. Rev. Lett. 69 Ž1992. 233; Phys. Rev. D 50 Ž1994. 3913; Phys. Lett. B 337 Ž1994. 271. w3x E. Benedict, R. Jackiw, H.-J. Lee, Phys. Rev. D 54 Ž1996. 6213; D. Cangemi, R. Jackiw, B. Zwiebach, Ann. Physics ŽNY. 245 Ž1995. 408. w4x A.T. Filippov, Mod. Phys. Lett. A 11 Ž1996. 1691; Int. J. Mod. Phys. A 12 Ž1997. 13.
w5x K.V. Kuchar, ˇ J.D. Romano, M. Varadarajan, Phys. Rev. D 55 Ž1997. 795. w6x T. Strobl, Procs. of the Second Meeting on Constrained Dynamics and Quantum Gravity, Nucl. Phys. ŽSuppl.. B 57 Ž1997. 330. w7x M. Cavaglia, ` V. de Alfaro, A.T. Filippov, Int. J. Mod. Phys. D 4 Ž1995. 661; D 5 Ž1996. 227. w8x J. Cruz, J. Navarro-Salas, Mod. Phys. Lett. A 12 Ž1997. 2345; J. Cruz, J.M. Izquierdo, D.J. Navarro, J. Navarro-Salas, Preprint FTUV-97-10, e-Print Archive: hep-thr9704168. w9x R. Jackiw, Procs. of the Second Meeting on Constrained Dynamics and Quantum Gravity, Nucl. Phys. ŽSuppl.. B 57 Ž1997. 160. w10x K.V. Kuchar, ˇ Phys. Rev. D 50 Ž1994. 3961. w11x D. Louis-Martinez, J. Gegenberg, G. Kunstatter, Phys. Lett. B 321 Ž1994. 193; D. Louis-Martinez, G. Kunstatter, Phys. Rev. D 52 Ž1995. 3494. w12x W. Kummer, S.R. Lau, Ann. Phys. 258 Ž1997. 37. w13x See for instance: J.M. Jauch, F. Rohrlich, The Theory of Photons and Electrons, Addison-Wesley, 1955, p. 103.
Physics Letters B 424 Ž1998. 265–270
Quantization of the string inspired dilaton gravity and the Birkhoff theorem Marco Cavaglia` a , Vittorio de Alfaro a
b,1
, Alexandre T. Filippov
c
Max-Planck-Institut fur ¨ GraÕitationsphysik, Albert-Einstein-Institut, Schlaatzweg 1, D-14473 Potsdam, Germany b Dipartimento di Fisica Teorica dell’UniÕersita` di Torino, Via Giuria 1, I-10125 Torino, Italy INFN, Sezione di Torino, Italy c Joint Institute for Nuclear Research, R-141980 Dubna, Moscow Region, Russia Received 9 May 1997; revised 20 February 1998 Editor: L. Alvarez-Gaume´
Abstract We show that, for the pure dilaton CGHS model, the Gupta-Bleuler quantization is viable because the constraints can be linearized due to positivity conditions present in the model Žand not in the general string case.. There is no ambiguity nor anomalies in the quantization. The expectation values of the metric and dilaton fields obey the classical requirements, thus exhibiting at the quantum level the Birkhoff theorem. q 1998 Published by Elsevier Science B.V. PACS: 04.60.-m; 04.60.Kz; 03.70.q k Keywords: Quantum gravity; Two-dimensional models; Canonical quantization; Field theory
Gravitational models in 1 q 1 dimensions may hopefully clarify some difficult problems of quantum black holes and, more generally, of quantum gravity in higher dimensions w1–6x. Most of the models studied in some detail in the literature are dilatongravity ŽDG. models coupled to gauge and scalar fields Žfor a very compact overview and references see w4x.. In general, these models are not integrable and their solution cannot be found even classically. As a consequence, not much is known about their quantization: only the so-called CGHS model w1x has been studied in detail w2,3,5x. Two-dimensional pure DG is classically inte-
1
E-mail: [email protected].
