Nelson Pinto-Neto- Quantum Cosmology: How to Interpret and Obtain Results

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330 Brazilian Journal of Physics, vol. 30, no. 2, June, 2000

Quantum Cosmology: How to Interpretand Obtain Results

Nelson Pinto-NetoCentro Brasileiro de Pesquisas Fisicos,

Rua Dr. Xavier Sigaud 150, Urea22290-180, Rio de Janeiro, RJ, Brazil

E-mails:[email protected] 7 January, 2000

We argue that the Copenhagen interpretation of quantum mechanics cannot be applied to quantumcosmology. Among the alternative interpretations, we choose to apply the Bohm-de Broglie inter-pretation of quantum mechanics to canonical quantum cosmology. For minisuperspace models, weshow that there is no problem of time in this interpretation, and that quantum effects can avoid theinitial singularity, create inflation and isotropize the Universe. For the general case, it is shown that,irrespective of any regularization or choice of factor ordering of the Wheeler-DeWitt equation, theunique relevant quantum effect which does not break spacetime is the change of its signature fromlorentzian to euclidean. The other quantum effects are either trivial or break the four-geometryof spacetime. A Bohm-de Broglie picture of a quantum geometrodynamics is constructed, whichallows the investigation of these latter structures.

I IntroductionAlmost all physicists believe that quantum mechanics isa universal and fundamental theory, applicable to anyphysical system, from which classical physics can be re-covered. The Universe is, of course, a valid physicalsystem: there is a theory, Standard Cosmology, whichis able to describe it in physical terms, and make pre-dictions which can be confirmed or refuted by obser-vations. In fact, the observations until now confirmthe standard cosmological scenario. Hence, suppos-ing the universality of quantum mechanics, the Uni-verse itself must be described by quantum theory, fromwhich we could recover Standard Cosmology. However,the Copenhagen interpretation of quantum mechanics[1 , 2, 3]1, which is the one taught in undergraduatecourses and employed by the majority of physicists inall areas (specially the von Neumann's approach), can-not be used in a Quantum Theory of Cosmology. Thisis because it imposes the existence of a classical domain.In von Neumann's view, for instance, the necessity of aclassical domain comes from the way it solves the mea-surement problem (see Ref. [4] for a good discussion).In an impulsive measurement of some observable, thewave function of the observed system plus macroscopicapparatus splits into many branches which almost donot overlap (in order to be a good measurement), each

one containing the observed system in an eigenstate ofthe measured observable, and the pointer of the appa-ratus pointing to the respective eigenvalue. However,in the end of the measurement, we observe only one ofthese eigenvalues, and the measurement is robust in thesense that if we repeat it immediately after, we obtainthe same result. So it seems that the wave functioncollapses, the other branches disappear. The Copen-hagen interpretation assumes that this collapse is real.However, a real collapse cannot be described by theunitary Schrodinger evolution. Hence, the Copenhageninterpretation must assume that there is a fundamentalprocess in a measurement which must occur outside thequantum world, in a classical domain. Of course, if wewant to quantize the whole Universe, there is no placefor a classical domain outside it, and the Copenhageninterpretation cannot be applied. Hence, if someoneinsists with the Copenhagen interpretation, she or hemust assume that quantum theory is not universal, orat least try to improve it by means of further concepts.One possibility is by invoking the phenomenon of de-coherence [5]. In fact, the interaction of the observedquantum system with its environment yields an effec-tive diagonalization of the reduced density matrix, ob-tained by tracing out the irrelevant degrees of freedom.Decoherence can explain why the splitting of the wave

1Although these three authors have different views from quantum theory, the first emphasizing the indivisibility of quantum phe-nomena, the secondwith his notion of potentiality, and the third with the concept of quantum states, for all of them the existence of aclassical domain is crucial. That is why we group their approaches under the same name "Copenhagen interpretation".
mailto:E-mails:[email protected]:E-mails:[email protected]


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function is given in terms of the pointer basis states,and why we do not see superpositions of macroscopicobjects. In this way, classical properties emerge fromquantum theory without the need of being assumed. Inthe framework of quantum gravity, it can also explainhow a classical background geometry can emerge in aquantum universe [6]. In fact, it is the first quantityto become classical. However, decoherence is not yet acomplete answer to the measurement problem [7,8]. Itdoes not explain the apparent collapse after the mea-surement is completed, or why all but one of the diag-onal elements of the density matrix become null whenthe measurement is finished. The theory is unable togive an account of the existence of facts, their unique-ness as opposed to the multiplicity of possible phenom-ena. Further developments are still in progress, likethe consistent histories approach [9], which is howeverincomplete until now. The important role played bythe observers in these descriptions is not yet explained[10], and still remains the problem on how to describea quantum universe when the background geometry isnot yet classical.

Nevertheless, there are some alternative solutionsto this quantum cosmological dilemma which, togetherwith decoherence, can solve the measurement problemmaintaining the universality of quantum theory. Onecan say that the Schrodinger evolution is an approxi-mation of a more fundamental non-linear theory whichcan accomplish the collapse [11, 12], or that the col-lapse is effective but not real, in the sense that theother branches disappear from the observer but do notdisappear from existence. In this second category wecan cite the Many-Worlds Interpretation [13] and theBohm-de Broglie Interpretation [14, 15]. In the former,all the possibilities in the splitting are actually realized.In each branch there is an observer with the knowl-edge of the corresponding eigenvalue of this branch,but she or he is not aware of the other observers andthe other possibilities because the branches do not in-terfere. In the latter, a point-particle in configurationspace describing the observed system and apparatus issupposed to exist, independently on any observations.In the splitting, this point particle will enter into one ofthe branches (which one depends on the initial positionof the point particle before the measurement, which isunknown), and the other branches will be empty. Itcan be shown [15]that the empty waves can neither in-teract with other particles, nor with the point particlecontaining the apparatus. Hence, no observer can beaware of the other branches which are empty. Againwe have an effective but not real collapse (the emptywaves continue to exist), but now with no multiplica-tion of observers. Of course these interpretations can beused in quantum cosmology. Schrodinger evolution isalways valid, and there is no need of a classical domainoutside the observed system.

