Nelson Pinto-Neto 331 funtion is given in terms of the pointer basis states, an why we o not see superpositions of marosopi objets. In this way, lassi

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1 330 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 Quantum Cosmology: How to Interpret an Obtain Results Nelson Pinto-Neto Centro Brasileiro e Pesquisas F sias, Rua Dr. Xavier Sigau 150, Ura , Rio e Janeiro, RJ, Brazil s: nelsonpn@lafex.bpf.br Reeive 7 January, 2000 We argue that the Copenhagen interpretation of quantum mehanis annot be applie to quantum osmology. Among the alternative interpretations, we hoose to apply the Bohm-e Broglie interpretation of quantum mehanis to anonial quantum osmology. For minisuperspae moels, we show that there is no problem of time in this interpretation, an that quantum effets an avoi the initial singularity, reate inflation an isotropize the Universe. For the general ase, it is shown that, irrespetive of any regularization or hoie of fator orering of the Wheeler-DeWitt equation, the unique relevant quantum effet whih oes not break spaetime is the hange of its signature from lorentzian to euliean. The other quantum effets are either trivial or break the four-geometry of spaetime. A Bohm-e Broglie piture of a quantum geometroynamis is onstrute, whih allows the investigation of these latter strutures. I Introution Almost all physiists believe that quantum mehanis is auniversal an funamental theory, appliable to any physial system, from whih lassial physis an be reovere. The Universe is, of ourse, a vali physial system: there is a theory, Stanar Cosmology, whih is able to esribe it in physial terms, an make preitions whih an be onfirme or refute by observations. In fat, the observations until now onfirm the stanar osmologial senario. Hene, supposing the universality of quantum mehanis, the Universe itself must be esribe by quantum theory, from whih we oul reover Stanar Cosmology. However, the Copenhagen interpretation of quantum mehanis [1, 2, 3] 1, whih is the one taught in unergrauate ourses an employe by the majority ofphysiists in all areas (speially the von Neumann's approah), annotbeuseinaquantum Theory of Cosmology. This is beause it imposes the existene of a lassial omain. In von Neumann's view, for instane, the neessity ofa lassial omain omes from the way it solves the measurement problem (see Ref. [4] for a goo isussion). In an impulsive measurement of some observable, the wave funtion of the observe system plus marosopi apparatus splits into many branhes whih almost o not overlap (in orer to be a goo measurement), eah one ontaining the observe system in an eigenstate of the measure observable, an the pointer of the apparatus pointing to the respetive eigenvalue. However, in the en of the measurement, we observe only one of these eigenvalues, an the measurement is robust in the sense that if we repeat it immeiately after, we obtain the same result. So it seems that the wave funtion ollapses, the other branhes isappear. The Copenhagen interpretation assumes that this ollapse is real. However, a real ollapse annot be esribe by the unitary Shröinger evolution. Hene, the Copenhagen interpretation must assume that there is a funamental proess in a measurement whihmust our outsie the quantum worl, in a lassial omain. Of ourse, if we want to quantize the whole Universe, there is no plae for a lassial omain outsie it, an the Copenhagen interpretation annot be applie. Hene, if someone insists with the Copenhagen interpretation, she or he must assume that quantum theory is not universal, or at least try to improve itby means of further onepts. One possibility is by invoking the phenomenon of eoherene [5]. In fat, the interation of the observe quantum system with its environment yiels an effetive iagonalization of the reue ensity matrix, obtaine by traing out the irrelevant egrees of freeom. Deoherene an explain why the splitting of the wave 1 Although these three authors have ifferent views from quantum theory, the first emphasizing the inivisibility of quantum phenomena, the seon with his notion of potentiality, an the thir with the onept of quantum states, for all of them the existene of a lassial omain is ruial. That is why we group their approahes uner the same name Copenhagen interpretation".

2 Nelson Pinto-Neto 331 funtion is given in terms of the pointer basis states, an why we o not see superpositions of marosopi objets. In this way, lassial properties emerge from quantum theory without the nee of being assume. In the framework of quantum gravity, it an also explain how alassial bakgroun geometry an emerge in a quantum universe [6]. In fat, it is the first quantity to beome lassial. However, eoherene is not yet a omplete answer to the measurement problem [7, 8]. It oes not explain the apparent ollapse after the measurement is omplete, or why all but one of the iagonal elements of the ensity matrix beome null when the measurement is finishe. The theory is unable to give anaount of the existene of fats, their uniqueness as oppose to the multipliity of possible phenomena. Further evelopments are still in progress, like the onsistent histories approah [9], whih ishowever inomplete until now. The important role playe by the observers in these esriptions is not yet explaine [10], an still remains the problem on how to esribe a quantum universe when the bakgroun geometry is not yet lassial. Nevertheless, there are some alternative solutions to this quantum osmologial ilemma whih, together with eoherene, an solve the measurement problem maintaining the universality of quantum theory. One an say that the Shröinger evolution is an approximation of a more funamental non-linear theory whih an aomplish the ollapse [11, 12], or that the ollapse is effetive but not real, in the sense that the other branhes isappear from the observer but o not isappear from existene. In this seon ategory we an ite the Many-Worls Interpretation [13] an the Bohm-e Broglie Interpretation [14, 15]. In the former, all the possibilities in the splitting are atually realize. In eah branh there is an observer with the knowlege of the orresponing eigenvalue of this branh, but she or he is not aware of the other observers an the other possibilities beause the branhes o not interfere. In the latter, apoint-partile in onfiguration spae esribing the observe system an apparatus is suppose to exist, inepenently onany observations. In the splitting, this point partile will enter into one of the branhes (whih one epens on the initial position of the point partile before the measurement, whih is unknown), an the other branhes will be empty. It an be shown [15] that the empty waves an neither interat with other partiles, nor with the point partile ontaining the apparatus. Hene, no observer an be aware of the other branhes whih are empty. Again we have an effetive but not real ollapse (the empty waves ontinue to exist), but now withno multipliation of observers. Of ourse these interpretations an be use in quantum osmology. Shröinger evolution is always vali, an there is no nee of a lassial omain outsie the observe system. In this paper we review some results on the appliation of the Bohm-e Broglie interpretation to quantum osmology [16, 17, 18, 19, 20]. In this approah, the funamental objet of quantum gravity, the geometry of 3-imensional spaelike hypersurfaes, is suppose to exist inepenently on any observation or measurement, as well as its anonial momentum, the extrinsi urvature of the spaelike hypersurfaes. Its