Review Article Conceptual Problems in Quantum …downloads.hindawi.com/archive/2013/509316.pdfReview...

Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 509316 17 pageshttpdxdoiorg1011552013509316

Review ArticleConceptual Problems in Quantum Gravityand Quantum Cosmology

Claus Kiefer

Institute for Theoretical Physics University of Cologne Zulpicher Strasse 77 50937 Koln Germany

Correspondence should be addressed to Claus Kiefer kieferthpuni-koelnde

Received 26 May 2013 Accepted 28 June 2013

Academic Editors M Montesinos and M Sebawe Abdalla

Copyright copy 2013 Claus Kiefer This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The search for a consistent and empirically established quantum theory of gravity is among the biggest open problems offundamental physics The obstacles are of formal and of conceptual nature Here I address the main conceptual problems discusstheir present status and outline further directions of research For this purpose the main current approaches to quantum gravityare briefly reviewed and compared

1 Quantum Theory and GravitymdashWhat Isthe Connection

According to our current knowledge the fundamental inter-actions of nature are the strong the electromagnetic theweak and the gravitational interactions The first three aresuccessfully described by the Standard Model of particlephysics in which a partial unification of the electromagneticand the weak interactions has been achieved Except for thenonvanishing neutrino masses there exists at present noempirical fact that is clearly at variance with the StandardModel Gravity is described by Einsteinrsquos theory of generalrelativity (GR) and no empirical fact is known that is inclear contradiction to GR From a pure empirical point ofview we thus have no reason to search for new physicallaws From a theoretical (mathematical and conceptual) pointof view however the situation is not satisfactory Whereasthe Standard Model is a quantum field theory describingan incomplete unification of interactions GR is a classicaltheory

Let us have a brief look at Einsteinrsquos theory see forexample Misner et al [1] It can be defined by the Einstein-Hilbert action

119878EH =1198884

16120587119866intM

1198894119909 radicminus119892 (119877 minus 2Λ) minus

1198884

8120587119866int120597M

1198893119909 radicℎ119870

(1)

where 119892 is the determinant of the metric 119877 is the Ricciscalar and Λ is the cosmological constant In additionto the two main terms which consist of integrals over aspacetime region M there is a term that is defined on theboundary 120597M (here assumed to be space like) of this regionThis term is needed for a consistent variational principlehere ℎ is the determinant of the three-dimensional metricand 119870 is the trace of the second fundamental form

In the presence of nongravitational fields (1) is aug-mented by a ldquomatter actionrdquo 119878m From the sum of theseactions one finds Einsteinrsquos field equations by variation withrespect to the metric

119866120583] = 119877120583] minus

1

2119892120583]119877 =

8120587119866

1198884119879120583] minus Λ119892120583] (2)

The right-hand side displays the symmetric (Belinfante)energy-momentum tensor

119879120583] =

2

radicminus119892

120575119878m120575119892120583]

(3)

plus the cosmological-constant term which may itself beaccommodated into the energy-momentum tensor as a con-tribution of the ldquovacuum energyrdquo If fermionic fields areadded onemust generalize GR to the Einstein-Cartan theoryor to the Poincare gauge theory because spin is the source oftorsion a geometric quantity that is identically zero in GR(see eg [2])

2 ISRNMathematical Physics

As one recognizes from (2) these equations can no longerhave exactly the same form if the quantum nature of the fieldsin 119879

120583] is taken into account For then we have operators inHilbert space on the right-hand side and classical functionson the left-hand side A straightforward generalization wouldbe to replace 119879

120583] by its quantum expectation value

119877120583] minus

1

2119892120583]119877 + Λ119892120583] =

8120587119866

1198884⟨Ψ

10038161003816100381610038161003816120583]10038161003816100381610038161003816Ψ⟩ (4)

These ldquosemiclassical Einstein equationsrdquo lead to problemswhen viewed as exact equations at the most fundamentallevel compare Carlip [3] and the references therein Theyspoil the linearity of quantum theory and even seem to be inconflict with a performed experiment [4] They may never-theless be of some value in an approximate way Independentof the problems with (4) one can try to test them in a simplesetting such as the Schrodinger-Newton equation it seemshowever that such a test is not realizable in the foreseeablefuture [5]This poses the question of the connection betweengravity and quantum theory [6]

Despite its name quantum theory is not a particulartheory for a particular interaction It is rather a general frame-work for physical theories whose fundamental concepts haveso far exhibited an amazing universality Despite the ongoingdiscussion about its interpretational foundations (which weshall address in the last section) the concepts of states inHilbert space and in particular the superposition principlehave successfully passed thousands of experimental tests

It is in fact the superposition principle that pointstowards the need for quantizing gravity In the 1957 ChapelHill Conference Richard Feynman gave the following argu-ment [7 pages 250ndash260] see also Zeh [8] He considersa Stern-Gerlach type of experiment in which two spin-12particles are put into a superposition of spinup and spindownand is guided to two counters He then imagines a connectionof the counters to a ball of macroscopic dimensions Thesuperposition of the particles is thereby transferred to asuperposition of the ball being simultaneously at two posi-tions But this means that the ballrsquos gravitational field is in asuperposition too In Feynmanrsquos own words [7 p 251])

ldquoNow how do we analyze this experiment accord-ing to quantummechanicsWe have an amplitudethat the ball is up and an amplitude that the ballis down That is we have an amplitude (from awave function) that the spin of the electron in thefirst part of the equipment is either up or downAnd if we imagine that the ball can be analyzedthrough the interconnections up to this dimension(asymp1 cm) by the quantum mechanics then beforewe make an observation we still have to give anamplitude when the ball is up and an amplitudewhen the ball is down Now since the ball is bigenough to produce a real gravitational field wecould use that gravitational field to move anotherball and amplify that and use the connections tothe second ball as the measuring equipment Wewould then have to analyze through the channel

provided by the gravitational field itself via thequantum mechanical amplitudesrdquo

In other words the gravitational field must then bedescribed by quantum states subject to the superpositionprinciple The same argument can of course be applied tothe electromagnetic field (for which we have strong empiricalsupport that it is of quantum nature)

Feynmanrsquos argument is of course not an argument oflogic that would demand the quantization of gravity withnecessity It is an argument based on conservative heuristicideas that proceed from the extrapolation of establishedand empirically confirmed concepts (here the superpositionprinciple) beyond their present range of application It is inthis way that physics usually evolves Albers et al [9] containa detailed account of arguments that demand the necessityof quantizing the gravitational field It is shown that all thesearguments are of a heuristic value and that they do notlead to the quantization of gravity by a logical conclusionIt is conceivable for example that the linearity of quantumtheory breaks down in situations where the gravitational fieldbecomes strong see for example Penrose [10] Singh [11] andBassi et al [12] But one has to emphasize that no empiricalhint for such a drastic modification exists so far

Alternatives to the direct quantization of the gravitationalfield include what is called ldquoemergent gravityrdquo comparePadmanabhan [13] and the references therein Motivated bythe thermodynamic properties of black holes (see below)one might get the impression that the gravitational field isan effective thermodynamic entity that does not demand itsdirect quantization but points to the existence of new so farunknownmicroscopic degrees of freedomunderlying gravityEven if this was the case (which is far from clear) theremay existmicroscopic degrees of freedom forwhich quantumtheory would apply Whether this leads to a quantized metricor not is not clear In string theory (see below) gravity is anemergent interaction but still the metric is quantized andthe standard approach of quantum gravitational perturbationtheory is naturally implemented In the following we restrictourselves to approaches in which a quantized metric makessense

Besides the general argument put forward by Feynmanthere exist a couple of further arguments that suggest thatthe gravitational interaction be quantized [6] Let me brieflyreview three of them

The first motivation comes from the continuation of thereductionist programme In physics the idea of unificationhas been very successful culminating so far in the StandardModel of strong and electroweak interactions Gravity actsuniversally to all forms of energies A unified theory of allinteractions including gravity should thus not be a hybridtheory in using classical and quantum concepts A coherentquantum theory of all interactions (often called ldquotheory ofeverythingrdquo or TOE) should thus also include a quantumdescription of the gravitational field

A second motivation is the unavoidable presence ofsingularities in Einsteinrsquos theory of GR see for exampleHawking and Penrose [14] and Rendall [15] Prominentexamples are the cases of the big bang and the interior of

ISRNMathematical Physics 3

black holes One would thus expect that a more fundamentaltheory encompassing GR does not predict any singularitiesThis would be similar to the case of the classical singularitiesfrom electrodynamics which are avoided in quantum elec-trodynamics The fate of the classical singularities in someapproaches to quantum gravity will be discussed below

A third motivation is known as the ldquoproblem of timerdquo Inquantummechanics time is absoluteThe parameter 119905 occur-ring in the Schrodinger equation has been directly inher-ited from Newtonian mechanics and is not turned into anoperator In quantum field theory time by itself is no longerabsolute but the four-dimensional spacetime is it constitutesthe fixed background structure onwhich the dynamical fieldsact GR is of a very different nature According to the Einsteinequations (2) spacetime is dynamical acting in a complicatedmannerwith energymomentumofmatter andwith itselfTheconcepts of time (spacetime) in quantum theory and GR arethus drastically different and cannot both be fundamentallytrue One thus needs a more fundamental theory with acoherent notion of time The absence of a fixed backgroundstructure is also called ldquobackground independencerdquo (cf [16])and often used a leitmotiv in the search for a quantum theoryof gravity although it is not quite the same as the problem oftime as we shall see below

A central problem in the search for a quantum theoryof gravity is the current lack of a clear empirical guidelineThis is partly related to the fact that the relevant scaleson which quantum effects of gravity should definitely berelevant are far remote from being directly explorable Thescale is referred to as the Planck scale and consists of thePlanck length 119897P Planck time 119905P and Planck mass 119898Prespectively They are given by the expressions

119897P = radicℎ119866

1198883asymp 162 times 10

minus33 cm

119905P =119897P119888= radic

ℎ119866


minus44 s

119898P =ℎ

119897P119888= radic

ℎ119888


minus5 g

asymp 122 times 1019 GeVc2

(5)

It must be emphasized that units of length time and masscannot be formed out of 119866 and 119888 (GR) or out of ℎ and 119888

(quantum theory) aloneIn view of the Planck scale the Standard Model provides

an additional motivation for quantum gravity it seems thatthe Standard Model does not exist as a consistent quantumfield theory up to arbitrarily high energies see for exampleNicolai [17] The reasons for this failure may be a potentialinstability of the effective potential and the existence ofLandau poles The Standard Model can thus by itself not bea fundamental theory although with the recently measuredHiggsmass119898H of about 126GeV it is in principle conceivablethat it holds up to the Planck scale It is so far an openissue why the Higgs mass is stabilized at such a low value

and not driven to high energies by quantum loop corrections(ldquohierarchy problemrdquo) In fact one has

(119898P119898H

)

2

sim 1034 (6)

Possible solutions to the hierarchy problem include super-symmetric models andmodels with higher dimensions but aclear solution is not yet available

In astrophysics the Planck scale is usually of no relevanceThe reason is that structures in the Universe with theexception of black holes occur at (length time and mass)scales that are different from the Planck scale by many ordersof magnitude This difference is quantified by the ldquofine-structure constant of gravityrdquo defined by

120572119892=

1198661198982

pr

ℎ119888= (

119898pr

119898P)

2

asymp 591 times 10minus39 (7)

where 119898pr denotes the proton mass The Chandrasekharmass for example which gives the correct order ofmagnitudefor main sequence stars like the Sun is given by 120572minus32

119892119898pr

Let us have at the end of this section a brief look at theconnection between quantum theory and gravity at the levelwhere gravity is treated as a classical interaction [6] Thelowest level is quantum mechanics plus Newtonian gravityat which many experimental tests exist Most of them employatom or neutron interferometry and can be described by theSchrodinger equation with the Hamiltonian given by

119867 =p2

2119898+ 119898gr minus 120596L (8)

where the second term is the Newtonian potential in thelimit of constant gravitational acceleration and the last termdescribes a coupling between the rotation of the Earth (oranother rotating system) and the angular momentum of theparticle An interesting recent suggestion in this context is thepossibility to see the general relativistic time dilatation in theinterference pattern produced by the interference of two par-tial particle beams at different heights in the gravitational field[18] Amore general treatment is based on the Dirac equationand its nonrelativistic (ldquoFoldy-Wouthuysenrdquo) expansion

The next level is quantum field theory in a curvedspacetime (or in a flat spacetime but in noninertial coor-dinates) Here concrete predictions are available althoughthey have so far not been empirically confirmed Perhapsthe most famous prediction is the Hawking effect accordingto which every stationary black hole is characterized by thetemperature [19]

119879BH =ℎ120581

2120587119896B119888 (9)

where 120581 is the surface gravity of a stationary black holewhich by the no-hair theorem is uniquely characterized by itsmass119872 its angular momentum 119869 and (if present) its electriccharge 119902 In the particular case of the spherically symmetricSchwarzschild black hole one has 120581 = 11988844119866119872 = 119866119872119877

2

S


where 119877S = 21198661198721198882 is the Schwarzschild radius and

therefore

119879BH =ℎ119888

3

8120587119896B119866119872asymp 617 times 10

minus8(119872

⊙

119872) K (10)