In this paper we review some results on the applica-

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tion of the Bohm-de Broglie interpretation to quantumcosmology [16, 17, 18, 19, 20]. In this approach, thefundamental object of quantum gravity, the geometryof 3-dimensional spacelike hypersurfaces, is supposedto exist independently on any observation or measure-ment, as well as its canonical momentum, the extrin-sic curvature of the spacelike hypersurfaces. Its evolu-tion, labeled by some time parameter, is dictated by aquantum evolution that is different from the classicalone due to the presence of a quantum potential whichappears naturally from the Wheeler-De Witt equation.This interpretation has been applied to many minisu-perspace models [16, 19, 21, 22, 23, 24], obtained bythe imposition of homogeneity of the spacelike hyper-surfaces. The classical limit, the singularity problem,the cosmological constant problem, and the time issuehave been discussed. For instance, in some of these pa-pers it was shown that in models involving scalar fieldsor radiation, which are nice representatives of the mat-ter content of the early universe, the singularity can beclearly avoided by quantum effects. In the Bohm-deBroglie interpretation description, the quantum poten-tial becomes important near the singularity, yieldinga repulsive quantum force counteracting the gravita-tional field, avoiding the singularity and yielding infla-tion. The classical limit (given by the limit where thequantum potential becomes negligible with respect tothe classical energy) for large scale factors are usuallyattainable, but for some scalar field models it dependson the quantum state and initial conditions. In fact itis possible to have small classical universes and largequantum ones [24]. About the time issue, it was shownthat for any choice of the lapse function the quantumevolution of the homogeneous hypersurfaces yield thesame four-geometry [19]. What remained to be studiedis if this fact remains valid in the full theory, where weare not restricted to homogeneous spacelike hypersur-faces. The question is, given an initial hypersurfacewith consistent initial conditions, does the evolutionof the initial three-geometry driven by the quantumbohmian dynamics yields the same four-geometry forany choice of the lapse and shift functions, and if it does,what kind of spacetime structure is formed? We knowthat this is true if the three-geometry is evolved by thedynamics of classical General Relativity (GR), yieldinga non degenerate four geometry, but it can be false ifthe evolving dynamics is the quantum bohmian one.The idea was to put the quantum bohmian dynamicsin hamiltonian form, and then use strong results pre-sented in the literature exhibiting the most general formthat a hamiltonian should have in order to form a nondegenerate four-geometry from the evolution of three-geometries [25]. Our conclusion is that, in general,the quantum bohmian evolution of the three-geometriesdoes not yield any non degenerate four-geometry atall [20]. Only for very special quantum states a rel-evant quantum non degenerate four-geometry can be


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obtained, and it must be euclidean. In the general case,the quantum bohmian evolution is consistent (still inde-pendent on the choice of the lapse and shift functions)but yielding a degenerate four-geometry, where specialvector fields, the null eigenvectors of the four geometry,are present''. We arrived at these conclusions withoutassuming any regularization and factor ordering of theWheeler-DeWitt equation. As we know, the Wheeler-DeWitt equation involves the application of the prod-uct of local operators on states at the same space point,which is ill defined [27] . Hence we need to regularize itin order to solve the factor ordering problem, and havea theory free of anomalies (for some proposals, see Refs[28, 29, 30]). Our conclusions are completely indepen-dent on these issues.

This paper is organized as follows: in the next sec-tion we review the Bohm-de Broglie interpretation ofquantum mechanics for non-relativistic particles andquantum field theory in flat spacetime. In section II I weapply the Bohm-de Broglie interpretation to canonicalquantum gravity in the minisuperspace case. We showthat there is no problem of time in this interpretation,and that quantum effects can avoid the initial singu-larity, create inflation, and isotropize the Universe. Insection IV we treat the general case. We prove thatthe bohmian evolution of the 3-geometries, irrespec-tive of any regularization and factor ordering of theWheeler-DeWitt equation, can be obtained from a spe-cific hamiltonian, which is of course different from theclassical one. We then use this hamiltonian to obtain apicture of these new quantum structures. We end withconclusions and many perspectives for future work.

II The Bohm-de Broglie Inter-pretation

In this section we will review the Bohm-de Broglie in-terpretation of quantum mechanics. We will first showhow this interpretation works in the case of a singleparticle described by a Schrodinger equation, and thenwe will obtain, by analogy, the causal interpretation ofquantum field theory in flat spacetime.

Let us begin with the Bohm-de Broglie interpre-tation of the Schrodinger equation describing a singleparticle. In the coordinate representation, for a non-relativistic particle with Hamiltonian H = p2/2m +V(x), the Schrodinger equation is

. aw(x, t) ['h2 2 ]z'h at = - 2m \7 + V(x) w(x, t).We can transform this differential equation over a com-plex field into a pair of coupled differential equations

Brazilian Journal of Physics, vol. 30, no. 2, June, 2000

over real fields, by writing W = Aexp(i5/'h) , where Aand 5 are real functions, and substituting it into (1).We obtain the following equations.

aA2 + \7. (A 2 \75) = O . (3)at mThe usual probabilistic interpretation, i.e. the Copen-hagen interpretation, understands equation (3) as acontinuity equation for the probability density A2 forfinding the particle at position x and time t. All phys-ical information about the system is contained in A2,and the total phase 5of the wave function is completelyirrelevant. In this interpretation, nothing is said about5 and its evolution equation (2). Now suppose that theterm !}~" \ 7 :A is not present in Eqs. (2 ) and (3). Thenwe could interpret them as equations for an ensembleof classical particles under the influence of a classicalpotential V through the Hamilton-Jacobi equation (2) ,whose probability density distribution in space A2 (x , t)satisfies the continuity equation (3), where \7 5(x , t)/mis the velocity field v(x, t) of the ensemble of particles.When the term !}~" \ 7 :A is present, we can still under-stand Eq. (2 ) as a Hamilton-Jacobi equation for anensemble of particles. However, their trajectories areno more the classical ones, due to the presence of thequantum potential term in Eq. (2).

The Bohm-de Broglie interpretation of quantummechanics is based on the two equations (2 ) and (3)in the way outlined above, not only on the last one asit is the Copenhagen interpretation. The starting ideais that the position x and momentum p are always welldefined, with the particle's path being guided by a newfield, the quantum field. The field W obeys Schrodingerequation (1), which can be written as the two real equa-tions (2 ) and (3). Equation (2 ) is interpreted as aHamilton-Jacobi type equation for the quantum par-ticle subjected to an external potential, which is theclassical potential plus the new quantum potential

(4 )

(1 )

Once the field W , whose effect on the particle trajectoryis through the quantum potential (4), is obtained fromSchrodinger equation, we can also obtain the particletrajectory, x(t), by integrating the differential equationp = mx = \7 5(x, t) , which is called the guidance re-lation (a dot means time derivative). Of course, fromthis differential equation, the non-classical trajectoryx(t) can only be known if the initial position of theparticle is given. However, we do not know the initial


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position of the particle because we do not know howto measure it without disturbances (it is the hiddenvariable of the theory). To agree with quantum me-chanical experiments, we have to postulate that, for astatistical ensemble of particles in the same quantumfield w , the probability density distribution of initialpositions Xo is P (xo, to) = A2(XO ' t = to). Equation(3) guarantees that P(x, t) = A2 (x, t) for all times. Inthis way, the statistical predictions of quantum theoryin the Bohm-de Broglie interpretation are the same asin the Copenhagen interpretation".