evolution, labele by some time parameter, is itate by a quantum evolution that is ifferent from the lassial one ue to the presene of a quantum potential whih appears naturally from the Wheeler-DeWitt equation. This interpretation has been applie to many minisuperspae moels [16, 19, 21, 22, 23, 24], obtaine by the imposition of homogeneity ofthespaelike hypersurfaes. The lassial limit, the singularity problem, the osmologial onstant problem, an the time issue have been isusse. For instane, in some of these papers it was shown that in moels involving salar fiels or raiation, whih are nie representatives of the matter ontent of the early universe, the singularity anbe learly avoie by quantum effets. In the Bohm-e Broglie interpretation esription, the quantum potential beomes important near the singularity, yieling a repulsive quantum fore ounterating the gravitational fiel, avoiing the singularity an yieling inflation. The lassial limit (given by the limit where the quantum potential beomes negligible with respet to the lassial energy) for large sale fators are usually attainable, but for some salar fiel moels it epens on the quantum state an initial onitions. In fat it is possible to have small lassial universes an large quantum ones [24]. About the time issue, it was shown that for any hoie of the lapse funtion the quantum evolution of the homogeneous hypersurfaes yiel the same four-geometry [19]. What remaine to be stuie is if this fat remains vali in the full theory, where we are not restrite to homogeneous spaelike hypersurfaes. The question is, given an initial hypersurfae with onsistent initial onitions, oes the evolution of the initial three-geometry riven by the quantum bohmian ynamis yiels the same four-geometry for anyhoie of the lapse an shift funtions, an if it oes, what kin of spaetime struture is forme? We know that this is true if the three-geometry is evolve by the ynamis of lassial General Relativity (GR), yieling a non egenerate four geometry, but it an be false if the evolving ynamis is the quantum bohmian one. The iea was to put the quantum bohmian ynamis in hamiltonian form, an then use strong results presente in the literature exhibiting the most general form that a hamiltonian shoul have in orer to form a non egenerate four-geometry from the evolution of threegeometries [25]. Our onlusion is that, in general, the quantum bohmian evolution of the three-geometries oes not yiel any non egenerate four-geometry at all [20]. Only for very speial quantum states a relevant quantum non egenerate four-geometry an be

3 332 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 obtaine, an it must be euliean. In the general ase, the quantum bohmian evolution is onsistent (still inepenent on the hoie of the lapse an shift funtions) but yieling a egenerate four-geometry, where speial vetor fiels, the null eigenvetors of the four geometry, are present 2. We arrive at these onlusions without assuming any regularization an fator orering of the Wheeler-DeWitt equation. As weknow, the Wheeler- DeWitt equation involves the appliation of the prout of loal operators on states at the same spae point, whih is ill efine [27]. Hene we nee to regularize it in orer to solve the fator orering problem, an have a theory free of anomalies (for some proposals, see Refs [28, 29,30]). Our onlusions are ompletely inepenent on these issues. This paper is organize as follows: in the next setion we review the Bohm-e Broglie interpretation of quantum mehanis for non-relativisti partiles an quantum fiel theory in flat spaetime. In setion III we apply the Bohm-e Broglie interpretation to anonial quantum gravity in the minisuperspae ase. We show that there is no problem of time in this interpretation, an that quantum effets an avoi the initial singularity, reate inflation, an isotropize the Universe. In setion IV we treat the general ase. We prove that the bohmian evolution of the 3-geometries, irrespetive of any regularization an fator orering of the Wheeler-DeWitt equation, an be obtaine from a speifi hamiltonian, whih is of ourse ifferent from the lassial one. We then use this hamiltonian to obtain a piture of these new quantum strutures. We en with onlusions an many perspetives for future work. II The Bohm-e Broglie Interpretation In this setion we will review the Bohm-e Broglie interpretation of quantum mehanis. We will first show how this interpretation works in the ase of a single partile esribe by a Shröinger equation, an then we will obtain, by analogy, the ausal interpretation of quantum fiel theory in flat spaetime. Let us begin with the Bohm-e Broglie interpretation of the Shröinger equation esribing a single partile. In the oorinate representation, for a nonrelativisti partile with Hamiltonian H = p 2 =2m + V (x); the Shröinger equation t) iμh =» 2m r2 + V (x) Ψ(x; t): (1) We an transform this ifferential equation over a omplex fiel into a pair of ouple ifferential equations over real fiels, by writing Ψ = A exp(is=μh), where A an S are real funtions, an substituting it into (1). We obtain the + (rs)2 2m + V μh2 r 2 A =0; (2) 2m + r A 2 rs m =0: (3) The usual probabilisti interpretation, i.e. the Copenhagen interpretation, unerstans equation (3) as a ontinuity equation for the probability ensity A 2 for fining the partile at position x an time t. All physial information about the system is ontaine in A 2, an the total phase S of the wave funtion is ompletely irrelevant. In this interpretation, nothing is sai about S an its evolution equation (2). Now suppose that the term μh2 r 2 A 2m A is not present in Eqs. (2) an (3). Then we oul interpret them as equations for an ensemble of lassial partiles uner the influene of a lassial potential V through the Hamilton-Jaobi equation (2), whose probability ensity istribution in spae A 2 (x; t) satisfies the ontinuity equation (3), where rs(x; t)=m is the veloity fiel v(x; t) oftheensemble of partiles. When the term μh2 r 2 A 2m A is present, we an still unerstan Eq. (2) as a Hamilton-Jaobi equation for an ensemble of partiles. However, their trajetories are no more the lassial ones, uetothepresene of the quantum potential term in Eq. (2). The Bohm-e Broglie interpretation of quantum mehanis is base on the two equations (2) an (3) in the way outline above, not only on the last one as it is the Copenhagen interpretation. The starting iea is that the position x an momentum p are always well efine, with the partile's path being guie by anew fiel, the quantum fiel. The fiel Ψ obeys Shröinger equation (1), whih an be written as the two real equations (2) an (3). Equation (2) is interprete as a Hamilton-Jaobi type equation for the quantum partile subjete to an external potential, whih is the lassial potential plus the new quantum potential Q μh2 r 2 A 2m A : (4) One the fiel Ψ, whose effet on the partile trajetory is through the quantum potential (4), is obtaine from Shröinger equation, we an also obtain the partile trajetory, x(t); by integrating the ifferential equation p = m _x = rs(x; t), whih is alle the guiane relation (a ot means time erivative). Of ourse, from this ifferential equation, the non-lassial trajetory x(t) an only be known if the initial position of the partile is given. However, we o not know the initial 2 For instane, the four geometry of Newtonian spaetime is egenerate [26], an its single null eigenvetor is the normal of the absolute hypersurfaes of simultaneity, the time. As we know, it oes not form a single spaetime struture beause it is broken in absolute spae plus absolute time.

