The canonical quantization of the essentially nonlinear midisuperspace model
describing cylindrically symmetric gravitational waves with two polarizations
is presented. A Fock space type representation is constructed. It is based on a
complete set of quantum observables. Physical expectation values may be
calculated in arbitrary excitations of the vacuum. Our approach provides a
non-linear generalization of the quantization of the collinearly polarized
Einstein-Rosen gravitational waves.Comment: 8 pages, LaTeX2

A. Ashtekar

A. Einstein

A. Izergin

A. Kompaneets

B. Julia

D. Korotkin

D. Maison

D. Marolf,

F. Ernst

G. A. Mena Marugán

H. Kastrup

H. Nicolai

H. Samtleben

K. Kuchař

L. Faddeev

R. Geroch

V. Belinskii

V. Husain

English

arXiv

Korotkin, D.

Samtleben, H.

MPG.PuRe

Canonical quantization of cylindrically symmetric gravitational waves with two polarizations

Korotkin, D. A.

Samtleben, Henning

Max Planck Society eDoc Server

VOLUME 80, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JANUARY 1998yy14Canonical Quantization of Cylindrical Gravitational Waves with Two PolarizationsD. Korotkin*II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, GermanH. Samtleben†Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Schlaatzweg 1, D-14473 Potsdam, German(Received 5 May 1997)The canonical quantization of the minisuperspace model describing cylindrically symmetric gravita-tional waves with two polarizations is presented. A Fock space-type representation is constructed. It isbased on a complete set of quantum observables. Physical expectation values may be calculated in ar-bitrary excitations of the vacuum. Our approach provides a nonlinear generalization of the quantizationof the collinearly polarized Einstein-Rosen gravitational waves. [S0031-9007(97)04861-8]PACS numbers: 04.60.Ds, 04.30.–w, 04.60.KzsertceltThe quantization of dimensionally reduced models o4D Einstein gravity serves as interesting testing grounfor many issues of quantum gravity. The physical outpof this approach to an understanding of characterisfeatures of the full theory, however, strongly depends othe complexity of the model under consideration.Probably, the simplest and best understood exampare the minisuperspace models [1] which contain only anite number of physical degrees of freedom and thus hithe field effects of quantum gravity. A more complicateexample of steady interest is given by the minisuperspamodel of cylindrically symmetric gravitational waves withone polarization [2,3]. This model already involves an infinite number of degrees of freedom. It becomes treatabwith the methods of flat space quantum field theory, bcause the Einstein field equations essentially reduce toaxisymmetric 3D wave equation. This underlying lineaity, on the other hand, may conceal typical nonlinear fetures of quantum gravity.It is the purpose of this Letter to generalize the resulof Refs. [2,3] to cylindrical gravitational waves withtwo polarizations, where the Einstein equations becomnonlinear. We achieve a consistent canonical quantizatin terms of a complete set of quantum observableCreation and annihilation operators are constructed intype of Fock space representation of these observabThe full space of physical quantum states is then bufrom excitations of the vacuum. This allows one tcalculate, in principle, all physical expectation values iarbitrary quantum states.We start from a general space-time with cylindricasymmetry, i.e., we assume the existence of two comming Killing vector fields, one of which has closed orbitsWe choose coordinates such that the Killing vector fieldare given by≠z and ≠w associated with the axis of sym-metry z and the azimuthal anglew, respectively. Furthergauge fixing brings the metric into the standard form [4]ds2 ­ eGsr, tds2dt2 1 dr2d 1 rg̃absr, tddxadxb ,a, b ­ 2, 3 ,(1)0031-9007y98y80(1)y14(4)$15.00fdutticnlesfi-dedce-lee-ther-a-tseions.ales.iltonlut-.s,with x2 ; z, x3 ; w, radial coordinater, and timet. The symmetric2 3 2 matrix g̃ is restricted by thecondition det̃g ­ 1.The essential part of the Einstein field equations igiven by the Ernst equation for the matrixg̃ or, equiva-lently, for the2 3 2 matrix g which carries the dualizedpotentials of the Ernst picture [5];≠rsrg21≠rgd 2 ≠tsrg21≠tgd ­ 0 . (2)The conformal factorGsr, td is then a functional ofg. Atspatial infinity,r ­ ` it is given byG` ­12Z `0rdr trfsg21≠rgd2 1 sg21≠tgd2g . (3)This factor generates evolution with respect to the timcoordinatet; its exponential measures the total energy peunit length in thez direction Ht and the deficit angle inthe asymptotic regionw0,Ht ­1pGw0 ­2Gs1 2 e2G`y2d . (4)The reduction of the metric to the form (1) can beperformed within the canonical formalism, such thathe Poisson bracket of the reduced model is the Dirabracket of the original structure after appropriate gaugfixing [3]. The resulting canonical Poisson structureis easily extracted from the effectively two-dimensionaLagrangian densityL s2d that comes from reduction viaKilling symmetries and gauge fixing of the originalLagrangianLs4dEH ­ s1yGdpjgmnj Rs4d:L s2dsr, td ­12Gr tr£sg21≠rgd2 2 sg21≠tgd2§.In matrix componentsgab , the Poisson brackets readhgabsrd, sg21≠tgg21dcdsr0dj ­Grdaddbcdsr 2 r0d .The restrictions of symmetry and the unit determinanof g require some additional technical effort and havebeen taken into account in the derivation of the followingresults.© 1997 The American Physical SocietyVOLUME 80, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JANUARY 1998tleof,tlyainra,dicoffCollinear polarizations.—Among the simplest nontriv-ial metrics of this model are the collinearly polarizedgravitational waves discovered by Einstein and Ros[6]. They correspond to a diagonal form of the matrig ; diagsef, e2fd, i.e., the number of degrees of freedom reduces to one. The Ernst equation (2) in this careduces to the cylindrical wave equation,2≠2tf 1 r21≠rf 1 ≠2rf ­ 0 ,with the general solution,fsr, td ­Z `0dlfA1sldJ0slrdeilt1 A2sldJ0slrde2iltg ,whereJ0 denotes Bessel functions of the first kind. Thcoefficients A1 ­ A2 build a complete set of observ-ables with canonical Poisson brackets,hA1sld, A2sl0dj ­ Gdsl 2 l0d . (5)Thus, quantization of this structure is straightforward [3and gives rise to a representation in terms of creation aannihilation operators,A2j0l ­ 0 with A1 ­ Ay2 . (6)In particular, coherent quantum states may be construcin the same way as in flat space quantum field theoRecent discussion, however, has shown that these stado not provide coherence of all essential physical quatities [7].As the first step towards the general case, we cathe truncated model of collinear polarization into a formthat will allow proper generalization. We introduce newvariables,T6swd ; expZ `0dlA6slde6iwl, (7)which build an equivalent complete set of observableIn the Fock space representation (6),T2swd is repre-sented as identity, whereasT1swd generates the coherenstate associated with a classical field that, on the symetry axisr ­ 0, is peaked as ad function att0 ­ w.In terms of these new variables, the Poisson structurebecomeshT2syd, T1swdj ­ 2Gy 2 wT2sydT1swd . (8)We shall see in the sequel that it is this quadratic formPoisson brackets which generically appears in the casetwo polarizations. Linearization to (5) is a special featurof the truncated model but not possible in the genercase.Two polarizations.