PHYSICAL REVIEW D, VOLUME 59, 084011
Geometrodynamical formulation of two-dimensional dilaton gravity
Marco Cavaglia`* Max-Planck-Institut fu¨r Gravitationsphysik, Albert-Einstein-Institut, Schlaatzweg 1, D-14473 Potsdam, Germany ͑Received 31 July 1998; published 12 March 1999͒ Two-dimensional matterless dilaton gravity with an arbitrary dilatonic potential can be discussed in a unitary way, both in the Lagrangian and canonical frameworks, by introducing suitable field redefinitions. The new fields are directly related to the original spacetime geometry and in the canonical picture they generalize the well-known geometrodynamical variables used in the discussion of the Schwarzschild black hole. So the model can be quantized using the techniques developed for the latter case. The resulting quantum theory exhibits the Birkhoff theorem at the quantum level. ͓S0556-2821͑99͒01708-7͔
PACS number͑s͒: 04.60.Ds, 04.20.Fy, 04.60.Kz
I. INTRODUCTION According to their integrability properties, dilaton gravity models can be roughly divided into three classes. Recently, a lot of attention has been devoted to the inves- ͑i͒ Completely integrable models, i.e., models that can be tigation of lower-dimensional gravity ͓1͔. The interest in di- expressed in terms of free fields by a canonical transforma- mensionally reduced theories of gravity relies essentially on tion. Some remarkable examples are matterless dilaton grav- their connection to string theory, black hole physics, and ity with an arbitrary potential ͓3͔ and the CGHS model ͓5,6͔. gravitational collapse. In this context, two-dimensional mod- ͑ii͒ Completely solvable models, i.e., models that cannot els of dilaton gravity play a very important role because of be analytically solved in terms of free fields but whose gen- their relation to higher-dimensional gravity and integrable eral solution is known. The two-dimensional effective gen- systems. eralized theory of 2ϩ1 cylindrical gravity minimally Two-dimensional dilaton gravity is described by the ac- coupled to a massless scalar field ͓8͔ and dilaton gravity with tion ͓2͔ constant or linear dilatonic potential minimally coupled to massless Dirac fermions ͓9,10͔ belong to this class. 2 ͑2͒ 2 -SDGMϭ d xͱϪ␥ ͓ ͑͒ ͑␥͒ϩ ͑͒ϩ ٌ͑͒͑͒ ͔ ͑iii͒ Partially integrable models, i.e., models that are in ͵⌺ U R V W tegrable in a (0ϩ1)-dimensional sector only, namely, after ϩSM͓,␥ ,fi͔, ͑1͒ reduction to a finite number of degrees of freedom. In this where , , and are arbitrary functions of the dilaton, category we find, for example, dilaton gravity minimally (2) U V W coupled to massless Dirac fermions with arbitrary potential is the two-dimensional Ricci scalar, and SM represents theR contribution of matter fields f which include any field ͓9͔ and two-dimensional effective models describing un- i charged black p-branes in N dimensions ͓13͔. but the dilaton and the graviton ␥ . Most of the models studied in some detail in the literature Completely integrable models are of particular interest are special cases of the model described by Eq. ͑1͒ where from the quantum point of view. In this case we are able to dilaton gravity is coupled to scalar, gauge, and fermion quantize the theory ͑in the free-field representation͒ and, fields. ͑See, for instance, Refs. ͓3–13͔ and references hopefully, to discuss quantization subtleties and nonpertur- bative quantum effects. ͑See, e.g., Refs. ͓5,6,18͔ for the therein.͒ For a given SM , Eq. ͑1͒ describes a family of mod- els whose elements are identified by the choice of the dila- CGHS model.͒ In particular, matterless dilaton gravity—Eq. tonic potential. Indeed, classically we may always choose ͑2͒—can be used to describe black holes and, in the case of ()ϭ and locally set ()ϭ0 by a Weyl rescaling of coupling with scalar matter, gravitational collapse. So the theU metric ͓2͔. ͑In this paperW we will always make this choice quantization program is worth exploring. for simplicity ͓14͔.