4 Nelson Pinto-Neto 333 position of the partile beause we o not know how to measure it without isturbanes (it is the hien variable of the theory). To agree with quantum mehanial experiments, we have to postulate that, for a statistial ensemble of partiles in the same quantum fiel Ψ, the probability ensity istribution of initial positions x 0 is P (x 0 ;t 0 ) = A 2 (x 0 ;t = t 0 ). Equation (3) guarantees that P (x; t) =A 2 (x; t) for all times. In this way, the statistial preitions of quantum theory in the Bohm-e Broglie interpretation are the same as in the Copenhagen interpretation 3. It is interesting to note that the quantum potential epens only on the form of Ψ, not on its absolute value, as an be seen from equation (4). This fat brings home the non-loal an ontextual harater of the quantum potential 4. This is a neessary feature beause Bell's inequalities together with Aspet's experiments show that, in general, aquantum theory must be either non-loal or non-ontologial. As the Bohme Broglie interpretation is ontologial, than it must be non-loal, as it is. The non-loal an ontextual quantum potential auses the quantum effets. It has no parallel in lassial physis. The funtion A plays a ual role in the Bohm-e Broglie interpretation: it gives the quantum potential an the probability ensity istribution of positions, but this last role is seonary. If in some moel there is no notion of probability, we an still get information from the system using the guiane relations. In this ase, A 2 oes not nee to be normalizable. The Bohme Broglie interpretation is not, in essene, a probabilisti interpretation. It is straightforwar to apply it to a single system. The lassial limit an be obtaine in a very simple way. We only have to fin the onitions for having Q = 0 5. The question on why in a real measurement we see an effetive ollapse of the wave funtion is answere by noting that, in a measurement, the wave funtion splits in a superposition of non-overlapping branhes. Hene the point partile (representing the partile being measure plus the marosopi apparatus) will enter into one partiular branh, whih one epens on the initial onitions, an it will be influene by the quantum potential relate only to this branh, whih is the only one that is not negligible in the region where the point partile atually is. The other empty branhes ontinue to exist, but they neither influene on the point partile nor on any other partile [15]. There is an effetive but not real ollapse. The Shröinger equation is always vali. There is no nee to have a lassial omain outsie the quantum system to explain a measurement, neither is the existene of observers ruial beause this interpretation is objetive. For quantum fiels in flat spaetime, we an apply a similar reasoning. As an example, take the Shröinger funtional equation for a quantum salar fiel: t) iμh ρ 1 3 x 2 Writing again the wave funtional as Ψ = A exp(is=μh), we + ρ 1 3 x ff μh 2 ffi 2 2 +(rffi)2 +U(ffi) Ψ(ffi; t): (5)» ff ffis 2+(rffi) 2 +U(ffi)+Q(ffi) =0; (6) Z + 3 x ffi A 2 ffis =0; (7) where Q(ffi) = μh 2 1 ffi 2 A 2A is the orresponing (unregulate) quantum potential. The first equation is 2 viewe as a moifie Hamilton-Jaobi equation governing the evolution of some initial fiel onfiguration through time, whih will be ifferent from the lassial one ue to the presene of the quantum potential. The guiane relation is now given by Π ffi = _ ffi = ffis : (8) 3 It has been shown that uner typial haoti situations, an only within the Bohm-e Broglie interpretation, a probability istribution P 6= A 2 woul rapily approah the value P = A 2 [32, 33]. In this ase, the probability postulate woul be unneessary, an we oul have situations, in very short time intervals, where this moifie Bohm-e Broglie interpretation woul iffer from the Copenhagen interpretation. 4 The non-loality ofqbeomes evient whenwe generalize the ausal interpretation to a many partiles system. 5 It shoul be very interesting to investigate the onnetion between this bohmian lassial limit an the phenomenon of eoherene. To our knowlege, no work has ever been one on this issue, whih may illuminate both the Bohm-e Broglie interpretation an the omprehension of eoherene.

5 334 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 The seon equation is the ontinuity equation for the probability ensity A 2 [ffi(x);t 0 ]ofhaving the initial fiel onfiguration at time t 0 given by ffi(x). A etaile analysis of the Bohm-e Broglie interpretation of quantum fiel theory is given in Ref. [34] for the ase of quantum eletroynamis. III The Bohm-e Broglie Interpretation of Minisuperspae Canonial Quantum Cosmology In this setion, we summarize the rules of the Bohme Broglie interpretation of quantum osmology in the ase of homogeneous minisuperspae moels. When we are restrite to homogeneous moels, the supermomentum onstraint of GR is ientially zero, an the shift funtion an be set to zero without loosing any of the Einstein's equations. The hamiltonian is reue to general minisuperspae form: H GR = N(t)H(p ff (t);q ff (t)); (9) where p ff (t) an q ff (t) represent the homogeneous egrees of freeom oming from Π ij (x; t) an h ij (x; t). The minisuperspae Wheeler-De Witt equation is: H(^p ff (t); ^q ff (t))ψ(q) =0: (10) Writing Ψ = R exp(is=μh), an substituting it into (10), we obtain the following equation: where 1 2 f fffi(q @q fi + U(q μ )+Q(q μ )=0; (11) Q(q μ )= 1 R 2 R fffi ; fi an f fffi (q μ )anu(q μ ) are the minisuperspae partiularizations of the DeWitt metri G ijkl [36] an of the salar urvature ensity h 1=2 R (3) (h ij ) of the spaelike hypersurfaes, respetively. The ausal interpretation applie to quantum osmology states that the trajetories q ff (t) are real, inepenently of any observations. Eq. (11) is the Hamilton-Jaobi equation for them, whih is the lassial one amene with a quantum potential term (12), responsible for the quantum effets. This suggests to efine: p ff ; (13) where the momenta are relate to the veloities in the usual way: p ff = f fffi fi : (14) To obtain the quantum trajetories we have to solve the following system of first orer ifferential ff ff = f fffi 1 : (15) Eqs. (15) are invariant uner time reparametrization. Hene, even at the quantum level, ifferent hoies of N(t) yiel the same spaetime geometry for a given non-lassial solution q ff (t). There is no problem of time in the ausal interpretation of minisuperspae quantum osmology. Let us now apply these rules, as examples, to minisuperspae moels with a free massless salar fiel. Take the lagrangian: L = p ;ρ ge ffi R wffi ;ρ ffi : (16) For w = 1 wehave effetive string theory without the Kalb-Rammon fiel. For w = 3=2 wehave a onformally ouple salar fiel. Performing the onformal transformation g μν = e ffi μg μν we obtain the following lagrangian: L = p g»r (! + 32 )ffi ;ρffi ;ρ ; (17) where the bars have been omitte. C w (! + 3). 