Semiclassical Universe from First Principles

a r X i v :h e p -t h /0411152v 1 16 N o v 2004SPIN-2004/22

ITP-UU-04/40

Semiclassical Universe from First Principles

J.Ambj?rn a,c ,J.Jurkiewicz b and R.Loll c

a

The Niels Bohr Institute,Copenhagen University

Blegdamsvej 17,DK-2100Copenhagen ?,Denmark.

email:ambjorn@nbi.dk

b Mark Ka

c Complex Systems Research Centre,

Marian Smoluchowski Institute of Physics,Jagellonian University,

Reymonta 4,PL 30-059Krakow,Poland.

email:jurkiewicz@https://www.wendangku.net/doc/e51015511.html,.pl

c Institute for Theoretical Physics,Utrecht University,Leuvenlaan 4,NL-3584CE Utrecht,The Netherlands.email:j.ambjorn@phys.uu.nl,r.loll@phys.uu.nl Abstract Causal Dynamical Triangulations in four dimensions provide a background-independent de?nition of the sum over space-time geometries in nonperturbative quantum gravity.We show that the macroscopic four-dimensional worl

d which

emerges in the Euclidean sector of this theory is a bounce which satis?es a semi-classical equation.After integrating out all degrees of freedom except for a global scale factor,we obtain the ground state wave function of the universe as a function of this scale factor.

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1.Introduction

One important application of any theory of quantum gravity is a description of the quantum evolution of the very early universe.This is also the realm of quan-tum cosmology,which tries to capture the essence of the gravitational dynamics by quantizing only a?nite number of degrees of freedom characterizing the uni-verse as a whole.The path integral formulation of quantum cosmology came to prominence with the work of the Cambridge group and others on Euclidean quantum gravity[1],and in particular that of Hartle and Hawking[2].Central in this and related approaches is the construction of a“wave function of the uni-verse”,either as a solution of the Wheeler-DeWitt equation or a propagator for the theory(see,for example,[3,4,5,6,7]).In attempting to do this,a variety of technical and conceptual issues has to be addressed,including the choice of boundary conditions for the wave function,the unboundedness of the gravita-tional action and ensuing divergence of the Euclidean cosmological path integral, the appropriateness of the minisuperspace and/or semiclassical approximations, and the physical interpretation of the construction(see[8]for a recent concise review).

One could hope that a nonperturbative path integral formulation which does not impose any a priori symmetry restrictions on the geometry of the universe would help resolve some of these issues.“Causal Dynamical Triangulations”provide exactly such a background-independent,nonperturbative de?nition of quantum gravity,in which the sum over all space-time geometries is constructively de?ned and the causal(Lorentzian)structure of space-time plays a crucial role [9,10,11,12,13].It can be viewed as a realization of an idea of Teitelboim’s, who argued that in a(continuum)proper-time formulation of the Lorentzian gravitational path integral one should integrate over positive lapse functions only, thereby building a notion of causality into the quantum dynamics[14].

In the context of quantum cosmology it has been argued[6,15]that a tun-neling wave function`a la Vilenkin,where a universe tunnels from a vanishing to an extended three-geometry,is a special case of Teitelboim’s causal propagator between two three-geometries.In the present work,where the wave function of the universe will be constructed from?rst principles,we will indeed observe a similar phenomenon,although our interpretation in the end will be somewhat di?erent.

In[13]we reported that the approach of Causal Dynamical Triangulations,de-spite its background independence,generates a four-dimensional universe around which(small)quantum?uctuations take place.The purpose of this letter is to identify the e?ective action which determines the shape of this macroscopic4d world.Rather surprisingly we?nd that the e?ective action which describes the infrared,long-distance part of the universe is closely related to a simple minisuper-

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space action frequently considered in quantum cosmology.The only di?erences in our full quantum treatment are that(i)the unboundedness problem of the conformal mode in the Euclidean sector is cured,and(ii)the ultraviolet,short-distance part of the e?ective action is such that the solution to the Euclidean action describes a bounce from a universe of no spatial extension to one of?nite spatial size.While resembling Vilenkin’s picture of a“universe from nothing”[3], the interpretation in the present context is rather in terms of the ground state wave function of the universe with everything but the scale factor integrated out. We describe how one can determine this wave function from?rst principles.