͒ So the matterless sector of Eq. ͑1͒ reads Although the classical properties of the model based upon ͓2͔ Eq. ͑2͒ are well known, a conclusive word about its quanti- zation is not known ͓2,19,20͔, even in the simplest ͑CGHS͒ S ϭ d2xͱϪg R͑2͒ g͒ϩV ͒ , ͑2͒ DG ͵ ͓ ͑ ͑ ͔ case. Let us recall the two most fruitful attempts to construct ⌺ a quantum theory of the CGHS model that are described in (2) where g and R are the two-dimensional, Weyl-rescaled, Refs. ͓5,6,18͔ and Ref. ͓21͔, respectively. metric and Ricci scalar, respectively. Different choices of The first approach is based on a canonical transformation V() identify different theories. Some remarkable examples mapping the original system to a system described by free- are the Callan-Giddings-Harvey-Strominger ͑CGHS͒ model fields. Then the theory is quantized in the free-field represen- ͓15͔ (Vϭconst), the Jackiw-Teitelboim model ͓16͔ (Vϭ), tation. The main drawback of this approach is that the new and the dimensionally reduced theory of four-dimensional canonical variables are not directly related to the original spherically symmetric Einstein gravity integrated on a two- spacetime geometry and important physical quantities cannot sphere of area 16 ͓Vϭ1/(2ͱ)͔ ͓2,17͔. be expressed in terms of the new fields ͓18͔. Further, it is not clear how to generalize the canonical transformation for an arbitrary dilatonic potential. ͑Recently, a proof of the exis- *Email address: [email protected] tence of a canonical transformation that generalizes the ca-
0556-2821/99/59͑8͒/084011͑8͒/$15.0059 084011-1 ©1999 The American Physical Society MARCO CAVAGLIA` PHYSICAL REVIEW D 59 084011
nonical transformation used in the CGHS case has been de- The outline of the paper is as follows. In the next section rived by Cruz and Navarro-Salas; see Ref. ͓22͔. Even though we present the classical theory of two-dimensional dilaton it seems reasonable to guess the existence of a canonical gravity ͑see Refs. ͓2,19,26,25͔͒ following the approach de- transformation in the general case, the relation between the veloped by Filippov ͓3͔. Although this section reviews es- new fields and the original geometrical variables remains a sentially previous work, its content is useful to set up the puzzle.͒ notation and make the paper more self-contained and read- The ‘‘geometrodynamical approach’’ was originally de- able. In Sec. III we introduce the point transformation and veloped by Kucharˇ for a canonical description of the the new Lagrangian. This result follows from the integrabil- Schwarzschild black hole ͓23͔. This approach uses variables ity properties of the system and constitutes the main contri- that are directly related to the spacetime geometry and does bution of the paper. Indeed, the Lagrangian formulation is at not make use of the field redefinitions of Refs. ͓5,6,18͔. the basis of the canonical formalism and leads straightfor- Again, only the CGHS model has been quantized using this wardly to the geometrodynamical variables. The canonical formalism ͓21͔. framework is discussed in Sec. IV. First, we derive the ca- In this paper we assume a different attitude and quantize nonical transformation to the geometrodynamical variables the general matterless dilaton gravity model described by for the general model. Then we present a careful treatment of Eq. ͑2͒ using a transformation of the configuration space falloff conditions which are essential in establishing the performed at the Lagrangian level. The transformation is Hamiltonian quantization. The discussion of falloff condi- suggested by the topological nature of two-dimensional grav- tions involves subtleties related to the definition of boundary ity and by the existence of a local integral of motion inde- conditions for arbitrary spacetimes. In Sec. V we quantize pendent of the coordinates first discussed by Filippov ͓3͔. the system. Thanks to the geometrodynamical variables the The new fields have clear physical meaning—they are the quantization of the general model can be achieved by imple- dilaton and the ‘‘mass’’ of the system—thus avoiding prob- menting the formalism developed by Kucharˇ for the quanti- lems related to their interpretation in terms of the geometri- zation of the Schwarzschild black hole ͓23͔. Finally, we cal variables. show that the ensuing quantum theory is equivalent to the In the canonical framework the new fields generalize the quantum