It is interesting to note that the quantum poten-tial depends only on the form of W , not on its abso-lute value, as can be seen from equation (4). This factbrings home the non-local and contextual character ofthe quantum potential". This is a necessary feature be-cause Bell's inequalities together with Aspect's exper-iments show that, in general, a quantum theory mustbe either non-local or non-ontological. As the Bohm-de Broglie interpretation is ontological, than it must benon-local, as it is. The non-local and contextual quan-tum potential causes the quantum effects. It has noparallel in classical physics.

The function A plays a dual role in the Bohm-deBroglie interpretation: it gives the quantum potentialand the probability density distribution of positions,but this last role is secondary. If in some model thereis no notion of probability, we can still get informationfrom the system using the guidance relations. In this

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case, A2 does not need to be normalizable. The Bohm-de Broglie interpretation is not, in essence, a proba-bilistic interpretation. It is straightforward to apply itto a single system.

The classical limit can be obtained in a very simpleway. We only have to find the conditions for havingQ = 05. The question on why in a real measurementwe see an effective collapse of the wave function is an-swered by noting that, in a measurement, the wavefunction splits in a superposition of non-overlappingbranches. Hence the point particle (representing theparticle being measured plus the macroscopic appara-tus) will enter into one particular branch, which onedepends on the initial conditions, and it will be influ-enced by the quantum potential related only to thisbranch, which is the only one that is not negligible inthe region where the point particle actually is. Theother empty branches continue to exist, but they nei-ther influence on the point particle nor on any otherparticle [15]. There is an effective but not real collapse.The Schrodinger equation is always valid. There is noneed to have a classical domain outside the quantumsystem to explain a measurement, neither is the exis-tence of observers crucial because this interpretation isobjective.

For quantum fields in flat spacetime, we can apply asimilar reasoning. As an example, take the Schrodingerfunctional equation for a quantum scalar field:

ih8W~:, t) = f d3X{~[ -h28~2 + (\7)2]+U()}W(, t).Writing again the wave functional as W = Aexp(iSjh), we obtain:

(5 )

(6 )

8A2 f 3 8 ( 28S)at + d x 8 A 8 = 0,where Q() = _h2 2~ ~ : 1is the corresponding (un-regulated) quantum potential. The first equation isviewed as a modified Hamilton-Jacobi equation gov-erning the evolution of some initial field configurationthrough time, which will be different from the classical

(7 )

one due to the presence of the quantum potential. Theguidance relation is now given by

(8 )31 t has been shown that under typical chaotic situations, and only within the Bohm-de Broglie interpretation, a probability dis-

tribution P o F A2 would rapidly approach the value P = A2 [32, 33]. In this case, the probability postulate would be unnecessary,and we could have situations, in very short time intervals, where this modified Bohm-de Broglie interpretation would differ from theCopenhagen interpretation.

4The non-locality of Q becomes evident when we generalize the causal interpretation to a many particles system.51 t should be very interesting to investigate the connection between this bohmian classical limit and the phenomenon of decoherence.

To our knowledge, no work has ever been done on this issue, which may illuminate both the Bohm-de Broglie interpretation and thecomprehension of decoherence.


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The second equation is the continuity equation for theprobability density A2 [ (x ), to ] of having the initial fieldconfiguration at time to given by (x).

A detailed analysis of the Bohm-de Broglie interpre-tation of quantum field theory is given in Ref. [34]forthe case of quantum electrodynamics.

III The Bohm-de Broglie In-terpretation of Minisuper-space Canonical QuantumCosmology

In this section, we summarize the rules of the Bohm-de Broglie interpretation of quantum cosmology in thecase of homogeneous minisuperspace models. When weare restricted to homogeneous models, the supermo-mentum constraint of GR is identically zero, and theshift function can be set to zero without loosing any ofthe Einstein's equations. The hamiltonian is reducedto general minisuperspace form:

where pa(t) and qa(t) represent the homogeneous de-grees of freedom coming from rrij (x , t) and hij (x , t) .The minisuperspace Wheeler-De Witt equation is:

1i(fP(t),fia(t))'I!(q) = O. (10)Writing 'I! = Rexp(iS/Ti), and substituting it into (10),we obtain the following equation:

where 1 fPRQ(qM) = --Rfa /3 8 8 'qa q/3and fa/3(qM) and U(qM) are the minisuperspace partic-ularizations of the DeWitt metric Gijkl [36] and of thescalar curvature density _h1/2 R(3) (hij) of the spacelikehypersurfaces, respectively. The causal interpretationapplied to quantum cosmology states that the trajecto-ries qa(t) are real, independently of any observations.Eq. (11) is the Hamilton-Jacobi equation for them,which is the classical one amended with a quantum po-tential term (12), responsible for the quantum effects.This suggests to define:

(12)

a 8Sp = 8qa' (13)where the momenta are related to the velocities in theusual way:

a _ fa/3 _ ! _ 8q/3p - N8t (14)


To obtain the quantum trajectories we have to solve thefollowing system of first order differential equations:

8S(qa) _ r/3 18q/3- - - - - - a q ; - - N 8t . (15)Eqs. (15) are invariant under time reparametriza-

tion. Hence, even at the quantum level, differentchoices of N(t) yield the same spacetime geometry for agiven non-classical solution qa(t). There is no problemof time in the causal interpretation of minisuperspacequantum cosmology.

Let us now apply these rules, as examples, to min-isuperspace models with a free massless scalar field.Take the lagrangian:

L = He- ( R - w;p;p)For w = -1 we have effective string theory without theKalb-Rammond field. For w = -3/2 we have a confor-mally coupled scalar field. Performing the conformaltransformation 9M V = egMv we obtain the following la-

(16)

grangian:(9 )

L = H [ R - (w + ~);p;p]where the bars have been omitted. We will defineew = = (w+ ~).