—In general, the Ernst equation (2)does not admit explicit solution. However, it is possiblto construct the analog of the quantitiesT6 defined above.Inspired by the auxiliary linear system associated with thenx-see]ndtedry.tesn-sts.tm-(5)ofofealeeErnst equation [8], we defineT6sw, td ; P expZ `0dr2µg2g21≠rg1 2 g22gg21≠tg1 2 g2∂,for w [ H6 , (9)where H6 denote the upper and lower half of thecomplex plane, respectively andg ; 2s1yrd fw 2 t 1psw 2 td2 2 r2g. For diagonalg, this definition indeedreduces to (7) above. The variablesT6 are still constantsof motion, i.e.,≠tT6sw, td ­ 0 . (10)They turn out to be holomorphic inH6, respectively, andgenerically do not coincide on the realw-axis. Definition(9) further implies detT6 ­ 1 andT1swd ­ T2swd.Interestingly, it may be shown that the matrix producM ­ T1Tt2 on the real axis has a well-defined physicameaning, namely, it coincides with the values of thoriginal matrixg on the symmetry axis:Msw [ Rd ; T1swdTt2swd ­ gsr ­ 0, t ­ wd , (11)In particular, it is symmetric and real:Mswd ­ Mtswd and Mswd ­ Mswd . (12)Since theT6 contain the initial values of the metric andthe Ernst potential on the symmetry axisr ­ 0, theycontain sufficient information to recoverg everywhereby means of (2) [note that≠rgsr ­ 0d ­ 0 for solutionswhich are regular on the symmetry axis]. Thus, the setT6swd builds acompleteset of observables for the Ernstequation.Continuing the program of canonical quantizationwe next calculate the Poisson algebra to subsequenquantize it. A direct but lengthy calculations revealsquadraticPoisson algebra for the matrix entriesTab6 swd:hTab6 syd, Tcd6 swdj ­Gy 2 wfTad6 sydTcb6 swd2 Tcb6 sydTad6 swdg , (13)hTab2 syd, Tcd1 swdj ­Gy 2 wfTab2 sydTcd1 swd2 T cb2 sydTad1 swd2 dbdTam2 sydTcm1 swdg .(14)which consistently encloses the scalar algebra (8)the componentsT116 swd. Quantization of this quadraticstructure is rather more subtle than that of a linear algebsincea priori there appear ambiguities on the right-hanside (rhs.) due to different orderings of the quadratexpressions. Fortunately, the proper quantum analogthe Poisson brackets (13) is known in the theory o15VOLUME 80, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JANUARY 1998ce bothofintegrable systems [9] as the so-called SL(2)-Yangian algebra,fTab6 syd, Tcd6 swdg ­ih̄Gy 2 wfTcb6 swdTad6 syd 2 Tcb6 sydTad6 swdg . (15)The problem is to translate the Poisson brackets (14) and the symmetry relation (12) into the quantum picture sininvolve nonlinear expressions in the fieldsT6. Their consistent quantization is uniquely given by the following setmixed relations:fTab2 syd, Tcd1 swdg ­ih̄Gy 2 w 1 ih̄GTcd1 swdTab2 syd2ih̄Gsy 2 wdsy 2 w 1 ih̄Gd sy 2 w 2 ih̄GdfTad1 swdTcb2 syd 1 dbdT cm1 swdTamsydg1sih̄Gd2sy 2 w 1 ih̄Gd sy 2 w 2 ih̄Gddbd fTam1 swdTcm2 syd 2 Tcm1 swdTam2 sydg , (16)la-nlybutch,caltennnre.alaly,sanntshetesand the symmetry conditionT1swdTt2swd ­ fT1swdTt2swdgt . (17)Apart from the proper ordering of the quadratic expressions and the quantum corrections of orderh̄2 in (16),the essential content of these equations is the shift of tdenominator on the rhs. in (16). This provides a centrextension of (14), which is required for consistency of thquantum model.Classically,Mswd contains the essential physical ob-jects according to (11). In the quantum model, the defintion Mswd ­ T1swdTt2swd ensures that the commutationrelations (15) and (16) actually yield aclosedcommutatoralgebra of the matrix entries ofMswd. Moreover, theseare Hermitian operators, provided thatT ab1 swd ­ fTab2 sw̄dgy , (18)in accordance with the classical relations. Finally, thclassical condition of unit determinant detT6swd ­ 1 re-quires quantum corrections because of the nonlinear terand is substituted by the “quantum determinant” [10],T116 sw 1 ih̄GdT226 swd 2 T126 sw 1 ih̄GdT216 swd ­ 1 ,(19)which is indeed compatible with the relations (15) an(16) and may as such be imposed as an operator identitIn summary, we have formulated the consistent quantualgebra in terms of the operatorsTab6 swd, subject to thecommutation relations (15) and (16), as well as to unquantum determinant (19), hermiticity (18), and symmetr(17). We are now in a position to introduce a Fock spacetype representation of this algebra, inspired by the scalcase (6). Therefore, letT2swd act trivially on the vacuum,Tab2 swd j0l ­ dabj0l , (20)andT1swd generate the Fock space spanned by the bastates,mYi­1Tai bi1 swid j0l, wi [ H1, m ­ 0, 1, . . . .