mechanical theory which is obtained by imposing geometrodynamical variables of Kucharˇ ͓23͔ and Varadara- the Birkhoff theorem at the classical level. In Sec. VI we jan ͓21͔ to a generic dilatonic potential. Thus the quantiza- state our conclusions. tion is straightforward and can be completed along the lines of Refs. ͓23,21͔. The quantum theory reduces to quantum II. CLASSICAL THEORY mechanics and the Hilbert space coincides with the Hilbert space obtained by quantizing the theory first reducing it to a Let us consider Eq. ͑2͒. Varying the action with respect to 0ϩ1 dynamical system with a finite number of degrees of the metric and the dilaton we obtain ͓25͔ freedom and then imposing the quantization algorithm †see ϭ 1 Ref. ͓27͔ for the case V 1/(2ͱ)‡. This result represents ٌ͑ ٌ Ϫg ٌ ٌ͒ϩ g V͑͒ϭ0, ͑4͒ the quantum generalization of the well-known Birkhoff theo- ͑ ) 2 rem ͓24,25͔ for spherically symmetric gravity in four dimen- sions. ͑A somewhat different derivation of the so-called dV Rϩ ϭ0, ͑5͒ quantum Birkhoff theorem for the CGHS model is discussed d in Ref. ͓18͔. The approach of Ref. ͓18͔ makes use of the canonical transformation to free fields. Here we extend the where the symbol ٌ represents covariant derivatives with results of Ref. ͓18͔ to the general model with an arbitrary respect to the metric g . dilatonic potential using a different and more powerful ap- It is easy to prove that Eq. ͑5͒ is satisfied if Eq. ͑4͒ is proach.͒ satisfied provided that The quantum Birkhoff theorem is schematically defined by the following diagram: H͑g ,͒Þ0, ͑6͒
where H(g ,)ϭٌٌ . This condition can be lifted if one requires the continuity of the fields and of their deriva- tives at any spacetime point. We will see in a moment—see Eq. ͑17͒ below—that the equation H(g ,)ϭ0 defines the horizon͑s͒ of the two-dimensional metric. So by requiring the continuity of the fields and their derivatives across the horizon͑s͒ Eq. ͑4͒ implies Eq. ͑5͒ everywhere. The field equations ͑4͒,͑5͒ can be solved performing a Ba¨cklund transformation ͑see Ref. ͓3͔͒. In covariant lan- guage the Ba¨cklund transformation reads
MϭN͑͒Ϫٌ ٌ, N͑͒ϭ dЈV͑Ј͒, ͑7͒ ͑3͒ ͵
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theory. In Sec. V we will see how this property influences ٌ .ϭ , ͑8͒ the quantization of the theory ٌ ٌ Let us briefly discuss the local geometrical properties ofٌ the solution ͑16͒. The horizon͑s͒ of the metric are deter- where M(t,x) and (t,x) are the transformed fields. (M and mined by the equation N coincides—apart from constant factors – to C and J of Refs. ͓19,2͔ respectively.͒ Note that the transformation is N͑͒ϪMϵH͑g ,͒ϭ0. ͑17͒ singular for H(g ,)ϭ0. Using the new fields Eq. ͑4͒ reads For a given choice of the dilatonic potential, Eq. ͑17͒ is an algebraic equation in whose solutions ͕ ͖ determine the ٌ ϭ0, ͑9͒ iٌ values of the radial coordinate where the horizon͑s͒ are lo- cated. So the request of continuity of the solution—and of its Mϭ0. ͑10ٌ͒ derivatives ͑see Ref. ͓23͔͒—across the horizons enforces the Since the transformation ͑7͒,͑8͒ is defined when Eq. ͑6͒ continuity of the fields and at the points H(g ,) holds, Eqs. ͑9͒,͑10͒ are equivalent to the original field equa- ϭ0 and vice versa. This justifies a posteriori the assumption tions ͑4͒,͑5͒ except at the horizon͑s͒. Equations ͑9͒,͑10͒ have of continuity made below Eq. ͑6͒. With this assumption Eqs. a deep significance. The first equation implies that is a free ͑4͒,͑5͒ are equivalent to Eqs. ͑9͒,͑10͒ everywhere. ͑D’Alembert͒ field. From the second equation we find that M The local asymptotic structure of the solution ͑16͒ and the is a locally conserved quantity. existence of singularities depend on the choice of the dila- In two dimensions any metric is locally conformally flat tonic potential. In particular, from Eq. ͑5͒ one finds that sin- ͓28͔. So there exists a coordinate transformation which gularities of the metric are determined by singular points of brings the metric into the form the first derivative of V() with respect to . The local asymptotic structure can be also roughly investigated using ds2ϭ4͑u,v͒dudv, ͑11͒ Eq. ͑5͒. For instance, let us suppose that the asymptotic