The presence of a temperature for black holes means thatthese objects have a finite lifetime Upon radiating awayenergy they become hotter releasing evenmore energy untilall the mass (or almost all the mass) has been radiated awayFor the lifetime of a Schwarzschild black hole one finds theexpression [20]

120591BH asymp 407(119891(119872

0)

1535)

minus1

(119872

1010 g)

3

s

asymp 624 times 10minus27119872

3

0[g] 119891minus1

(1198720) s

(11)

where 119891(1198720) is a measure of the number of emitted particle

species it is normalized to 119891(1198720) = 1 for 119872

0≫ 10

17 gwhen only effectively massless particles are emitted If onesums over the contributions from all particles in the StandardModel up to an energy of about 1TeV one finds 119891(119872) =

1535 which motivates the occurrence of this number in(11) Because the temperature of black holes that result fromstellar collapse is too small to be observable the hope is thatprimordial black holes exist for which the temperature canbecome high [21]

Since black holes have a temperature they also have anentropy It is called ldquoBekenstein-Hawking entropyrdquo and isfound from thermodynamic arguments to be given by theuniversal expression

119878BH =119896B119860

41198972

P (12)

where 119860 is the surface of the event horizon For the specialcase of the Schwarzschild black hole it reads

119878BH =119896B120587119877

2

S119866ℎ

asymp 107 times 1077119896B(

119872

119872⊙

)

2

(13)

All these expressions depend on the fundamental constantsℎ 119866 and 119888Theymay thus provide a key for quantumgravity

In flat spacetime an effect exists that is analogous tothe black-hole temperature (9) If an observer moves withconstant acceleration through the standard Minkowski vac-uum heshe will perceive this state not as empty but as filledwith thermal particles (ldquoUnruh effectrdquo) see Unruh [22] Thetemperature is given by the ldquoDavies-Unruh temperaturerdquo

119879DU =ℎ119886

2120587119896B119888asymp 405 times 10

minus23119886 [

cms2] K (14)

The similarity of (14) and (9) is connected with the presenceof an event horizon in both cases A suggestion for anexperimental test of (14) can be found inThirolf et al [23]

Many of the open questions to be addressed in any theoryof quantum gravity are connected with the temperature andthe entropy of black holes [24] The two most importantquestions concern the microscopic interpretation of entropy

and the final fate of a black hole the latter is deeplyintertwined with the problem of information loss

The Bekenstein-Hawking entropy (12) was derived fromthermodynamic considerations without identifying appro-priate microstates and performing a counting in the sense ofstatistical mechanics Depending on the particular approachto quantum gravity various ways of counting states have beendeveloped In most cases one can get the behaviour that thestatistical entropy 119878stat prop 119860 although not necessarily with thedesired factor occurring in (12)

Hawkingrsquos calculation leading to (9) breaks down whenthe black hole becomes small and effects of full quantumgravity are expected to come into play This raises thefollowing question According to (9) the radiation of theblack hole is thermal What then happens in the final phaseof the black-hole evolution If only thermal radiation wasleft all initial states that lead to a black hole would endup in one and the same final statemdasha thermal state that isthe information about the initial state would be lost Thisis certainly in contradiction with standard quantum theoryfor a closed system for which the von Neumann entropy119878 = minus119896B tr(120588 ln 120588) is constant where 120588 denotes the densitymatrix of the systemThis possible contradiction is called theldquoinformation-loss paradoxrdquo or ldquoinformation-loss problemrdquofor black holes [25]

Much has been said since 1976 about the information-loss problem (see eg [26 27]) without final consensusThis is not surprising because the final solution will only beobtained if the final theory of quantum gravity is availableSome remarks can be made though Hawkingrsquos originalcalculation does not show that black-hole radiation is strictlythermal It only shows that the expectation value of theparticle-number operator is strictly thermal There certainlyexist pure quantum states which lead to such a thermalexpression [28 29] A relevant example is a two-modesqueezed state which after tracing out one mode leads to thedensity matrix of a canonical ensemble such a state can betaken as a model for a quantum state entangling the interiorand exterior of the black hole see Kiefer [28 29] and thereferences therein

The same can be said about the Unruh effect Thetemperature (14) can be understood as arising from tracingout degrees of freedom in the total quantum state which ispure [30] In the case of a moving mirror the pure quantumstate can exhibit thermal behaviour without any informationloss (see eg [31 Section 44])

For a semiclassical black hole therefore the information-loss problem does not arise The black hole can and in factmust be treated as an open quantum system so that a mixedstate emerges from the process of decoherence [32] The totalstate of system and environment (which means everythinginteracting with the black hole) can be assumed to stay in apure state Still there is some discussion whether somethingunusual happens at the horizon even for a large black hole[33] but this is a contentious issue

If the black hole approaches the Planck regime thequestion about the information-loss problem is related to thefate of the singularity If the singularity remains in quantumgravity (which is highly unlikely) information will indeed be


destroyed Most approaches to quantum gravity indicate thatentropy is conserved for the total system so there will notbe an information-loss problem in accordance with standardquantum theory How the exact quantum state during thefinal evaporation phase looks like is unclear One can makeoversimplified models with harmonic oscillators [34] butan exact solution from an approach to quantum gravity iselusive

2 Main Approaches to Quantum Gravity

Following Isham [35] one can divide the approaches roughlyinto two classes In the first class one starts from a givenclassical theory of gravity and applies certain quantizationrules to arrive at a quantum theory of gravity In mostcases the starting point is GR This does not yet lead to aunification of interactions one arrives at a separate quantumtheory for the gravitational field in analogy to quantumelectrodynamics (QED)Most likely the resulting theory is aneffective theory only valid only in certain situations and forcertain scales ldquoit is generally believed today that the realistictheories that we use to describe physics at accessible energiesare what are known ldquoeffective field theoriesrdquordquo [36 p 499]Depending on the method used one distinguishes betweencovariant and canonical quantum gravity

The second class consists of approaches that seek toconstruct a unified theory of all interactions Quantumaspects of gravity are then seen only in a certain limitmdashin thelimit where the various interactions become distinguishableThe main representative of this second class is string theory

In the rest of this section I shall give a brief overview ofthe main approaches For more details I refer to Kiefer [6]In most expressions units are chosen with 119888 = 1

21 Covariant QuantumGravity In covariant quantum grav-ity one employs methods that make use of four-dimensionalcovariance Today this is usually done by using the quantumgravitational path integral see for example Hamber [37]Formally the path integral reads

119885 [119892] = intD119892120583] (119909) e

i119878[119892120583](119909)] (15)

where the sum runs over all metrics on a four-dimensionalmanifoldM quotiented by the diffeomorphism groupDiffMIn addition one may wish to perform a sum over all topolo-gies because they may also be subject to the superpositionprinciple This is however not possible in full generalitybecause four manifolds are not classifiable Considerable caremust be taken in the treatment of the integration measure Inorder to make it well defined one has to apply the Faddeev-Popov procedure known from gauge theories

One application of the path integral (15) is the derivationof Feynman rules for the perturbation theory One makes theansatz

119892120583] = 119892120583] +

radic32120587119866119891120583] (16)

where 119892120583] denotes the background field with respect to which

covariance is implemented in the formalism and 119891120583] denotes

the quantized field (the ldquogravitonsrdquo) with respect towhich theperturbation theory is performed Covariance with respect tothe background metric means that no particular backgroundis distinguished in this sense ldquobackground independencerdquo isimplemented into the formalism

There is an important difference in the quantum grav-itational perturbation theory as compared to the StandardModel the theory is nonrenormalizable This means thatone encounters a new type of divergences at each orderof perturbation theory resulting in an infinite number offree parameters For example at two loops the followingdivergence in the Lagrangian is found [38]

L(div)2-loop =

209ℎ2

2880

32120587119866

(161205872)2

120598

radicminus119892119877120572120573

120574120575119877120574120575

120583]119877120583]

120572120573 (17)

where 120598 = 4 minus 119863 with 119863 being the number of spacetimedimensions and 119877

120583]

120572120573 and so forth denotes the Riemann

tensor corresponding to the background metricNew developments have given rise to the hope that a

generalization of covariant quantum general relativity maynot even be renormalizable but even finitemdashthis is 119873 = 8

supergravity Bern et al [39] have found that the theory isfinite at least up to four loops This indicates that a hithertounknown symmetry may be responsible for the finiteness ofthis theory Whether such a symmetry really exists and whatits nature could be are not known at present

Independent of the problem of nonrenormalizability onecan study covariant quantum gravity at an effective leveltruncating the theory at for example the one-loop levelAt this level concrete predictions can be made because theambiguity from the free parameters at higher order doesnot enter One example is the calculation of the quantumgravitational correction to the Newtonian potential betweentwo masses [40] The potential at one-loop order reads

119881 (119903) = minus119866119898

1119898

2

119903

times (1 + 3119866 (119898

1+ 119898

2)

1199031198882+

41

10120587

119866ℎ

11990321198883+ O (119866

2))

(18)

where the first correction term is an effect from classical GRand only the second term is a genuine quantum gravitationalcorrection term (which is however too small to be measur-able)

Apart from perturbation theory expressions such as (15)for the path integral can in four spacetime dimensionsand higher only be defined and evaluated by numericalmethods One example is Causal Dynamical Triangulation(CDT) [41] Here spacetime is foliated into a set of four-dimensional simplices and Monte Carlo methods are usedfor the path integral It was found that spacetime appearsindeed effectively as four-dimensional in the macroscopiclimit but becomes two-dimensional when approaching thePlanck scale

This effective two-dimensionality is also seen in anotherapproach of covariant quantum gravity-asymptotic safety(see eg [42]) A theory is called asymptotically safe if all


essential coupling parameters approach for large energiesa fixed point where at least one of them does not vanish(if they all vanish one has the situation of asymptoticfreedom) Making use of renormalization group equationsstrong indications have been found that quantum GR isasymptotically safe If true this would be an example fora theory of quantum gravity that is valid at all scales Thesmall-scale structure of spacetime is among themost excitingopen problems in quantum gravity see Carlip [43] and thearticles collected in the volume edited by Amelino-Cameliaand Kowalski-Glikman [44]

22 Canonical Approaches In canonical approaches to quan-tum gravity one constructs a Hamiltonian formalism atthe classical level before quantization In this procedurespacetime is foliated into a family of spacelike hypersurfacesThis leads to the presence of constraints which are connectedwith the invariances of the theory One has four (local)constraints associated with the classical diffeomorphismsOne is the Hamiltonian constraint H

perp which generates

hypersurface deformations (many-fingered time evolution)the three other constraints are the momentum or diffeomor-phism constraints H

119886 which generate three-dimensional

coordinate transformations If one uses tetrads instead ofmetrics three additional constraints (ldquoGauss constraintsrdquo)associated with the freedom of performing local Lorentztransformations are present Classically the constraints obeya closed (but not Lie) algebra For the exact relation of theconstraints to the classical spacetime diffeomorphisms seefor example Pons et al [45] and Barbour and Foster [46]

By quantization the constraints are turned into quantumconstraints for physically admissible wave functionals Theexact form depends on the choice of canonical variables Ifone uses the three-dimensional metric as the configurationvariable one arrives at quantum geometrodynamics If oneuses a certain holonomy as the configuration variable onearrives at loop quantum gravity

221 Quantum Geometrodynamics In geometrodynamicsthe canonical variables are the three-metric ℎ

119886119887(119909) and its

conjugate momentum 119901119888119889(119910) which is linearly related to the

second fundamental form In the quantum theory they areturned into operators that obey the standard commutationrules

[ℎ119886119887(119909) 119901

119888119889(119910)] = iℎ120575119888

(119886120575119889

119887)120575 (119909 119910) (19)

Adopting a general procedure suggested by Dirac the con-straints are implemented as quantum constraints on the wavefunctionals

HperpΨ = 0

H119886Ψ = 0

(20)

Thefirst equation is calledWheeler-DeWitt equation [47 48]the three other equations are called quantum momentumor diffeomorphism constraints The latter guarantee that thewave functional is invariant under three dimensional coor-dinate transformations The configuration space of all three-metrics divided by three-dimensional diffeomorphisms iscalled superspace

In the vacuum case the above equations assume theexplicit form

HperpΨ = (minus16120587119866ℎ

2119866119886119887119888119889

1205752

120575ℎ119886119887120575ℎ

119888119889

minusradicℎ

16120587119866((3)119877 minus 2Λ))Ψ = 0

(21)

H119886Ψ = minus2119863

119887ℎ119886119888

ℎ

i120575Ψ

120575ℎ119887119888

= 0 (22)

Here119863119887is the three-dimensional covariant derivative 119866

119886119887119888119889

is the DeWitt metric (which is an ultralocal function of thethree-metric) and (3)

119877 is the three-dimensional Ricci scalarIn the presence of nongravitational fields their Hamiltoniansare added to the expressions (21) and (22)

There are various problems connected with these equa-tions One is the problem to make mathematical sense outof them and to look for solutions This includes the imple-mentation of the positive definiteness of the three-metricwhich may point to the need of an ldquoaffine quantizationrdquo[49] Other problems are of conceptual nature and will bediscussed below Attempts to derive the entropy (12) existin this framework see for example Vaz et al [50] but thecorrect proportionality factor between the entropy and 119860 hasnot yet been reproduced

