"Behaviour of the Gravitational Field Around the Initial Singularity"

"BEHAVIOUR OF THE GRAVITATIONAL FIELD AROUND THE INITIAL SINGULARITY"

Gerson Francisco.

t ? õ

INSTIT0T0 01 flSICA TEÓRICA "BEHAVIOUR OF THE GRAVITATIONAL FIELD AROUND THE INITIAL SINGULARITY"

Gerson Francisco Instituto de Física Teórica Rua Pamplona, 145 01405 - São Paulo-SP BRASIL. Abstract

The ultralocal representation of the canonical^ ly quantized gravitational field is used to obtain the evolution of coherent states in the immediate neighborhood of the singularity. It is shown that smearing functions play the role of classical fields since they correspond to cosrnological solutions around the singularity. A special class of ultralocal coherent states is shown to contain the essential aspects of the dynamics of the system when we choose a simple representation for the field operators of the theory. When the ultralocality condition is broken a conjecture will be made about the quantum evolution of coherent states in the classical limit. i-í . Intiüd^': I ion

Cosmologica.1 solutions to Einstein equations

a-o'jnd the Initial singularity have been thouroughly investigated

_y b^Lin^kii, KhaiaLnikov and Lifshitz |1—3| (or BKL t\u short).

[n fiLs connection ue also mention the u»ork of Eardley et al. |4|

inn thu Hami ltoninr' tiaatment of Liang |5|. The main point of theso

st'ciRs is ttip far:t thac a large class of solutions to Einstein rtn,iaf,ions \n r.ne asymptotic region close to the singularity exhi-

;•>: ' :. kifnj cr ?\.>antaneously decoupled dynamics in the sense that

field ewoJution at a given initial point on a Cauchy hypersur-

5^ is totally independent from what occurs in its neighborhood. This oxtrerpeJy localized behaviour of the dynamics is often

•T>rrPr! to as ul tr alocality |6|.

The gravitational field Hamlltonian constraint

Q., ic; tho ^ridtidl iisLric on 2Ü > Qa dttu'i » r\ is the scalar Ri_L

L .i;/,i ., - it O,, , JllJ is the momentum canonically cunjuga- :" •' Í» ar"! X c l6TtGI& uihere (7 is Newton's constant. In na'1 / '-;on~in i o'jiral ^ociwis it happens that the kinetic term , nr jii.iuloL.3l '' i '• 11 tonian conatrainti in (l) containing the momen domiratas |l-5| over the potential term |ll|

Since the momentum variables can be expressed in terms of the "tinia" derivative of 4JJ (see (9)), cosmologists have called those cosmologicel solutions, that evolve according to (3) velocity dominated. On the other hand all spatial derivatives responsi- ble for the dynamical correlation between spatial points ara contained in the potential (4) and thus the velocity dominated Hamil- tonian Jfc generates an ultralocal dynamics.

In this paper we will quantize the velocity d£ minated Hamiltonian (3) using the ultralocal techniques as develo£ ed by KlauderJ4,l2|. Liang |5| remarked that only three components of the metric tensor are necessary to specify the dynamics around the singularity and it is these components, together with their canon^ cally conjugate momenta, that we propose to quantize* Having obtain ed the field operators of the theory we are able to introduce cohe rent states and to discuss dynamical evolution. Briefly the paper 1B organized as follows.

In §2 we outline the Hamiltonian approach to velocity dominatod cosmological models. In §3 the ultralocal quantization of the model is presented and coherent states introduced. Dynamical evolution and the role of smearing functions is the ob- ject of §4 while §5 contains more discussion about classical models and a conjecture about what we should expect the evolution to be when ultralocality is violated; around the singularity.

Throughout the paper we assume a 3*1 split of spacetimo P\mTj*ff{ » where) T~ is compact and without boundary; the four dimensional metric &(£ m 0,1,2,3 , has signature -•++. All four dinensional objects have a superscript * while three dimensional objects on have no superscript.