grable and both the metric and dilaton fields may be expressed in terms of one free ŽD’Alembert. field and of one invariant parameter which is a local integral of motion independent of the coordinates Ž‘‘mass’’ operator.. Using the gauge in which this free field is one of the coordinates, any solution depends actually on one coordinate. These properties constitute a generalization of the classical Birkhoff theorem for spherically symmetric gravity in any dimension Žwe thus call, somewhat improperly, ‘‘ static’’ these solutions.. The drastic reduction of DG to a finite-dimensional dynamical system signals that DG is actually a topological theory. Unlike the theories coupled to scalars, the pure DG model can be quantized by first reducing it to a finite dimensional dynamical system
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 2 2 6 - 3
266
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
with a single constraint. ŽThis can be regarded as a particular gauge fixing.. The resulting Hilbert space is spanned by the eigenvectors of the Žgauge invariant. mass operator. This quantization approach, that uses first the Birkhoff theorem and then the quantization algorithm, has been introduced in Ref. w7x in the case of the 3 q 1 dimensional Schwarzschild black hole; the procedure can be generalized to any pure two dimensional DG model. A further approach to the quantization of the pure DG model does not use the topological nature of the models nor the Birkhoff theorem. Since the classical solution can be written as a function of a single free field, one may try and find a canonical transformation of DG to constrained free fields. This transformation has been implemented in the CGHS case w3,5x, and recently an interesting new proposal in this direction has been advanced w8x. ŽThe main results of these investigations are clearly summarized in the report w9x.. However, up to now an explicit canonical transformation has not been constructed for the general case. Furthermore, the investigation of the CGHS model revealed two main obstacles to quantization: first, the quantum canonical transformations to free fields are highly nontrivial and difficult to construct; second, the quantization is obstructed by anomalies in the commutator of the constraints, and the theory needs to be modified. Due to the presence of anomalies, different quantum theories corresponding to the same classical model can be found. In particular, in the Schrodinger representation there exists a quantiza¨ tion scheme in which the anomaly in the CGHS model is cancelled w3x. However, this scheme is based on the use of negative energy states for the string fields instead of the standard string quantization that introduces negative norm states, and the result obtained has no evident connection to the quantum version of the reduced theory via the Birkhoff theorem. For these reasons we return to the subject and consider a further quantization scheme for the pure dilaton CGHS model, that directly proves the Birkhoff reduction at the quantum level and does not produce any anomalies. The quantization is carried out by use of the standard Gupta-Bleuler ŽGB. method. This is possible because the model has actually more information in it than the canonically transformed version described by a couple of string
fields. In the first place, there exists a gauge invariant local mass that cannot be expressed purely in terms of the string fields. Moreover, further information comes from the canonical transformation between the original fields and the string ones, under the form of positivity conditions for certain functions. This allows an essential step: the linearization of the constraints. There is no anomaly in the algebra of the linearized constraints Žthe algebra of the quadratic constraints does not concern us any more.. This quantization of the string-like fields is similar to the ‘‘Schrodinger’’ quantization that uses positive ¨ norms and negative energy states w3x. The only gauge invariant operators are the mass and its conjugate momentum Žconjugate operators were discussed in Refs. w10,11x. and the ground state of the string must be labeled by the eigenvalue of the mass operator Žas discussed in a different context in Refs. w2,11x.. Our quantization procedure explicitly shows how the reduction to quantum mechanics is achieved by the GB quantization of the field theory. The quantized field theory has the same content as scalar-longitudinal electrodynamics and is pure gauge. The set of GB states corresponds to reparametrization of the coordinates, expectation values of the metric and dilaton fields agree with the corresponding classical quantities and the ‘‘Birkhoff reduced’’ quantum theory is recovered. Our starting point is the two-dimensional action S s d 2 x y g Ž w R y lr2 . ,
H '
Ž 1.
where w is the dilaton field. The two-dimensional metric gmn can be written w3x gmn s e f
ž
a 2-b 2 b
b . y1
/
Ž 2.
Here a and b play the role of the lapse function and of the shift vector respectively; f Ž x 0 , x 1 . represents the dynamical gravitational degree of freedom. The coordinates x 0 , x 1 are both defined on R. Eq. Ž1. can be cast in the canonical form S s d 2 x f˙p f q wp ˙ w yaHyb P ,
H
Ž 3.
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
where the super-Hamiltonian and super-momentum are respectively H s p f pw q f Xw X y 2 w XX q l e fr2 s 0 , P s 2p fX y pw w X y p f f X s 0 .
Ž 4.