222 Loop Quantum Gravity In loop quantum gravityvariables are used that are conceptually closer to Yang-Millstype of variables The loop variables have grown out ofldquoAshtekarrsquos new variablesrdquo [51] which are defined as followsThe role of themomentumvariable is played by the densitizedtriad (dreibein)

119864119886

119894(119909) = radicℎ (119909) 119890

119886

119894(119909) (23)

while the configuration variable is the connection

119866119860119894

119886(119909) = Γ

119894

119886(119909) + 120573119870

119894

119886(119909) (24)

Here 119886(119894) denotes a space index (internal index) Γ119894119886(119909) is

the spin connection and 119870119894

119886(119909) is related to the second

fundamental form The parameter 120573 is called Barbero-Immirzi parameter and can assume any nonvanishing realvalue In loop quantum gravity it is a free parameter Itmay be fixed by the requirement that the black-hole entropycalculated from loop quantum cosmology coincides withthe Bekenstein-Hawking expression (12) One thereby finds(from numerically solving an equation) the value 120573 =

023753 see Agullo et al [52] and the references therein Itis of interest to note that the proportionality between entropyand area can only be obtained if the microscopic degrees offreedom (the spin networks) are distinguishable (cf [53])

Classically Ashtekarrsquos variables obey the Poisson-bracketrelation

119860119894

119886(119909) 119864

119887

119895(119910) = 8120587120573120575

119894

119895120575119887

119886120575 (119909 119910) (25)


The loop variables are constructed from these variables ina nonlocal fashion The new connection variable is theholonomy 119880[119860 120572] which is a path-ordered exponential ofthe integral over the connection (24) around a loop 120572 In thequantum theory it acts on wave functionals as

[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)

Thenewmomentumvariable is the flux of the densitized triadthrough a two-dimensional surface S bounded by the loopIts operator version reads

119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)

where the embedding of the surface is given by (1205901 1205902) equiv

997891rarr 119909119886(120590

1 120590

2) The variables obey the commutation

relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)

where 120580(120572S) = plusmn1 0 is the ldquointersection numberrdquo whichdepends on the orientations of 120572 and S Given certain mildassumptions the holonomy-flux representation is uniqueand gives rise to a unique Hilbert-space structure at thekinematical level that is before the constraints are imposed

At this level one can define an area operator for whichone obtains the following spectrum

A [S] Ψ119878[119860] = 8120587120573119897

2

P sum

Pisin119878capSradic119895P (119895P + 1)Ψ119878

[119860]

= 119860 [S] Ψ119878[119860]

(29)

Here P denotes the intersection points between the spin-network 119878 and the surface S and the 119895P can assume integerand half-integer values (arising from the use of the groupSU(2) for the triads) There thus exists a minimal ldquoquantumof actionrdquo of the order of 120573 times the Planck-length squaredComprehensive discussions of loop quantum gravity can befound in Gambini and Pullin [54] Rovelli [55] Ashtekar andLewandowski [56] andThiemann [57]

Here we have been mainly concerned with the canonicalversion of loop quantumgravity But there exists also a covari-ant version it corresponds to a path-integral formulationthrough which the spin networks are evolved ldquoin timerdquo It iscalled the spin-foam approach compare Rovelli [58] and thereferences therein

23 String Theory String theory is fundamentally differentfrom the approaches discussed so far It is not a directquantization of GR or any other classical theory of gravity Itis an example for a unified quantum theory of all interactionsGravity as well as the other known interactions only emergesin an appropriate limit (this is why string theory is anexample of ldquoemergent gravityrdquo) Strings are one-dimensionalobjects characterized by a dimensionful parameter 1205721015840 or thestring length 119897s = radic21205721015840ℎ In spacetime it forms a two-dimensional surface the worldsheet Closer inspection of

the theory exhibits also the presence of higher-dimensionalobjects called D-branes which are as important as the stringsthemselves compare Blumenhagen et al [59]

String theory necessarily contains gravity because thegraviton appears as an excitation of closed strings It isthrough this appearance that a connection to covariantquantum gravity discussed above can be made String theoryalso includes gauge theories since the corresponding gaugebosons are found in the spectrum It also requires thepresence of supersymmetry for a consistent formulationFermions are thus an important ingredient of string theoryOne recognizes that gravity other fields and matter appearon the same footing

Because of reparametrization invariance on the world-sheet string theory also possesses constraint equations Theconstraints do however not close but contain a central termon the right-hand side This corresponds to the presenceof an anomaly (connected with Weyl transformations) Thevanishing of this anomaly can be achieved if ghost fields areadded that gain a central termwhich cancels the original oneThe important point is that this works only in a particularnumber 119863 of dimensions 119863 = 26 for the bosonic stringand 119863 = 10 for the superstring (or 119863 = 11 in M-theory)The presence of higher spacetime dimensions is an essentialingredient of string theory

Let us consider for simplicity the bosonic string Itsquantization is usually performed through the Euclidean pathintegral

119885 = intD119883Dℎeminus119878P (30)

where 119883 and ℎ are a shorthand for the embedding variablesand the worldsheet metric respectively The action in theexponent is the ldquoPolyakov actionrdquo which is an action definedon the worldsheet Besides the dynamical variables 119883 and ℎit contains various background fields on spacetime amongthem the metric of the embedding space and a scalar fieldcalled dilaton It is obvious that this formulation is notbackground independent In as much string theory (or 119872-theory) can be formulated in a fully background independentway is a controversial issue It has been argued that partialbackground independence is implemented in the context ofthe AdSCFT conjecture see Section 4 below

If the string propagates in a curved spacetime withmetric119892120583] the demand for the absence of a Weyl anomaly leads to

consistency equations that correspond (up to terms of order1205721015840) to Einstein equations for the background fields These

equations can be obtained froman effective action of the form

119878eff prop int119889119863119909 radicminus119892 eminus2Φ

times (119877 minus2 (119863 minus 26)

31205721015840minus1

12119867

120583]120588119867120583]120588

+4nabla120583Φnabla

120583Φ + O (120572

1015840) )

(31)


where Φ is the dilaton 119877 is the Ricci scalar correspondingto 119892

120583] and 119867120583]120588 is the field strength associated with an

antisymmetric tensor field (which in 119863 = 4 would bethe axion) This is the second connection of string theorywith gravity after the appearance of the graviton as a stringexcitation A recent comprehensive overview of string theoryis Blumenhagen et al [59]

Attempts were made in string theory to derive theBekenstein-Hawking entropy from amicroscopic counting ofstates In this context theD-branes turned out to be of centralimportance Counting D-brane states for extremal and close-to-extremal string black holes one was able to derive (12)including the precise prefactor [60 61] For the Schwarzschildblack hole however such a derivation is elusive

Originally string theory was devised as a ldquotheory ofeverythingrdquo in the strict sense This means that the hope wasentertained to derive all known physical laws including allparameters and coupling constants from this fundamentaltheory So far this hope remains unfulfilled Moreover thereare indications that this may not be possible at all due to theldquolandscape problemrdquo string theory seems to lead to manyground states at the effective level the number exceeding10

500 [62] Without any further idea it seems that a selectioncan be made only on the basis of the anthropic principlecompare Carr [63] If this was the case string theory wouldno longer be a predictive theory in the traditional sense

24 Other Approaches Besides the approaches mentioned sofar there exist further approaches which are either meantto be separate quantum theories of the gravitational fieldor candidates for a fundamental quantum theory of allinteractions Some of them have grown out of one of theabove approaches others have been devised from scratchIt is interesting to note that most of these alternatives startfrom discrete structures at the microscopic level Amongthem are such approaches as causal sets spin foams groupfield theory quantum topology and theories invoking sometype of noncommutative geometry Most of them have notbeen developed as far as the approaches discussed above Anoverview can be found in Oriti [64]

3 Quantum Cosmology

Quantum cosmology is the application of quantum theory tothe universe as a whole Conceptually this corresponds to theproblem of formulating a quantum theory for a closed systemfrom within without reference to any external observersor measurement agencies In its concrete formulation itdemands a quantum theory of gravity since gravity is thedominating interaction at large scales On the one handquantum cosmology may serve as a testbed for quantumgravity in a mathematically simpler setting This concernsin particular the conceptual questions which are of concernhere On the other hand quantum cosmologymay be directlyrelevant for an understanding of the real Universe Generalintroductions into quantum cosmology include Coule [65]Halliwell [66] Kiefer [6] Kiefer and Sandhofer [67] and

Wiltshire [68] A discussion in the general context of cos-mology can be found in Montani et al [69] Supersymmetricquantum cosmology is discussed at depth in Moniz [70] Anintroduction to loop quantum cosmology is Bojowald [71]A comparison of standard quantum cosmology with loopquantum cosmology can be found in Bojowald et al [72]

Quantum cosmology is usually discussed for homoge-neous models (the models are then called minisuperspacemodels) The simplest case is to assume also isotropy Thenthe line element for the classical spacetime metric is given by

1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2

3is the line element of a constant curvature space

with curvature index 119896 = 0 plusmn1 In order to consider amatter degree of freedom a homogeneous scalar field 120601 withpotential 119881(120601) is added

In this setting the momentum constraints (22) are iden-tically fulfilled The Wheeler-DeWitt equation (21) becomesa two-dimensional partial differential equation for a wavefunction 120595(119886 120601)

(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)

where Λ is the cosmological constant and 1205812

= 8120587119866Introducing120572 equiv ln 119886 (which has the advantage to have a rangefrom minusinfin to +infin) one obtains the following equation

(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)

Most versions of quantumcosmology take Einsteinrsquos theory asthe classical starting point More general approaches includesupersymmetric quantum cosmology [70] string quantumcosmology (2003) noncommutative quantum cosmology(see eg [73]) Ho rava-Lifshitz quantum cosmology [74]and third-quantized cosmology [75]

In loop quantum cosmology features from full loopquantum gravity are imposed on cosmological models [71]Since one of the main features is the discrete nature ofgeometric operators the Wheeler-DeWitt equation (33) isreplaced by a difference equation This difference equationbecomes indistinguishable from the Wheeler-DeWitt equa-tion at scales exceeding the Planck length at least in certainmodels Because the difference equation is difficult to solve ingeneral one makes heavy use of an effective theory [76]

Many features of quantum cosmology are discussed inthe limit when the solution of the Wheeler-DeWitt equationassumes a semiclassical or WKB form [66] This holds inparticular when the no-boundary proposal or the tunnelingproposal is investigated for concrete models see below It haseven been suggested that the wave function of the universe


be interpreted only in the WKB limit because only then atime parameter and an approximate (functional) Schrodingerequation are available [77]

This can lead to a conceptual confusion Implications forthe meaning of the quantum cosmological wave functionsshould be derived as much as possible from exact solutionsThis is because the WKB approximation breaks down inmany interesting situations even for a universe of macro-scopic size One example is a closed Friedmann universewith a massive scalar field [78] The reason is the followingFor a classically recollapsing universe one must impose theboundary condition that the wave function goes to zero forlarge scale factors Ψ rarr 0 for large 119886 As a consequencenarrow wave packets do not remain narrow because of theensuing scattering phase shifts of the partial waves (that occurin the expansion of the wave function into basis states) fromthe turning point The correspondence to the classical modelcan only be understood if the quantum-to-classical transitionin the sense of decoherence (see below) is invoked

Another example is the case of classically chaotic cos-mologies see for example Calzetta and Gonzalez [79] andCornish and Shellard [80] Here one can see that the WKBapproximation breaks down in many situations This is ofcourse a situation well known from quantum mechanicsOne of the moons of the planet Saturn Hyperion exhibitschaotic rotational motion Treating it quantummechanicallyone recognizes that the semiclassical approximation breaksdown and that Hyperion is expected to be in an extremelynonclassical state of rotation (in contrast towhat is observed)This apparent conflict between theory and observation can beunderstood by invoking the influence of additional degrees offreedom in the sense of decoherence see Zurek and Paz [81]and Section 3343 in [82]The samemechanism should curethe situation for classically chaotic cosmologies [83]

4 The Problem of Time

The problem of time is one of the major conceptual issuesin the search for a quantum theory of gravity [84ndash86] Aswas already emphasized above time is treated differentlyin quantum theory and in general relativity Whereas it isabsolute in the first case it is dynamical in the second Thisis the reason why GR is background independentmdasha majorfeature to consider in the quantization of gravity

Background independence is only part of the problemof time If one applies the standard quantization rules toGR spacetime disappears and only space remains Thiscan be understood in simple terms Classically spacetimecorresponds to what is a particle trajectory in mechanicsUpon quantization the trajectory vanishes and only theposition remains In GR spacetime vanishes and only thethree-metric remains Time has disappeared

This is explicitly seen by the timeless nature of theWheeler-DeWitt equation (21) or the analogous equationin loop quantum gravity How can one interpret such asituation Equation (21) results from a classical constraintin which all momenta occur quadratically If one could

reformulate this constraint in a way where one canonicalmomentumappears linearly its quantized formwould exhibita Schrodinger-type equation Concretely one would haveclassically

P119860+ ℎ

119860= 0 (35)

where P119860

is a momentum for which one can solve theconstraint ℎ

119860simply stands for the remaining terms Upon

quantization one obtains

iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)

This is of a Schrodinger form The dynamics of this equationis in general inequivalent to the Wheeler-DeWitt equationUsually ℎ