§2. /elocity Dominated Cosmologies! Models

The variational principle leading to the sour- ce free Einstein field equations is based on the classical action

where f^cfet^flU* » f\ is tnB scalar Ricci curvature Arnouiitt, Deser and Pilsner |0 |, or ADM for short (see also Oirac |7 I)» have rewritten (5) in terms of three dimensional objects d£ fined on

where fll*(m $„)" * and N • i^X) J* . Observe that the G#* components are sssentially Lagrange multiplier and only nas the spatial tensor on 2Ü, 9/i • f

•**» where the bar means couariant derivative u/ith respact to Qy , Fxom the ADM action we can see that the quantity uftp (H^H^ f plays the rola of tha Hamiltonian of the system» Varying (6) with respect to Qu and K* gives Hamilton equationa for these quart titles («ore on this below). Varying H and 0? one obtains that JL and jh must vanish. ADM show that this corresponds respec- tively to the four constraint Einstein equations J|## =O and

^P ,S 0 . A geometric interpretation for JÇ, and fa is as follows |l9»15|t j£ is the generator of infinitesimal deforma tions in the normal direction away from JEL while Jfcg corresponds to the generator of infinitesimal coordinate transformations on£. Thus all possible variations of the metric and corresponding mnmen ta are encoded in the contraints

In any physical theory the existence of cons - traints is a sign that there are redundant variables that can be eliminated without affecting the essential aspects of the dynamics (however in this elimination process one will surely loose explicit covariance). The canonical approach to gravity provides us with a clear insight into this problem since there are three constraint generators jj^i of coordinate transformations and the geometry des cribed by the six components Q.. should not depend on coordinate system's. The full ADM reduction process is a way of eliminat- ing all spurious variablas leaving only the "true" degrees of free dom. It consists in imposing four "coordinate conditions" on the metric tensor and in solving all the four constraints for the momenta reducing the system from six to two degrees of freedom* This prescription is rather vague but the process is considered success- ful if the action in terms of the two remaining pairs of variables with their conjugate momenta can be expressed in canonical form and if there is compatibility between the choice of variables and the Lagrange multipliers. Below u/e will use a remarkable property oF velocity dominated models to cast the ADM action into canonical

form (see (17)) without actually solving the constraints jjj . In

doing this we have two possibilities regarding the constraint V .

The full reduction demands that JjT, be solved classically and this

leads to a choice of a time variable, the intrinsic time |16| of

the system. This time is "intrinsic" because it should depend only on the geometry of 4j» and not on coordinate systems (for example any monotonic function Of fls dcff» can in principle be usedte.g. the variable Qs-T'&ft in (17) conjugate to P ). Another possibly lity, the one we choose, is not to solve V • This partial ADM re duction process allows us to introduce the concept of proper time evolution in the quantum theory developed in §4.

In this paper we will assume that a synchro- nous frame |l7| has been chosen that is, the Lagrangean multipliers H vanish. Also we impose that the remaining multiplier is constant and we choose flr «f ; this means that the parameter T can be identified with the proper time of the system henceforth denoted by 0 . Thus ths line element has the simple form

..' that (8) is time reversal invariant.

In the frame described above the momenta canoni caJly conjugate to Qn are related to the extrinsic curvature % °f

l spatial indices art» raisad or lowered by the rrtbtric 0.» ). Relationship (9) corresponds precisely to the equation of motion that would be obtained from (6) upon arbitrary variations of & * If tue combine (9) with the remaining equations of motion obtained upon variations with respect to A» then the six dynamical Einstein equations are reproduced (we use the mixed Ricci tensor for conve- nience |l?|)s

(10)

From (10) a very interesting property of velocity denominated models can be deduced.

Suppose we are studying a cosmological situation tuhuro the spatial derivatives play no significant role (e.g. around the initial spacelike singularity). This means that we can drop J^' from (10) since this is the only term where spatial deri vativiis occur. From

(11) it follows that if K]\.; is diagonalized on the initial surface than it will remain diagonal when evolved out of ^L, . This will auraly be the case for Ú.. as well since 9u9'^%$\s\ ' The ccm elusion is thet only three components of the metric tensor will be necessary to specify the dynamics and we write wit»- XCLsO • Th» conjugate «ementa asauma tha form

«ith 2. Q »O «The *£.{£) arc tine independent fields and serve to diagonallze tha ««trie tanaor on £ (BKL |l*3| and Liang |*| Meue)eX||Lf<|*Ho form an orthonormal coveetor triad frama on^[). Wo will eoncantrata our dlacuaaion on tho quantitiaa Ô^CL and Jti 9 Bk «inca thaaa are tha variablaa wo intand to quantize while (MJL. do not ovolva and-contain no dynamics. Although our approach to quantization in §4 uses different techniques fro» that of Liang (ha uses the Wheeler-DeWitt method |lO|) we fellow him in leaving aside the effects of the constraints J^j in trie quantum theory. Classically however these constraints impose some conditions on the fields Qt QÃt Jtí',& ,ü)êi and their spatial derivatives |5|.