(17)

IIL1. The isotropic caseWe consider now the Robertson-Walker metric

ds2 = -N2dt2 + 1~tl:2 [dr2 + r2(d02 + sin2(O)d


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(21)(22)

Usually the scale factor has dimensions of length be-cause we use angular coordinates in closed spaces.Hence we will define a dimensionless scale factor a = =a / (3 . In that case the hamiltonian becomes, omittingthe tilde:

H = N ( _ P~ + 6 p ! - m ) (23)2(3 a Cwa3As (3 appears as an overall multiplicative constant inthe hamiltonian, we can set it equal to one without anyloss of generality, keeping in mind that the scale fac-tor which appears in the metric is (3a, not a. We canfurther simplify the hamiltonian by defining 0: = = In(a)obtaining

N [ 2 6 2 ]H = () - Pa + -C P - E exp (40:)2exp 30: w . (2 4)where

e3aaPa Ne3a; p

P c; 6N(25)(26)

The momentum P is a constant of motion whichwe will call k . We will restrict ourselves to the physi-cally interesting case, due to observations, of E = 0 andc; > o .

The classical solutions in the gauge N = 1 are,(27 )

where Cl is an integration constant. In term of cosmictime they are:

a

(29)(28)

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The solutions contract or expand forever from a singu-larity, depending on the sign of k , without any infla-tionary epoch.

Let us now quantize the model. With a particu-lar choice of factor ordering, we obtain the followingWheeler- DeWitt equation

(30)Employing the separation of variables method, we ob-tain the general solution

w(o:, ) = f F(k)Ado:)Bd)dk (31 )where k is a separation constant, and

(32)and

Ak(O:) = al exp(iko:) + a2 exp( -iko:) (33)We will now make gaussian superpositions of these

solutions and interpret the results using the causal in-terpretation of quantum mechanics. The function F(k)is:

(34)

We take the wave function:

w(o:, ) = f F(k)[Ado:)Bd) + A-do:)Bd)dk](35)

with a2 = b2 = o .Performing the integration in k we obtain for W (we

will define ( j ) = = j Q j and omit the bars from now on)

{ [ (0:+)2a2] [ (0:_ )2a2] }W = aV'7f exp - 4 exp[id(o: + )] + exp - 4 exp[-id(o: - )] (36)


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In order to obtain the Bohmian trajectories, we have tocalculate the phase S of the above wave function andsubstitute it into the guidance formula

Pa (37)(38)P

where

Pa(40)(39)

P

We will work in the gauge N = 1. These equationsconstitute a planar system which can be easily studied:

exp(So') { 2[cos(2do:) + cosh(a20 : 1 / ] }[ o : a 2 sin(2do:) - 2dcos(2do:) - 2COSh(a20 : 1 / ]

exp(So') { 2[cos(2do:) + COSh(a20 : 1 / ] }

The line 0 : = 0 divides configuration space in two sym-metric regions. The line I /> = 0 contains all singu-lar points of this system, which are nodes and cen-ters. The nodes appear when the denominator of theabove equations, which is proportional to the norm


(41)

of the wave function, is zero. No trajectory can passthrough these points. They happen when I /> = 0 andcos(do:) = 0, or 0: = (2n + 1)7f/2d, n an integer, withseparation 7f/ d. The center points appear when thenumerators are zero. They are given by I /> = 0 and0: = 2d[cotan(do:)]/a2 They are intercalated withthe node points. As 1 0: 1 - + 00 these points tend ton a: / d, and their separations cannot exceed 7f/ d. As onecan see from the above system, the classical solutions(a(t) e x : tl/3) are recovered when 1 0: 1 - + 00 or 1 I /> 1 - + 00,the other being different from zero.

There are plenty of different possibilities of evolu-tion, depending on the initial conditions. Near the cen-ter points we can have oscillating universes without sin-gularities and with amplitude of oscillation of order 1.For negative values of 0:, the universe arise classicallyfrom a singularity but quantum effects become impor-tant forcing it to recollapse to another singularity, re-covering classical behaviour near it. For positive valuesof 0 : , the universe contracts classically but when 0 : issmall enough quantum effects become important creat-ing an inflationary phase which avoids the singularity.The universe contracts to a minimum size and afterreaching this point it expands forever, recovering theclassical limit when 0: becomes sufficiently large. Wecan see that for 0 : negative we have classical limit forsmall scale factor while for 0: positive we have classicallimit for big scale factor.(42)111.2. The anisotropic case

To exemplify the quantum isotropization of theUniverse, let us take now, instead of the Friedman-Robertson-Walker of Eq. (18), the homogeneous andanisotropic Bianchi I line element

-N (t)dt2 + exp[2,80(t) + 2,8+(t) + 2V3,8-(t)] dx2 +exp[2,80(t) + 2,8+(t) - 2V3,8-(t)] dy2 +exp[2,80(t) - 4,8+ ( t ) ] dz2

This line element will be isotropic if and only if,8+(t) and ,8_(t) are constants. Inserting Eq. (43) intothe lagrangian (17), supposing that the scalar field I />depends only on time, discarding surface terms, andperforming a Legendre transformation, we obtain thefollowing minisuperspace classical hamiltonian

H N ( 2 2 2 2 )= 24exp (3,80) -P o + P+ + p_ + P , (44)where (Po,P+,P_,p) are canonically conjugate to

(43)

(,80,,8+, ,8_, 1 / , respectively, and we made the trivialredefinition I /> - + y?J;;J6 1 / > .

We can write this hamiltonian in a compact form bydefining y M = (,80,,8+,,8 _, 1 / and their canonical mo-menta PM = (Po,p+,p_,p), obtaining

H- N M l/- 24exp(3yO)17 PMPl/, (45)where l7Ml/ is the Minkowski metric with signature(- + ++). The equations of motion are the constraintequation obtained by varying the hamiltonian with re-


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spect to the lapse function N1 -l = = 'f}IWPMPl/ = 0,

and the Hamilton's equations(46)

N-------:----,--'f}Ml/p12 exp (3yo) l/, (47)

PM = _Oo1-l = o .yMThe solution to these equations in the gauge N =12 exp(3yo) is

(48)

(49)

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where the momenta Pl / are constants due to the equa-tions of motion and the C M are integration constants.We can see that the only way to obtain isotropy inthese solutions is by making PI = P+ = 0 and P2 =p: = 0, which yield solutions that are always isotropic,the usual Friedman-Robertson-Walker (FRW) solutionswith a scalar field. Hence, there is no anisotropicsolution in this model which can classically becomeisotropic during the course of its evolution. Onceanisotropic, always anisotropic. If we suppress the >degree of freedom, the unique isotropic solution is flatspacetime because in this case the constraint (46) en-forces Po = o .