ŽWe neglect boundary terms as they are irrelevant for the following discussion. See e.g. Ref. w5x and references therein.. We may also define a functional M of the canonical variables which is conserved under time and space translations Ž M˙ s M X s 0. w4,11,12x: yf
M s lwr2 q e
ž
p f2 y w X 2
/.
Ž 5.
The Birkhoff reduction may be stated as follows. Setting a s 1 and b s 0 and introducing the coordinates u s Ž x 0 q x 1 .r2, Õ s Ž x 0 y x 1 .r2, the two-dimensional line element corresponding to the metric tensor Ž2. becomes ds 2 s 4 r Ž u,Õ . dudÕ Ž r s e f .. A metric is called static iff it can be cast in the form w4x
r Ž u,Õ . s h Ž C . aX Ž u . bX Ž Õ . , C ' aŽ u . q b Ž Õ . ,
Ž 6.
where a and b are arbitrary functions. If, in addition, w depends only on C , the solution will be called static. In the pure DG case the general solution can be written
r ' e f s F X Ž C . EuCEÕC , wsFŽ c . ,
Eu EÕC s 0 .
Ž 7.
So any solution of the generic DG model is static. In particular, for the CGHS case we find F ŽC . s C0 eylC r2 q 2 Mrl. Here C0 is an integration constant; the second integration constant was expressed in terms of the constant M by using Ž4.. M appears thus as a zero mode of the field w . Now, let us use the canonical transformation w3x
l A 0 s 2 eyf r2 Ž p f cosh S y w X sinh S . , p 0 s yl e f r2 cosh S y l AX1 , l A1 s 2 e
yf r2 f r2
w Ž x 0 ,c .. To make the transformation invertible, one has to supplement the new field variables Aa ,pa by a pair of conjugate variables, e.g. w Ž x 0 ,c s `. and 2 S Ž`. Ž w 0 will commute with all the field variables if we choose c s `.. Using Ž8., Ž9. the two constraints become 1 Hs
2l
l
p apa q
AX aAXa s 0 ,
2
P s yp aAXa s 0 ,
Ž 10 .
where the indices are lowered and raised by the two-dimensional Minkowski metric h s Ž1,y 1.. The functional M is
l M s M0 q
x1
H 4 b
dxX1 Ž ´ a bpa Ab q l A aAXa . ,
Ž 11 .
where ´ ab is the two-dimensional Levi-Civita tensor. The value of b is irrelevant and M0 must be a constant since M is independent of t. On the equations of motion we have M s M0 . Let us introduce the operators C a and D a C a s p a y l´ abAXb , D a s p a q l´ abAXb .
Ž 12 .
M˙ and M X are proportional to C a . The classical quantities H , P, r , w , M can be written as functions of C a , D a and Aa . They read 2 l H s C a Da ,
2 l P s ´a b D a C b ,
l2r s D a Da ,
Ž 13 . a
w s 2 Mrl y l Aa A r2, M s M0 y Ž lr4 .
x1
Hb
dxX1 ´ a bAa Cb .
Ž 14 .
The Poisson brackets of C a and D a are
Ž 8.
X
Ž p f sinh S y w cosh S . ,
p s l e sinh S q l AX0 , x1 where S Ž x 1 . s 12 Hy` dxX1pw Ž xX1 .. 1
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C a Ž x 0 , x . , D b Ž x 0 , y . s 2 l´ a bE y d Ž x y y . .
Ž 15 . Ž 9.
The transformation Ž8., Ž9. is canonical for the field variables Žyw X , S , f,p f ; Aa ,pa .. Note that the inverse transformation only defines w up to a zero mode w 0 Ž x 0 . '
The constraints can be linearized, using a suitable redefinition of the Lagrange multipliers. The resulting linearized constraints are the functions C a. In this form of the theory the generators are linear in the string variables Aa .