It should be emphasized that unlike in standard minisuperspace models for cosmology,we do not assume homogeneity or isotropy,nor do we impose any other a priori symmetry conditions on the gravitational degrees of freedom.We perform the full path integral and determine the e?ective Lagrangian which describes the dynamics of the global scale factor,as well as the ground state wave function of the universe as a function of this scale parameter.

The rest of this article is organized as follows:in the next section we recall some salient features of the Causal Dynamical Triangulations approach,includ-ing the set-up of the numerical simulations and recent numerical results in four dimensions.We then demonstrate in Sec.3that the numerical data are perfectly described by a simple minisuperspace action.Sec.4outlines how this result re-lates to the ground state wave function of the universe,and the?nal Sec.5is devoted to a discussion.

2.Observing the bounce

The idea to construct a quantum theory of gravity by using Causal Dynamical Triangulations was motivated by the desire to formulate a quantum gravity theory with the correct Lorentzian signature and causal properties[14],and to have a path integral formulation which may be closely related to attempts to quantize the theory canonically.For the purposes of this letter,we will only summarize the main properties of this approach;more details on the rationale and techniques can be found elsewhere[9,10,11,12].

We insist that only causally well-behaved geometries appear in the path in-tegral,which is regularized by summing over a particular class of triangulated, piecewise?at(i.e.piecewise Minkowskian)geometries.All causal simplicial space-times contributing to the path integral are foliated by a version of“proper time”t,and each geometry can be obtained by gluing together four-simplices in a way that respects this foliation.Each four-simplex has time-like links of negative length-squared?a2t and space-like links of positive length-squared a2s,with all of the latter located in spatial slices of constant(integer in lattice units)proper time t.These slices consist of purely space-like tetrahedra,forming a three-dimensional

3

piecewise ?at manifold,whose topology we choose for simplicity to be that of a three-sphere S 3.A necessary condition for obtaining a well-de?ned continuum limit from this regularized setting is that the lattice spacing a ∝a t ∝a s goes to zero while the number N 4of four-simplices goes to in?nity in such a way that the continuum four-volume V 4:=a 4N 4stays ?xed.Let us emphasize that the parameter a therefore does not play the role of a fundamental discrete length.A further property of our explicit construction is that each con?guration can be rotated to Euclidean signature,a necessary prerequisite for discussing the con-vergence properties of the sum over geometries,as well as for using Monte Carlo techniques.

The partition function for quantum gravity is

Z (Λ,G )= D [g ]e iS [g ],S [g ]=1|det g |(R ?2Λ),(1)

where S [g ]is the Einstein-Hilbert action including a cosmological-constant term Λ,and G the gravitational constant https://www.wendangku.net/doc/e51015511.html,ing our simplicial regularization this becomes Z (Λ,G )CDT =

T

1G ΛV 4?Z E (V 4,G ),(3)

where the partition function ?Z

E (V 4,G )for ?xed four-volume is de?ned as ?Z E (V 4,G )= D [g ]e ??S E [g ]δ( d 4x G

d 4x 1W

e ignore a numerical constant multiplying G in

(1).

4

Whenever V4is kept?xed we will use?Z E(V4,G)as our partition function.It is related to Z E(Λ,G)by the Laplace transformation(3).

The speci?c(Euclidean)partition function we will consider is the so-called quantum-gravitational proper-time propagator de?ned by

G EΛ,G(g3(0),g3(t))= D[g E]e?S E[g E].(5) where the integration is over all four-dimensional(Euclidean)geometries g E of topology S3×[0,1],each with proper time running from0to t,and with spatial boundary geometries g3(0)and g3(t)at proper times0and t.

Figure1:Monte Carlo snapshot of a“typical universe”of discrete volume91.100 four-simplices and total time extent(vertical direction)t=40.The circumference at integer proper time s is proportional to the spatial three-volume V3(s).The surface represents an interpolation between adjacent spatial volumes,without capturing the actual4d connectivity between neighbouring spatial slices.

While it may be di?cult to?nd an explicit analytic expression for the full propagator(5)of the four-dimensional theory,Monte Carlo simulations are read-

5

ily available,using standard techniques from Euclidean dynamically triangulated quantum gravity[16].For convenience of the computer simulations,we keep the total four-volume V4of space-time?xed and also often use periodic rather than ?xed boundary conditions,i.e.sum over space-times with topology S3×S1rather than S3×[0,1].This periodicity does not a?ect the results reported below,as is illustrated by Fig.1.This shows the typical“shape”(spatial three-volume V3(s) as a function of proper time s)of a space-time con?guration generated by the computer.2For given space-time volume V4,as long as t is chosen su?ciently large,the con?guration will develop a thin stalk like the one shown in Fig.1.It will then not matter for the analysis of the large-scale geometry whether or not time is periodically identi?ed.