2 III.1. The isotropi ase We will efine We onsier now the Robertson-Walker metri s 2 = N 2 t 2 + a(t)2 1+ ffl [r2 + r 2 ( 2 + sin 2 ( )' 2 )]; 4 r2 (18) where the spatial urvature ffl takes the values 0, 1, 1. Inserting this in the lagrangian (17), an using the units where μh = =1,we obtain the following ation: S = 3V Z Na 3 _a 2 4ßlp 2 2 N 2 a + C _ffi 2 2 w 6N + ffl t ; (19) 2 a 2 where V is the total volume ivie by a 3 of the spaelike hypersurfaes, whih are suppose to be lose, an l p is the Plank length. V epens on the value of ffl an on the topology of the hypersurfaes. For ffl = 0 it an be as large as we want beause their funamental polyhera an have arbitrary size. In the ase of ffl = 1 an topology S 3, V = 2ß 2. Defining fi 2 = 4ßl2 p 3V, the hamiltonian turns out to be: where H = N fi 2 p2 p 2 a ffi 2a +3fi2 C w a ffl a 3 2fi 2 : (20)

6 Nelson Pinto-Neto 335 p a = a_a fi 2 N ; (21) p ffi = C w a 3 _ ffi 6fi 2 N : (22) Usually the sale fator has imensions of length beause we use angular oorinates in lose spaes. Hene we will efine a imensionless sale fator ~a a=fi. In that ase the hamiltonian beomes, omitting the tile: H = N ffla p2 a 2fi a +6 p2 ffi C w a : (23) 3 As fi appears as an overall multipliative onstant in the hamiltonian, we an set it equal to one without any loss of generality, keeping in min that the sale fator whih appears in the metri is fia, not a. We an further simplify the hamiltonian by efining ff ln(a) obtaining H = where» N p 2 ff + 6 p 2 ffi 2 exp(3ff) C ffl exp(4ff) w p ff = e3ff _ff N : (24) ; (25) e 3ff ffi p ffi = C _ w 6N : (26) The momentum p ffi is a onstant of motion whih we will all k. μ We will restrit ourselves to the physially interesting ase, ue to observations, of ffl = 0 an C w > 0. The lassial solutions in the gauge N =1are, r 6 ffi = ± ff + 1 ; (27) C w where 1 is an integration onstant. In term of osmi time they are: r 6 a = e ff =3 kt μ 1=3 ; (28) C r w 2 ffi = ln(t)+ 2 : (29) 3C w The solutions ontrat or expan forever from a singularity, epening on the sign of k, μ without any inflationary epoh. Let us now quantize the moel. With a partiular hoie of fator orering, we obtain the following Wheeler-DeWitt equation Ψ ffff 6 C w Ψ ffle 4ff Ψ=0 : (30) Employing the separation of variables metho, we obtain the general solution Z Ψ(ff; ffi) = F (k)a k (ff)b k (ffi)k ; (31) where k is a separation onstant, an an B k (ffi) =b 1 exp(i r Cw 6 kffi)+b 2 exp( i r Cw 6 kffi) ; (32) A k (ff) =a 1 exp(ikff)+a 2 exp( ikff) ; (33) We will now make gaussian superpositions of these solutions an interpret the results using the ausal interpretation of quantum mehanis. The funtion F (k) is:» (k )2 F (k) = exp ff 2 We take the wave funtion: : (34) Z Ψ(ff; ffi) = F (k)[a k (ff)b k (ffi)+a k (ff)b k (ffi)k] ; (35) with a 2 = b 2 =0. Performing the integration in k we obtain for Ψ (we will efine ffi μ q Cw ffi an omit the bars from now on) 6 Ψ=ff p ρ ß exp» (ff + ffi)2 ff 2 4 exp[i(ff + ffi)] + exp» (ff ffi)2 ff 2 4 ff exp[ i(ff ffi)] : (36)

7 336 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 In orer to obtain the Bohmian trajetories, we haveto alulate the phase S of the above wave funtion an substitute it into the guiane formula where p ff = S ff ; (37) p ffi = S ffi ; (38) p ff = e3ff _ff N p ffi = e3ff ffi _ N ; (39) : (40) We will work in the gauge N = 1. These equations onstitute a planar system whih an be easily stuie: _ff = _ffi =» ffiff 2 sin(2ff)+2sinh(ff 2 ffffi) ρ ff ; (41) exp(3ff) 2[os(2ff) + osh(ff 2 ffffi)]» ffff 2 sin(2ff) 2 os(2ff) 2 osh(ff 2 ffffi) ρ ff : exp(3ff) 2[os(2ff) + osh(ff 2 ffffi)] (42) The line ff = 0 ivies onfiguration spae in two symmetri regions. The line ffi = 0 ontains all singular points of this system, whih are noes an enters. The noes appear when the enominator of the above equations, whih is proportional to the norm of the wave funtion, is zero. No trajetory an pass through these points. They happen when ffi = 0 an os(ff) =0,or ff =(2n +1)ß=2, n an integer, with separation ß=. The enter points appear when the numerators are zero. They are given by ffi = 0 an ff = 2[otan(ff)]=ff 2. They are interalate with the noe points. As j ff j! 1 these points ten to nß=, an their separations annot exee ß=. As one an see from the above system, the lassial solutions (a(t) / t 1=3 ) are reovere when j ff j! 1 or j ffi j! 1, the other being ifferent from zero. There are plenty of ifferent possibilities of evolution, epening on the initial onitions. Near the enter points we an have osillating universes without singularities an with amplitue of osillation of orer 1. For negative values of ff, the universe arise lassially from a singularity but quantum effets beome important foring it to reollapse to another singularity, reovering lassial behaviour near it. For positive values of ff, the universe ontrats lassially but when ff is small enough quantum effets beome important reating an inflationary phase whih avois the singularity. The universe ontrats to a minimum size an after reahing this point it expans forever, reovering the lassial limit when ff beomes suffiiently large. We an see that for ff negative we have lassial limit for small sale fator while for ff positive wehave lassial limit for big sale fator. III.2. The anisotropi ase To exemplify the quantum isotropization of the Universe, let us take now, instea of the Frieman- Robertson-Walker of Eq. (18), the homogeneous an anisotropi Bianhi I line element s 2 = N 2 (t)t 2 + exp[2fi 0 (t)+2fi + (t)+2 p 3fi (t)] x 2 + exp[2fi 0 (t)+2fi + (t) 2 p 3fi (t)] y 2 + exp[2fi 0 (t) 4fi + (t)] z 2 : (43) This line element will be isotropi if an only if fi + (t) an fi (t) are onstants. Inserting Eq. (43) into the lagrangian (17), supposing that the salar fiel ffi epens only on time, isaring surfae terms, an performing a Legenre transformation, we obtain the following minisuperspae lassial hamiltonian H = N 24 exp (3fi 0 ) ( p2 0 + p p 2 + p 2 ffi); (44) where (p 0 ;p + ;p ;p ffi ) are anonially onjugate to (fi 0 ;fi + ;fi ;ffi), respetively, an we mae the trivial reefinition ffi! p C w =6 ffi. We an write this hamiltonian in a ompat form by efining y μ = (fi 0 ;fi + ;fi ;ffi) an their anonial momenta p μ =(p 0 ;p + ;p ;p ffi ), obtaining H = N 24 exp (3y 0 ) μν p μ p ν ; (45) where μν is the Minkowski metri with signature ( + ++). The equations of motion are the onstraint equation obtaine by varying the hamiltonian with re-