Now it is convenient to introduce Misner's |1B, 19| redefinition of the variables introduced above:

With these variables the (ultralocal) ADM action takes the following canonical form

(15, where. /§ is the ultralocal Hamilton!an constraint h

We could solve the constraint generator (16) so that (15) becomes the reduced action

with

* * *• * ~ ' (18)

Clearly from (17) only two components of the metric are left and Hamilton equations of motion for the pairs of canonical variables (Q«,/») » CQ,R) can De obtained upon arbitrary variations of Q. A ; P will play the rola of the Hamiltonian and evolution takes place with respect to the intrinsic time Q canonically conjugate to

Although (16) is constrained to zero we prefer not to solve it explicitly and thus we concentrate on the dynamics as given by (15). On the classical level u/e can drop all the factors in front of (16) since this u/ill not effect the dynamics and corresponds to a reescaling of the P-parameter. However previous studies in the ultralocal quantization of the gravitational field |20| have revealed that it is the quantum generator associated to

that transforms as a scalar density and not the quantum counter- 10

part of X Oj&iM^ •• one »iQM hav> anticipated. Thus» rearranging the numerical factors in front of 1£ for latar eonve nience, we will quantize in the next section the generator

Then the cosmological solutions around the singularity obtained from (IS) and (19) are

Q.. P. £*0 (20) Ó•P *'°

where the dot means derivative with respect to the proper time.

§3. quantization

The quantum theory starts with the introduction of creation and annihilation operators AvS»$* ÍU^3 » A(Z,fa $.I2)t AL A n real parameters, acting on a Hilbert space // possess ing an unit vector foS satisfying

(22)

{23) 11

The next step is to introduce an over complete set of states in

(24)

where the "wave function" y belongs to the Hubert space of squ£ re integrable complex valued functions on £*m « henceforth called snail Hubert space ' and denoted by Z » An important pro» perty is that |^^ is eignristate of the annihilation operator:

The inner product in Z. , ( T»T /"/ Y T » can be used to induce the following (normalized) inner product in rj

«her. /^(,y)

A fundamental step in this quantization ucheme is provided by the introduction of a translated annihilation opera tor

The model function Q is real and libels the representations de- tsrmlning many of their properties |6|. If ^ it not square integrable than ft and 3 are not unitarlly equivalent tinea there is no vector in U that is annihilated by 3 • *n addition the canoni- 12

cal momentum generator (see below) is not a wall defined operator, for the cosmological applications we have in mind we choose the easiest representation Ç*| } other choices are possible (see Ref. |20|). We will see that Q can be interpreted as a kind of vacuum state in the small Hilbert space (this is an overstatement since it is not square integrable» however see aquation (35b)). Since we want to quantize a velocity dominated cosmological model we expect that the ultra localized nature of the dynamical evolution must lead to a quantum theory in which all the field operators be reduced to $ -independent operators acting on some Hubert space. This is precisely the role of the small MU

bert space / • All the components Q , Qm , Q of the metric tensor will be represented as multiplication operators while the conjtj gate momenta ij ,JR * P will be expressed as differential momentum operators acting on £ . More specificaly the representations have the following structure |6,20,21|

(28>

, Xf m J 9/dfe , fo £ P/PÜ . A full justification for theés representations can be found in Klauder's paper |6j. He proves for a scalar field that the vacuum expectation value of products of field operators smeared by test functions with disjoint compact support must factoriza. This factorization of spji tially disjoint volumes ia the essence of ultralocality. Below as we proceed further in this investigation ona will get more and more convinced that we are really dealing with ultralocel operators.

Notice that acting with any of the operators (28) - (31) on H reduces to acting with A , J2 , ^ , f on l » e.9. $(i>lfy*flfàt V W> (to »•• this just use (25)). Also the commutator between two arbitrary operators

where W acts only on the variableB&>A J2 , reduce to the com 0 mutator in 4. t &jW>l*WK,qQB .