119860is referred to as a ldquophysical Hamiltonianrdquo because

it actually describes an evolution in a physical parameternamely the coordinate 119902

119860conjugate to P

119860

There are many problems with such a ldquochoice of timebefore quantizationrdquo [86] The constraints of GR cannot beput globally into the form (35) [87] In most cases theoperator ℎ

119860cannot be defined rigorously It has therefore

been suggested to introduce dustmatterwith the sole purposeto define a standard of time [88] More recently a masslessscalar field is used in loop quantum cosmology for thispurpose compare Ashtekar and Singh [89] One can alsodefine such an internal time from an electric field [90] Atan effective level one can choose different local internaltimes within the same model [91] It is however not verysatisfactory to adopt the concept of time to the particularmodel under consideration

Most of these discussions make use of the Schrodingerpicture of quantum theory An interesting perspective on theproblem of time using the Heisenberg picture is presented inRovelli [92] It leads to the notion of an lsquoevolving constantof motionrsquo which unifies the intuitive idea of an evolutionwith the timelessness of quantum gravity the evolutionproceeds with respect to a physical ldquoclockrdquo variable Anexplicit construction of evolving constants in a concrete non-trivial model can be found for example in Montesinos et al[93]

A recent approach to treat the problem of time is shapedynamics see Barbour et al [94] and the references thereinIn this approach three-dimensional conformal invarianceplays the central role The configuration space is conformalsuperspace (the geometrodynamic shape space) timesR+ thesecond part coming from the volume of three-space Space-time foliation invariance at the classical level has been lostIn shape dynamics the variables are naturally separated intodimensionless true degrees of freedom and a single variablethat serves the role of time This approach is motivated by asimilar approach in particle mechanics [95] see also Barbour[96] and Anderson [97] The consequences of this for thequantum version of shape dynamics have still to be explored

Independent of these investigations addressing the con-cept of time in full quantum gravity it is obvious that thelimit of quantum field theory in curved spacetime (in whichtime as part of spacetime exists) must be recovered in anappropriate limit How this is achieved is not clear in all of


the above approaches It is most transparent for theWheeler-DeWitt equation as I will be briefly explaining now

In the semiclassical approximation to the Wheeler-DeWitt equation one starts with the ansatz

1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)

and performs an expansion with respect to the inverse Planckmass squared 119898

minus2

P This is inserted into (21) and (22) andconsecutive orders in this expansion are considered seeKiefer and Singh [98] Bertoni et al [99] and Barvinskyand Kiefer [100] This is close to the Born-Oppenheimer(BO) approximation scheme in molecular physics In (37)ℎ119886119887

denotes again the three-metric and the Dirac bra-ketnotation refers to the nongravitational fields for which theusual Hilbert-space structure is assumed

The highest orders of the BO scheme lead to the fol-lowing picture One evaluates the ldquomatter wave functionrdquo|120595[ℎ

119886119887]⟩ along a solution of the classical Einstein equations

ℎ119886119887(x 119905) which corresponds to a chosen solution 119878[ℎ

119886119887] of the

Hamilton-Jacobi equations One can then define

ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)

where 119873 is the lapse function and 119873119886 is the shift vector

their choice reflects the chosen foliation and coordinateassignment In this way one can recover a classical spacetimeas an approximation a spacetime that satisfies Einsteinrsquosequations in this limitThe time derivative of the matter wavefunction is then defined by

120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)

where the notation |120595(119905)⟩means |120595[ℎ119886119887]⟩ evaluated along the

chosen spacetime with the chosen foliation This leads to afunctional Schrodinger equation for quantized matter fieldsin the chosen external classical gravitational field

iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)

where m denotes the matter-field Hamiltonian in the

Schrodinger picture which depends parametrically on the(generally nonstatic) metric coefficients of the curved space-time background recovered from 119878[ℎ

119886119887] The ldquoWKB timerdquo

119905 controls the dynamics in this approximationIn the semiclassical limit one thus finds a Schrodinger

equation of the form (36) without using an artificial scalarfield or dust matter that is often introduced with the solepurpose of defining a time at the exact level compare(36) From an empirical point of view nothing more isneeded because we do not have any access so far to aregime where the semiclassical approximation is not validIn this limit standard quantum theory with all its machinery(Hilbert space probability interpretation) emerges and it is

totally unclear whether this machinery is needed beyond thesemiclassical approximation

Proceeding to the next order of the 119898minus2

119875-expansion one

can derive quantum gravitational corrections to the func-tional Schrodinger equation (40) These may in principlelead to observable contributions to the anisotropy spectrumof the cosmic microwave background (CMB) anisotropyspectrum [101 102] although they are at present too tiny

A major conceptual issue concerns the arrow of time[103] Although our fundamental laws as known so far aretime-reversal invariant (or a slight generalization thereof)there are classes of phenomena that exhibit a definite tem-poral direction This is expressed by the Second Law ofthermodynamics Is there a hope that the origin of thisirreversibility can be found in quantum gravity

Before addressing this possibility let us estimate howspecial our Universe really is that is how large its entropy iscompared with its maximal possible entropy Roger Penrosehas pointed out that the maximal entropy for the observableUniverse would be obtained if all its matters were assembledinto one black hole [104] Taking the most recent observa-tional data this gives the entropy [105]

119878max asymp 18 times 10121 (42)

(Here and below 119896B = 1) This may not yet be themaximal possible entropy Our Universe exhibits currentlyan acceleration caused by a cosmological constant Λ or adynamical dark energy If it was caused by Λ it wouldexpand forever and the entropy in the far future would bedominated by the entropy of the cosmological event horizonThis entropy is called the ldquoGibbons-Hawking entropyrdquo [106]and leads to [105]

119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)

which is about one order of magnitude higher than (42)Following the arguments in Penrose [104] the ldquoprobabil-

ityrdquo for our Universe can then be estimated as

exp (119878)exp (119878max)

asymp

exp (31 times 10104)exp (29 times 10122)

asymp exp (minus29 times 10122)

(44)

Our Universe is thus very special indeed It is much morespecial than what would be estimated from the anthropicprinciple

Turning to quantum gravity the question arises howone can derive an arrow of time from a framework that isfundamentally timeless A final answer is not yet available butvarious ideas exist [103 105] The Wheeler-DeWitt equation(21) does not contain any external time parameter but onecan define an intrinsic time from the hyperbolic nature ofthis equation which is entirely constructed from the three-metric [107] In quantum cosmology this intrinsic time isthe scale factor compare (33) and (34) If one adds smallinhomogeneous degrees of freedom to a Friedmann model


[108] the Wheeler-DeWitt equation turns out to be of theform Ψ

= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

rarr0 for 120572rarrminusinfin

]

]

)Ψ = 0

(45)

where the 119909119894 denotes the inhomogeneous degrees of free-

dom as well as neglected homogeneous ones119881119894(120572 119909

119894) are the

corresponding potentials One recognizes immediately thatthis Wheeler-DeWitt equation is hyperbolic with respect tothe intrinsic time 120572 Initial conditions are thus most naturallyformulated with respect to constant 120572

The important property of (45) is that the potentialbecomes small for120572 rarr minusinfin (where the classical singularitieswould occur) but complicated for increasing 120572 In the generalcase (not restricting to small inhomogeneities) this may befurther motivated by the BKL conjecture according to whichspatial gradients become small near a spacelike singularity[109]TheWheeler-DeWitt equation thus possesses an asym-metrywith respect to ldquointrinsic timerdquo120572 One can in particularimpose the simple boundary condition [103]

Ψ120572997888rarrminusinfin

997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)

which means that the degrees of freedom are initially notentangled They will become entangled for increasing 120572

because then the coupling in the potential between 120572 and the119909

119894 becomes importantThis leads to a positive entanglement

entropy In the semiclassical limit where a WKB time 119905 canbe defined there is a correlation between 119905 and 120572 which canexplain the standard Second Law at least in principle Inthis scenario entropy increase is correlated with scale-factorincreaseTherefore the arrow of time would formally reverseat the turning point of a classically recollapsing universealthough in a quantum scenario the classical evolution wouldend there and no transition to a recollapsing phase couldever be observed [110] Whether a boundary condition ofthe form (46) follows as a necessary requirement from themathematical structure of the full theory or not is an openissue Alternative ideas to the recovery of the arrow oftime can be found in Penrose [111] Vilenkin [112] and thereferences therein

Our discussion about the problem of time was per-formed in the framework of quantum geometrodynamicsIts principle features should also be applicable with somemodifications to loop quantumgravity Butwhat about stringtheory

String theory contains general relativity As we haveseen above the quantum equation that gives back Einsteinrsquosequation in the semiclassical limit is the Wheeler-DeWittequation (21) One would thus expect that the Wheeler-DeWitt equation can be recovered from string theory in thelimit where the string constant 1205721015840 is small Unfortunately thishas not been shown so far in any explicit manner

It is clear that the problem of time is the same in stringtheory New insights may be obtained in addition for the

concept of space This is best seen in the context of theAdSCFT correspondence see for example Maldacena [113]for a review In short words this correspondence states thatnonperturbative string theory in a background spacetimewhich is asymptotically anti-de Sitter (AdS) is dual to aconformal field theory (CFT) defined in a flat spacetimeof one fewer dimension (the boundary of the backgroundspacetime) What really corresponds here are certain matrixelements and symmetries in the two theories an equiva-lence at the level of the quantum states is not shown Thiscorrespondence can be considered as an intermediate steptowards a background-independent formulation of stringtheory because the background metric enters only throughboundary conditions at infinity compare Blau and Theisen[114]

The AdSCFT correspondence can also be interpretedas a realization of the ldquoholographic principlerdquo which statesthat the information of a gravitating system is located on theboundary of this system the most prominent example is theexpression (12) for the entropy which is given by the surfaceof the event horizon In a particular case laws includinggravity in 119889 = 3 are equivalent to laws excluding gravity in119889 = 2 In a loose sense space has then vanished too [115]

Our discussion of the problem of time presented inthis section is far from complete There exist for exampleinteresting suggestions of a thermodynamical origin of timein a gravitational context [116] It has been shown thatstatistical mechanics of generally covariant quantum theoriescan be developed without a preferred time and thus withouta preferred notion of temperature [117]

5 Singularity Avoidance andBoundary Conditions

As we have mentioned at the beginning classical GR predictsthe occurrence of singularities [14] What can the aboveapproaches say about their fate in the quantum theory Inorder to discuss this one first has to agree about a definitionof singularity avoidance in quantum gravity Since such anagreement does not yet exist one has to study heuristicexpectations Already DeWitt [47] has speculated that aclassical singularity is avoided if the wave function vanishesat the corresponding region in configuration space

Ψ[(3)Gsing] = 0 (47)

One example where this condition can be implementedconcerns the fate of a singularity called ldquobig brakerdquo [118]Thisoccurs for an equation of state of the form 119901 = 119860120588 119860 gt 0called ldquoanti-Chaplygin gasrdquo For a Friedmann universe withscale factor 119886(119905) and a scalar field 120601(119905) this equation of statecan be realized by the potential

119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)


where 1205812 = 8120587119866 The classical dynamics develops a pressuresingularity (only 119886(119905) becomes singular) and comes to anabrupt halt in the future ldquobig brakerdquo

TheWheeler-DeWitt equation for this model reads

ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)

where120572 = ln 119886 andLaplace-Beltrami factor ordering has beenused The vicinity of the big-brake singularity is the region ofsmall 120601 we can therefore use the approximation

ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581

2It was shown in Kamenshchik et al [118] that all normal-

izable solutions are of the form

Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)

where1198700is a Bessel function1198711

119896minus1are Laguerre polynomials

and 119881120572equiv

01198906120572 They all vanish at the classical singularity

This model therefore implements DeWittrsquos criterium aboveThe same holds in more general situations of this type [119]In a somewhat different approach the big brake singularityis not avoided [120] Singularity avoidance in the case of aldquobig riprdquo which can occur for phantom fields is discussed inDąbrowski et al [121]

The vanishing of the wave function at the classicalsingularity plays also a role in the treatment of supersym-metric quantum cosmological billiards [122] In 119863 = 11

supergravity one can employ near a spacelike singularitysuch a description based on the Kac-Moody group 119864

10and

derive the corresponding Wheeler-DeWitt equation It wasfound there too that Ψ rarr 0 near the singularity and thatDeWittrsquos criterium is fulfilled Singularity avoidance for theWheeler-DeWitt equation is also achieved when the Bohminterpretation is used [123] Singularity avoidance was alsodiscussed using ldquowavelet quantizationrdquo instead of canonicalquantization [124] there the occurrence of a repulsive poten-tial was found

Singularity avoidance also occurs in the framework ofloop quantum cosmology although in a somewhat differentway [71] The big-bang singularity can be avoided by solu-tions of the difference equation that replaces the Wheeler-DeWitt equation The avoidance can also be achieved by theoccurrence of a bounce in the effective Friedmann equationsThese results strongly indicate that singularities are avoidedin loop quantum cosmology although no general theorems

exist [89] Possible singularity avoidance can also be dis-cussed for black-hole singularities and for naked singularities[125]

The question of singularity avoidance is closely connectedwith the role of boundary conditions in quantum cosmologyLet us thus here have a brief look at the conceptual side of thisissue