8 Nelson Pinto-Neto 337 spet to the lapse funtion N an the Hamilton's equations H μν p μ p ν =0; (46) _y μ N μ 12 exp (3y 0 ) μν p ν ; (47) _p μ =0: The solution to these equations in the gauge N = 12 exp(3y 0 )is y μ = μν p ν t + C μ ; (49) where the momenta p ν are onstants ue to the equations of motion an the C μ are integration onstants. We an see that the only way to obtain isotropy in these solutions is by making p 1 = p + = 0 an p 2 = p = 0, whih yiel solutions that are always isotropi, the usual Frieman-Robertson-Walker (FRW) solutions with a salar fiel. Hene, there is no anisotropi solution in this moel whih an lassially beome isotropi uring the ourse of its evolution. One anisotropi, always anisotropi. If we suppress the ffi egree of freeom, the unique isotropi solution is flat spaetime beause in this ase the onstraint (46)enfores p 0 =0. To isuss the appearane of singularities, we nee the Weyl square tensor W 2 W fffiμν W fffiμν. It reas W 2 = e 12fi0 (2p 0 p 3 + 6p 0p 2 p + + p 4 +2p 2 + p2 + p p 2 0 p2 + + p 2 0 p2 ): (50) Hene, the Weyl square tensor is proportional to exp ( 12fi 0 ) beause the p's are onstants (see Eq. (48)) an the singularity is at t = 1. The lassial singularity an be avoie only if we set p 0 = 0. But then, ue to equation (46), we woul also have p i = 0, whih orrespons to the trivial ase of flat spaetime. Hene, the unique lassial solution whih is non-singular is the trivial flat spaetime solution. The Dira quantization proeure yiels the Wheeler-DeWitt equation, whih in the present ase reas μ y ν Ψ(yμ )=0 : (51) Let us now investigate spherial-wave solutions of Eq. (51). They rea Ψ 3 = 1 y q P3» f(y 0 + y)+g(y 0 y) ; (52) where y i=1 (yi ) 2. The guiane relations in the gauge N = 12 exp(3y 0 ) are (see Eqs. (47)) Ψ 3 p 0 0 S =Im = _y 0 ; Ψ 3 Ψ 3 p i i S =Im = _y i ; Ψ 3 (54) where S is the phase of the wave funtion. In terms of f an g the above equations rea f 0 (y 0 + y)+g 0 (y 0 y) _y 0 = Im ; (55) f(y 0 + y)+g(y 0 y) _y i = yi f 0 y Im (y 0 + y) g 0 (y 0 y) ; (56) f(y 0 + y)+g(y 0 y) where the prime means erivative with respet to the argument of the funtions f an g, an Im(z) is the imaginary part of the omplex number z. From Eq. (56) we obtain that y i y j = yi y j ; (57) whih implies that y i (t) = i j yj (t), with no sum in j, where the i j are real onstants an 1 1 = 2 2 = 3 3 = 1. Hene, apart some positive multipliative onstant, knowing about one of the y i means knowing about all y i. Consequently,we an reue the four equations (55) an (56) to a planar system by writing y = Cjy 3 j, with C > 1, an working only with y 0 an y 3, say. The planar system now reas ; (58) f 0 (y 0 + Cjy 3 j)+g 0 (y 0 Cjy 3 j) _y 0 = Im f(y 0 + Cjy 3 j)+g(y 0 Cjy 3 j) _y 3 = sign(y3 ) f 0 (y 0 + Cjy 3 j) g 0 (y 0 Cjy 3 j) Im C f(y 0 + Cjy 3 j)+g(y 0 Cjy 3 j) (59) Note that if f = g, y 3 stabilizes at y 3 = 0 beause _y 3 as well as all other time erivatives of y 3 are zero at this line. As y i (t) = i j yj (t), all y i (t) beome zero, an the osmologial moel isotropizes forever one y 3 reahes this line. Of ourse one an fin solutions where y 3 never reahes this line, but in this ase there must be some region where _y 3 = 0, whih implies _y i = 0, an this is an isotropi region. Consequently, quantum anisotropi osmologial moels with f = g always :

9 338 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 have an isotropi phase, whih an beome permanent in many ases. IV The Bohm-e Broglie Interpretation of Superspae Canonial Quantum Cosmology In this setion, we will quantize General Relativity Theory (GR) without making any simplifiations or utting of egrees of freeom. The matter ontent is a minimally ouple salar fiel with arbitrary potential. All subsequent results remain essentially the same for any matter fiel whih ouples uniquely with the metri, not with their erivatives. The lassial hamiltonian of full GR with a salar fiel is given by: Z H = 3 x(nh + N j H j ) (60) where H =»G ijkl Π ij Π kl h 1=2 Π 2 ffi +» +h 1=2» 1 (R (3) 2Λ) i ffi@ j ffi + U(ffi) (61) H j = 2D i Π i j +Π j ffi: (62) In these equations, h ij is the metri of lose 3- imensional spaelike hypersurfaes, an Π ij is its anonial momentum given by Π ij = h 1=2 (K ij h ij K)=G ijkl ( _ h kl D k N l D l N k ); (63) where K ij = 1 2N (_ h ij D i N j D j N i ); (64) is the extrinsi urvature of the hypersurfaes (inies are raise an lowere by the 3-metri h ij an its inverse h ij ). The anonial momentum of the salar fiel is now Π ffi = h1=2 _ffi N i ffi : (65) N The quantity R (3) is the intrinsi urvature of the hypersurfaes an h is the eterminant ofh ij. The lapse funtion N an the shift funtion N j are the Lagrange multipliers of the super-hamiltonian onstraint Hß0 an the super-momentum onstraint H j ß 0, respetively. They are present ue to the invariane of GR uner spaetime oorinate transformations. The quantities G ijkl an its inverse G ijkl (G ijkl G ijab = ffikl ab) are given by G ijkl = 1 2 h1=2 (h ik h jl + h il h jk 2h ij h kl ); (66) G ijkl = 1 2 h 1=2 (h ik h jl + h il h jk h ij h kl ); (67) whih is alle the DeWitt metri. The quantity D i is the i-omponent of the ovariant erivative operator on the hypersurfae, an» =16ßG= 4. The lassial 4-metri s 2 = (N 2 N i N i )t 2 +2N i x i t + h ij x i x j (68) an the salar fiel whih are solutions of the Einstein's equations an be obtaine from the Hamilton's equations of motion _h ij = fh ij ;Hg; (69) _Π ij = fπ ij ;Hg; (70) _ffi = fffi; Hg; (71) _ Π ffi = fπ ffi ;Hg; (72) for some hoie of N an N i, an if we impose initial onitions ompatible with the onstraints Hß0; (73) H i ß 0: (74) It is a feature of the hamiltonian of GR that the 4- metris (68) onstrute in this way, with the same initial onitions, esribe the same four-geometry for any hoie of N an N i. The algebra of the onstraints lose in the following form (we follow the notation of Ref. [25]):

10 Nelson Pinto-Neto 339 fh(x); H(x 0 )g = H i (x)@ i ffi 3 (x; x 0 ) H i (x 0 )@ i ffi 3 (x 0 ;x) fh i (x); H(x 0 )g = H(x)@ i ffi 3 (x; x 0 ) (75) fh i (x); H j (x 0 )g = H i (x)@ j ffi 3 (x; x 0 )+H j (x 0 )@ i ffi 3 (x; x 0 ) To quantize this onstraine system, we follow the Dira quantization proeure. The onstraints beome onitions impose on the possible states of the quantum system, yieling the following quantum equations: ^H i j Ψ> = 0 (76) ^H jψ> = 0 (77) In the metri an fiel representation, the first equation is 2h li D j ffiψ(h ij ;ffi) ffih lj + ffiψ(h i ffi =0; (78) whih implies that the wave funtional Ψ is an invariant uner spae oorinate transformations. The seon equation is the Wheeler-DeWitt equation [35, 36]. Writing it unregulate in the oorinate representation we get ρ μh 2»»G ijkl ffi ffih ij ffi + 1 ffi2 ffih kl 2 h 1=2 2 ff +V Ψ(h ij ;ffi)=0; (79) where V is the lassial potential given by» V = h 1=2» 1 (R (3) 2Λ) i ffi@ j ffi + U(ffi) : (80) This equation involves prouts of loal operators at the same spae point, hene it must be regularize. After oing this, one shoul fin a fator orering whih makes the theory free of anomalies, in the sense that the ommutator of the operator version of the onstraints lose in the same way as their respetive lassial Poisson brakets (75). Hene, Eq. (79) is only a formal one whih must be worke out [28, 29,30]. Let us now see what is the Bohm-e Broglie interpretation of the solutions of Eqs. (76) an (77) in the metri an fiel representation. First we write the wave funtional in polar form Ψ = A exp(is=μh), where A an S are funtionals of h ij an ffi. Substituting it in Eq. (78), we gettwo equations saying that A an S are invariant uner general spae oorinate transformations: 2h li D j ffis(h ij ;ffi) ffih lj 2h li D j ffia(h ij ;ffi) ffih lj + ffis(h i ffi =0; (81) + ffia(h i ffi =0: (82) The two equations we obtain for A an S when we substitute Ψ = A exp(is=μh) into Eq. (77) will of ourse epen on the fator orering we hoose. However, in any ase, one of the equations will have the form ffis ffis»g ijkl ffis ffih ij ffih kl 2 h 1=2 +V + Q =0; (83) where V is the lassial potential given in Eq. (80). Contrary to the other terms in Eq. (83), whih are alreay well efine, the preise form of Q epens on the regularization an fator orering whih are presribe for the Wheeler-DeWitt equation. In the unregulate form given in Eq. (79), Q is Q = μh 2 1 A ffi 2 A»G ijkl + h 1=2 ffi 2 A : (84) ffih ij ffih kl 2 2 Also, the other equation besies (83) in this ase is ffi»g ijkl A ffis ffi h 1=2 A ffis 2 =0: (85) ffih ij ffih kl 2 Let us now implement the Bohm-e Broglie interpretation for anonial quantum gravity. First of all we note that Eqs. (81) an (83), whih are always vali irrespetive of any fator orering of the Wheeler-DeWitt equation, are like the Hamilton-Jaobi equations for GR, supplemente by an extra term Q in the ase of Eq. (83), whih we will all the quantum potential. By analogy with the ases of non-relativisti partile an

11 340 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 quantum fiel theory in flat spaetime, we will postulate that the 3-metri of spaelike hypersurfaes, the salar fiel, an their anonial momenta always exist, inepenent onany observation, an that the evolution of the 3-metri an salar fiel an be obtaine from the guiane relations Π ij = ffis(h ab;ffi) ffih ij ; (86) Π ffi = ffis(h ij;ffi) ; (87) with Π ij an Π ffi given by Eqs. (63) an (65), respetively. Like before, these are first orer ifferential equations whih anbeintegrate to yiel the 3-metri an salar fiel for all values of the t parameter. These solutions epen on the initial values of the 3-metri an salar fiel at some initial hypersurfae. The evolution of these fiels will of ourse be ifferent from the lassial one ue to the presene of the quantum potential term Q in Eq. (83). The lassial limit is one more oneptually very simple: it is given by the limit where the quantum potential Q beomes negligible with respet to the lassial energy. The only ifferene from the previous ases of the non-relativisti partile an quantum fiel theory in flat spaetime is the fat that the equivalent of Eqs. (3) an (7) for anonial quantum gravity, whih in the naive orering is Eq. (85), annot be interprete as a ontinuity equation for a probability ensity A 2 beause of the hyperboli nature of the DeWitt metri G ijkl. However, even without a notion of probability, whih in this ase woul mean the probability ensity istribution for initial values of the 3-metri an salar fiel in an initial hypersurfae, we an extrat a lot of information from Eq. (83) whatever is the quantum potential Q, as will see now. After we get these results, we will return to this probability issue in the last setion. First we note that, whatever is the form of the quantum potential Q, itmust be a salar ensity ofweight one. This omes from the Hamilton-Jaobi equation (83). From this equation we an express Q as ffis ffis Q =»G ijkl 1 2 ffis ffih ij ffih kl 2 h 1=2 V: (88) As S is an invariant (see Eq. (81)), then ffis=ffih ij an ffis= must be a seon rank tensor ensity an a salar ensity, both of weight one, respetively. When their prouts are ontrate with G ijkl an multiplie by h 1=2, respetively, they form a salar ensityofweight one. As V is also a salar ensity ofweight one, then Q must also be. Furthermore, Q must epen only on h ij an ffi beause it omes from the wave funtional whih epens only on these variables. Of ourse it an be non-loal (we show an example in the appenix), i.e., epening on integrals of the fiels over the whole spae, but it annot epen on the momenta. Now we will investigate the following important problem. From the guiane relations (86) an (87), whih will be written in the form an Φ ij Π ij ffis(h ab;ffi) ffih ij ß 0; (89) Φ ffi Π ffi ffis(h ij;ffi) ß 0: (90) we obtain the following first orer partial ifferential equations: an _h ij =2NG ijkl ffis ffih kl + D i N j + D j N i (91) 1=2 ffis _ffi = Nh + N i ffi: (92) The question is, given some initial 3-metri an salar fiel, what kin of struture o we obtain when we integrate this equations in the parameter t? Does this struture form a 4-imensional geometry with a salar fiel for any hoie of the lapse an shift funtions? Note that if the funtional S were a solution of the lassial Hamilton-Jaobi equation, whih oes not ontain the quantum potential term, then the answer woul be in the affirmative beause we woul be in the sope of GR. But S is asolution of the moifie Hamilton-Jaobi equation (83), an we annot guarantee that this will ontinue to be true. We may obtain a omplete ifferent struture ue to the quantum effets riven by the quantum potential term in Eq. (83). To answer this question we will move from this Hamilton-Jaobi piture of quantum geometroynamis to a hamiltonian piture. This is beause many strong results onerning geometroynamis were obtaine in this later piture [25, 37]. We will onstrut a hamiltonian formalism whih is onsistent with the guiane relations (86) an (87). It yiels the bohmian trajetories (91) an (92) if the guiane relations are satisfie initially. One we have this hamiltonian, we anusewell known results in the literature to obtain strong results about the Bohme Broglie view of quantum geometroynamis. Examining Eqs. (81) an (83), we an easily show [20] that the hamiltonian whih generates the bohmian trajetories, one the guiane relations (86) an (87) are satisfie initially, isgiven by: Z H Q = 3 x»n(h + Q)+N i H i (93) where we efine H Q H+ Q: (94) The quantities H an H i are the usual GR superhamiltonian an super-momentum onstraints given by Eqs. (61) an (62). In fat, the guiane relations (86) an (87) are onsistent with the onstraints H Q ß 0