re- gion is defined by ϱ and that the behavior of the dila- where uϭ(tϩx)/2, vϭ(tϪx)/2. Using conformal light- tonic potential at infinity→ is V()Ϸk, where k is a constant cone coordinates Eqs. ͑9͒,͑10͒ can be explicitly integrated. parameter. Thus the two-dimensional spacetime is asymp- The general solution is totically flat for ϱ if kϽ1, and has constant curvature for ϱ if kϭ1. → ϭU͑u͒ϩV͑v͒, MϭM 0 . ͑12͒ Let→ us conclude this section with a concrete example and derive the Schwarzschild solution using the formalism de- The original fields and are can be written as functions of scribed above. The dimensional reduction of the four- and M using Eqs. ͑7͒,͑8͒. With a little algebra one finds dimensional vacuum Einstein gravity, d 1 ϭ , ͑13͒ 1 d N ϪM 4 ͑4͒ ͑͒ SEHϭ d xͱϪgR ͑g͒, ͑18͒ 16 ͵⌺ v. ͑14͒ץuץϭ͓N͑͒ϪM͔ can be obtained using the ansatz Equations ͑12͒–͑14͒ imply that the general solution of the model is actually (0ϩ1) dimensional, i.e., that any solution 1 possesses a Killing vector ͓29͔. Indeed, using the coordinates 2 2 ds͑4͒ϭ gdx dx ϩ4 d⍀ , у0, ͑19͒ (U,V) the general solution reads ͱ
ds2ϭ4͓N͑͒ϪM͔dUdV, ϵ͑UϩV͒, ͑15͒ where g is a two-dimensional metric with signature (Ϫ1,1) and d⍀2 is the line element of the unit two-sphere. or, using the coordinates (,TϵUϪV), Using Eq. ͑19͒, and integrating on the two-sphere, the four- dimensional Einstein-Hilbert action can be cast into the form ds2ϭϪ͓N͑͒ϪM͔dT2ϩ͓N͑͒ϪM͔Ϫ1d2. ͑16͒ ͑2͒ with V()ϭ1/(2ͱ). Using Eq. ͑16͒ the line element Thus the general solution depends on the single variable . ͑19͒ reads ͑With a somewhat improper terminology we call these solu- tions static, even though the Killing vector may not be time- M d 2 2 2 2 like and hypersurface orthogonal on the entire manifold.͒ ds͑4͒ϭϪ 1Ϫ dT ϩ ϩ4 d⍀2 . ͩ ͱͪ M This result constitutes a generalization of the classical 1Ϫ Birkhoff theorem ͓25,24,3,29͔. ͑For spherically symmetric ͩ ͱͪ Einstein gravity the ‘‘local integral of motion independent of ͑20͒ the coordinates’’ is just the Schwarzschild mass.͒ The reduc- tion of the theory to a finite-dimensional dynamical system Clearly Eq. ͑20͒ reduces to the standard Schwarzschild signals that pure dilaton gravity is actually a topological solution with the substitution 4ϭR2.
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III. LAGRANGIAN FORMALISM volves derivatives with respect to t and x. We call it a ‘‘point transformation’’ because it can be implemented at the La- The Ba¨cklund transformation introduced in the previous grangian level.͔ section can be used to find a transformation from the original Varying Eq. ͑22͒ we find fields, (g ,), to new fields (Xi), iϭ1,...,4, where one of the new fields Xi coincides with M. Since M is a locally ٌ ٌϪV͑͒ϭ0, ͑25͒ conserved quantity, this transformation simplifies drastically the dilaton gravity Lagrangian in Eq. ͑2͒. 1 The key of the construction is the observation that in two ٌ)MϪ gٌٌ Mϭ0, ͑26ٌ͒͑ dimensions the Ricci scalar R is a total divergence and can 2 be locally written as Mٌ Mϩٌٌ ٌٌ Mϭ0. ͑27ٌ͒ R ٌ ٌ ٌϪٌٌ ٌ ϭٌ A, Aϭ , ͑21͒ - ٌ ٌ Equations ͑25͒–͑27͒ are equivalent to the field equations ob 2 tained from Eq. ͑2͒. Equation ͑25͒ corresponds to the trace of where is an arbitrary, nonconstant, function of the coordi- Eq. ͑4͒. Further, by differentiation of Eq. ͑7͒ one finds that nates. Equation ͑21͒ can be easily checked using conformal Eqs. ͑26͒,͑27͒ are satisfied if Eq. ͑4͒ is satisfied because Eq. coordinates. Since Eq. ͑21͒ is a generally covariant expres- ͑4͒ implies ٌMϭ0. The converse latter statement is also sion, and any two-dimensional metric can be locally cast in true provided that Eq. ͑6͒ is satisfied. When this condition the form ͑11͒ by a coordinate transformation ͓28͔, Eq. ͑21͒ is holds Eq. ͑26͒ implies ٌMϭ0. By requiring the continuity valid in any system of coordinates. of the fields Eqs. ͑25͒–͑27͒ and Eqs. ͑4͒,͑5͒ are equivalent. Differentiating Eq. ͑7͒, and choosing ϭ, both V() The equivalence of Eqs. ͑25͒–͑27͒ and the original field equations can also be directly checked using the metric pa- and R can be written as functions of M and ٌ. Finally, by an integration per parts we find rametrization defined in Eq. ͑24͒.