To represent the Hamiltonian one must spBcify the meaning of product of ultralocal operators at the same point on y" . The regularized product is defined to be

«jUQS'i^B. (32) from (32), (30), (31), (19) the ultralocal Hamiltonian operator should be 14

where the Hamiltonian in the small Hilbert space reads

(3*)

Here ^* is the "wave operator" #/a^*3Vi|C -d/ftfl and

is a kind of 11 V(êttá.tQ) "potential in * and it has bean in eluded to guarantee that

(35a)

with jf»Jd*/b(j) . Clearly (35») is equivalent to

(35b)

and we interpret {, as the "vacuum state" in the small Hilbert space since it is annihilated by fl .In order that (331) holds we see that V(0;0.,Q) oust satisfy

tv*c- zc v. (35c)

Thus when (^ is coqftant V*0 .

The commutation relations between all the operators of the theory are now readily obtained (rhooss

(J6) 15

(38)

We have thus eccomplished tha taak of quanti* zing coamologicel models around tha singularity in tha context of uitralocal representations. Having obtained tha field operators we can now define coherent states.

Smear «ia quantu» Meld* (},(*>, ().{*),(}(*) and the conju gate momenta £,($),£(*),£*(%) •** differentiates, real teat functiona on £ /<$*and and define the following overcomplete eet of statee

. (39)

In the smell Hubert space we can show that fftt/ corresponda to

(40) since Al?»*}}* ^ft.- 1?*%) 'Nex t ws inve8ti9ats the role of the coherent states (jo), (40) in the study of ths evolution process.

$4. Quantum Evolution

The usual way to implement a classical constraint in quantum theory is to imposs that tha corresponding quantum generator should annihilate a class of states, called physical states» When we implement (19), (33) this way, 16

Ztf) W>*O , (41) we obtain a kind of Klein-Gordon aquation in tha smell Hubert, < «O , that is •°- («)

We do not write tha jg -dependence of y due to ultralocality: there ia a copy of (42) at each g#ZL and tha "potential" V * always * -independent. Recall that ü representa the intrinsic time operator QC& in the swell Hubert space and thus (42) gives the ii -tine evolution of the "wave function"

Well known problems elwaya arise in connection with Klein-Gordon equatione in relativiatic quantum mechanics since time dependent potentials are very awkward do deal with and square root operators ara unavoidable. One can circunvent these difficul- ties by introducing the proper time evolution aa proposed by Stueckelberg |22| (see also Feynmen |23|). Since (42) is a relativistic wave equation in the variables /5#/áL#j2 ** seems natural to extrapolate the proper time method into the realms of ultralocal quantum gravity» So instead of (4l)y (42) we prescribe the following form for the evolution of the state vector:

or, equivalently in I , 1?

A which resembles a Schroedinger equation with Hasdltonian & gene- rating 0 -time evolution. If we are given an initial condition ^(JLI Q)*.ty(tyf+,f*,&) tne unique solution to (44) is written formally as

* J üf C WJ (45)

with an analogous expression for Recently many, authors have contributed to cla- rifying the use of the proper tine formal!tá in relativistic problems (for exemple |24| and references therein; for a gravity orien ted approach see |25|). In the present context the evolution as prescribed by (45) yields an interesting connection between classical and quantum evolution. Below we discuss how coherent states enable us to find such a connection.

It is a well known result that coherent states

are in general deformed |2év27| during the process of quantum evolu, tion. In other words call X the space of all coherent states (40) in the small Hilbert space I , Clearly l*C £ , but 9+1 , and

if we are gi/nn an initial state yM 1 2fr then whether or not the solution family parametrized by 0 and evolved from yL by £ still remains in Ji will depend on the form of the "potential" in the Hamilton!an (34) (we will not discuss possible enlargements of X so as to contain the evolution), for instance in one dimensional quantum mechanics the harmonic os- cillator Hamiltoniar will evolve a coherent state into another coherent state for all times* Any system with this property is called exact. Examples of quantum mechanical systems which are not exact 18

are those with cubic or higher order potentials. In this connection Klauder J26| introduced a way of finding an optimal aet of coherent states that approximate» in the &-»O limit, the evolution of systems that are not exact. In the case under diacuesion this means that it is possible to find e set of smearing functions such that

where | is a positive number (for an estimate of § in quantum mechanics see |27|). Next we will find 9 , J for the case where the model function is Cs* *n<* snow that the evolution is exact, thus no Jt-+9 approximation is necessary (in |2l| it was show*that the coherent states Iftf/ so constructed are compatible with Klau- der's optimal set).