The Wheeler-DeWitt equation (34) is of hyperbolicnature with respect to 120572 equiv ln 119886 It makes thus sense to specifythe wave function and its derivative at constant 120572 In thecase of an open universe this is fine In the case of a closeduniverse however one has to impose the boundary conditionthat the wave function vanishes at 120572 rarr infin The existence ofa solution in this case may then lead to a restriction on theallowed values for the parameters of the model such as thecosmological constant or the mass of a scalar field or mayallow no solution at all

Another type of boundary conditions makes use of thepath-integral formulation (15) In 1982 Hawking formulatedhis ldquono-boundary conditionrdquo for the wave function [126]which was then elaborated in Hartle and Hawking [127] andHawking [128] The wave function is given by the Euclideanpath integral

Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)

The sum over M expresses the sum over all four manifoldswith measure ](M) (which actually cannot be performed)The no-boundary condition states thatmdashapart from theboundary where the three metric ℎ

119886119887is specifiedmdashthere is

no other boundary on which initial conditions have to bespecified

Originally the hope was entertained that the no-boundary proposal leads to a unique wave function (or asmall class of wave functions) and that the classical big-bang singularity is smoothed out by the absence of the initialboundary It was however later realized that there are in factmany solutions and that the integration has to be performedover complexmetrics (see eg [129 130])Moreover the pathintegral can usually only be evaluated in a semiclassical limit(using the saddle-point approximation) so it is hard to makea general statement about singularity avoidance

In a Friedmannmodel with scale factor 119886 and a scalar field120601 with a potential119881(120601) the no-boundary condition gives thesemiclassical solution [128]

120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)

We note that the no-boundary wave function is alwaysreal The form (53) corresponds to the superposition of anexpanding and a recollapsing universe

Another prominent boundary condition is the tunnelingproposal (see [131] and the references therein) It was orig-inally defined by the choice of taking ldquooutgoingrdquo solutions


at singular boundaries of superspace The term ldquooutgoingrdquois somewhat misleading because the sign of the imaginaryunit has no absolute meaning in the absence of external time[107] What one can say is that a complex solution is chosenin contrast to the real solution found from the no-boundaryproposal In the same model one obtains for the tunnellingproposal instead of (53) the expression

120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))

times exp(minus i3119881 (120601)

(1198862119881 (120601) minus 1)

32

)

(54)

Considering the conserved Klein-Gordon type of current

119895 =i2(120595

lowastnabla120595 minus 120595nabla120595

lowast) nabla119895 = 0 (55)

where nabla denotes the derivatives in minisuperspace one findsfor a WKB solution of the form 120595 asymp 119862 exp(i119878) the expression

119895 asymp minus|119862|2nabla119878 (56)

The tunnelling proposal states that this current should pointoutwards at large 119886 and 120601 (provided of course that 120595 is ofWKB form there) If 120595 was real (as is the case in the no-boundary proposal) the current would vanish Again thewave function is of semiclassical form It has been suggestedthat it may only be interpreted in this limit [77] Otherboundary conditions include the SIC proposal put forwardby Conradi and Zeh [132]

An interesting application of these boundary conditionsconcerns the prediction of an inflationary phase for the earlyuniverse Here it seems that the tunnelling wave functionfavours such a phase while the no-boundary conditiondisfavors it [133] In the context of the string theory landscapethe no-boundary proposal was applied to inflation evenleading to a prediction for the spectral index of the CMBspectrum [134]

6 General Interpretation of Quantum Theory

Quantum cosmology can shed some light on the problem ofinterpreting quantum theory in general After all a quantumuniverse possesses by definition no external classical mea-suring agency It is not possible to apply the Copenhageninterpretation which presumes the existence of classicalrealms from the outset

Since all the approaches discussed above preserve the lin-earity of the formalism the superposition principle remainsvalid and with it the measurement problem In most investi-gations the Everett interpretation [135] is applied to quantumcosmology In this interpretation all components of the wavefunction are equally real This view is already reflected by thewords of Bryce DeWitt in DeWitt [47]

ldquoEverettrsquos view of the world is a very natural one toadopt in the quantum theory of gravity where oneis accustomed to speak without embarrassment of theldquowave function of the universerdquo It is possible thatEverettrsquos view is not only natural but essentialrdquo

Alternative interpretations include the Bohm interpretationsee Pinto-Neto et al [123] and the references therein

Quantum cosmology (and quantum theory in general)can be consistently interpreted if the Everett interpretationis used together with the process of decoherence [82] Deco-herence is the irreversible emergence of classical propertiesfrom the unavoidable interaction with the ldquoenvironmentrdquo(meaning irrelevant or negligible degrees of freedom inconfiguration space) In quantum cosmology the irrelevantdegrees of freedom include tiny gravitational waves anddensity fluctuations [136] Their interaction with the scalefactor and global matter fields transform them into variablesthat behave classically see for example Kiefer [137] andBarvinsky et al [138] An application of decoherence tothe superposition of triad superpositions in loop quantumcosmology can be found in Kiefer and Schell [139]

Once the ldquobackground variablesrdquo such as 119886 or 120601 havebeen rendered classical by decoherence the question ariseswhat happens to the primordial quantum fluctuations out ofwhichmdashaccording to the inflationary scenariomdashall structurein the Universe emerges Here again decoherence plays thedecisive role see Kiefer and Polarski [140] and the referencesthereinThe quantumfluctuations assume classical behaviourby the interaction with small perturbations that arise eitherfrom other fields or from a self-coupling interaction Thedecoherence time typically turns out to be of the order

119905119889sim119867I119892 (57)

where 119892 is a dimensionless coupling constant of the inter-action with the other fields causing decoherence and 119867Iis the Hubble parameter of inflation The ensuing coarsegraining brought about by the decohering fields causes anentropy increase for the primordial fluctuations [141] All ofthis is in accordance with current observations of the CMBanisotropies

These considerations can in principle be extended to theconcept of the multiverse as it arises for example from thestring landscape or from inflation (see eg [63 142]) butadditional problems emerge (such as the measure problem)that have so far not been satisfactorily been dealt with

As we have mentioned in Section 1 the linearity of quan-tum theorymay break down if gravity becomes important Inthis case a collapse of the wave function may occur whichdestroys all the components of the wave function exceptone see for example Landau et al [143] But as long asthere is no empirical hint for the breakdown of linearity theabove scenario provides a consistent andminimalistic (in thesense of mathematical structure) picture of the quantum-to-classical transition in quantum cosmology

In addition to the many formal and mathematical prob-lems conceptual problems form a major obstacle for thefinal construction of a quantum theory of gravity and itsapplication to cosmology They may however also providethe key for the construction of such a theory Whether orwhen this will happen is however an open question


References

[1] C W Misner K S Thorne and J A Wheeler GravitationFreeman San Francisco Calif USA 1973

[2] F Gronwald and F W Hehl ldquoOn the gauge aspects of gravityrdquoErice 1995 Quantum Gravity vol 10 pp 148ndash198 1996

[3] S Carlip ldquoIs quantum gravity necessaryrdquo Classical and Quan-tum Gravity vol 25 no 15 Article ID 154010 6 pages 2008

[4] D N Page and C D Geilker ldquoIndirect evidence for quantumgravityrdquo Physical Review Letters vol 47 no 14 pp 979ndash9821981

[5] D Giulini and A Groszligardt ldquoGravitationally induced inhi-bitions of dispersion according to the Schrodinger-NewtonequationrdquoClassical andQuantumGravity vol 28 no 10 ArticleID 195026 2011

[6] C Kiefer Quantum Gravity Oxford University Press OxfordUK 3rd edition 2012

[7] CMDeWitt andDRickles ldquoThe role of gravitation in physicsrdquoReport from the 1957 Chapel Hill Conference Berlin Germany2011

[8] H D Zeh ldquoFeynmanrsquos quantum theoryrdquo European PhysicalJournal H vol 36 pp 147ndash158 2011

[9] M Albers C Kiefer and M Reginatto ldquoMeasurement analysisand quantum gravityrdquo Physical Review D vol 78 no 6 ArticleID 064051 17 pages 2008

[10] R Penrose ldquoOn gravityrsquos role in quantum state reductionrdquoGeneral Relativity and Gravitation vol 28 no 5 pp 581ndash6001996

[11] T P Singh ldquoQuantum mechanics without spacetime a case fornoncommutative geometryrdquo Bulgarian Journal of Physics vol33 p 217 2006

[12] A Bassi K Lochan S Satin T P Singh and H UlbrichtldquoModels of wave-function collapse underlying theories andexperimental testsrdquo Reviews of Modern Physics vol 85 pp 471ndash527 2013

[13] T Padmanabhan ldquoThermodynamical aspects of gravity newinsightsrdquo Reports on Progress in Physics vol 73 Article ID046901 2010

[14] S W Hawking and R Penrose The Nature of Space and TimePrinceton University Press Princeton NJ USA 1996

[15] A D Rendall ldquoThe nature of spacetime singularitiesrdquo in 100Years of RelativitymdashSpace-Time Structure Einstein and BeyondA Ashtekar Ed pp 76ndash92 World Scientific Singapore 2005

[16] J Anderson Principles of Relativity Physics Academic PressNew York NY USA 1967

[17] H Nicolai ldquoQuantum Gravity the view from particle physicsrdquoIn press httparxivorgabs13015481

[18] M Zych F Costa I Pikovski T C Ralph and C BruknerldquoGeneral relativistic effects in quantum interference of pho-tonsrdquo Classical and Quantum Gravity vol 29 no 22 Article ID224010 2012

[19] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975Erratum ibid vol 46 pp 206 1976

[20] J HMacGibbon ldquoQuark- and gluon-jet emission fromprimor-dial black holes II The emission over the black-hole lifetimerdquoPhysical Review D vol 44 pp 376ndash392 1991

[21] B J Carr ldquoPrimordial black holes as a probe of cosmologyand high energy physicsrdquo in Quantum Gravity FromTheory toExperimental Search D Giulini C Kiefer and C Lammerzahl

Eds vol 631 of Lecture Notes in Physics pp 301ndash321 SpringerBerlin Germany 2003

[22] W G Unruh ldquoNotes on black-hole evaporationrdquo PhysicalReview D vol 14 pp 870ndash892 1976

[23] P G Thirolf D Habs A Henig et al ldquoSignatures of the Unruheffect via highpower short-pulse lasersrdquoThe European PhysicalJournal D vol 55 no 2 pp 379ndash389 2009

[24] A Strominger ldquoFive problems in quantum gravityrdquo NuclearPhysics B vol 192-193 pp 119ndash125 2009

[25] S W Hawking ldquoBreakdown of predictability in gravitationalcollapserdquo Physical Review D vol 14 no 10 pp 2460ndash2473 1976

[26] D N Page ldquoBlack hole informationrdquo in Proceedings of the5th Canadian Conference on General Relativity and RelativisticAstrophysics R Mann and R McLenaghan Eds pp 1ndash41World Scientific 1994

[27] D N Page ldquoTime dependence of Hawking radiation entropyrdquoIn press httparxivorgabs13014995

[28] C Kiefer ldquoHawking radiation from decoherencerdquo Classical andQuantum Gravity vol 18 no 22 pp L151ndashL154 2001

[29] C Kiefer ldquoIs there an information-loss problem for blackholesrdquo in Decoherence and Entropy in Complex Systems H-TElze Ed vol 633 of Lecture Notes in Physics Springer BerlinGermany 2004

[30] K Freese C T Hill and M T Mueller ldquoCovariant functionalSchrodinger formalism and application to the Hawking effectrdquoNuclear Physics B vol 255 no 3-4 pp 693ndash716 1985

[31] N D Birrell and P C W Davies Quantum Fields in CurvedSpace vol 7 CambridgeUniversity Press Cambridge UK 1982

[32] HD Zeh ldquoWhere has all the information gonerdquoPhysics LettersA vol 347 pp 1ndash7 2005

[33] A Almheiri D Marolf J Polchinski and J Sully ldquoBlack holescomplementarity or firewallsrdquo Journal of High Energy Physicsvol 2 article 62 2013

[34] C Kiefer J Marto and P V Moniz ldquoIndefinite oscillators andblack-hole evaporationrdquo Annalen der Physik vol 18 no 10-11pp 722ndash735 2009

[35] C J Isham ldquoQuantum gravityrdquo in General Relativity andGravitation M A H MacCallum Ed pp 99ndash129 CambridgeUniversity Press Cambridge UK 1987

[36] S Weinberg The Quantum Theory of Fields vol 1 CambridgeUniversity Press Cambridge UK 1995

[37] H W Hamber Quantum GravitationmdashThe Feynman PathIntegral Approac Springer Berlin Germany 2009

[38] M H Goroff and A Sagnotti ldquoQuantum gravity at two loopsrdquoPhysics Letters B vol 160 pp 81ndash86 1985

[39] Z Bern J J M Carrasco L J Dixon H Johansson and RRoiban ldquoUltraviolet behavior of 119873 = 8 supergravity at fourloopsrdquo Physical Review Letters vol 103 no 8 Article ID 0813012009

[40] N E J Bjerrum-Bohr J F Donoghue and B R HolsteinldquoQuantum gravitational corrections to the nonrelativistic scat-tering potential of twomassesrdquo Physical Review D vol 67 no 8Article ID 084033 12 pages 2003