To discuss the appearance of singularities, we needthe Weyl square tensor W2 = = W",,sMl/W,,,,sMl/. It reads

1W2 =_e-2,so (2p p3 _ 6p p2 P + p4 + 2p2 p2 + p4 + p2p2 + p2p2 ).432 0 + 0 - + - + - + 0 + 0-

Hence, the Weyl square tensor is proportional toexp (-12,80) because the p's are constants (see Eq.(48)) and the singularity is at t = - 00. The classi-cal singularity can be avoided only if we set Po = o .But then, due to equation (46), we would also havePi = 0, which corresponds to the trivial case of flatspacetime. Hence, the unique classical solution whichis non-singular is the trivial flat spacetime solution.

The Dirac quantization procedure yields theWheeler-DeWitt equation, which in the present casereads 02'f}M l/_ _ W(yM ) = 0oyMyl/

Let us now investigate spherical-wave solutions ofEq. (51) . They read

(51)

W 3 = ~ [f(yO + y) + g(yO - y)], (52)where y = = l' ,;= 1 (yi)2.

The guidance relations in the gauge N12exp(3yo) are (see Eqs. (47)) read

(00W3) .0Po = oo S = 1m ~ = -y , (53)O S (OiW3).iPi = i = 1m ~ = y ,

where S is the phase of the wave function. In terms off and 9 the above equations read

(54)

. ( f' (yO + y) + gl (yO - y) )yO - -1m- f(yo+y)+g(yO_y)'

(55)

(50)

.. yi (f '(YO + y) _ gl(yO _ y))y' - -1m- y f (yO + y) + 9 (yO - y) , (56)where the prime means derivative with respect to theargument of the functions f and g, and Im(z) is theimaginary part of the complex number z.

From Eq. (56) we obtain that

(57 )which implies that yi(t) = c}yj(t), with no sum in j,where the c} are real constants and d = c~ = c~ =1. Hence, apart some positive multiplicative constant,knowing about one of the yi means knowing about allyi . Consequently, we can reduce the four equations (55)and (56) to a planar system by writing y = Cly31, withC > 1, and working only with yO and y3 , say. Theplanar system now reads

. (f '(YO + Cly31) + gl(yO _ C1y31))yO - -1m- f(yO + Cly31) + g(yO - Cly31) ,. sign(y3) (f'(YO + Cly31) _ gl(yO - C1y31))~= ~ .C f(yO + Cly31) + g(yO - Cly31) (59)

Note that if f = g, y3 stabilizes at y3 = 0 becauseil as well as all other time derivatives of y3 are zeroat this line. As yi(t) = c}yj(t), all yi(t) become zero,and the cosmological model isotropizes forever once y3reaches this line. Of course one can find solutions wherey3 never reaches this line, but in this case there mustbe some region where il = 0, which implies y i = 0,and this is an isotropic region. Consequently, quan-tum anisotropic cosmological models with f = 9 always

(58)


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have an isotropic phase, which can become permanentin many cases.

IV The Bohm-de Broglie In-terpretation of SuperspaceCanonical Quantum Cos-mology

In this section, we will quantize General Relativity The-ory (GR) without making any simplifications or cutting


of degrees of freedom. The matter content is a mini-mally coupled scalar field with arbitrary potential. Allsubsequent results remain essentially the same for anymatter field which couples uniquely with the metric,not with their derivatives.

The classical hamiltonian of full GR with a scalarfield is given by:

where(60)

""G.. IIijIIkl + !h-1/2II2 +,~ 2Jkl 2 +h1/2 [ - ",-1R(3) - 2A ) + ~hij{M)OA)+ U()]-2DiII; + IIoj.

In these equations, hij is the metric of closed 3-dimensional spacelike hypersurfaces, and IIij is itscanonical momentum given byIIij = _h1/2(Kij - hij K) = Gijkl (h kl - DkNI - DINk),(63)where

(64)is the extrinsic curvature of the hypersurfaces (indicesare raised and lowered by the 3-metric hij and its in-verse hij). The canonical momentum of the scalar fieldis now

(65)The quantity R(3) is the intrinsic curvature of the hy-persurfaces and h is the determinant of hij. The lapsefunction N and the shift function Nj are the Lagrangemultipliers of the super-hamiltonian constraint 'H ~ 0and the super-momentum constraint 1 - l j ~ 0, respec-tively. They are present due to the invariance of GR un-der spacetime coordinate transformations. The quan-tities Gijkl and its inverse Gijkl (GijklGijab = 1 5 k ? ) aregiven by

Gijkl = ~h1/2(hikhjl + hi1hjk _ 2hijhkl), (66)1 -1/2Gijkl = 2h (hikhjl + hilhjk - hijhkl), (67)

(61)(62 )

which is called the DeWitt metric. The quantity D, isthe i-component ofthe covariant derivative operator onthe hypersurface, and", = 1 67r G / c4.

The classical 4-metric

and the scalar field which are solutions of the Einstein'sequations can be obtained from the Hamilton's equa-tions of motion hij = {hij,H}, (69)

iIij = {IIij, H}, (70)={,H}, (71)

IT= {II, H}, (72 )for some choice of Nand Ni, and if we impose initialconditions compatible with the constraints

1 -l ~ 0, (73)(74)

It is a feature of the hamiltonian of GR that the 4-metrics (68) constructed in this way, with the sameinitial conditions, describe the same four-geometry forany choice of N and Ni. The algebra of the constraintsclose in the following form (we follow the notation ofRef. [25]):


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Nelson Pinto-Neto 339

{ 1- l ( x ), 1 -l (X l) }{1-li(X),1-l(x l)}

{1-li(X),1-lj(xl)}

1 -l i (x)8 i83 (x , z" ) -1- li(x l)8 i83 (z", x)1-l(x )8i83 (x , z" )1- l i (x)8j83 (x , z" ) + 1 - lj ( xl)8 i83 (x , z" )

(75)

To quantize this constrained system, we follow theDirac quantization procedure. The constraints becomeconditions imposed on the possible states of the quan-tum system, yielding the following quantum equations:

oo

(76)(77)

In the metric and field representation, the first equation

is-2h .D. 8' I! (hij , + 8'I!(h ij'8.A.=0u J 8h 1j 8 > ''I' ,

which implies that the wave functional 'I! is an invariantunder space coordinate transformations.