A convenient“observable”is the spatial volume-volume correlator de?ned by

C V

4(?)≡ V3(0)V3(?) V

4

=

1

V1/2

3

,V3=V3(s)+V3(s+δ).(7)

V3(s) dV3(s)

5Again we have applied?nite size scaling techniques,starting out with an arbitrary power Vα3in the denominator in(7),and have determinedα=1/2from the principle of maximal overlap of the distributions for various V3’s.

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3.Minisuperspace

Let us now consider the simplest minisuperspace model for a closed universe in quantum cosmology,as for instance used by Hartle and Hawking in their semiclassical evaluation of the wave function of the universe[2].In Euclidean signature and proper-time coordinates,the metrics are of the form

ds2=dt2+a2(t)d?23,(14) where the scale factor a(t)is the only dynamical variable and d?23denotes the metric on the three-sphere.The corresponding Einstein-Hilbert action is

S eff=1

dt 2?a(t)+λa3(t)

.(15)

If no four-volume constraint is imposed,λis the cosmological constant.If the four-volume is?xed to V4,such that the discussion parallels the computer simu-lations reported above,λshould be viewed as a Lagrange multiplier enforcing a given size of the universe.In the latter case we obtain the same e?ective action as in(13)up to an overall sign,due to the infamous conformal divergence of the classical Einstein action.Let us for the moment ignore this overall minus sign and compare the two potentials relevant for the calculation of semiclassical Euclidean solutions associated with the actions(15)and(13).The“potential”6is

V(a)=?a+λa3,(16) and is shown in Fig.4,without and with small-a modi?cation,for the stan-dard minisuperspace model and our e?ective model,respectively.The quantum-induced di?erence for small a is important since the action(13)allows for a classi-cally stable solution a(t)=0which explains the“stalk”observed in the computer simulations.Moreover,it is appropriate to speak of a Euclidean“bounce”be-cause a=0is a local maximum.If one therefore naively turns the potential upside down when rotating back to Lorentzian signature,the metastable state

a(t)=0can tunnel to a state where a(t)～V1/4

4,with a probability amplitude

per unit time which is(the exponential of)the Euclidean action.We will discuss this further in the next section.

In order to understand how well the semiclassical action(13)can reproduce the Monte Carlo data,that is,the correlator C V

4

(?)of Fig.2,we have solved for the semiclassical bounce using(13),and presented the result as the continuous

2for which the kinetic term in the actions assumes the standard quadratic form. It is the resulting potential?V(x)=?x2/3+λx2which in the case of(13)should be modi?ed for small x such that?V′(0)=0.

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Figure4:The potential V(a)of(16)underlying the standard minisuperspace dynamics (left)and the analogous potential in the e?ective action obtained from the full quantum gravity model,with small-a modi?cation due to quantum?uctuations(right).

black curve in Fig.2.7.The agreement with the real data generated by the Monte Carlo simulations is clearly perfect.

4.The wave function of the universe

The picture emerging from the above for the e?ective dynamics of the scale factor resembles that of a universe created by tunneling from nothing(see,for example, [3,4,5]),although the presence of a preferred notion of time makes our situ-ation closer to conventional quantum mechanics.In the set-up analyzed here, there is apparently a state of vanishing spatial extension which can“tunnel”to a

universe of?nite linear extension of order a～V1/4

4.Adopting such a tunneling

interpretation,the action of the bounce is

S eff V4～

V1/2

4

7More precisely,we solved the classical equation of motion corresponding to the potential shown in Fig.4on the right,with an energy slightly below zero(the closer to zero the longer the stalk),and used this solution to create an arti?cial distribution of three-volumes V3(s) analogous to the one generated by Monte Carlo simulation from?rst principles.We then treated this arti?cial distribution precisely as if it had come from real Monte Carlo data.

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lation possesses a well-de?ned Hamiltonian8which is bounded below,the ground state wave function can be chosen real and positive.In view of this,calling(18)a tunneling probability is misleading,since it would imply an oscillating behaviour for a?V1/4

4

.The correct interpretation of(18)is rather that of the square of

the ground state wave function for a～V1/4

4.