(21)16-healei-emsdy.mity-arsicThese excitations are not independent but obey the retions (15), (17), and (19) forT1. The intuitive idea that theT ab1 swd generate the complete spectrum of states is not osupported by the exactly solved scalar case from above,even stronger by the fact that the conserved chargesT1swdcanonically Poisson generate the Geroch group [11] whias a symmetry group, acts transitively among the classisolutions of the field equations [12].It is straightforward to further extract the relevanphysical information from the quantum model. Thhermiticity relations (18), together with the commutatiorelations (16), allow one to calculate the expectatiovalues of arbitrary polynomials in theTab6 swd in arbitraryexcitations of the vacuum. Indeed, the commutatiorelations (16) show that theT ab2 swd may be shuffledthrough to the right in any sequence of operators, whethey finally “annihilate” the vacuum according to (20)The rhs. of (16) may be viewed accordingly as a normordering of the quadratic expressions.We can also derive expectation values of the conformfactorG` and its exponentialeG` , related to energy, deficitangle, and metric components at infinity (4). Namelclassically one may calculate the Poisson bracketshG`, T6swdj ­ G≠wT6swd . (22)In the quantum theory, the conformal factor can thube represented as derivation operatorih̄G ≠y≠w; itsexponentialeG` becomes the shift operatorw ° w 1ih̄G. These operators may be shown to be Hermitiin the representations (18), (20), and (21). It is aelementary exercise to calculate their matrix elemenbetween arbitrary quantum states. In particular, tconformal factor G` exhibits a positive spectrum inaccordance with its classical form (3); e.g., its eigenstaof the first level are of the formZRdw exps2ilwyh̄GdT1swd j0l ,such that, due to holomorphy ofT1swd in H1, the integralvanishes for negativel.VOLUME 80, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 5 JANUARY 1998ialls-3d..c/..tyuprfs..sn.The presented quantum model provides the exact qutization of a minisuperspace model of quantum graviwith nonlinear characteristics. The complete set of quatum observables and a complete spectrum of physicquantum states are at hand. The techniques are suciently developed to start exploring the properties of thspectrum and relevant observables.It would be of great interest to identify some kind ocoherent states in this model, i.e., quantum states wcertain semiclassical properties. Because of the nolinear setting, it is reasonable to suspect that not allthe standard properties of the usual coherent states cansatisfied. The insufficiency of the traditional frameworof coherent states for the description of quantized gravitional waves is actually supported by recent observatioin the simpler model of collinear polarizations [7].Another exciting feature of our quantum model emergefrom the quantum analog of the determinant (19): In vieof the physical interpretation ofMswd (11), which suppliesthe spectral parameterw with a spacetime meaning, itis tempting to consider (19) as a sign of arising discrestructures and nonlocality of the quantum operatorsthe Planck scale. Let us recall that we have alreaencountered two other discrete effects showing up in tquantum model: The classical singularities in the Poissalgebra (14) have been shifted away from the real linean amount ofih̄G in (16). They may hence affect theholomorphy of the action of the quantum operatorsT6swdin a corresponding domain of Planck size. Secondlthe Hamiltonian (4) has been shown to be representby a discrete Planck length shift operator in the quantutheory.Since the presented quantization employs mainly tgroup-theoretical properties of the model, it will allownatural generalization to other and more complicatemodels of dimensionally reduced gravity, including highedimensional supergravity as well as Einstein-Maxwesystems [13]. Similarly, it should find application to theGowdy model, wherer becomes a timelike variable [14].The weak field limit of the nonlinear Poisson structurin this case is isomorphic to the isomonodromic Poissostructure quantized in [15]. With a different norm othe reducing Killing vector fields, the whole schemmay futhermore be applied to stationary axisymmetrspacetimes, providing an exact quantization of the blahole solutions in a vast class of models. A detaileaccount of the presented results will follow.We would like to thank H. Nicolai and V. Schome-rus for valuable discussions. The work of D. K. wasupported by DFG Contract No. Ni 290y5-1. H. S.an-tyn-alffi-efithn-ofbekta-nsswteondyheonbyy,edmhedrllenfeicckdsthanks Studienstiftung des deutschen Volkes for financsupport.*On leave of absence from Steklov MathematicaInstitute, Fontanka, 27, St. Petersburg 191011 Russia.Present address: Max-Planck-Institut für Gravitationphysik, Albert-Einstein-Institut, Schlaatzweg 1, D-1447Potsdam, Germany.Electronic address: korotkin@aei-potsdam.mpg.de†Electronic address: henning@aei-potsdam.mpg.de[1] See, e.g., A. Ashtekar, R. Tate, and C. Uggla, Int. J. MoPhys. D2, 15 (1993); H. Kastrup and T. Thiemann, NuclPhys. B425, 665 (1994); D. Marolf, Phys. Rev. D53,6979 (1996); G. A. Mena Marugán, Phys. Rev. D53, 3156(1996).[2] K. Kuchǎr, Phys. Rev. D4, 955 (1971).[3] A. Ashtekar and M. Pierri, J. Math. Phys.37, 6250 (1996).[4] A. Kompaneets, Sov. Phys. JETP7, 659 (1958).[5] F. Ernst, Phys. Rev.167, 1175 (1968). The relationbetweeng̃ and g is explicitly given by parameterizingg̃ ; Dr≥ 1 B̃B̃ B̃2 1 r2yD2¥, g ; 1D≥ 1 BB B2 1 D2¥withthe dual potentialsdB ­ D2rpdB̃.[6] A. Einstein and N. Rosen, J. Franklin Inst.223, 43 (1937).[7] A. Ashtekar, Phys. Rev. Lett.77, 4864 (1996); R. Gambiniand J. Pullin, Reports No. CGPG-97/4-1 and No. gr-q9703088, 1997 (to be published).[8] V. Belinskii and V. Zakharov, Sov. Phys. JETP48, 985(1978); D. Maison, Phys. Rev. Lett.41, 521 (1978).[9] L. Faddeev, E. Sklyanin, and L. Takhtajan, Theor. MathPhys.40, 194 (1979); L. Faddeev, N. Reshetikhin, and LTakhtajan, Leningrad Math. J.1, 193 (1990).[10] A. Izergin and V. Korepin, Sov. Math.26, 653 (1981).[11] This has been shown for a related model (with timeliker)in D. Korotkin and H. Samtleben, Class. Quantum Gravi14, L151 (1997). To be precise, the classical Geroch groin that model is generated by operatorsT21swdadT swd,where theT swd satisfy (13) but, in contrast to (11), carryno physical information.[12] R. Geroch, J. Math. Phys.13, 394 (1972).[13] B. Julia, in Unified Field Theories in More than 4Dimensions,edited by V. D. Sabbata and E. Schmutze(World Scientific, Singapore, 1983); H. Nicolai, inRecentAspects of Quantum Fields,edited by H. Mitter and H.Gausterer (Springer-Verlag, Berlin, 1991).[14] A recent treatment of the Gowdy model in terms oAshtekar variables has been given in V. Husain, PhyRev. D53, 4327 (1996); G. A. Mena Marugán, Phys. RevD 56, 908 (1997).[15] D. Korotkin and H. Nicolai, Nucl. Phys.B475, 397(1996); D. Korotkin and H. Samtleben, ReportNo. DESY 96-130 and No. hep-th/9607095, CommuMath. Phys. (to be published).17

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