The second equation is the Wheeler-DeWitt equa-tion [35, 36]. Writing it unregulated in the coordinaterepresentation we get

(78)

{ _ 1;2 [ , , " G . . kl_ 8 _ _ 8 _ + ~h-1 /2_ f_ ] +V},T '(h .. A .) - 0Ib ,~ 'J 8hij 8hkl 2 8>2 ~ 'J ' 'I' - ,where V is the classical potential given by

(79)

This equation involves products of local operators atthe same space point, hence it must be regularized. Af-ter doing this, one should find a factor ordering whichmakes the theory free of anomalies, in the sense that thecommutator of the operator version of the constraintsclose in the same way as their respective classical Pois-son brackets (75). Hence, Eq. (79) is only a formal onewhich must be worked out [28, 29, 30].

Let us now see what is the Bohm-de Broglie inter-pretation of the solutions of Eqs. (76) and (77) in themetric and field representation. First we write the wavefunctional in polar form 'I! = A exp(i5j'h) , where A and5 are functionals of hij and > . Substituting it in Eq.(78), we get two equations saying that A and 5are in-variant under general space coordinate transformations:

(81)

(82)

The two equations we obtain for A and 5when wesubstitute 'I! = A exp (i5j 'h) into Eq. (77) will of coursedepend on the factor ordering we choose. However, in

(80)

any case, one of the equations will have the form.. 85 85 ~ -1/2 ( 85)2 _",G'JkI8hij 8h kl + 2 h 8 > +V + Q - 0, (83)

where V is the classical potential given in Eq. (80).Contrary to the other terms in Eq. (83), which are al-ready well defined, the precise form of Q depends on theregularization and factor ordering which are prescribedfor the Wheeler-De Witt equation. In the unregulatedform given in Eq. (79), Q is

2 1( 82 A h - 1/2 82 A )Q = -'h A ",G ijkl 8hij8h

kl+ -2- 8>2 . (84)

Also, the other equation besides (83) in this case is.. _ 8 ( 2}_ _ ) ~ -1/2_ !_ ( 285)_",G'JkI8hij A 8h kl + 2 h 8 > A 8> - O. (85)

Let us now implement the Bohm-de Broglie inter-pretation for canonical quantum gravity. First of all wenote that Eqs. (81) and (83), which are always valid ir-respective of any factor ordering of the Wheeler-De Wittequation, are like the Hamilton-Jacobi equations forGR, supplemented by an extra term Q in the case ofEq. (83), which we will call the quantum potential. Byanalogy with the cases of non-relativistic particle and


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quantum field theory in flat spacetime, we will postu-late that the 3-metric of spacelike hypersurfaces, thescalar field, and their canonical momenta always exist,independent on any observation, and that the evolutionof the 3-metric and scalar field can be obtained from theguidance relations

(86)

88(hij,~ ~nII = 8> ,with IIij and II given by Eqs. (63) and (65), re-spectively. Like before, these are first order differentialequations which can be integrated to yield the 3-metricand scalar field for all values of the t parameter. Thesesolutions depend on the initial values of the 3-metricand scalar field at some initial hypersurface. The evo-lution of these fields will of course be different from theclassical one due to the presence of the quantum poten-tial term Q in Eq. (83). The classical limit is once moreconceptually very simple: it is given by the limit wherethe quantum potential Q becomes negligible with re-spect to the classical energy. The only difference fromthe previous cases of the non-relativistic particle andquantum field theory in flat spacetime is the fact thatthe equivalent of Eqs. (3) and (7) for canonical quan-tum gravity, which in the naive ordering is Eq. (85),cannot be interpreted as a continuity equation for aprobability density A2 because ofthe hyperbolic natureof the DeWitt metric Gijkl. However, even without anotion of probability, which in this case would mean theprobability density distribution for initial values of the3-metric and scalar field in an initial hypersurface, wecan extract a lot of information from Eq. (83) whateveris the quantum potential Q , as will see now. After weget these results, we will return to this probability issuein the last section.

First we note that, whatever is the form of the quan-tum potential Q , it must be a scalar density of weightone. This comes from the Hamilton-Jacobi equation(83). From this equation we can express Q as

88 88 1 -1/2 (88) 2_Q = -",Gijkl 8hij 8hkl - 2h 8 > V.As 8 is an invariant (see Eq. (81)), then 88j8hij and88j8>must be a second rank tensor density and a scalardensity, both of weight one, respectively. When theirproducts are contracted with Gijkl and multiplied byh-1/2, respectively, they form a scalar density of weightone. As V is also a scalar density of weight one, thenQ must also be. Furthermore, Q must depend only onhij and > because it comes from the wave functionalwhich depends only on these variables. Of course itcan be non-local (we show an example in the appendix),i.e., depending on integrals of the fields over the wholespace, but it cannot depend on the momenta.

(88)


Now we will investigate the following importantproblem. From the guidance relations (86) and (87),which will be written in the form

.. .. 88(hab, (89)h'J = II 'J - ~ 0~ - 8hij ,and

h - II 88 (hij , >) 0~ = - 8> ~ . (90)we obtain the following first order partial differential

equations:

and(92)

The question is, given some initial 3-metric and scalarfield, what kind of structure do we obtain when we in-tegrate this equations in the parameter t? Does thisstructure form a 4-dimensional geometry with a scalarfield for any choice of the lapse and shift functions?Note that if the functional 8were a solution of the clas-sical Hamilton-Jacobi equation, which does not containthe quantum potential term, then the answer would bein the affirmative because we would be in the scope ofGR. But 8 is a solution of the modified Hamilton-Jacobiequation (83), and we cannot guarantee that this willcontinue to be true. We may obtain a complete differ-ent structure due to the quantum effects driven by thequantum potential term in Eq. (83). To answer thisquestion we will move from this Hamilton-Jacobi pic-ture of quantum geometrodynamics to a hamiltonianpicture. This is because many strong results concern-ing geometrodynamics were obtained in this later pic-ture [25,37]. We will construct a hamiltonian formalismwhich is consistent with the guidance relations (86) and(87). It yields the bohmian trajectories (91) and (92)if the guidance relations are satisfied initially. Once wehave this hamiltonian, we can use well known results inthe literature to obtain strong results about the Bohm-de Broglie view of quantum geometrodynamics.