12 Nelson Pinto-Neto 341 an H i ß 0beauseS satisfies (81) an (83). Furthermore, they are onserve by the hamiltonian evolution given by (93) [20]. We now have a hamiltonian, H Q, whih generates the bohmian trajetories one the guiane relations (86) an (87) are impose initially. In the following, we an investigate if the the evolution of the fiels riven by H Q forms a four-geometry like in lassial geometroynamis. First we reall a result obtaine by Clauio Teitelboim [37]. In this paper, he shows that if the 3-geometries an fiel onfigurations efine on hypersurfaes are evolve by some hamiltonian with the form Z μh = 3 x(n H μ + N i Hi μ ); (95) an if this evolution an be viewe as the motion" of a 3-imensional ut in a 4-imensional spaetime (the 3- geometries an be embee in a four-geometry), then the onstraints H μ ß 0 an Hi μ ß 0 must satisfy the following algebra f μ H(x); μ H(x 0 )g = ffl[ μ H i (x)@ i ffi 3 (x 0 ;x)] μ H i (x 0 )@ i ffi 3 (x; x 0 ) (96) f μ Hi (x); μ H(x 0 )g = μ H(x)@i ffi 3 (x; x 0 ) (97) f μ Hi (x); μ Hj (x 0 )g = μ Hi (x)@ j ffi 3 (x; x 0 ) μ Hj (x 0 )@ i ffi 3 (x; x 0 ) (98) The onstant ffl in (96) an be ±1 epening if the fourgeometry in whih the 3-geometries are embee is euliean (ffl = 1) or hyperboli (ffl = 1). These are the onitions for the existene of spaetime. The above algebra is the same as the algebra (75) of GR if we hoose ffl = 1. But the hamiltonian (93) is ifferent from the hamiltonian of GR only by the presene of the quantum potential term Q in H Q. The Poisson braket fh i (x); H j (x 0 )g satisfies Eq. (98) beause the H i of H Q efine in Eq. (93) is the same as in GR. Also fh i (x); H Q (x 0 )g satisfies Eq. (97) beause H i is the generator of spatial oorinate transformations, an as H Q is a salar ensity ofweight one (remember that Q must be a salar ensity of weight one), then it must satisfies this Poisson braket relation with H i. What remains to be verifie is if the Poisson braket fh Q (x); H Q (x 0 )g loses as in Eq. (96). We now reall the result of Ref. [25]. There it is shown that a general super-hamiltonian μ H whih satisfies Eq. (96), is a salar ensity of weight one, whose geometrial egrees of freeom are given only by the three-metri h ij an its anonial momentum, an ontains only even powers an no non-loal term in the momenta (together with the other requirements, these last two onitions are also satisfie by H Q beause it is quarati in the momenta an the quantum potential oes not ontain any non-loal term on the momenta), then μ H must have the following form: where μh =»G ijkl Π ij Π kl h 1=2 ß 2 ffi + V G ; (99) V G fflh 1=2»» 1 (R (3) 2μΛ)+ 1 2 i ffi@ j ffi + μ U(ffi) : (100) With this result we an now establish two possible senarios for the Bohm-e Broglie quantum geometroynamis, epening on the form of the quantum potential: IV.1. Quantum geometroynamis evolution is onsistent an forms a non egenerate four-geometry In this ase, the Poisson braket fh Q (x); H Q (x 0 )g must satisfy Eq. (96). Then Q must be suh that V + Q = V G with V given by (80) yieling: Q = h»(ffl 1=2 +1)» 1 R (3) i ffi@ j ffi + 2» (ffl μλ+λ)+ffl U(ffi)+U(ffi) μ : (101)

13 342 Brazilian Journal of Physis, vol. 30, no. 2, June, 2000 Then we havetwo possibilities: IV.1.1. The spaetime is hyperboli (ffl = 1) In this ase Q is» 2 Q = h 1=2» ( μλ+λ) U(ffi)+U(ffi) μ : (102) Hene Q is like a lassial potential. Its effet is to renormalize the osmologial onstant an the lassial salar fiel potential, nothing more. The quantum geometroynamis is inistinguishable from the lassial one. It is not neessary to require the lassial limit Q = 0 beause V G = V + Q alreay may esribe the lassial universe we live in. IV.1.2. The spaetime is euliean (ffl =1) In this ase Q is Q = h»2 1=2» 1 R (3) i ffi@ j ffi + 2» ( μλ+λ)+ U(ffi)+U(ffi) μ : (103) Now Q not only renormalize the osmologial onstant an the lassial salar fiel potential but also hange the signature of spaetime. The total potential V G = V + Q may esribe some era of the early universe when it ha euliean signature, but not the present era, when it is hyperboli. The transition between these two phasesmusthappeninahypersurfae where Q = 0, whih is the lassial limit. We an onlue from these onsierations that if a quantum spaetime exists with ifferent features from the lassial observe one, then it must be euliean. In other wors, the sole relevant quantum effet whih maintains the non-egenerate nature of the four-geometry of spaetime is its hange of signature to a euliean one. The other quantum effets are either irrelevant or break ompletely the spaetime struture. This result points in the iretion of Ref. [38]. IV.2. Quantum geometroynamis evolution is onsistent but oes not form a non egenerate four-geometry In this ase, the Poisson braket fh Q (x); H Q (x 0 )g oes not satisfy Eq. (96) but is weakly zero in some other way. Some examples are given in Ref. [40]. They are real solutions of the Wheeler-DeWitt equation, where Q = V, an non-loal quantum potentials. It is very important to use the guiane relations to lose the algebra in these ases. It means that the hamiltonian evolution with the quantum potential is onsistent only when restrite to the bohmian trajetories. For other trajetories, it is inonsistent. Conluing, when restrite to the bohmian trajetories, an algebra whih oes not lose in general may lose, as shown in the above example. This is an important remark on the Bohm-e Broglie interpretation of anonial quantum osmology, whih sometimes is not notie. In the examples above, we have expliitly obtaine the "struture onstants" of the algebra that haraterizes the pre-four-geometry" generate by H Q i.e., the foam-like struture pointe long time ago in early works of J. A. Wheeler [35, 42]. V Conlusion an Disussions The Bohm-e Broglie interpretation of anonial quantum osmology yiels a quantum geometroynamial piture where the bohmian quantum evolution of threegeometries may form, epening on the wave funtional, a onsistent non egenerate four geometry whih must be euliean (but only for a very speial loal form of the quantum potential), an a onsistent but egenerate four-geometry iniating the presene of speial vetor fiels an the breaking of the spaetime struture as a single entity (in a wier lass of possibilities). Hene, in general, an always when the quantum potential is non-loal, spaetime is broken. The threegeometries evolve uner the influene of a quantum potential o not in general stik together to form a non egenerate four-geometry, a single spaetime with the ausal struture of relativity. This is not surprising, as it was antiipate long ago [42]. Among the onsistent bohmian evolutions, the more general strutures that are forme are egenerate four-geometries with alternative ausal strutures. We obtaine these results taking a minimally ouple salar fiel as the matter soure of gravitation, but it an be generalize to any matter soure with non-erivative ouplings with the metri, like Yang-Mills fiels. As shown in the previous setion, a non egenerate four-geometry an be attaine only if the quantum potential have the speifi form (101). In this ase, the sole relevant quantum effet will be a hange of signature of spaetime, something pointing towars Hawking's ieas. In the ase of onsistent quantum geometroynam-