We want to show that when Q*t we can find 9 ,0* such that

(47)

where 0 . In other words we want to solve the Schroedinger equation

(40)

The method» of R»f»32 can be readily extended to yield (46) where the model function can be more general than Ol. However see com- ments on §6* 19

with initial condition %Z • Including a phase Jtf with <*<*)« fCfS+p-Sfr) in the states Ifo^ «^ we compute both sides of (48) (from now on we drop tne -):

(49a)

Now the equality (48) is satisfied whenever #fe* , ú(0) are solu tions of the system ie ft

A quick glance at (20) reveals that the classical velocity domina^ ed solutions around the singularity have the same structure as (50) Thus iti this sense the classical solutions have been recovered in terms of test functions that smear the field operators and the co£ responding coherent state is said to be defined on the classical orbit.

The approach developed so far to describe the behaviour of the quantum gravitational field around the initial 20

singularity servas to important ganaralizations. The main obstacle to effectively carrying out these generalizations stems from the impossibility to represent the potential tarn (4) (not to ba con- fused with the "potential" Víf^A) (35c) inf) as an ultralocal operator. Several attempts at finding such an operator have not yet succeeded and it nay wall ba that we need a weaker concept |28| than that of ultralocality but still keeping the sana simplicity of ultralocal representations. In the next section wt describe some relevant classical resulta and than a proposal concerning the quasielaasical evolution of ™-states under the action of W is made.

§5. Quasiclasaical Evolution whan Ultralocality is Broken

Further contact with classical cosmology can be obtained if we relate system (5*) with the BKL solutions around the singularity |l-3|. Notice that BKL doea not uae Kamiltonian methods and that the ADH action was written in the form (15) only after we parametrized the metric tensor in terma of exponential components (12). If we now raeacale the proper time parameter cfc-* £•«/ A system (50) is solved as 21

where •*lt,m\t0ft t*^ are 0 -independent (due to ultralocality we do not write explicitly the J|" dependence of the sneering func tions). So we can reconstruct the spatial metric tensor in terms of test functions as

h<*>0*\ «*•>,] <52)

where the indices «,y refer to some orthonormal basis on Following BKL we locate the singularity in the past at £>«© and propagate the system "backwards" as &•+<> > Thus, around the sin, gularity #•,. , »fl can be dropped and (51) gives

(5J.)

where

f**&(*** satisfy the BKL conditions 22

The metric (53a) was called Kasner metric by BKL and it can be shown |l| to constitute a solution to Einstein equations in those models where the spatial derivatives are negligible as compared with 0 -derivatives.

An additional discussion of the relation (53c) needs to be made. The only way to obtain Zfí »í is to impose the condition

(54)

on the "momentum11 smearing functions. As a consequence the corres» ponding coherent atate ia called Kasner coherent state and satisfy £ in tne reprBsentation C* ' • So we are forced to work with physical (42), i.e. Kasner, states if the solution (B) around the singularity is to be recovered. This causes us no trouble since system (SO) still holds true but the ultralocal wave function 3* does not evolve in 0-time. Below we will see that the action of ff'^R. breaks ultralocality and induces transitions between these stationary Kasner coherent states.

Several remarks bearing on the results obtained so far are in order:

(i) The numbers f£%) satisfying (53c) are called Kasner indices. If we assume they are written in the order fêCft^fè they can be uniquely described by a number £€ fòf) f called Kasner parameter )l|,

at each 23

(ii) Notice that we still have the freedom to reescale in (53a) by a 0-independent parameter &-+A& so that ijt Thus to fully define an ultralocal classical solution we need to especify two functions *(f> and A fa) on £ . Comparing (50) with (20) we immediately identify-CU with P and, from (14), (9) a relationship between tU and /\ is found

(56)

Thus an ultralocal solution can also be sp«eíj»â«l by 3BÍX) and LUCZ) ' (iii) System (53b) cannot be inverted to give ftfX) in terms of ECsJ alone: we must include another parameter which clearly has to be

ft«ft(*>,*). (57)

(iv) In the small Hilbert spaced, an ultralocal cosmology is dnscribed by a Kasner coherent state W whose fl4 &-,-& dependence in the (,*f representation is simply