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
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It is useful to introduce the conjugate variables Ž P ",X X " . w5x 2'l P "s C 0 " C 1 , X D 0 y D1 , 2'l X " s "D
Ž 16 .
and define the constraints Hq and Hy that generate reparametrization in u and Õ; they can be written as functions of the constraints P " and of the space X derivative of the conjugate constraints, X " : 0
1
0
X s y2 Xy Py ,
s yhab
Ž 18 . X X"
A crucial point is that have definite signs, as a consequence of Eqs. Ž8., Ž9.. Then it is easy to control that the constraints H s 0 and P s 0 are satisfied iff C a s 0. Accordingly, the coefficients X X" can be reabsorbed in the Lagrange multipliers. We have L s X˙q Pqq X˙y Pyy rq Pqy ry Py, where the Lagrange multipliers have been defined as X X rqs 2 Xq ly, rys y2 Xy lq. Finally, since C a Ž x0 , y . r Ž x 0 , x . ,C s 4´ ably1 Db Ž x 0 , x . E y d Ž x y y . ,
H 2p d Ž k
2
. ´ Ž k 0 . e i kŽ xyy. .
Ž 22 .
Aa s
dk
yi v x 0 qi k x 1
Hy` 2'pv b Ž k . e a
q h.c. ,
Ž 23 .
where v s < k <, or
Hy' H y P s Ž D 0 y D 1 . Ž C 0 q C 1 . r2 l X s 2 Xq Pq .
d2k
The field expansion is `
Ž 17 .
Ž 19 .
the linear constraints, P " or Ca , generate the reparametrizations of the metric. As a consequence of the linearization, the theory can be quantized in the GB scheme. Before doing that, let us see the form that the classical solution takes in terms of the fields Aa . The equations of motion are
Ž 20 .
and the solution is Aa s Ua Ž u. q Va Ž Õ .. The linear constraints C a s 0 correspond to h abEa Ab s 0, ´ abEa Ab s 0, or U0 Ž u. s U1Ž u., V0 Ž Õ . s yV1Ž Õ .. One can prove that the above solution coincides with Ž7.. Substituting Aa into Ž13., Ž14. one obtains
r ' e f s 4 U0X Ž u . V0X Ž Õ . , w s 2 Mrl y 2 lU0 Ž u . V0 Ž Õ . .
Aa Ž x . , Ab Ž y .
1
Hq' H q P s Ž D q D . Ž C y C . r2 l
l A˙a s ha b p b , p˙ a s lh a bAXXb ,
The quantization is achieved by imposing the commutation relations Žfrom now on we set l s 1, there is no real loss of generality while the formulae become more elegant.
Ž 21 .
This solution coincides with Ž7. if C is defined by C0 eylC r2 s y2 lU0 Ž u.V0 Ž Õ .. The ‘‘staticity’’ of the classical solution is thus embodied in Eqs. Ž21..
`
Ua s
H0
Va s
H0
`
dk 2'p k dk 2'p k
aa Ž k . ey2 i k u q h.c. , ba Ž k . ey2 i k Õ q h.c. ,
Ž 24 .
and aa Ž k . s ba Žyk ., k ) 0. Consequently Ž k ) 0. the non-vanishing commutators are aa Ž k . ,ab† Ž kX . s ha b d Ž k y kX . , ba Ž k . ,bb† Ž kX . s ha b d Ž k y kX . .
Ž 25 .
X " . can be expressed The canonical quantities Ž P ",X linearly in function of q a, b Ž k . and pa, b Ž k . respectively, where q a s a 0 y a1 ,
qb s b 0 q b 1 ,
pa s a 0 q a1 ,
p b s b 0 y b1 .
Ž 26 .
Their non vanishing commutators are q a Ž k . , p†a Ž kX . s qb Ž k . , p†b Ž kX . s 2 d Ž k y kX . .
Ž 27 .
The classical quadratic constraints Ž10. – Ž17., Ž18. cannot be implemented operatorially, since as operator equations they are in contrast with the quantization rules Ž22., Ž25. and further they exhibit the usual bosonic string anomaly, c s 2 Žsee e.g. w3x.. However this does not concern us: in the present scheme the generators of reparametrizations are the C a and there is no anomaly for them because they commute. So we may carry out a different quantization scheme, quantizing the theory with the linear constraints Ž12.. Then the GB procedure can be carried out following the lines of QED w13x. The vacuum is chosen as
M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
Žstring vacuum. aa <0: s 0, ba <0: s 0. This leads to negative norm states. Now we implement the constraints a` la GB by requiring that, for each oscillation mode, physical states be selected by q a
a
. qb† Ž kX1 . . . . qb† Ž kXn . <0: b
Ž 28 . satisfy the vacuum condition and have zero norm if n a / 0 or n b / 0. The general solution of the constraints is then
Ý
na
Hd kHd
nb X
k
na, nb
=Cn a n bŽ k 1 , . . . k n a ;kX1 , . . . kXn b . < n a ,n b 4 : .