[41] J Ambjoslashrn A Goerlich J Jurkiewicz and R Loll ldquoQuantumgravity via causal dynamical triangulationsrdquo httparxivorgabs13022173

[42] A Nink andM Reuter ldquoOn quantum gravity asymptotic safetyand paramagnetic dominancerdquo In press httparxivorgabs12124325


[43] G Ellis JMurugan andAWeltman Eds Foundations of Spaceand Time Cambridge University Press 2012

[44] G Amelino-Camelia and J Kowalski-Glikman Eds PlanckScale Effects in Astrophysics and Cosmology vol 669 of LectureNotes in Physics Springer Berlin Germany 2005

[45] J M Pons D C Salisbury and K A Sundermeyer ldquoRevisitingobservables in generally covariant theories in the light of gaugefixing methodsrdquo Physical Review D vol 80 Article ID 08401523 pages 2009

[46] J B Barbour and B Foster ldquoConstraints and gauge trans-formations diracrsquos theorem is not always validrdquo In presshttparxivorgabs08081223

[47] B S DeWitt ldquoQuantum theory of gravity I The canonicaltheoryrdquo Physical Review vol 160 pp 1113ndash1148 1967

[48] J A Wheeler ldquoSuperspace and the nature of quantumgeometrodynamicsrdquo in Battelle Rencontres C M DeWitt and JA Wheeler Eds pp 242ndash307 Benjamin New York NY USA1968

[49] J R Klauder ldquoAn affinity for affine quantum gravityrdquo Proceed-ings of the Steklov Institute of Mathematics vol 272 no 1 pp169ndash176 2011

[50] C Vaz S Gutti C Kiefer T P Singh and L C R Wijeward-hana ldquoMass spectrum and statistical entropy of the BTZ blackhole from canonical quantum gravityrdquo Physical Review D vol77 no 6 Article ID 064021 9 pages 2008

[51] A Ashtekar ldquoNew variables for classical and quantum gravityrdquoPhysical Review Letters vol 57 no 18 pp 2244ndash2247 1986

[52] I Agullo G Barbero E F Borja J Diaz-Polo and E J SVillasenor ldquoDetailed black hole state counting in loop quantumgravityrdquo Physical Review D vol 82 no 8 Article ID 084029 31pages 2010

[53] C Kiefer and G Kolland ldquoGibbsrsquo paradox and black-holeentropyrdquo General Relativity and Gravitation vol 40 no 6 pp1327ndash1339 2008

[54] R Gambini and J Pullin A First Course in Loop QuantumGravity Oxford University Press Oxford UK 2011

[55] C Rovelli Quantum Gravity Cambridge University PressCambridgeUK 2004

[56] A Ashtekar and J Lewandowski ldquoBackground independentquantum gravity a status reportrdquo Classical and QuantumGravity vol 21 no 15 pp R53ndashR152 2004

[57] T Thiemann Modern Canonical Quantum General RelativityCambridge University Press Cambridge UK 2007

[58] C Rovelli ldquoCovariant loop gravityrdquo in Quantum Gravity andQuantum Cosmology G Calcagni L Papantonopoulos GSiopsis and N Tsamis Eds vol 863 of Lecture Notes in Physicspp 57ndash66 Springer Berlin Germany 2013

[59] R Blumenhagen D Lust and S Theisen Basic Concepts ofString Theory Springer Berlin Germany 2013

[60] A Strominger and C Vafa ldquoMicroscopic origin of theBekenstein-Hawking entropyrdquo Physics Letters B vol 379 no 1ndash4 pp 99ndash104 1996

[61] G T Horowitz ldquoQuantum states of black holesrdquo in BlackHoles and Relativistic Stars R M Wald Ed pp 241ndash266 TheUniversity of Chicago Press Chicago Ill USA 1998

[62] M R Douglas ldquoThe statistics of stringM theory vacuardquo Journalof High Energy Physics vol 5 article 046 2003

[63] B J Carr Universe or Multiverse Cambridge University PressCambridge UK 2007

[64] D Oriti Ed Approaches to Quantum Gravity CambridgeUniversity Press Cambridge UK 2009

[65] D H Coule ldquoQuantum cosmological modelsrdquo Classical andQuantum Gravity vol 22 no 12 pp R125ndashR166 2005

[66] J J Halliwell ldquoIntroductory lectures on quantum cosmologyrdquoin Quantum Cosmology and Baby Universes S Coleman J BHartle T Piran and S Weinberg Eds pp 159ndash243 WorldScientific Singapore 1991

[67] C Kiefer and B Sandhoefer ldquoQuantum cosmologyrdquo In presshttparxivorgabs08040672

[68] D L Wiltshire ldquoAn introduction to quantum cosmologyrdquoin Cosmology The Physics of the Universe B Robson NVisvanathan and W S Woolcock Eds pp 473ndash531 WorldScientific Singapore 1996

[69] G Montani M V Battisti R Benini and G ImponentePrimordial Cosmology World Scientific Singapore 2011

[70] P V Moniz Quantum CosmologymdashThe Supersymmetric Per-Spective vol 804 of Lecture Notes in Physics Springer BerlinGermany 2010

[71] M Bojowald Quantum Cosmology vol 835 of Lecture Notes inPhysics Springer Berlin Germany 2011

[72] M Bojowald C Kiefer and P V Moniz ldquoQuantumcosmology for the 21st century a debaterdquo In presshttparxivorgabs10052471

[73] C BastosO BertolamiN CDias and JN Prata ldquoPhase-spacenoncommutative quantum cosmologyrdquo Physical Review D vol78 no 2 Article ID 023516 10 pages 2008

[74] O Bertolami and C A D Zarro ldquoHorava-Lifshitz quantumcosmologyrdquo Physical ReviewD vol 85 no 4 Article ID 04404212 pages 2011

[75] S P Kim ldquoMassive scalar field quantum cosmologyrdquo In presshttparxivorgabs13047439

[76] M Bojowald ldquoQuantum cosmology effective theoryrdquo Classicaland Quantum Gravity vol 29 no 21 Article ID 213001 58pages 2012

[77] A Vilenkin ldquoInterpretation of the wave function of the Uni-verserdquo Physical Review D vol 39 pp 1116ndash1122 1989

[78] C Kiefer ldquoWave packets inminisuperspacerdquo Physical Review Dvol 38 no 6 pp 1761ndash1772 1988

[79] E Calzetta and J J Gonzalez ldquoChaos and semiclassical limit inquantum cosmologyrdquo Physical Review D vol 51 pp 6821ndash68281995

[80] N J Cornish and E P S Shellard ldquoChaos in quantum cosmol-ogyrdquo Physical Review Letters vol 81 pp 3571ndash3574 1998

[81] W H Zurek and J P Paz ldquoQuantum chaos a decoherentdefinitionrdquo Physica D vol 83 pp 300ndash308 1995

[82] E Joos H D Zeh C Kiefer D Giulini J Kupsch and I-O Stamatescu Decoherence and the Appearance of a ClassicalWorld in Quantum Theory Springer Berlin Germany 2ndedition 2003

[83] E Calzetta ldquoChaos decoherence and quantum cosmologyrdquoClassical andQuantumGravity vol 29 no 14 Article ID 14300130 pages 2012

[84] E Anderson ldquoProblem of time in quantum gravityrdquo Annalender Physik vol 524 no 12 pp 757ndash786 2012

[85] C J Isham ldquoCanonical quantum gravity and the problem oftimerdquo in Integrable Systems Quantum Groups and QuantumField Theories L A Ibort and M A Rodrıguez Eds vol 409pp 157ndash287 Kluwer Dordrecht The Netherlands 1993

[86] K V Kuchar ldquoTime and interpretations of quantum gravityrdquo inProceedings of the 4th Canadian Conference on General Relativ-ity and Relativistic Astrophysics G Kunstatter D Vincent andJ Williams Eds pp 211ndash314 World Scientific 1992


[87] C G Torre ldquoIs general relativity an ldquoalready parametrizedrdquotheoryrdquo Physical Review D vol 46 pp 3231ndash324 1993

[88] J D Brown and K V Kuchar ldquoDust as a standard of space andtime in canonical quantum gravityrdquo Physical Review D vol 51no 10 pp 5600ndash5629 1995

[89] A Ashtekar and P Singh ldquoLoop quantum cosmology a statusreportrdquo Classical and Quantum Gravity vol 28 no 21 ArticleID 213001 p 122 2011

[90] SAlexanderM BojowaldAMarciano andD Simpson ldquoElec-tric time in quantumcosmologyrdquo In press httparxivorgabs12122204

[91] M Bojowald P A Hohn and A Tsobanjan ldquoAn effectiveapproach to the problem of timerdquo Classical and QuantumGravity vol 28 no 3 Article ID 035006 18 pages 2011

[92] C Rovelli ldquoTime in quantum gravity an hypothesisrdquo PhysicalReview D vol 43 no 2 pp 442ndash456 1991

[93] M Montesinos C Rovelli and T Thiemann ldquo119878119871(2R) modelwith two Hamiltonian constraintsrdquo Physical Review D vol 60no 4 Article ID 044009 10 pages 1999

[94] J B Barbour T Koslowski and F Mercati ldquoThe solu-tion to the problem of time in shape dynamicsrdquo In presshttparxivorgabs13026264

[95] J B Barbour and B Bertotti ldquoMachrsquos principle and the structureof dynamical theoriesrdquo Proceedings of the Royal Society A vol382 no 1783 pp 295ndash306 1982

[96] J BarbourTheEnd of Time OxfordUniversity Press NewYorkNY USA 2000

[97] E Anderson ldquoThe problem of time and quantum cosmol-ogy in the relational particle mechanics arenardquo In presshttparxivorgabs11111472

[98] C Kiefer and T P Singh ldquoQuantum gravitational correctionsto the functional Schrodinger equationrdquo Physical Review D vol44 no 4 pp 1067ndash1076 1991

[99] C Bertoni F Finelli and G Venturi ldquoThe Born-Oppenheimerapproach to the matter-gravity system and unitarityrdquo Classicaland Quantum Gravity vol 13 no 9 pp 2375ndash2383 1996

[100] A O Barvinsky and C Kiefer ldquoWheeler-DeWitt equation andFeynman diagramsrdquo Nuclear Physics B vol 526 no 1ndash3 pp509ndash539 1998

[101] C Kiefer andMKramer ldquoQuantumgravitational contributionsto the CMB anisotropy spectrumrdquo Physical Review Letters vol108 no 2 Article ID 021301 4 pages 2012

[102] D Bini G Esposito C Kiefer M Kramer and F PessinaldquoOn the modification of the cosmic microwave backgroundanisotropy spectrum from canonical quantum gravityrdquo PhysicalReview D vol 87 Article ID 104008 2013

[103] H D ZehThe Physical Basis of the Direction of Time SpringerBerlin Germany 5th edition 2007

[104] R Penrose ldquoTime-asymmetry and quantum gravityrdquo in Quan-tum Gravity C J Isham R Penrose and D W Sciama Edsvol 2 pp 242ndash272 Clarendon Press Oxford UK 1981

[105] C Kiefer ldquoCan the arrow of time be understood from quantumcosmologyrdquo The Arrows of Time Fundamental Theories ofPhysics vol 172 pp 191ndash203 2009

[106] G W Gibbons and S W Hawking ldquoCosmological event hori-zons thermodynamics and particle creationrdquo Physical ReviewD vol 15 no 10 pp 2738ndash2751 1977

[107] HD Zeh ldquoTime in quantum gravityrdquo Physics Letters A vol 126pp 311ndash317 1988

[108] J J Halliwell and S W Hawking ldquoOrigin of structure in theUniverserdquo Physical Review D vol 31 no 8 pp 1777ndash1791 1985

[109] V A Belinskii I M Khalatnikov and E M Lifshitz ldquoA generalsolution of the Einstein equations with a time singularityrdquoAdvances in Physics vol 31 pp 639ndash667 1982

[110] C Kiefer and H-D Zeh ldquoArrow of time in a recollapsingquantum universerdquo Physical Review D vol 51 no 8 pp 4145ndash4153 1995

[111] R Penrose ldquoBlack holes quantum theory and cosmologyrdquoJournal of Physics vol 174 no 1 Article ID 012001 2009

[112] A Vilenkin ldquoArrows of time and the beginning of the universerdquoIn press httparxivorgabs13053836

[113] J Maldacena ldquoThe gaugegravity dualityrdquo in Black Holes inHigher Dimensions G Horowitz Ed Cambridge UniversityPress 2012

[114] M Blau and S Theisen ldquoString theory as a theory of quantumgravity a status reportrdquo General Relativity and Gravitation vol41 no 4 pp 743ndash755 2009

[115] J Maldacena ldquoThe illusion of gravityrdquo Scientific American pp56ndash63 2005

[116] C Rovelli ldquoStatistical mechanics of gravity and the thermody-namical origin of timerdquo Classical and Quantum Gravity vol 10no 8 pp 1549ndash1566 1993

[117] M Montesinos and C Rovelli ldquoStatistical mechanics of gener-ally covariant quantum theories a Boltzmann-like approachrdquoClassical and QuantumGravity vol 18 no 3 pp 555ndash569 2001