To illustrate what we have in mind,let us consider a quantum-mechanical system with Hamiltonian H=p2/2+V(x),where the minimum of the potential V is at x=0.The ground state wave function has the path integral representation

Ψ0(a)～ x(0)=a x(?∞)=0D x(t)e?S E[x(t)],(19) where S E[x(t)]is the classical Euclidean action

S E[x(t)]= 0?∞dt 1

2V(x).(21)

As an example,for the harmonic oscillator we have x cl(t)=a eωt and(21)is exact.For a general potential the semiclassical approximation will of course not be exact.Nevertheless,we have presented strong evidence that in the case of quantum gravity,integrating over all degrees of freedom except the three-volume

and de?ning a=V1/3

3,the semiclassical approximation is excellent.If we assume

that it is equally good in the absence of the four-volume constraint,we compute in a straightforward way from(21)that

Ψ0(a)～e?c

8In the framework of Causal Dynamical Triangulations one has a transfer matrix between consecutive proper-time slices whose logarithm gives in principle the full quantum Hamiltonian, see[12]for details.

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It is important to understand that the wave functionΨ0(a)can be addressed via computer simulations using a decomposition analogous to(3),namely,

Ψ0(a)= dV4e?Λ

G d4x√det g?V4),(24) where the functional integration in(24)is over four-geometries with V3(?∞)=0 and V3(t=0)=a3.The computer simulations reported here were done for the special case V3(t=0)=0.9Using(23)and(24)one can now check whether the semiclassical approximation reported here is valid for all values of a and V4.If so,one will be led to(22).

5.Discussion

Causal Dynamical Triangulations constitute a framework for de?ning quantum gravity nonperturbatively as the continuum limit of a well-de?ned regularized sum over geometries.We reported recently on the outcome of the?rst Monte Carlo simulations in four dimensions[13].Very encouragingly,we observed the dynamical generation of a macroscopic four-dimensional(Euclidean)world,with small quantum?uctuations superimposed.In this letter we showed that the scale factor characterizing the macroscopic shape of this ground state of geometry is well described by an e?ective action similar to that of the simplest minisuperspace model used in quantum cosmology.However,in our case such a result has for the ?rst time–we believe–been derived from?rst principles.

The negative sign of the kinetic term in the standard minisuperspace action (15)re?ects the well-known unboundedness of the conformal mode in the Eu-clidean Einstein-Hilbert action.Amazingly,after integrating out all variables except the scale factor(which is simply the global conformal mode),we obtain a positive kinetic term in our e?ective action(13).This is consistent with a contin-uum formulation of the gravitational path integral in proper-time gauge,where a strong argument was made for the nonperturbative cancellation of the confor-mal divergence by a measure factor coming from a Faddeev-Popov determinant [19].Similar ideas were pursued in[20],although in that case matter?elds were needed to change the sign of the conformal mode term.The phenomenon of sign change of the conformal kinetic term is also familiar from the2d Euclidean quan-tum theory where,due to the conformal anomaly,integrating out unitary matter with central charge0≤c≤1yields an e?ective term?c(?φ)2for the conformal

factorφin the action,making it unbounded below.Again it is a Faddeev-Popov determinant,arising from the requirement to integrate not over metrics,but only geometries,which adds a26(?φ)2and ensures that the combined kinetic term is positive.

A number of open issues remain to be addressed,including the details of the renormalization mechanism.Here Causal Dynamical Triangulations gives us the possibility to study Weinberg’s scenario of“asymptotic safety”[21]in the context of an explicit quantum-gravitational model.As indicated in earlier work on Causal Dynamical Triangulations in space-time dimension three,the renormalization may be non-standard[22],which in a way would be welcome. This is also supported by the present computer simulations,in the sense that no ?ne-tuning of the bare gravitational coupling constant seems to be necessary to reach the continuum limit.Details of this will be discussed elsewhere[18].

A most interesting question is of course how the above semiclassical cosmolog-ical picture is changed by the inclusion of matter?elds.We have now the chance to investigate a number of possible scenarios suggested in quantum cosmology from?rst principles.

Acknowledgment

We thank Y.Watabiki for discussions.J.A.and J.J.were supported by“Ma-PhySto”,the Center of Mathematical Physics and Stochastics,?nanced by the National Danish Research Foundation.J.J.acknowledges support by the Polish Committee for Scienti?c Research(KBN)grant2P03B09622(2002-2004). References

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