M IV. CANONICAL FORMALISM ٌٌ 2 22͒͑ , ץSϭ d x ͱϪg ϩS ͵⌺ N͑͒ϪM The canonical formalism is an essential step in the quan- tization procedure. Starting from Eq. ͑2͒, and using the met- is the surface term: ric parametrization, Eq. ͑24͒, the action can be cast in the ץwhere S Hamiltonian form ͓2͔ 2 ϭ2 d x ͱϪg ٌٌ͓ ϩA ͔. ͑23͒ץS ͵ xb ⌺ ˙ ˙ Sϭ ͵ dt ͵ dx ͓ϩϪ␣ 0Ϫ 1͔, ͑28͒ xa H H Let us check that Eq. ͑22͒ has the same number of degrees of freedom ͑DOF͒ of the original action ͑2͒. In two dimensions where the overdots represent derivatives with respect to the a generic metric can be written timelike coordinate t,(,,,) are the phase space variables, and , are the Arnowitt-Deser-Misner 2 2 0 1 ␣ Ϫ  ͑ADM͒ super-HamiltonianH H and supermomentum, respec- g ϭ . ͑24͒ ͩ  Ϫ1 ͪ tively:
In the canonical formalism ␣(t,x) and (t,x) play the role Ј ϭ ϩ ЈϪ2ЉϪV͑͒, of the lapse function and of the shift vector respectively; H0 (t,x) is the dynamical DOF. As a result of the chosen pa- rametrization, the Lagrangian in Eq. ͑2͒ is a functional of the 1ϭϪЈϩЈϩ2Ј. ͑29͒ two dynamical fields (,) and of the two nondynamical H variables (␣,). Now let us use Eq. ͑24͒ in Eq. ͑22͒ and Here primes represent derivatives with respect to the spatial neglect the surface term. The new Lagrangian is again a coordinate x. Equations ͑29͒ include, as particular cases, the functional of two fields (M,) and of two nondynamical models discussed in Ref. ͓21͔ and Ref. ͓23͔. Denoting with variables (␣,). Indeed, since Eq. ͑22͒ only contains the subscripts v and k the canonical variables of Ref. ͓21͔ and Weyl-invariant combinations ͱϪgg, the transformed ac- ͓23͔, respectively, we have tion is invariant under changes of coordinates which belong 2 to the conformal group and g does not contribute any dy- R P Ϫ⌳ P Rv v Rv v ⌳v namical DOF to the action. As a consequence, the transfor- ϭ , ϭ2 , 4 2 mation (,,␣,) (M,,␣,) is a ‘‘point transforma- Rv tion’’ with MϵM(→,,␣,) defined by Eq. ͑7͒. ͓Quotation marks are due to the fact that the transformation P ⌳v (,,␣,) (M,,␣,) should not be regarded as a point ϭR2⌳2 , ϭ , → v v 2 transformation according to the usual lore because it in- 2Rv⌳v
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Nv r ¯ Ј Ј ␣ϭϪ , ϭϪNv , ͑30͒ 2 ⌳v ͩ ͓N͑¯ ͒ϪM͔ ͪ ¯ M ϭ¯ ϩV͑͒Mϩ 2 . ͑34͒ for the CGHS model (Vϭconst), and ¯ Ј 1Ϫ ͫ ͩ ͓N͑¯ ͒ϪM͔ ͪ ͬ 2 2R P Ϫ⌳ P M Rk k Rk k ⌳k ϭ , ϭ , 4 R2 After some tedious calculations one can check that the only k nonvanishing Poisson brackets at equal time t are
2 R ⌳ P⌳ k k k ͓M͑t,x͒,M͑t,xЈ͔͒ϭ␦͑xϪxЈ͒, ϭ , ϭ , 2 Rk⌳k ¯ ͓͑t,x͒,¯ ͑t,xЈ͔͒ϭ␦͑xϪxЈ͒; ͑35͒ Nk r ␣ϭϪ , ϭϪNk , ͑31͒ so Eqs. ͑33͒,͑34͒ define a canonical map. Finally, the differ- ⌳k ence of the Liouville forms reads for the Schwarzschild black hole VϭϪ1/(2ͱ). ͓The mi- x x nus sign of V is due to the choice of the metric signature in b ˙ ¯˙ b ˙ ˙ ͵ dx ͑M Mϩ ¯͒Ϫ͵ dx ͑ ϩ ͒ Eq. ͑24͒ that is opposite to the signature used in Eq. ͑19͒.͔ xa xa Starting from Eq. ͑22͒ the super-Hamiltonian and super- ¯ momentum read ͑for later convenience we set ϭ¯ ) ϭF͑,¯ ,M,M͒, ͑36͒