[118] A Y Kamenshchik C Kiefer and B Sandhofer ldquoQuantumcosmology with a big-brake singularityrdquo Physical Review D vol76 no 6 Article ID 064032 13 pages 2007

[119] M Bouhmadi-Lopez C Kiefer B Sandhoefer and P V MonizldquoQuantum fate of singularities in a dark-energy dominateduniverserdquo Physical Review D vol 79 no 12 Article ID 12403516 pages 2009

[120] A Y Kamenshchik and S Manti ldquoClassical and quantum bigbrake cosmology for scalar field and tachyonicmodelsrdquoPhysicalReview D vol 85 Article ID 123518 11 pages 2012

[121] M P Dąbrowski C Kiefer and B Sandhofer ldquoQuantumphantom cosmologyrdquo Physical Review D vol 74 no 24 ArticleID 044022 12 pages 2006

[122] A Kleinschmidt M Koehn and H Nicolai ldquoSupersymmetricquantum cosmological billiardsrdquo Physical Review D vol 80 no6 Article ID 061701 5 pages 2009

[123] N Pinto-Neto F T Falciano R Pereira and E Sergio SantinildquoWheeler-DeWitt quantization can solve the singularity prob-lemrdquo Physical Review D vol 86 no 6 Article ID 063504 12pages 2012

[124] H Bergeron ADapor J P Gazeau and PMalkiewicz ldquoWaveletquantum cosmologyrdquo In press httparxivorgabs13050653

[125] P S Joshi ldquoThe rainbows of gravityrdquo In press httparxivorgabs13051005

[126] SWHawking ldquoThe boundary conditions of the universerdquo Pon-Tificia Academiae Scientarium Scripta Varia vol 48 pp 563ndash574 1982

[127] J B Hartle and SW Hawking ldquoWave function of the universerdquoPhysical Review D vol 28 no 12 pp 2960ndash2975 1983

[128] S W Hawking ldquoThe quantum state of the universerdquo NuclearPhysics B vol 239 no 1 pp 257ndash276 1984

[129] J J Halliwell and J Louko ldquoSteepest-descent contours in thepath-integral approach to quantum cosmology III A generalmethod with applications to anisotropic minisuperspace mod-elsrdquo Physical Review D vol 42 no 12 pp 3997ndash4031 1990

[130] C Kiefer ldquoOn the meaning of path integrals in quantumcosmologyrdquo Annals of Physics vol 207 no 1 pp 53ndash70 1991


[131] A Vilenkin ldquoQuantum cosmology and eternal inflationrdquo inTheFuture of the Theoretical Physics and Cosmology pp 649ndash666Cambridge University Press Cambridge UK 2003

[132] HD Conradi andHD Zeh ldquoQuantum cosmology as an initialvalue problemrdquo Physics Letters A vol 154 pp 321ndash326 1991

[133] A O Barvinsky A Y Kamenshchik C Kiefer and CSteinwachs ldquoTunneling cosmological state revisited origin ofinflation with a nonminimally coupled standard model Higgsinflatonrdquo Physical Review D vol 81 Article ID 043530 2010

[134] J B Hartle SWHawking and THertog ldquoLocal observation ineternal inflationrdquoPhysical Review Letters vol 106 no 14 ArticleID 141302 4 pages 2011

[135] H Everett III ldquoldquoRelative staterdquo formulation of quantummechanicsrdquo Reviews of Modern Physics vol 29 pp 454ndash4621957

[136] H D Zeh ldquoEmergence of classical time from a universalwavefunctionrdquo Physics Letters A vol 116 no 1 pp 9ndash12 1986

[137] C Kiefer ldquoContinuous measurement of mini-superspace vari-ables by highermultipolesrdquoClassical andQuantumGravity vol4 no 5 pp 1369ndash1382 1987

[138] A O Barvinsky A Y Kamenshchik C Kiefer and I VMishakov ldquoDecoherence in quantum cosmology at the onset ofinflationrdquo Nuclear Physics B vol 551 no 1-2 pp 374ndash396 1999

[139] C Kiefer and C Schell ldquoInterpretation of the triad orientationsin loop quantum cosmologyrdquo Classical and Quantum Gravityvol 30 no 3 Article ID 035008 9 pages 2013

[140] C Kiefer and D Polarski ldquoWhy do cosmological perturbationslook classical to usrdquo Advanced Science Letters vol 2 pp 164ndash173 2009

[141] C Kiefer I Lohmar D Polarski andA A Starobinsky ldquoPointerstates for primordial fluctuations in inflationary cosmologyrdquoClassical and Quantum Gravity vol 24 no 7 pp 1699ndash17182007

[142] A Vilenkin ldquoGlobal structure of the multiverse and the mea-sure problemrdquo inAIPConference Proceedings vol 1514 pp 7ndash13Szczecin Poland 2012

[143] S Landau C G Scoccola andD Sudarsky ldquoCosmological con-straints on nonstandard inflationary quantum collapsemodelsrdquoPhysical Review D vol 85 Article ID 123001 2012

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


As one recognizes from (2) these equations can no longerhave exactly the same form if the quantum nature of the fieldsin 119879

120583] is taken into account For then we have operators inHilbert space on the right-hand side and classical functionson the left-hand side A straightforward generalization wouldbe to replace 119879

120583] by its quantum expectation value

119877120583] minus

1

2119892120583]119877 + Λ119892120583] =

8120587119866

1198884⟨Ψ

10038161003816100381610038161003816120583]10038161003816100381610038161003816Ψ⟩ (4)

These ldquosemiclassical Einstein equationsrdquo lead to problemswhen viewed as exact equations at the most fundamentallevel compare Carlip [3] and the references therein Theyspoil the linearity of quantum theory and even seem to be inconflict with a performed experiment [4] They may never-theless be of some value in an approximate way Independentof the problems with (4) one can try to test them in a simplesetting such as the Schrodinger-Newton equation it seemshowever that such a test is not realizable in the foreseeablefuture [5]This poses the question of the connection betweengravity and quantum theory [6]

Despite its name quantum theory is not a particulartheory for a particular interaction It is rather a general frame-work for physical theories whose fundamental concepts haveso far exhibited an amazing universality Despite the ongoingdiscussion about its interpretational foundations (which weshall address in the last section) the concepts of states inHilbert space and in particular the superposition principlehave successfully passed thousands of experimental tests

It is in fact the superposition principle that pointstowards the need for quantizing gravity In the 1957 ChapelHill Conference Richard Feynman gave the following argu-ment [7 pages 250ndash260] see also Zeh [8] He considersa Stern-Gerlach type of experiment in which two spin-12particles are put into a superposition of spinup and spindownand is guided to two counters He then imagines a connectionof the counters to a ball of macroscopic dimensions Thesuperposition of the particles is thereby transferred to asuperposition of the ball being simultaneously at two posi-tions But this means that the ballrsquos gravitational field is in asuperposition too In Feynmanrsquos own words [7 p 251])

ldquoNow how do we analyze this experiment accord-ing to quantummechanicsWe have an amplitudethat the ball is up and an amplitude that the ballis down That is we have an amplitude (from awave function) that the spin of the electron in thefirst part of the equipment is either up or downAnd if we imagine that the ball can be analyzedthrough the interconnections up to this dimension(asymp1 cm) by the quantum mechanics then beforewe make an observation we still have to give anamplitude when the ball is up and an amplitudewhen the ball is down Now since the ball is bigenough to produce a real gravitational field wecould use that gravitational field to move anotherball and amplify that and use the connections tothe second ball as the measuring equipment Wewould then have to analyze through the channel

provided by the gravitational field itself via thequantum mechanical amplitudesrdquo

In other words the gravitational field must then bedescribed by quantum states subject to the superpositionprinciple The same argument can of course be applied tothe electromagnetic field (for which we have strong empiricalsupport that it is of quantum nature)

Feynmanrsquos argument is of course not an argument oflogic that would demand the quantization of gravity withnecessity It is an argument based on conservative heuristicideas that proceed from the extrapolation of establishedand empirically confirmed concepts (here the superpositionprinciple) beyond their present range of application It is inthis way that physics usually evolves Albers et al [9] containa detailed account of arguments that demand the necessityof quantizing the gravitational field It is shown that all thesearguments are of a heuristic value and that they do notlead to the quantization of gravity by a logical conclusionIt is conceivable for example that the linearity of quantumtheory breaks down in situations where the gravitational fieldbecomes strong see for example Penrose [10] Singh [11] andBassi et al [12] But one has to emphasize that no empiricalhint for such a drastic modification exists so far

Alternatives to the direct quantization of the gravitationalfield include what is called ldquoemergent gravityrdquo comparePadmanabhan [13] and the references therein Motivated bythe thermodynamic properties of black holes (see below)one might get the impression that the gravitational field isan effective thermodynamic entity that does not demand itsdirect quantization but points to the existence of new so farunknownmicroscopic degrees of freedomunderlying gravityEven if this was the case (which is far from clear) theremay existmicroscopic degrees of freedom forwhich quantumtheory would apply Whether this leads to a quantized metricor not is not clear In string theory (see below) gravity is anemergent interaction but still the metric is quantized andthe standard approach of quantum gravitational perturbationtheory is naturally implemented In the following we restrictourselves to approaches in which a quantized metric makessense

Besides the general argument put forward by Feynmanthere exist a couple of further arguments that suggest thatthe gravitational interaction be quantized [6] Let me brieflyreview three of them

The first motivation comes from the continuation of thereductionist programme In physics the idea of unificationhas been very successful culminating so far in the StandardModel of strong and electroweak interactions Gravity actsuniversally to all forms of energies A unified theory of allinteractions including gravity should thus not be a hybridtheory in using classical and quantum concepts A coherentquantum theory of all interactions (often called ldquotheory ofeverythingrdquo or TOE) should thus also include a quantumdescription of the gravitational field

A second motivation is the unavoidable presence ofsingularities in Einsteinrsquos theory of GR see for exampleHawking and Penrose [14] and Rendall [15] Prominentexamples are the cases of the big bang and the interior of





119897P = radicℎ119866


minus33 cm

119905P =119897P119888= radic

ℎ119866


minus44 s

119898P =ℎ

119897P119888= radic

ℎ119888


minus5 g


(5)





(119898P119898H

)

2

sim 1034 (6)



120572119892=

1198661198982

pr

ℎ119888= (

119898pr

119898P)

2



119892119898pr


119867 =p2

2119898+ 119898gr minus 120596L (8)



119879BH =ℎ120581

2120587119896B119888 (9)


2

S



therefore

119879BH =ℎ119888

3

8120587119896B119866119872asymp 617 times 10

minus8(119872

⊙

119872) K (10)


120591BH asymp 407(119891(119872

0)

1535)

minus1

(119872

1010 g)

3

s


3

0[g] 119891minus1

(1198720) s

(11)



0≫ 10




119878BH =119896B119860

41198972

P (12)


119878BH =119896B120587119877

2

S119866ℎ

asymp 107 times 1077119896B(

119872

119872⊙

)

2

(13)



119879DU =ℎ119886

2120587119896B119888asymp 405 times 10

minus23119886 [

cms2] K (14)

















119885 [119892] = intD119892120583] (119909) e

i119878[119892120583](119909)] (15)



119892120583] = 119892120583] +

radic32120587119866119891120583] (16)





L(div)2-loop =

209ℎ2

2880

32120587119866

(161205872)2

120598

radicminus119892119877120572120573

120574120575119877120574120575

120583]119877120583]

120572120573 (17)


120583]






119881 (119903) = minus119866119898

1119898

2

119903

times (1 + 3119866 (119898

1+ 119898

2)

1199031198882+

41

10120587

119866ℎ

11990321198883+ O (119866

2))

(18)













119886119887(119909) and its



[ℎ119886119887(119909) 119901

119888119889(119910)] = iℎ120575119888

(119886120575119889

119887)120575 (119909 119910) (19)


HperpΨ = 0

H119886Ψ = 0

(20)




2119866119886119887119888119889

1205752

120575ℎ119886119887120575ℎ

119888119889

minusradicℎ

16120587119866((3)119877 minus 2Λ))Ψ = 0

(21)

H119886Ψ = minus2119863

119887ℎ119886119888

ℎ

i120575Ψ

120575ℎ119887119888

= 0 (22)


119886119887119888119889





119864119886

119894(119909) = radicℎ (119909) 119890

119886

119894(119909) (23)


119866119860119894

119886(119909) = Γ

119894

119886(119909) + 120573119870

119894

119886(119909) (24)







119860119894

119886(119909) 119864

119887

119895(119910) = 8120587120573120575

119894

119895120575119887

119886120575 (119909 119910) (25)



[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)


119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)


997891rarr 119909119886(120590

1 120590


relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)



A [S] Ψ119878[119860] = 8120587120573119897

2

P sum


[119860]

= 119860 [S] Ψ119878[119860]

(29)








119885 = intD119883Dℎeminus119878P (30)







31205721015840minus1

12119867

120583]120588119867120583]120588


120583Φ + O (120572

1015840) )

(31)









3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119897P = radicℎ119866


minus33 cm

119905P =119897P119888= radic

ℎ119866


minus44 s

119898P =ℎ

119897P119888= radic

ℎ119888


minus5 g


(5)





(119898P119898H

)

2

sim 1034 (6)



120572119892=

1198661198982

pr

ℎ119888= (

119898pr

119898P)