This resembles a plane wave solution to O in i mlOi ,#net9y. propagating in the direction (frfL) at each £é2Z. (Observe that the "momentum operators* in ,£ do not appear in (58)

since TCtC*O * KC )• The ultralocal classical and quantum andeis studied so far provide a starting point for «or* general and interesting situations where the potential ter* WmíQf\ does play • role (this is a necessary step since many important models are not velocity dominated in the asymptotic region around the singularity). In this respect a remarkable discovery of BKL |2,J| enables us to construct a general inhomogeneous solution close to the spacelike singularity. The key idea is that such a solution can be decomposed as a sequence of Kasner metric tensors of the form (53a). The action of the potential term }pj on a general solution can be shown to induce transitions between these Kasner configurations as the system is propagated towards the singularity» The effect of the spatial derivatives contained in yf is coded in a law (see |2,?| for a complete description of this interesting mechanism) that ex- presses the transitions in terms of a shift of the parameters S and CO (from (57) we could use ft and 4L as well). Sine* these parameters completely characterize the Kasner (ultralocal) metric we effectively have a succession of Kasner configurations on approach to the singularity and BKL showed that this is a general feature of Relativistic Cosmology in that region. It is important to mention that the action of W i* rather brieft the universe moves most of the time "freely" as a Kasner universe parametrized by (•»«>.) or by tytf') and «uddtnly this state is changed abruptly to som» other value of these parameters when a new Kasner regime takes place. There is an infinite succession of Kasner configurations as &-+O at each 25

On the quantum level we can use the BKL construction to see what the evolution of a quantum state towards the singularity under the action of tyr should look like. Suppose that yj could be represented in X by a differential operator W C* $* li-Ú) involving partial derivatives with respect to X*(x'.

(59)

Although this rY has not yet been found» as pointed out in the last paragraph, we can make the following conjecture |2l| about the evolution under (59). It was shown earlier (47) that the evolution under fL«-$*P1' is exact and that the presence of a "potential" |^JU3L,J)) in fv disturbs the evolution in the sense that coherent states are deformed and can only be used in the K"*° limit (46), Similarly we expect that the presence of jy will yield an evolution which it not exact and we art left With the problem of guessing what is the set of smearing functions that would approximate the evolution in the queeiclassical regime. Since in the absence of ^f in the Qu-i representation the coherent state giving the evolution was parametrized by velocity dominated (i.e. ultralocal, i.e. Kasner) classical solutions, the evolution under jV is expected to be given by the coherent state whose smearing functions are constructed from the BKL general solution around the singularity* If this be true then the evolutldn of art initial •*•*• Viu**//*» " in ^58' towardjthe singularity it approximated by a family of plane waves where the dependence on £ Of tht directions (fi,f-) vary discern tlnuously during the transitions according to the BKL construction 26-

Every aspect of tha proposed approximation sch£ me can be shown to hold |29| in those homogeneous cosmological mo dels where the £ -dependence is removed fro» the metric tensor (in this rase ]V is expressed in terms of the structure constants of the homogeneity group |30|). The most important homogeneous cos- cosmology studied so far is tha Bianchi IX universe and much of tha BKL construction outlined above was patterned after this model. One can prove that tha asymptotic distribution of the (t>

(60)

giving the probability for a Kasner configuration § to be visited during che evolution. However it is still lacking in tha literature the joint probability in tha variables C and ft) or £ and fm . In a recent study |29| an ensemble of W-states around the Bianchi IX singularity based on (60) was defined and an attempt at introducing a new concept of entropy in the physics of the very early uni^ verse was put forward. The BKL construction and the conjecture dejs cribed above may lead to an extension of these ideas to any cosmol£ gical situation.

§<>. Conclusions and Discussion

The quantum evolution of states of the gravitational field around the singularity h»§ been described in tha context of ultralocal representations» In this approach it is transpa- rent how the classical evolution can be used in order to gain unds£ standing about the qu.ntum behaviour* 27

We have chosen the simplest representation since the velocity dominated cosmologies can be easily recovered as test functions. Using the methods of |32| we can extend the discussion to mora general representations and coherent states that approxima te(46) the quantum evolution can still be found (however in this case u»e are not able to find an easy connection with classical cosmology and in this paper only the Ç*l model was considered)* We conjectured what to expect from the action of the potential term in the quasiclassical evolution of the quantum state. This approach opens the way for nontrivial applications of classical results into * the realms of the quantum theory of the gravitational field as »-

Acknowledgement

The author wishes to express his gratitude to Prof. C.3.1 sham. 28

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