Quantum Einstein s equations and constraints algebra

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1 PRAMANA cfl Idia Academy of Scieces Vol. 58, No. 1 joural of Jauary 2002 physics pp Quatum Eistei s equatios ad costraits algebra FATIMAH SHOJAI 1;2 ad ALI SHOJAI 2 1 Physics Departmet, Ira Uiversity of Sciece ad Techology, P.O. Box , Narmak, Tehra, Ira 2 Istitute for Studies i Theoretical Physics ad Mathematics, P.O. Box , Tehra, Ira fatimah@theory.ipm.ac.ir; shojai@theory.ipm.ac.ir MS received 23 April 2001; revised 14 August 2001 Abstract. I this paper we shall address this problem: Is quatum gravity costraits algebra closed ad what are the quatum Eistei s equatios. We shall ivestigate this problem i the de- Broglie Bohm quatum theory framework. It is show that the costrait algebra is weakly closed ad the quatum Eistei s equatios are derived. Keywords. Quatum gravity; casual quatum theory; costraits algebra. PACS Nos Hw; Kz; Bz 1. Itroductio de-broglie Bohm causal quatum mechaics [1] has several positive poits: (a) First of all, it is causal ad so describes the system i a ordered way i time. (b) Perhaps the most importat cocept i the de-broglie Bohm theory is the lack of eed for the assumptio of the existece of a classical domai i the measuremet pheomea. (c) This theory provides a useful framework for quatum gravity. (d) It does ot suffer from the coceptual problems like the meaig of the wave fuctio for a sigle system ad so o. I fact there are a umber of such positive poits of this theory which ca be foud i the literature [2]. The applicatio of this theory to quatum gravity has bee ivestigated from differet aspects: The quatum force may be repulsive ad thus ca remove the iitial sigularity ad iflatio would be emerged. Sice the quatum domai is defied as the domai that the quatum force is smaller tha the classical force, it is possible to have the classical uiverse for the small scale factors ad coversely quatum uiverse for large scale factors [3]. As the scale factor represets the uiverse radius, there is ot a oe-to-oe correspodece betwee large ad classical uiverses. Because of the guidace formula, the time parameter appears automatically ad the time problem does ot exist. Moreover a ew approach based o the de-broglie Bohm theory has bee preseted that brigs out may iterestig physical results, such as uificatio of quatal ad gravitatioal behaviour of matter [4]. 13

2 Fatimah Shojai ad Ali Shojai I the ADM decompositio of the space time i geeral relativity the o-dyamical ature of shift ad lapse fuctios (these are fuctios used i the slicig of the space time ad will be itroduced later) must be cosistet with the evolutio. This is obviously satisfied if the costrait algebra is closed. Moreover, we kow the secodary costraits i the ADM decompositio of the space time are eergy ad mometum costraits. These are four o-dyamical Eistei s equatios which together with the closedess of costrait algebra provides ecessary ad sufficiet coditios for lapse ad shift fuctio to be o dyamical as the space time evolves. This is satisfied at the classical level as we kow that the geeral relativity is idepedet of the space time reparametrizatio. This property is guarateed by the geometrical Biachi idetities. I the preset work we wat to discuss this poit at the quatum level. I order to do so, first we shall derive the costrait algebra at the quatum level. We shall use the itegrated versio of diffeomorphism ad Hamiltoia costraits. I the other sectios we shall derive the quatum Eistei s equatios choosig arbitrary lapse ad shift fuctios, i.e. i a arbitrary gauge. These equatios are the exteded form of those previously discussed by us [5]. 2. Quatum costraits algebra I the ADM formulatio, the Hamiltoia of geeral relativity is i which where ad H = Z d 3 xh (1) H = N + N i H i ; (2) = G ijkl Π ij Π kl + p h(r 2Λ) (3) H i = 2 j Π j i : (4) I these equatios, ~N is the shift fuctio ad N is the lapse fuctio. These fuctios produce the time evolutio of space-like surfaces i ormal ad taget directios respectively. G ijkl is the superspace metric ad is give by G ijkl = 1 h 2 p h ik h jl + h il h jk h ij h kl ; (5) where h ij is the three dimesioal space-like hypersurface metric, Π ij the cojugate caoical mometum of h ij,r the itrisic curvature of the hypersurface adλ the cosmological costat. By Guass Codazzi relatios oe ca show that = 0 (6) 14 Pramaa J. Phys., Vol. 58, No. 1, Jauary 2002

