We give an explicit canonical transformation which transforms a generic
chiral 2D dilaton gravity model into a free field theory.Comment: LaTeX file, 4 pages, to appear in Phys. Rev. 

A. Mikovic

C. Crnovic

C. G. Callan

D. Cangemi

D. Louis-Martinez

E. Benedict

E. Verlinde

G. Barnich

J. Cruz

J. Gegenberg

J. M. Bardeen

J. Navarro-Salas

K. Schoutens

K. V. Kuchar

M. Navarro

S. Hirano

T. Banks

English

arXiv

We give an explicit canonical transformation which transforms a generic chiral 2D dilaton gravity model into a free field theory

Cruz Muñoz, José Luis

Navarro Salas, José

Navarro Navarro, Miguel

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:gr-qc/9806106v1  26 Jun 1998Free Fields for Chiral 2D Dilaton Gravity. ∗J. Cruza†, J. Navarro-Salasa‡ and M. Navarrob.§a) Departamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC. Facultad de F´ısica, Universidad de Valencia, Burjassot-46100, Valencia, Spain.b) Instituto Carlos I de F´ısica Teo´rica y Computacional, Facultad de Ciencias, Universidadde Granada. Campus de Fuentenueva, 18002, Granada, Spain.AbstractWe give an explicit canonical transformation which transforms a generic chiral2D dilaton gravity model into a free field theory.PACS number(s): 04.60.Kz, 04.60.DsKeywords: Canonical Transformations, 2D gravity, free fields.∗Work partially supported by the Comisio´n Interministerial de Ciencia y Tecnolog´ıa andDGICYT†cruz@lie.uv.es‡jnavarro@lie.uv.es§mnavarro@fismat2.ugr.esIn the last years, a great effort has been dedicated to study two-dimensionalgravity theories. They provide a simplified setting to study quantum aspects offour-dimensional gravity. The simplest one of these models was introduced byCallan, Giddigs, Harvey and Strominger (CGHS) This theory can be canonicallymapped into a theory of two free fields with a Minkowskian target space [2]. Inthe new variables it is possible to carry out a consistent functional Schro¨dingerquantization of the matter-coupled CGHS theory [3, 4]. This result has beenrecently extended to the Jackiw-Teitelboim model and the model with an ex-ponential potential by finding explicit canonical transformations which alsoconvert these theories into a theory of two free fields with a flat target space ofMinkowskian signature [5]. In fact, it has been proven that such a transforma-tion exists for an arbitrary model of 2D dilaton gravity [6] although the explicitform remains elusive except for the above mentioned cases. In this paper wewill give an explicit expression for the canonical transformation which maps ageneric 2D dilaton gravity model into a free field theory when chiral matter iscoupled.It is well-known that the action of a generic 2D dilaton gravity model canbe brought to the form [7]S =∫d2x√−g[Rφ+ V (φ) +12(∇f)2], (1)by a conformal reparametrization of the fields. The equations of motion derivedfrom (1) areR+ V ′ (φ) = 0 , (2)∇µ∇νφ−12gµνV (φ) =12T fµν =12(∇µf∇νf −12gµν (∇f)2), (3)⊔⊓f = 0 . (4)This theory is equivalent, via a canonical transformation, to a theory of threefree fields. However, as we have already mentioned the explicit form of the fieldtransformation is unknown, up to same particular cases. If we restrict the phasespace of the theory by impossing the condition T fµνTfµν = 0, which physicallymeans to restrict the matter degrees to freedom to (let say) left-movers, thecanonical transformation relating dilaton-gravity and free fields variables of the1modified (chiral) theory can be explicitly given in the general case. A convenientform for the metric, which is useful to study the chiral matter-coupled theory,is the following [8]ds2 = e2ρAdv2 + 2eρdvdφ , (5)where φ is the dilaton field. In this way the dilaton is taken to be a coordinateand the two degrees of freedom are ρ and A. The chirality condition is auto-matically satisfied if we take T fvv = Tfvv (v) as the only non-vanishing componentof the energy momentum tensor. It is easy to check that with this conditionthe covariant conservation of the energy momentum tensor (i.e, the equation ofmotion for the matter field) holds identically. The different components of theequation (3) areρ,φ = −12Tfφφ = 0 , (6)− eρA,v = T fvv =12f2,v , (7)A,φ + V (φ) = −Aρ,φ . (8)From equation (6) we see that ρ is a chiral field ρ = ρ (v) while equation (8)can be written as(A+ J (φ)),φ = 0 , (9)where dJ(φ)dφ= V (φ). So, the solution can be expressed asA+ J (φ) = −2E , (10)with E = E (v) an arbitrary chiral function. This function turns out to be thelocal energy [9]E =12[(∇φ)2 − J (φ)], (11)which is a conserved quantity when there is no matter at all. The constraintequation (7) becomes now2eρE,v = Tfvv . (12)Therefore the general solution can be written asds2 = −e2ρ(∫e−ρT fvvdv + J (φ))dv2 + 2eρdφdv , (13)2with ρ = ρ (v) an arbitrary function which is associated with reparametrizationsof the coordinate v. If we choose ρ = 0 the above expression is a generalizationof the Vaidya solution of spherically symmetric gravitational collapse for anarbitrary 2D dilaton gravity model. Our aim now is to employ this result toconstruct a canonical transformation which maps the theory into a free fieldtheory. To this end we shall work out the symplectic 2-form on the phase-spaceconstrained by the chirality condition. A useful way to achieve this is by usingthe covariant phase-space formalism [10] which automatically incorporates theconstraint in a consistent way. The symplectic form of the theory is given byω = δ∫Σjµdσµ , (14)where δ stands for the exterior differential on phase space and jµ is given byjµ = −φ(gαβ∇µδgαβ − gµα∇βδgαβ)+∇µφgαβδgαβ −∇αφδgαµ +∇µfδf . (15)To evaluate ω on the space of chiral solutions it is convenient to choose φ =constant as the initial data surface. The result isω =∫dv [δ(−2E)δeρ + δfδf,v] , (16)implying the following Poisson bracket of dilaton-gravity variables{−2E (v) , eρ(v˜)}= δ (v − v˜) . (17)This is a generalization of the reassoning used in Ref[11] for the particular caseof the CGHS theory.The phase-space of the chiral theory is, therefore, made out of two arbitrary(chiral) functions (E (v) , ρ (v)) in addition to the left-mover matter field f(v).DefiningX = −2E , (18)Π = eρ , (19)the constraint (12) becomes the constraint of a free field theoryT fvv +ΠX,v = 0 . (20)3The above formulas are the main result of this paper and provide the explicitrealization of the canonical transformation proposed in Ref[6] for the chiralmatter case. To finish this paper it is interesting to comment that in the absenceof matter(T fvv = 0), E is a constant of motion and the Poisson bracket (17)converts into {−2E,∫dv eρ}= 1 , (21)indicating that the only (global) degree of freedom is C = −2E and the canon-ically conjugated variable is P =∫dv eρ in agreement with [12].AcknowledgementsJ. C. acknowledges the Generalitat Valenciana for a FPI grant. M. N. acknowl-edges the Spanish MEC, CSIC and IMAFF (Madrid) for a research contract.References[1] C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger, Phys. Rev.D 45, 1005 (1992).[2] E. Verlinde and H. Verlinde, Nucl. Phys. B 406 (1993) 43.S. Hirano, Y. Kazama and Y. Satoh, Phys. Rev. D 48 (19939 1687.A. Mikovic and M. Navarro, Phys. Lett. B 315 (1993) 267.D. Cangemi, R. Jackiw and B. Zwiebach, Ann. Phys (N.Y) 245, 408 (1996).[3] E. Benedict, R. Jackiw and H.-J. Lee, Phys. Rev. D 54 (1996) 6213.[4] K. V. Kuchar, J. D. Romano and M. Varadarajan, Phys. Rev. D 55, 795(1997).[5] J. Cruz, J. M. Izquierdo, D. J. Navarro and J. Navarro-Salas, ” Free fieldsvia Canonical Transformations of Matter-coupled 2D Dilaton Gravity Mod-els ” hep-th/9704168 (Phys. Rev. D, in press)[6] J. Cruz and J. Navarro-Salas, Mod. Phys. Lett. A 12, 31 (1997) 2345.4[7] T. Banks and M. O′Loughlin, Nucl. Phys. B 362, 649 (1991).[8] J. M. Bardeen, Phys. Rev. Lett 46 (1981) 382.[9] J. Gegenberg, G. Kunstatter and D. Louis-Martines, Phys. Rev. D51 (1995)1781.[10] C. Crnovic and E. Witten, in ” Three Hundred Years of Gravitation”, edsS. W. hawking and W. Israel (Cambridge, 1987) p. 676. G. Barnich, M.Henneaux, and C. Schomblond, Phys. Rev. D 44, 939 (1991)[11] K. Schoutens, E. Verlinde and H. Verlinde, Phys. Rev. D 48 (1993) 2670.[12] D. Louis-Martinez, J. Gegenberg and G. Kunstatter, Phys. Lett. B 321,193 (1994).5

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