14 Nelson Pinto-Neto 343 ial evolution but with egenerate four-geometry, we have shown that any real solution of the Wheeler- DeWitt equation yiels a struture whih is the iealization of the strong gravity limit of GR. This type of geometry, whih is egenerate, has alreay been stuie [41]. Due to the generality of this piture (it is vali for any real solution of the Wheeler-DeWitt equation, whih is a real equation), it eserves further attention. It may well be that these egenerate four-metris were the orret quantum geometroynamial esription of the young universe. It woulbealsointeresting to investigate if these strutures have a lassial limit yieling the usual four-geometry of lassial osmology. For non-loal quantum potentials, we have shown that apparently inonsistent quantum evolutions are in fat onsistent if restrite to the bohmian trajetories satisfying the guiane relations (86) an (87). This is apoint whih is sometimes not taken into aount. If we want to be strit an impose that quantum geometroynamis oes not break spaetime, then we will have stringent bounary onitions. As sai above, a non egenerate four-geometry an be obtaine only if the quantum potential have the form (101). This is a severe restrition on the solutions of the Wheeler-DeWitt equation. These restritions on the form of the quantum potential o not our in minisuperspae moels [19] beause there the hypersurfaes are restrite to be homogeneous. The only freeom we have is in the time parametrization of the homogeneous hypersurfaes whih foliate spaetime. There is a single onstraint, whih of ourse always ommute with itself, irrespetive of the quantum potential. The theorem proven in Ref. [25], whih was essential in all the reasoning of the last setion, annot be use here beause minisuperspae moels o not satisfy one of their hypotheses. In setion 3 we stuie quantum effets in suh minisuperspae moels an we showe that they an avoi singularities, isotropize the Universe, an reate inflationary epohs. It shoul be very interesting to investigate if these quantum phases of the Universe may have left some traes whih oul be etete now, as in the anisotropies of the osmi mirowave bakgroun raiation. As we have seen, in the Bohm-e Broglie approah we an investigate further what kin of struture is forme in quantum geometroynamis by using the Poisson braket relation (96), an the guiane relations (91) an (92). By assuming the existene of 3- geometries, fiel onfigurations, an their momenta, inepenently on any observations, the Bohm-e Broglie interpretation allows us to use lassial tools, like the hamiltonian formalism, to unerstan the struture of quantum geometry. If this information is useful, we o not know. Alreay in the two-slit experiment innon- relativisti quantum mehanis, the Bohm-e Broglie interpretation allows us to say from whih slit the partile has passe through: if it arrive at the upper half of the sreen it must have ome from the upper slit, an vie-versa. Suh information we o not have in the many-worls interpretation. However, this information is useless: we an neither hek it nor use it in other experiments. In anonial quantum osmology the situation may be the same. The Bohm-e Broglie interpretation yiels a lot of information about quantum geometroynamis whih we annot obtain from the many-worls interpretation, but this information may be useless. However, we annot answer this question preisely if we onotinvestigate further, an the tools are at our isposal. We woul like to remark that all these results were obtaine without assuming any partiular fator orering an regularization of the Wheeler-DeWitt equation. Also, we i not use any probabilisti interpretation of the solutions of the Wheeler-DeWitt equation. Hene, it is a quite general result. However, we woul like to make some omments about the probability issue in quantum osmology. The Wheeler-DeWitt equation when applie to a lose universe oes not yiel a probabilisti interpretation for their solutions beause of its hyperboli nature. However, it has been suggeste many times [21, 43, 44, 45, 46] that at the semilassial level we an onstrut a probability measure with the solutions of the Wheeler-DeWitt equation. Hene, for interpretations where probabilities are essential, the problem of fining a Hilbert spae for the solutions of the Wheeler-DeWitt equation beomes ruial if someone wants to get some information above the semilassial level. Of ourse, probabilities are also useful in the Bohm-e Broglie interpretation. When we integrate the guiane relations (91) an (92), the initial onitions are arbitrary, an it shoul be nie to have some probability istribution on them. However, as we have seen along this paper, we an extrat a lot of information from the full quantum gravity level using the Bohm-e Broglie interpretation, without appealing to any probabilisti notion. It woul also be importanttoinvestigate the Bohme Broglie interpretation for other quantum gravitational systems, like blak holes. Attempts in this iretion have been mae, but within spherial symmetry in empty spae [47], where we have only a finite number of egrees of freeom. It shoul be interesting to investigate more general moels. The onlusions of this paper are of ourse limite by many strong assumptions we have taitly mae, as supposing that a ontinuous three-geometry exists at the quantum level (quantum effets oul also estroy it), or the valiity ofquantization of stanar GR, forgetting other evelopments like string theory. However, even if this approah is not the appropriate one, it is nie to see how far we an go with the Bohm-e Broglie interpretation, even in suh inomplete stage of anonial quantum gravity. It seems that the Bohm-