Examining Eqs. (81) and (83), we can easily show[20]that the hamiltonian which generates the bohmiantrajectories, once the guidance relations (86) and (87)are satisfied initially, is given by:

where we define1-lQ = 'H + Q . (94)

The quantities 'H and 1-li are the usual GR super-hamiltonian and super-momentum constraints given byEqs. (61) and (62). In fact, the guidance relations (86)and (87) are consistent with the constraints 1-lQ ~ 0


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and 1 -li ~ 0 because S satisfies (81) and (83). Further-more, they are conserved by the hamiltonian evolutiongiven by (93) [20].

We now have a hamiltonian, HQ , which generatesthe bohmian trajectories once the guidance relations(86) and (87) are imposed initially. In the following, wecan investigate if the the evolution of the fields drivenby HQ forms a four-geometry like in classical geometro-dynamics. First we recall a result obtained by ClaudioTeitelboim [37]. In this paper, he shows that if the

341

3-geometries and field configurations defined on hyper-surfaces are evolved by some hamiltonian with the form

(95)

and if this evolution can be viewed as the "motion" of a3-dimensional cut in a 4-dimensional spacetime (the 3-geometries can be embedded in a four-geometry), thenthe constraints il ~ 0 and ili ~ 0 must satisfy thefollowing algebra

{il(x ), il( x')}{ili(X ), il( x')}

{ il i( X) , il j( x') }

-E [ili (X )O i83 (z", x)] - ili (X I)O i83 ( x, X l)il(X )O i8 3 (x , z" )ili (x )O j83 (x , z") - ilj (X I) O i8 3 ( x, z" )

(96)(97)(98)

The constant E in (96) can be 1 depending if the four-geometry in which the 3-geometries are embedded iseuclidean (E = 1) or hyperbolic (E = -1). These arethe conditions for the existence of spacetime.

The above algebra is the same as the algebra (75)of GR if we choose E = -1. But the hamiltonian (93)is different from the hamiltonian of GR only by thepresence of the quantum potential term Q in 1- lQ. ThePoisson bracket {1-li(X),1-lj(xl)} satisfies Eq. (98) be-cause the 1-l i of HQ defined in Eq. (93) is the same as inGR. Also {1 -l i ( x ), 1 - lQ(X l) } satisfies Eq. (97) because 1-l iis the generator of spatial coordinate transformations,and as 1-lQ is a scalar density of weight one (rememberthat Q must be a scalar density of weight one), thenit must satisfies this Poisson bracket relation with 1-li.What remains to be verified is if the Poisson bracket{1 -lQ(x) , 1 - lQ(X I) } closes as in Eq. (96). We now recallthe result of Ref. [25]. There it is shown that a gen-eral super-hamiltonian il which satisfies Eq. (96), is ascalar density of weight one, whose geometrical degreesof freedom are given only by the three-metric hij andits canonical momentum, and contains only even pow-ers and no non-local term in the momenta (togetherwith the other requirements, these last two conditions

are also satisfied by 1-lQ because it is quadratic in themomenta and the quantum potential does not containany non-local term on the momenta), then il must havethe following form:

'1/ - " "G . . r r i j r r kl + ~h-1/2~2 + V;TL -,~ 2Ju 2" a, (99)where

1/2[ -1 (3) - 1 ii -]Va=-Eh -K (R -2A)+2hJOi>ojC/>+U(.(100)

With this result we can now establish two possible sce-narios for the Bohm-de Broglie quantum geometrody-namics, depending on the form of the quantum poten-tial:

IV.1. Quantum geometrodynamics evo-lution is consistent and forms a non de-generate four-geometryIn this case, the Poisson bracket {1-lQ(x),1-lQ(XI)}

must satisfy Eq. (96). Then Q must be such thatV + Q = Va with V given by (80) yielding:

(101)


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Then we have two possibilities:IV.1.1. The spacetime is hyperbolic ( E = -1)

In this case Q is

Q = _ h 1/2 [~( -A + A) - U ( + U ( ] . (102)Hence Q is like a classical potential. Its effect is torenormalize the cosmological constant and the classical


scalar field potential, nothing more. The quantum ge-ometrodynamics is indistinguishable from the classicalone. It is not necessary to require the classical limitQ = 0 because Va = V + Q already may describe theclassical universe we live in.

IV.1.2. The spacetime is euclidean ( E = 1)In this case Q is

Now Q not only renormalize the cosmological con-stant and the classical scalar field potential but alsochange the signature of spacetime. The total poten-tial Va = V + Q may describe some era of the earlyuniverse when it had euclidean signature, but not thepresent era, when it is hyperbolic. The transition be-tween these two phases must happen in a hypersurfacewhere Q = 0, which is the classical limit.

We can conclude from these considerations thatif a quantum spacetime exists with different featuresfrom the classical observed one, then it must be eu-clidean. In other words, the sole relevant quantum ef-fect which maintains the non-degenerate nature of thefour-geometry of spacetime is its change of signature toa euclidean one. The other quantum effects are eitherirrelevant or break completely the spacetime structure.This result points in the direction of Ref. [38].

IV.2. Quantum geometrodynamics evo-lution is consistent but does not form anon degenerate four-geometryIn this case, the Poisson bracket {1-lQ(x),1-lQ(x')}

does not satisfy Eq. (96) but is weakly zero in someother way. Some examples are given in Ref. [40].They are real solutions of the Wheeler-DeWitt equa-tion, where Q = -V, and non-local quantum poten-tials. It is very important to use the guidance relationsto close the algebra in these cases. It means that thehamiltonian evolution with the quantum potential isconsistent only when restricted to the bohmian trajec-tories. For other trajectories, it is inconsistent. Con-cluding, when restricted to the bohmian trajectories, analgebra which does not close in general may close, asshown in the above example. This is an important re-mark on the Bohm-de Broglie interpretation of canon-ical quantum cosmology, which sometimes is not no-ticed.

In the examples above, we have explicitly obtained

(103)

the "structure constants" of the algebra that charac-terizes the "pre-four-geometry" generated by HQ i.e.,the foam-like structure pointed long time ago in earlyworks of J. A. Wheeler [35, 42].

V Conclusion and DiscussionsThe Bohm-de Broglie interpretation of canonical quan-tum cosmology yields a quantum geometrodynamicalpicture where the bohmian quantum evolution of three-geometries may form, depending on the wave func-tional, a consistent non degenerate four geometry whichmust be euclidean (but only for a very special local formof the quantum potential), and a consistent but degen-erate four-geometry indicating the presence of specialvector fields and the breaking of the spacetime struc-ture as a single entity (in a wider class of possibilities).Hence, in general, and always when the quantum po-tential is non-local, spacetime is broken. The three-geometries evolved under the influence of a quantumpotential do not in general stick together to form a nondegenerate four-geometry, a single spacetime with thecausal structure of relativity. This is not surprising,as it was anticipated long ago [42]. Among the consis-tent bohmian evolutions, the more general structuresthat are formed are degenerate four-geometries with al-ternative causal structures. We obtained these resultstaking a minimally coupled scalar field as the mattersource of gravitation, but it can be generalized to anymatter source with non-derivative couplings with themetric, like Yang-Mills fields.