Ϫ1 where ϭ͓N͑¯ ͒ϪM͔¯ ϩ͓N͑¯ ͒ϪM͔ ¯ ЈMЈ, H0 M ¯ F͑,¯ ,M,M ͒ ϭϪ¯Ј¯ϪMЈ . ͑32͒ H1 M xb ¯ Ј Eventually, both canonical actions must be complemented by ϭ dx 2¯Ј arctanh ͵ ¯ xa ͫ ͓N͑͒ϪM͔ ͬ a boundary term at the spatial boundaries. This can be done ͭ M along the lines of Refs. ͓23,21͔ as we will see later in this – section. ¯ Ϫ2͑N͑͒ϪM ͒M ¯ ¯ ͮ The two charts (, ,,) and (, ,M,M) are re- lated by the transformation xb ¯ Ј Ј Ϫ dx 2¯˙ arctanh . ͑37͒ 2 2 2 ͵ ¯ ϪЈ xa ͫ ͓N͑͒ϪM͔ ͬ ͭ M ͮ MϭN͑͒Ϫ , ¯ The canonical variables (,¯ ,M,M) are a generaliza- 2 tion of the geometrodynamical variables introduced by Ku- ϭ , charˇ 23 and Varadarajan 21 . This can be easily proved M 2 2 2 ͓ ͔ ͓ ͔ ϪЈ using Eqs. ͑30͒,͑31͒ and Eqs. ͑33͒,͑34͒. Now we must take care of boundary terms and define ¯ ϭ, falloff conditions at the spatial boundaries. We set
͑a,b͒ 2 ¯ ϭ͓N͑¯ ͒ϪM͔͑1ϩ⑀¯ ͒, ͑38͒ Ј Ј Ј ¯ ϭ Ϫ V͑͒ϩ2 . ͑33͒ Ј 2 2 2ͫ ͩ ͪ ͬ ϪЈ ¯˙ ͑a,b͒ ϭ⑀¯˙ , The transformation given above is easily invertible. The re- sult is ͑a,b͒ ͑a,b͒ MϭM ͑t͒͑1ϩ⑀M ͒, 2 ¯ Ј 2 ¯ ͑a,b͒ ϭM͓N͑͒ϪM͔ 1Ϫ , ⑀ ͫ ͩ ͓N͑¯ ͒ϪM͔ ͪ ͬ ¯ M ¯ ϭ , N͑¯ ͒ϪM 1 ϭ 2 , ⑀͑a,b͒ ¯ Ј M M 1Ϫ Mϭ , ͫ ͩ ͓N͑¯ ͒ϪM͔ ͪ ͬ N͑¯ ͒ϪM M
͑a,b͒ ͑a,b͒ ϭ¯ , ␣ϭ␣ ͑t͒͑1ϩ⑀␣ ͒,
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͑a,b͒ Following ͓30͔ we must complement the action by a ϭ⑀ , boundary term to allow functional differentiability of the ac- where ⑀(a,b) are functions of t and x vanishing at the spatial tion. Using the falloff conditions ͑38͒ we have that the sole boundaries xa and xb , i.e., nonvanishing boundary term due to the variation of the ac- tion with respect to the canonical variables has the form lim ⑀͑a,b͒͑t,x͒ϭ0 ͑39͒ x xa ,xb → dt ͓Ϫ␣͑b͒͑t͒␦M͑b͒ϩ␣͑a͒͑t͒␦M͑a͔͒; ͑44͒ and ͵
͑a,b͒ so we add to the action the boundary term ⑀¯ lim ϭ0, ¯ ͑b͒ ͑b͒ ͑a͒ ͑a͒ x xa ,xbN͑͒ϪM S ϭ dt ͓␣ ͑t͒M ͑t͒Ϫ␣ ͑t͒M ͑t͔͒, → boundary ͵ ͑45͒ ⑀͑a,b͒ M lim ϭ0, where ␣(a,b)(t) parametrize the action at the boundary and x x ,x N͑¯ ͒ϪM → a b are interpreted as prescribed values of t ͑see Ref. ͓23͔ for a more detailed discussion about this point͒. ͑a,b͒Ј ⑀M The canonical field equations and the constraints lim ϭ0. ͑40͒ 0 ¯ ϭ0, ϭ0 are easily solved using the geometrodynamicalH x xa ,xbN͑͒ϪM 1 → H¯ chart (,¯ ,M,M). The general solution of the constraints The exact behavior of the ⑀(a,b) functions depends on the is particular potential under consideration. By requiring that (a,b) ⑀ go to zero rapidly enough, both the Liouville form and ¯ ϭ0, MЈϭ0. ͑46͒ the super-Hamiltonian and supermomentum are well defined ¯ Equations ͑46͒ have the same physical content of Eq. ͑10͒. and the difference of the Liouville forms F(,¯ ,M,M) reduces to an exact form. For instance, in the Schwarzschild ͑Note that M weakly commutes with the constraints, as ex- black hole case N()ϭͱ the spatial boundaries are located pected for a local integral of motion.͒ Equation ͑9͒ is a direct consequence of the canonical field equations. at xaϭϪϱ and xbϭϩϱ and with a little algebra one can check that the falloff conditions ͑38͒ become V. QUANTIZATION x2 ¯ ¯ Ϫ1Ϫ⑀ Ϯϱ Ϫ2Ϫ⑀ ϭ ͓1ϩϮ͉x͉ ϩO ͉͑x͉ ͔͒, The quantization of the full 1ϩ1 theory can be imple- 4 mented using the geometrodynamical canonical variables.