3 Quatum Eistei s equatios ad H i = 0 (7) are i fact the four o-dyamical (costraits) Eistei s equatios. So the Hamiltoia desity vaishes by the Eistei s equatios. The other dyamical Eistei s equatios ca be derived by differetiatig the first equatio with respect to h ij. Correspodig to the two directios of time evolutio of hypersurface (ormal ad taget), the Hamiltoia ca be separated as ad CI c (N)= Z d 3 xn (8) eci c (~N)= Z d 3 xn i H i : (9) Because of the above explaatios, e CI c ( ~ N) is called diffeomorphism costrait while CI c (N) is called Hamiltoia costrait. It is istructive to see the algebra of costraits. Usig the otatio of [6], oe obtais eci c ( ~ N); e CI c ( ~ N 0 ) o = e CI c (N i~ N 0 i N 0i~ N i ); (10) Φ CI c (N); CI c (N 0 ) Ψ = e CI c (N ~ N 0 N 0~ N); (11) eci c (~N); CI c (N) o = CI c (~N ~ N): (12) To quatize accordig to the stadard quatum mechaics, oe ca use the Dirac quatizatio procedure which leads to [CI (N)Ψ = 0; (13) [eci (~N)Ψ = 0: (14) These are quatum costrait ad limit the physical wave fuctio. The former is the WDW equatio ad the latter represets the ivariace uder geeral spatial trasformatio. Now we shall apply the de-broglie Bohm theory to caoical quatum gravity. I the Hamilto Jacobi laguage (which is suitable for our discussio), i de-broglie Bohm theory the desired quatum system is subjected to quatum potetial i additio to the classical oes. This term icludes all the quatum iformatio about the system. It is a o-local potetial ad obtaied by the orm of the wave fuctio. By this simple descriptio of this theory, we ca discuss the costraits algebra at the quatum level. As discussed i [2,5], the followig chage i aloe will be sufficiet for descriptio at quatum level. Pramaa J. Phys., Vol. 58, No. 1, Jauary

4 Fatimah Shojai ad Ali Shojai! + Q; (15) where Q is the quatum potetial. The costrait equatios correspodig to eqs (6) ad (7) are G ijkl Π ij Π kl + p h(r 2Λ)+Q=0; (16) 2 j Π j = 0: (17) i Dyamical equatios are derived by differetiatig the first equatio ad usig Gauss Codazzi relatios, as this is doe i the ext sectio. Equivaletly we have CI (N)=CI c (N)+Q(N); (18) eci (~N)= e CI c (~N); (19) where Q(N)= Z d 3 xnq = Z d 3 xn ψ! δ 2 jψj p G hjψj ijkl : (20) δh kl ~2 Sice at the quatum level e CI ( ~ N) has ot chaged relatio (10) is satisfied agai. Some calculatios leads to the followig algebraic relatios: eci (~N); e CI (~N 0 ) o = e CI (N i~ N 0 i N 0i~ N i ); (21) Φ CI (N); CI (N 0 ) Ψ = e CI c (N ~ N 0 N 0~ N) +2 Z d 3 zd 3 x p h(z)g ijkl (z)π kl (z) N(z)N 0 (x)+n(x)n 0 (z) δq(x) (z) ß 0; (22) eci ( ~ N); CI (N) o = CI ( ~ N ~ N): (23) The relatio (23) is the same as the classical oe ad the relatio (22) is weakly zero (ß 0), i.e. zero oly whe the equatio of motio is used. To see this, oe must evaluate the derivative of quatum potetial from the equatio of motio (16) as δq(x) (z) = 3 4 p h h kl Πij Π kl δ(x z) p h 2 hij (R 2Λ)δ(x z) p h δr : (24) 16 Pramaa J. Phys., Vol. 58, No. 1, Jauary 2002

5 Quatum Eistei s equatios Usig the kow idetity F δ R(x) (z) = FR ij + i j F h ij 2 F δ(x z); (25) where F is ay arbitrary fuctio, ad substitutig these relatios i the Poisso bracket (22), shows that it is weakly zero. This meas the oe parameter family of diffeomorphism of spatial slices is a symmetry of the quatum space time. But pushig spatial slices i the ormal directio is a symmetry satisfied oly at the level of quatum equatios of motio. This poit cofirms our previous result ([5] ad its refereces). Therefore, as at the classical level, we must choose the iitial coditios for spacelike metric, lapse ad shift fuctios to be cosistet with the symmetry of space-time. Sice the costrait algebra is ot closed strogly, the Hamiltoia is ot ivariat uder reparametrizatio of space time. The uder dyamical evolutio the symmetry does ot survive. I this way differet idetical coditios lead to differet solutios. Because of the lack of ivariace uder time reparametrizatio we expect the geeral covariace symmetry to be broke, but this is ot obvious at the level of the equatios of motio. If we are iterested i seeig the symmetry breakig at this level, we must look at quatum Eistei s equatios. This poit is explaied i the ext sectio. 3. Quatum Eistei s equatios Previously [5] i the Bohmia quatum gravity framework, we have studied the modificatios of Eistei s equatios i some special gauge. It was show that the correctio terms cotai the quatum potetial as we expected. Our discussio i [5] is based o ADM decompositio of the space time ad Gauss Codazzi equatios. But there it was assumed that the lapse fuctio is 1 ad the shift is zero for simplificatio. Here we derive the modified Eistei s equatios i the geeral case. The Gauss Codazzi equatios for ay choice of lapse ad shift fuctios are ad G µν µ ν = 1 2 R + K 2 K ij K ij (26) where G µi µ = j K j i i K j i ; (27) µ = 1 N (1; ~ N) (28) represets a field of time-like vectors ormal to a space-like slice. K ij ad R ij are extrisic curvature ad Riema tesor of three metric respectively. If we express the costraits quatities, CI ad e CI, i terms of the itrisic curvature, usig the followig formula for cojugate mometa Π ij = p h(k ij h ij K); (29) Pramaa J. Phys., Vol. 58, No. 1, Jauary