As shown in the previous section, a non degeneratefour-geometry can be attained only if the quantum po-tential have the specific form (101). In this case, thesole relevant quantum effect will be a change of signa-ture of spacetime, something pointing towards Hawk-ing's ideas.

In the case of consistent quantum geometrodynam-


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Nelson Pinto-Neto

ical evolution but with degenerate four-geometry, wehave shown that any real solution of the Wheeler-DeWitt equation yields a structure which is the ide-alization of the strong gravity limit of GR. This type ofgeometry, which is degenerate, has already been stud-ied [41]. Due to the generality of this picture (it is validfor any real solution of the Wheeler-DeWitt equation,which is a real equation), it deserves further attention.Itmay well be that these degenerate four-metrics werethe correct quantum geometrodynamical description ofthe young universe. It would be also interesting to in-vestigate if these structures have a classical limit yield-ing the usual four-geometry of classical cosmology.

For non-local quantum potentials, we have shownthat apparently inconsistent quantum evolutions are infact consistent if restricted to the bohmian trajectoriessatisfying the guidance relations (86) and (87). This isa point which is sometimes not taken into account.

If we want to be strict and impose that quantumgeometrodynamics does not break spacetime, then wewill have stringent boundary conditions. As said above,a non degenerate four-geometry can be obtained only ifthe quantum potential have the form (101). This is a se-vere restriction on the solutions of the Wheeler-DeWittequation.

These restrictions on the form of the quantum po-tential do not occur in minisuperspace models [19]because there the hypersurfaces are restricted to behomogeneous. The only freedom we have is in thetime parametrization ofthe homogeneous hypersurfaceswhich foliate spacetime. There is a single constraint,which of course always commute with itself, irrespec-tive of the quantum potential. The theorem proven inRef. [25], which was essential in all the reasoning ofthe last section, cannot be used here because minisu-perspace models do not satisfy one of their hypothe-ses. In section 3 we studied quantum effects in suchminisuperspace models and we showed that they canavoid singularities, isotropize the Universe, and createinflationary epochs. It should be very interesting to in-vestigate if these quantum phases of the Universe mayhave left some traces which could be detected now, asin the anisotropies of the cosmic microwave backgroundradiation.

As we have seen, in the Bohm-de Broglie approachwe can investigate further what kind of structure isformed in quantum geometrodynamics by using thePoisson bracket relation (96), and the guidance rela-tions (91) and (92). By assuming the existence of 3-geometries, field configurations, and their momenta, in-dependently on any observations, the Bohm-de Broglieinterpretation allows us to use classical tools, like thehamiltonian formalism, to understand the structure ofquantum geometry. If this information is useful, we donot know. Already in the two-slit experiment in non-relativistic quantum mechanics, the Bohm-de Broglieinterpretation allows us to say from which slit the par-

343

ticle has passed through: if it arrive at the upper halfof the screen it must have come from the upper slit,and vice-versa. Such information we do not have inthe many-worlds interpretation. However, this infor-mation is useless: we can neither check it nor use itin other experiments. In canonical quantum cosmologythe situation may be the same. The Bohm-de Broglieinterpretation yields a lot of information about quan-tum geometrodynamics which we cannot obtain fromthe many-worlds interpretation, but this informationmay be useless. However, we cannot answer this ques-tion precisely if we do not investigate further, and thetools are at our disposal.

We would like to remark that all these results wereobtained without assuming any particular factor order-ing and regularization of the Wheeler-De Witt equation.Also, we did not use any probabilistic interpretation ofthe solutions of the Wheeler-DeWitt equation. Hence,it is a quite general result. However, we would liketo make some comments about the probability issuein quantum cosmology. The Wheeler-De Witt equationwhen applied to a closed universe does not yield a prob-abilistic interpretation for their solutions because ofits hyperbolic nature. However, it has been suggestedmany times [21, 43, 44, 45, 46] that at the semiclassi-cal level we can construct a probability measure withthe solutions of the Wheeler-DeWitt equation. Hence,for interpretations where probabilities are essential, theproblem of finding a Hilbert space for the solutions ofthe Wheeler-De Witt equation becomes crucial if some-one wants to get some information above the semiclas-sicallevel. Of course, probabilities are also useful in theBohm-de Broglie interpretation. When we integrate theguidance relations (91) and (92), the initial conditionsare arbitrary, and it should be nice to have some prob-ability distribution on them. However, as we have seenalong this paper, we can extract a lot of informationfrom the full quantum gravity level using the Bohm-deBroglie interpretation, without appealing to any prob-abilistic notion.

Itwould also be important to investigate the Bohm-de Broglie interpretation for other quantum gravita-tional systems, like black holes. Attempts in this direc-tion have been made, but within spherical symmetryin empty space [47], where we have only a finite num-ber of degrees of freedom. It should be interesting toinvestigate more general models.

The conclusions of this paper are of course limitedby many strong assumptions we have tacitly made, assupposing that a continuous three-geometry exists atthe quantum level (quantum effects could also destroyit), or the validity of quantization of standard GR, for-getting other developments like string theory. How-ever, even if this approach is not the appropriate one,it is nice to see how far we can go with the Bohm-deBroglie interpretation, even in such incomplete stage ofcanonical quantum gravity. It seems that the Bohm-


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de Broglie interpretation may at least be regarded as anice "gauge" [48] to be used in quantum cosmology, as,probably, it will prove harder, or even impossible, toreach the detailed conclusions of this paper using otherinterpretations. Furthermore, if the finer view of theBohm-de Broglie interpretation of quantum cosmologycan yield useful information in the form of observationaleffects, then we will have means to decide between inter-pretations, something that will be very important notonly for quantum cosmology, but for quantum theoryitself.

AcknowledgmentsWe would like to thank CNPq of Brazil for financial

support.

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[48] We thank this image to Brandon Carter.[49] K. Kuchar, Phys. Rev. D 50, 3961 (1994).[50] J. Louko and S. N. Winters-Hilt, Phys. Rev. D 54, 2647

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Nelson Pinto-Neto- Quantum Cosmology: How to Interpret and Obtain Results

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