Ϯϱ Ϫ⑀ From Eqs. ͑46͒ we read that M does not depend on the space- MϭM Ϯ͑t͓͒1ϩO ͉͑x͉ )͔, like coordinate x. The effective Hamiltonian is simply given Ϯϱ Ϫ2Ϫ⑀ by the boundary term ͑45͒ and the reduced action reads ¯ ϭO ͉͑x͉ ͒, dm ϭOϮϱ͉͑x͉Ϫ1Ϫ⑀͒, S ϭ d p Ϫm , ͑47͒ M eff ͵ ͫ d m ͬ ␣ϭ␣ ͑t͓͒1ϩOϮϱ͑ x Ϫ⑀͔͒, Ϯ ͉ ͉ (b) (a) xb where mϵM (t)ϭM (t), pmϵ͐ dx M , and (t) xa ϭOϮϱ͉͑x͉Ϫ⑀͒,0Ͻ⑀р1. ͑41͒ ϭ͐tdtЈ͓␣(a)(tЈ)Ϫ␣(b)(tЈ)͔. The theory reduces formally to quantum mechanics and the quantization can be carried on as In this case the super-Hamiltonian and supermomentum fall usual. The Schro¨dinger equation is off as
ץ ⑀Ϯϱ Ϫ1Ϫ⑀ Ϯϱ Ϫ1Ϫ 0ϭO ͉͑x͉ ͒, 1ϭO ͉͑x͉ ͒, ͑42͒ i ⌿͑m;͒ϭ ⌿͑m;͒, ϵm. ͑48͒ Heff Heffץ H H and the Liouville form The stationary states are the eigenfunctions of m and the ϩϱ Hilbert space coincides with the Hilbert space obtained in the ˙ ¯˙ ˙ ¯˙ Ϯϱ Ϫ1Ϫ⑀ dx ͑MMϩ¯͒, MMϩ¯ϭO ͉͑x͉ ͒ 0ϩ1 approach. Let us see briefly this point in detail. ͵Ϫϱ ͑43͒ In the 0ϩ1 approach we take advantage that every solu- tion is static—according to the definition given below Eq. is well defined. Finally, using Eqs. ͑41͒ the second term on ͑16͒—and set from the beginning gϵg(t), ϵ(t). the right-hand side of Eq. ͑37͒ vanishes at spatial infinities The action ͑density͒ reads and the difference of the Liouville forms is an exact form. The above falloff conditions coincide with those used by S ϭ dt ͓˙ ϩ˙ Ϫ␣ ͔, ͑49͒ Kucharˇ in Ref. ͓23͔. 0ϩ1 ͵ H
084011-6 GEOMETRODYNAMICAL FORMULATION OF TWO- . . . PHYSICAL REVIEW D 59 084011 where ␣ is a Lagrange multiplier enforcing the constraint to any dilaton gravity model. We have seen that the general ϭ0. ( corresponds to the 0ϩ1 slice of 0 . The super- dilaton gravity action, Eq. ͑2͒, can be cast into the form ͑22͒ momentumH H constraint vanishes identically.H͒ and the system can be described both in the canonical and H1 So in the 0ϩ1 sector of the theory we can express the Lagrangian frameworks using the dilaton and the mass as field equations as a canonical system in a finite, (2ϫ2)- new variables. The quantization of the general dilaton grav- dimensional, phase space. The equations of motion are ana- ity model becomes straightforward and the resulting quan- lytically integrable and their solution coincides with the fi- tum theory exhibits at the quantum level the Birkhoff theo- nite gauge transformation generated by the constraint H rem. ϭ0. We can find a couple of gauge-invariant canonically We believe that Eq. ͑22͒ and Eqs. ͑32͒ can be used to look conjugate quantities m and pm corresponding to the 0ϩ1 at two-dimensional gravity from a new perspective. Up to sections of M and M introduced in Eqs. ͑33͒. The canonical now physicists have struggled themselves to find a canonical variables m and pm can be identified with the quantities de- transformation mapping the general dilaton gravity theory fined below Eq. ͑47͒. based upon Eq. ͑2͒ into a system described by free fields— Now we can construct the maximal gauge-invariant ca- see for instance, Ref. ͓22͔. Even though it seems reasonable nonical chart (m,p , , ) and use to fix the gauge. In- m H T T to assume the existence of such a canonical transformation, deed, the transformation properties of under the gauge we know from the CGHS case ͑the simplest possible case͒ transformation generated by imply thatT time defined by H that the relation between the free fields and the ‘‘physical’’ this variable covers once and only once the symplectic mani- fields ͑metric, dilaton, mass͒ is highly nonlinear. Further, the fold; i.e., time defined by is a global time. The quantization T canonical transformation may be pathological and subtleties becomes trivial and the Hilbert space is spanned by the and ambiguities may arise. For instance, in the CGHS case eigenvectors of the sole—apart from its conjugate the gauge-invariant operator M cannot be expressed as a momentum—gauge-invariant operator m corresponding to function of the new free fields, as we would expect imple- the mass of the system. menting a canonical transformation ͓18͔. This means that the This quantization program has been implemented in detail canonical transformation to free fields is ill defined. Indeed, a in Ref. ͓27͔ for the case of spherically symmetric Einstein careful analysis shows that the transformation cannot be in- gravity but can be easily generalized to an arbitrary V(). verted and, in order to make it invertible, one has to supple- ͑See, for instance, Ref. ͓31͔.