2



119892119898pr


119867 =p2

2119898+ 119898gr minus 120596L (8)



119879BH =ℎ120581

2120587119896B119888 (9)


2

S



therefore

119879BH =ℎ119888

3

8120587119896B119866119872asymp 617 times 10

minus8(119872

⊙

119872) K (10)


120591BH asymp 407(119891(119872

0)

1535)

minus1

(119872

1010 g)

3

s


3

0[g] 119891minus1

(1198720) s

(11)



0≫ 10




119878BH =119896B119860

41198972

P (12)


119878BH =119896B120587119877

2

S119866ℎ

asymp 107 times 1077119896B(

119872

119872⊙

)

2

(13)



119879DU =ℎ119886

2120587119896B119888asymp 405 times 10

minus23119886 [

cms2] K (14)

















119885 [119892] = intD119892120583] (119909) e

i119878[119892120583](119909)] (15)



119892120583] = 119892120583] +

radic32120587119866119891120583] (16)





L(div)2-loop =

209ℎ2

2880

32120587119866

(161205872)2

120598

radicminus119892119877120572120573

120574120575119877120574120575

120583]119877120583]

120572120573 (17)


120583]






119881 (119903) = minus119866119898

1119898

2

119903

times (1 + 3119866 (119898

1+ 119898

2)

1199031198882+

41

10120587

119866ℎ

11990321198883+ O (119866

2))

(18)













119886119887(119909) and its



[ℎ119886119887(119909) 119901

119888119889(119910)] = iℎ120575119888

(119886120575119889

119887)120575 (119909 119910) (19)


HperpΨ = 0

H119886Ψ = 0

(20)




2119866119886119887119888119889

1205752

120575ℎ119886119887120575ℎ

119888119889

minusradicℎ

16120587119866((3)119877 minus 2Λ))Ψ = 0

(21)

H119886Ψ = minus2119863

119887ℎ119886119888

ℎ

i120575Ψ

120575ℎ119887119888

= 0 (22)


119886119887119888119889





119864119886

119894(119909) = radicℎ (119909) 119890

119886

119894(119909) (23)


119866119860119894

119886(119909) = Γ

119894

119886(119909) + 120573119870

119894

119886(119909) (24)







119860119894

119886(119909) 119864

119887

119895(119910) = 8120587120573120575

119894

119895120575119887

119886120575 (119909 119910) (25)



[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)


119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)


997891rarr 119909119886(120590

1 120590


relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)



A [S] Ψ119878[119860] = 8120587120573119897

2

P sum


[119860]

= 119860 [S] Ψ119878[119860]

(29)








119885 = intD119883Dℎeminus119878P (30)







31205721015840minus1

12119867

120583]120588119867120583]120588


120583Φ + O (120572

1015840) )

(31)









3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











therefore

119879BH =ℎ119888

3

8120587119896B119866119872asymp 617 times 10

minus8(119872

⊙

119872) K (10)


120591BH asymp 407(119891(119872

0)

1535)

minus1

(119872

1010 g)

3

s


3

0[g] 119891minus1

(1198720) s

(11)



0≫ 10




119878BH =119896B119860

41198972

P (12)


119878BH =119896B120587119877

2

S119866ℎ

asymp 107 times 1077119896B(

119872

119872⊙

)

2

(13)



119879DU =ℎ119886

2120587119896B119888asymp 405 times 10

minus23119886 [

cms2] K (14)

















119885 [119892] = intD119892120583] (119909) e

i119878[119892120583](119909)] (15)



119892120583] = 119892120583] +

radic32120587119866119891120583] (16)





L(div)2-loop =

209ℎ2

2880

32120587119866

(161205872)2

120598

radicminus119892119877120572120573

120574120575119877120574120575

120583]119877120583]

120572120573 (17)


120583]






119881 (119903) = minus119866119898

1119898

2

119903

times (1 + 3119866 (119898

1+ 119898

2)

1199031198882+

41

10120587

119866ℎ

11990321198883+ O (119866

2))

(18)













119886119887(119909) and its



[ℎ119886119887(119909) 119901

119888119889(119910)] = iℎ120575119888

(119886120575119889

119887)120575 (119909 119910) (19)


HperpΨ = 0

H119886Ψ = 0

(20)




2119866119886119887119888119889

1205752

120575ℎ119886119887120575ℎ

119888119889

minusradicℎ

16120587119866((3)119877 minus 2Λ))Ψ = 0

(21)

H119886Ψ = minus2119863

119887ℎ119886119888

ℎ

i120575Ψ

120575ℎ119887119888

= 0 (22)


119886119887119888119889





119864119886

119894(119909) = radicℎ (119909) 119890

119886

119894(119909) (23)


119866119860119894

119886(119909) = Γ

119894

119886(119909) + 120573119870

119894

119886(119909) (24)







119860119894

119886(119909) 119864

119887

119895(119910) = 8120587120573120575

119894

119895120575119887

119886120575 (119909 119910) (25)



[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)


119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)


997891rarr 119909119886(120590

1 120590


relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)



A [S] Ψ119878[119860] = 8120587120573119897

2

P sum


[119860]

= 119860 [S] Ψ119878[119860]

(29)








119885 = intD119883Dℎeminus119878P (30)







31205721015840minus1

12119867

120583]120588119867120583]120588


120583Φ + O (120572

1015840) )

(31)









3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119885 [119892] = intD119892120583] (119909) e

i119878[119892120583](119909)] (15)



119892120583] = 119892120583] +

radic32120587119866119891120583] (16)





L(div)2-loop =

209ℎ2

2880

32120587119866

(161205872)2

120598

radicminus119892119877120572120573

120574120575119877120574120575

120583]119877120583]

120572120573 (17)


120583]






119881 (119903) = minus119866119898

1119898

2

119903

times (1 + 3119866 (119898

1+ 119898

2)

1199031198882+

41

10120587

119866ℎ

11990321198883+ O (119866

2))

(18)













119886119887(119909) and its



[ℎ119886119887(119909) 119901

119888119889(119910)] = iℎ120575119888

(119886120575119889

119887)120575 (119909 119910) (19)


HperpΨ = 0

H119886Ψ = 0

(20)




2119866119886119887119888119889

1205752

120575ℎ119886119887120575ℎ

119888119889

minusradicℎ

16120587119866((3)119877 minus 2Λ))Ψ = 0

(21)

H119886Ψ = minus2119863

119887ℎ119886119888

ℎ

i120575Ψ

120575ℎ119887119888

= 0 (22)


119886119887119888119889





119864119886

119894(119909) = radicℎ (119909) 119890

119886

119894(119909) (23)


119866119860119894

119886(119909) = Γ

119894

119886(119909) + 120573119870

119894

119886(119909) (24)







119860119894

119886(119909) 119864

119887

119895(119910) = 8120587120573120575

119894

119895120575119887

119886120575 (119909 119910) (25)



[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)


119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)


997891rarr 119909119886(120590

1 120590


relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)



A [S] Ψ119878[119860] = 8120587120573119897

2

P sum


[119860]

= 119860 [S] Ψ119878[119860]

(29)








119885 = intD119883Dℎeminus119878P (30)







31205721015840minus1

12119867

120583]120588119867120583]120588


120583Φ + O (120572

1015840) )

(31)









3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119886119887(119909) and its



[ℎ119886119887(119909) 119901

119888119889(119910)] = iℎ120575119888

(119886120575119889

119887)120575 (119909 119910) (19)


HperpΨ = 0

H119886Ψ = 0

(20)




2119866119886119887119888119889

1205752

120575ℎ119886119887120575ℎ

119888119889

minusradicℎ

16120587119866((3)119877 minus 2Λ))Ψ = 0

(21)

H119886Ψ = minus2119863

119887ℎ119886119888

ℎ

i120575Ψ

120575ℎ119887119888

= 0 (22)


119886119887119888119889





119864119886

119894(119909) = radicℎ (119909) 119890

119886

119894(119909) (23)


119866119860119894

119886(119909) = Γ

119894

119886(119909) + 120573119870

119894

119886(119909) (24)







119860119894

119886(119909) 119864

119887

119895(119910) = 8120587120573120575

119894

119895120575119887

119886120575 (119909 119910) (25)



[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)


119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)


997891rarr 119909119886(120590

1 120590


relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)



A [S] Ψ119878[119860] = 8120587120573119897

2

P sum


[119860]

= 119860 [S] Ψ119878[119860]

(29)








119885 = intD119883Dℎeminus119878P (30)







31205721015840minus1

12119867

120583]120588119867120583]120588


120583Φ + O (120572

1015840) )

(31)









3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[119860 120572]Ψ119878[119860] = 119880 [119860 120572]Ψ

119878[119860] (26)


119864119894[S] = minus8120587120573ℎiint

S

1198891205901119889120590

2119899119886()

120575

120575119860119894

119886[x ()]

(27)


997891rarr 119909119886(120590

1 120590


relations

[ [119860 120572] 119864119894[S]] = i1198972P120573120580 (120572S) 119880 [120572

1 119860] 120591

119894119880 [120572

2 119860]

(28)



A [S] Ψ119878[119860] = 8120587120573119897

2

P sum


[119860]

= 119860 [S] Ψ119878[119860]

(29)








119885 = intD119883Dℎeminus119878P (30)







31205721015840minus1

12119867

120583]120588119867120583]120588


120583Φ + O (120572

1015840) )

(31)









3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















3 Quantum Cosmology




1198891199042= minus119873(119905)

2119889119905

2+ 119886(119905)

2119889Ω

2

3 (32)

where 119889Ω2




(ℎ21205812

12119886120597

120597119886119886120597

120597119886minusℎ2

2

1205972

1205971206012+ 119886

6(119881 (120601) +

Λ

1205812)

minus3119896119886

4

1205812)Ψ (119886 120601) = 0

(33)



(ℎ21205812

12

1205972

1205971205722minusℎ2

2

1205972

1205971206012+ 119890

6120572(119881 (120601) +

Λ

1205812)

minus31198904120572 119896

1205812)Ψ (120572 120601) = 0

(34)













P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















P119860+ ℎ

119860= 0 (35)

where P119860




iℎ 120575Ψ120575119902

119860

= ℎ119860Ψ (36)





119860










1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1003816100381610038161003816Ψ [ℎ119886119887]⟩ = 119862 [ℎ

119886119887] ei119898

2

P119878[ℎ119886119887] 1003816100381610038161003816120595 [ℎ119886119887]⟩(37)


minus2






119886119887] of the


ℎ119886119887= 119873119866

119886119887119888119889

120575119878

120575ℎ119888119889

+ 2119863(119886119873

119887) (38)



120597

120597119905

1003816100381610038161003816120595 (119905)⟩ = int1198893119909 ℎ

119886119887(x 119905) 120575

120575ℎ119886119887(x)

1003816100381610038161003816120595 [ℎ119886119887]⟩ (39)



iℎ 120597120597119905

1003816100381610038161003816120595 (119905)⟩ = m 1003816100381610038161003816120595 (119905)⟩ (40)

m= int 119889

3119909 119873 (x) Hm

perp(x) + 119873119886

(x) Hm119886(x) (41)












119878max asymp 18 times 10121 (42)


119878GH =3120587

Λ1198972

Pasymp 29 times 10

122 (43)



exp (119878)exp (119878max)

asymp



(44)





= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= (2120587119866ℎ

2

3

1205972

1205971205722+sum

119894

[

[

minusℎ2

2

1205972

1205971199092

119894

+ 119881119894(120572 119909

119894)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


]

]

)Ψ = 0

(45)



119894) are the




997888997888997888997888997888997888rarr 1205950(120572)prod

119894

120595119894(119909

119894) (46)













Ψ[(3)Gsing] = 0 (47)


119881 (120601) = 1198810(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

1198810= radic

119860

4

(48)




ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601)

+ 11988101198906120572(sinh (radic31205812 1003816100381610038161003816120601

1003816100381610038161003816) minus1

sinh (radic31205812 10038161003816100381610038161206011003816100381610038161003816)

)

times Ψ (120572 120601) = 0

(49)


ℎ2

2(1205812

6

1205972

1205971205722minus

1205972

1205971206012)Ψ (120572 120601) minus

0

10038161003816100381610038161206011003816100381610038161003816

1198906120572Ψ (120572 120601) = 0 (50)

where 0= 119881

03120581



Ψ (120572 120601) =

infin

sum

119896=1

119860 (119896) 119896minus32

1198700(1

radic6

119881120572

ℎ2119896120581)

times (2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816) 119890

minus119881120572119896|120601|1198711

119896minus1(2119881120572

119896

10038161003816100381610038161206011003816100381610038161003816)

(51)



and 119881120572equiv





10and







Ψ [ℎ119886119887 Φ Σ] = sum

M

] (M) intM

D119892DΦeminus119878E[119892120583]Φ] (52)






120595NB prop (1198862119881(120601) minus 1)

minus14

exp( 1

3119881 (120601))

times cos((119886

2119881 (120601) minus 1)

32

3119881 (120601)minus120587

4)

(53)





120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120595119879prop (119886

2119881 (120601) minus 1)

minus14

exp(minus 1

3119881 (120601))


(1198862119881 (120601) minus 1)

32

)

(54)


119895 =i2(120595


lowast) nabla119895 = 0 (55)












119905119889sim119867I119892 (57)






References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References



























































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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