Ž 29 . The Žnormal ordered. constraints H s 0, P s 0 hold for matrix elements. Using the Lagrangian and Ž28. the expectation value of r is ²C < : r Ž u,Õ . :
Ž 30 .
where dk
) y2 i k u q c.c. , 00 C10 e
Fs
H 2'p k C
Gs
H 2'p k C
dk
) y2 i k Õ q c.c. 00 C 01 e
Ž 31 .
The above result coincides with the classical relation Ž21.; we have of course F Ž u . G Ž Õ . s ²C < U0 Ž u . V0 Ž Õ .
Ž 32 .
Note that Eq. Ž30. is the essence of classical staticity Žcompare with the classical definition Ž6... Let us now consider the operator M, Eq. Ž11.. The integrand in Ž11. classically vanishes. In the quantum case, each term contains one of the operators q a, b or q†a, b . Adopting a normal ordering, the matrix elements of I between physical states vanish. This corresponds to the classical property. So, ²C 2 < M
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eigenvalue of M0 . The only gauge invariant label of a state is m. This result is similar to the case of the Schwarzschild metric discussed in w7x, where staticity was imposed from the beginning, reducing the problem to quantum mechanics, and states were labeled by the eigenvalues of the mass operator. Finally, the expectation value of w reads ²C ;m <: w Ž u,Õ .:
Acknowledgements This investigation was supported in part by the grants RFBR97-01-01041 and INTAS 93-127-ext. One of us ŽM.C.. is supported by a Human Capital
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M. CaÕaglia` et al.r Physics Letters B 424 (1998) 265–270
and Mobility grant of the European Union, contract number ERBFMRX-CT96-0012.
References w1x C. Callan, S. Giddings, J. Harvey, A. Strominger, Phys. Rev. D 45 Ž1992. 1005; H. Verlinde, in: M. Sato, T. Nakamura ŽEds.., Sixth Marcel Grossmann Meeting on General Relativity, World Scientific, Singapore, 1992. w2x D. Cangemi, R. Jackiw, Phys. Rev. Lett. 69 Ž1992. 233; Phys. Rev. D 50 Ž1994. 3913; Phys. Lett. B 337 Ž1994. 271. w3x E. Benedict, R. Jackiw, H.-J. Lee, Phys. Rev. D 54 Ž1996. 6213; D. Cangemi, R. Jackiw, B. Zwiebach, Ann. Physics ŽNY. 245 Ž1995. 408. w4x A.T. Filippov, Mod. Phys. Lett. A 11 Ž1996. 1691; Int. J. Mod. Phys. A 12 Ž1997. 13.
w5x K.V. Kuchar, ˇ J.D. Romano, M. Varadarajan, Phys. Rev. D 55 Ž1997. 795. w6x T. Strobl, Procs. of the Second Meeting on Constrained Dynamics and Quantum Gravity, Nucl. Phys. ŽSuppl.. B 57 Ž1997. 330. w7x M. Cavaglia, ` V. de Alfaro, A.T. Filippov, Int. J. Mod. Phys. D 4 Ž1995. 661; D 5 Ž1996. 227. w8x J. Cruz, J. Navarro-Salas, Mod. Phys. Lett. A 12 Ž1997. 2345; J. Cruz, J.M. Izquierdo, D.J. Navarro, J. Navarro-Salas, Preprint FTUV-97-10, e-Print Archive: hep-thr9704168. w9x R. Jackiw, Procs. of the Second Meeting on Constrained Dynamics and Quantum Gravity, Nucl. Phys. ŽSuppl.. B 57 Ž1997. 160. w10x K.V. Kuchar, ˇ Phys. Rev. D 50 Ž1994. 3961. w11x D. Louis-Martinez, J. Gegenberg, G. Kunstatter, Phys. Lett. B 321 Ž1994. 193; D. Louis-Martinez, G. Kunstatter, Phys. Rev. D 52 Ž1995. 3494. w12x W. Kummer, S.R. Lau, Ann. Phys. 258 Ž1997. 37. w13x See for instance: J.M. Jauch, F. Rohrlich, The Theory of Photons and Electrons, Addison-Wesley, 1955, p. 103.