͒ In the 0ϩ1 approach one can ment the new field variables by an extra pair of conjugate go further and discuss the self-adjointness properties of the variables related to the value of the fields at the boundary. mass operator. It turns out that the Hermitian operator m in ͑See, for instance, ͓18͔ and ͓6͔.͒ In spite of these difficulties, the gauge-fixed, positive-norm, Hilbert space is not self- the CGHS model can still be managed and different ap- adjoint, while its square is a self-adjoint operator with posi- proaches to the quantization can be carried on, leading to a tive eigenvalues. This result is due to the fact that the con- consistent quantum theory ͓5,6,18͔. However, we find it very jugate variable to the mass, pm , has positive support, hard to believe that models with more complicated dilatonic analogously to what happens for the radial momentum in potentials can be dealt with using free fields. Eventually, one ordinary quantum mechanics. However, the relevant point is wants quantum operators corresponding to the physical that the mass m—or its square—is the only gauge-invariant quantities of the model, i.e., quoting Kucharˇ, Romano, and observable of the system ͑apart from the conjugate variable, Varadarajan, ‘‘...the interesting questions in dilatonic grav- of course͒ and the Hilbert space of the 0ϩ1 approach coin- ity are precisely those which are concerned with the physical cides with the Hilbert space of the full quantum 1ϩ1 theory spacetime...’’ ͓6͔. Free fields are very distant from this pic- obtained through the geometrodynamical formalism. This is ture. the essence of the quantum Birkhoff theorem. Note that our Conversely, the canonical variables defined in Eq. ͑33͒, definition of the quantum Birkhoff theorem differs from the being directly related to the spacetime geometry, do not suf- definition that can be found in the previous literature ͓2,19͔. fer from the problems outlined above. Thus a quantum ͑Actually, in the previous literature a precise definition of the theory in which quantum operators have a clear physical quantum Birkhoff theorem is missing.͒ We define the meaning is easily achieved. Finally, Eq. ͑22͒ may ͑hopefully͒ Birkhoff theorem as the equivalence of the (0ϩ1)- and (1 provide a completely new starting point in the investigation ϩ1)-dimensional quantization procedures—see the diagram of open issues as, for instance, the thermodynamics of black at the end of Sec. I. This definition gives a clear meaning to holes and gravitational collapse ͑when matter is coupled to the quantum Birkhoff theorem that up to now was simply the system͒. intended as the property that quantum states depend on a single parameter ͑the mass of the system͒. While conclusions are identical, conceptual differences are great. ACKNOWLEDGMENTS I am indebted to Vittorio de Alfaro, Alexandre T. Filip- VI. CONCLUSIONS AND PERSPECTIVES pov, Claus Kiefer, Jorma Louko, and Dmitri Vassilevich for many interesting discussions and useful suggestions about Let us conclude with few remarks. We have derived a the subject of this paper. This work has been supported by a canonical transformation to geometrodynamical variables Human Capital and Mobility grant of the European Union, that generalizes the transformation of Ref. ͓21͔ and Ref. ͓23͔ Contract No. ERBFMRX-CT96-0012.
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