6 Fatimah Shojai ad Ali Shojai we fid = 2G µν µ ν ; (30) H i = 2G µi µ : (31) Now accordig to de-broglie Bohm quatum theory of gravity we have 2G µν µ ν Q= 0; (32) G µi µ = 0: (33) I cotiuatio we use the form of geeral relativity actio i terms of three metric ad extrisic curvature Z A = 1 d 4 x p g(r + K 2 K 16πG ij K ij ): (34) Usig the relatios (34), (26), ad (32) the dyamical equatio of three metric is obtaied as G ij = δq : (35) As we expected, i the right had side of the above dyamical equatio the quatum force is appeared. From this relatio by usig eqs (32) ad (33) we get the other remaiig Eistei s equatios G 0i = NN i Q+ N j δq ; (36) G 00 = N N i N i N δ Q Q+N 2 i N j : (37) It is simply see that these equatios are geeral form of the results of [5]. Thus i the modified Eistei s equatios i a geeral case both shift ad lapse fuctios appear. This motivates us to doubt about covariace uder geeral coordiate trasformatios. A simple calculatio shows that these equatios are ot covariat. Settig the right had side of modified Eistei s equatios as the compoets of a matrix X µν, to show the lack of geeral covariace oe must ivestigate the trasformatio properties of X µν. Specifyig the trasformatio to oe i which t! t 0 (t) ad ~x does ot chage, we have (usig the trasformatio law of the metric) h 0 ij = h ij ; (38) N = 0 FN; (39) N 0i = FN i (40) 18 Pramaa J. Phys., Vol. 58, No. 1, Jauary 2002

7 Quatum Eistei s equatios with F = t (41) t 0 ad the quatum potetial would remai uchaged. So we have X 000 = FN F 2 N i N i FN 2 Q+ F 2 N i N j δq ; (42) X 00i = F 2 NN i Q+ FN j δq ; (43) X 0ij = X ij ; (44) which shows that X µν does ot trasform as a secod rak tesor. Thus the geeral covariace priciple is broke at the quatum level. 4. Coclusio The quatum effects ca be studied i gravity, as well as ay other theory, by itroducig the quatum potetial. Sice the quatum potetial modifies the Hamiltoia, there is o guaratee that the costraits algebra be closed as it is i the classical case. The costraits algebra is i fact closed oly weakly, i.e. by usig the equatios of motio. This shows that the associated symmetry should break dow. We saw how oe ca obtai the equatios of motio, i.e. the quatum Eistei s equatios ad how they are ot geeral covariat. Refereces [1] D Bohm, Phys. Rev. 85, 166 (1952) D Bohm, Phys. Rev. 85, 180 (1952) L de Broglie, Noliear wave mechaics, Tras. A J Kodel (Elsevier, 1960) [2] D Bohm ad J Hiley, The udivided Uiverse (Routledge, 1993) P R Hollad, The quatum theory of motio (Cambridge Uiversity Press, 1993) [3] R Colistete Jr., J C Fabris ad N Pito-Neto, Phys. Rev. D57, 4707 (1998) [4] F Shojai ad M Golshai, It. J. Mod. Phys. A13, 4, 677 (1998) F Shojai, A Shojai ad M Golshai, Mod. Phys. Lett. A13, 34, 2725 (1998) F Shojai, A Shojai ad M Golshai, Mod Phys. Lett. A13, 36, 2915 (1998) A Shojai, F Shojai ad M Golshai, Mod. Phys. Lett. A13, 37, 2965 (1998) F Shojai ad A Shojai, It. J. Mod. Phys. A15, 13, 1859 (2000) A Shojai, It. J. Mod. Phys. A15, 12, 1757 (2000) [5] F Shojai ad M Golshai, It. J. Mod. Phys. A13, 13, 2135 (1998) [6] J Baez ad J P Muiai, Gauge fields, kots ad gravity (World Scietific Publishig Co. Pte. Ltd., 1994) Pramaa J. Phys., Vol. 58, No. 1, Jauary

QUANTUM EINSTEIN S EQUATIONS AND CONSTRAINTS ALGEBRA

QUANTUM EINSTEIN S EQUATIONS AND CONSTRAINTS ALGEBRA FATIMAH SHOJAI Physics Department, Iran University of Science and Technology, P.O.Box 16765 163, Narmak, Tehran, IRAN and Institute for Studies in Theoretical