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PHYSICAL REVIEW D, VOLUME 58, 087501

Free fields for chiral 2D dilaton gravity

J. Cruz* and J. Navarro-Salas†

Departamento de Fı´sica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC. Facultad de Fı´sica, Universidad de Valencia,Burjassot-46100, Valencia, Spain

M. Navarro‡

Instituto Carlos I de Fı´sica Teo´rica y Computacional, Facultad de Ciencias, Universidad de Granada. Campus de Fuentenuev18002, Granada, Spain

~Received 18 June 1997; revised manuscript received 1 April 1998; published 31 August 1998!

We give an explicit canonical transformation which transforms a generic chiral 2D dilaton gravity modelinto a free field theory.@S0556-2821~98!02918-X#

PACS number~s!: 04.60.Kz, 04.60.Ds

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In the last years, great effort has been dedicated to stwo-dimensional gravity theories. They provide a simplifisetting to study quantum aspects of four-dimensional gravThe simplest one of these models was introduced by CaGiddings, Harvey, and Strominger~CGHS! @1#. This theorycan be canonically mapped into a theory of two free fiewith a Minkowskian target space@2#. In the new variables itis possible to carry out a consistent functional Schro¨dingerquantization of the matter-coupled CGHS theory@3,4#. Thisresult has been recently extended to the Jackiw-Teitelbmodel and the model with an exponential potential by fining explicit canonical transformations which also convthese theories into a theory of two free fields with a flat tarspace of Minkowskian signature@5#. In fact, it has beenproven that such a transformation exists for an arbitrmodel of 2D dilaton gravity@6# although the explicit formremains elusive except for the above-mentioned cases. Inpaper we will give an explicit expression for the canonictransformation which maps a generic 2D dilaton gravmodel into a free field theory when chiral matter is couple

It is well known that the action of a generic 2D dilatogravity model can be brought to the form@7#

S5E d2xA2gFRf1V~f!11

2~¹ f !2G ~1!

by a conformal reparametrization of the fields. The equatiof motion derived from Eq.~1! are

R1V8~f!50, ~2!

¹m¹nf21

2gmnV~f!5

1

2Tmn

f 51

2S ¹m f ¹n f 21

2gmn~¹ f !2D ,

~3!

h f 50. ~4!

*Email address: CRUZ@LIE.UV.ES†Email address: JNAVARRO@LIE.UV.ES‡Email address: MNAVARRO@FISMAT2.UGR.ES

0556-2821/98/58~8!/087501~2!/$15.00 58 0875

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This theory is equivalent, via a canonical transformation, ttheory of three free fields. However, as we have alreamentioned the explicit form of the field transformationunknown, up to some particular cases. If we restrictphase space of the theory by impossing the conditTmn

f Tf mn50, which physically means to restrict the mattdegrees of freedom to~say! left movers, the canonical transformation relating dilaton-gravity and free fields variablesthe modified~chiral! theory can be explicitly given in thegeneral case. A convenient form for the metric, whichuseful to study the chiral matter-coupled theory, is the flowing @8#:

ds25e2rAdv212erdvdf, ~5!

wheref is the dilaton field. In this way the dilaton is taketo be a coordinate and the two degrees of freedom arer andA. The chirality condition is automatically satisfied if wtake Tvv

f 5Tvvf (v) as the only nonvanishing component

the energy momentum tensor. It is easy to check that wthis condition the covariant conservation of the energy mmentum tensor~i.e., the equation of motion for the mattefield! holds identically. The different components of Eq.~3!are

r ,f521

2Tff

f 50, ~6!

2erA,v5Tvvf 5

1

2f ,v

2 , ~7!

A,f1V~f!52Ar ,f. ~8!

From Eq.~6! we see thatr is a chiral fieldr5r(v) whileEq. ~8! can be written as

@A1J~f!# ,f50, ~9!

wheredJ(f)/df5V(f). So the solution can be expresseas

A1J~f!522E, ~10!

with E5E(v) an arbitrary chiral function. This functionturns out to be the local energy@9#

© 1998 The American Physical Society01-1

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BRIEF REPORTS PHYSICAL REVIEW D 58 087501

E51

2@~¹f!22J~f!#, ~11!

which is a conserved quantity when there is no matter atThe constraint equation~7! becomes now

2erE,v5Tvvf . ~12!

Therefore the general solution can be written as

ds252e2rS E e2rTvvf dv1J~f! Ddv212erdfdv,

~13!

with r5r(v) an arbitrary function which is associated wireparametrizations of the coordinatev. If we chooser50,the above expression is a generalization of the Vaidya stion of spherically symmetric gravitational collapse forarbitrary 2D dilaton gravity model. Our aim now is to employ this result to construct a canonical transformation whmaps the theory onto a free field theory. To this end we swork out the symplectic two-form on the phase space cstrained by the chirality condition. A useful way to achiethis is by using the covariant phase-space formalism@10#which automatically incorporates the constraint in a content way. The symplectic form of the theory is given by

v5dES

j mdsm, ~14!

whered stands for the exterior differential on phase spaand j m is given by

j m52f~gab¹mdgab2gma¹bdgab!

1¹mfgabdgab2¹afdgam1¹m f d f . ~15!

To evaluatev on the space of chiral solutions it is convnient to choosef5const as the initial data surface. The rsult is

v5E dv@d~22E!der1d f d f ,v#, ~16!

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implying the following Poisson bracket of dilaton-gravitvariables:

$22E~v !,er~ v !%5d~v2 v !. ~17!

This is a generalization of the reassoning used in Ref.@11#for the particular case of CGHS theory.

The phase space of the chiral theory is, therefore, mout of two arbitrary~chiral! functions@E(v),r(v)# in addi-tion to the left-mover matter fieldf (v). Defining

X522E, ~18!

P5er, ~19!

the constraint~12! becomes the constraint of a free fietheory:

Tvvf 1PX,v50. ~20!

The above formulas are the main result of this paperprovide an explicit realization of the canonical transformtion proposed in Ref.@6# for the chiral matter case. To finisthis paper it is interesting to comment that in the absencematter (Tvv

f 50), E is a constant of motion and the Poissobrackets~17! convert into

H 22E,E dverJ 51, ~21!

indicating that the only~global! degree of freedom isC522E and the canonically conjugated variableP5*dver in agreement with@12#.

J.C. acknowledges the Generalitat Valenciana for agrant. M.N. acknowledges the Spanish MEC, CSIC, aIMAFF ~Madrid! for a research contract. This work was patially supported by the Comisio´n Interministerial de Cienciay Tecnologı´a and DGICYT.

ys.

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