AAspects of Quantum Cosmology
Doctor of Philosophy
SACHIN PANDEY
Roll no. 13IP016
Under the supervision ofProf. Narayan Banerjee
A thesis submitted in the partial fulfilment of the requirements for the award of the Degree ofDoctor of Philosophy to
Department of Physical SciencesIndian Institute of Science Education and Research KolkataDecember 2020 a r X i v : . [ g r- q c ] J a n eclaration Date : 30 December, 2020I,
Sachin Pandey
Registration No. dated 25 July 2013, a student of De-partment of Physical Sciences of the Integrated PhD Program of IISER Kolkata, herebydeclare that this thesis is my own work and, to the best of my knowledge, it neithercontains materials previously published or written by any other person, nor it has beensubmitted for any degree/diploma or any other academic award anywhere before.I also declare that all copyrighted material incorporated into this thesis is in com-pliance with the Indian Copyright (Amendment) Act, 2012 and that I have receivedwritten permission from the copyright owners for my use of their work. Chapter 2-6 ofthis thesis is based on work published in the peer-reviewed journals whose references arementioned in respective chapters.I hereby grant permission to IISER Kolkata to store the thesis in a database whichcan be accessed by others.
Sachin Pandey
Department of Physical SciencesIndian Institute of Science Education and Research KolkataMohanpur 741246, West Bengal, India iii ertificate
Date : 30 December, 2020This is to certify that the thesis entitled “Aspects of Quantum Cosmology" sub-mitted by Mr.
Sachin Pandey
Registration No. dated 25 July 2013, astudent of Department of Physical Sciences of the Integrated PhD Program of IISERKolkata, is based upon his own research work under my supervision. This is also to cer-tify that neither the thesis nor any part of it has been submitted for any degree/diplomaor any other academic award anywhere before. In my opinion, the thesis fulfils therequirement for the award of the degree of Doctor of Philosophy.
Prof. Narayan Banerjee
ProfessorDepartment of Physical SciencesIndian Institute of Science Education and Research KolkataMohanpur 741246, West Bengal, India v cknowledgement
First and foremost, I would like to express my sincere gratitude to my supervisorProf. Narayan Banerjee, for his constant support and advice throughout my doctoratestudies. I also thank him for all the useful discussions and offering invaluable advices.He is always open to new ideas, and encouraged and helped me to shape my own interestand ideas. One simply could not wish for a better or friendlier supervisor. His wordsof encouragement sustained me through my PhD studies and I am extremely gratefulfor that. I would never forget the stories he used to tell, especially the one where hementioned, "Everything seems simple once it is done".I would also like to thank my research progress committee members Dr. GolamMortuza Hossain and Prof. Rajesh Kumble Nayak, for their valuable advice and support.Thanks to the Department of Physical Sciences, of IISER Kolkata for providing all thelogistic support. I am thankful to Dr. Sridip Pal for insightful discussion. Thanksto CSIR(INDIA) for financial assistance through JRF-SRF(NET) fellowship during myPhD. I would like to thanks my teachers Dr. Rangeet Bhattacharyya from IISER Kolkataand Dr. Divya Haridas and Dr. Abhishek K. Singh and Prof. Patrick Das Gupta fromthe University of Delhi for their continuous support and valuable advice.Completing this work would have been all the more difficult without the support andcompany of wonderful friends and seniors like Aabir Mukhopadhyay, Aakash Anand,Abhinash Kumar, Ankit Kumar Singh, Dr. Anurag Banerjee, Avijit Chowdhury, Deban-gana Mukhopadhayay, Deepak Kumar Jha, Dipanjan Chakraborty, Kaustav Gangopad-hyay, Mayank Shreshtha, Nirbhay Kumar Bhadani, Pawan Kumar, Purba Mukherjee,Tanima Duary, Dr. Prateek Verma, Dr. Preethi Thomas, Shibendu Gupta Choudhury,Shivanand Mandraha, Dr. Shubham Chandel, Ravi Kumar, Dr. Sayak Ray, Dr. Sa-jal Mukherjee, Shashwat Kumar Singh, Soumik Mitra, Soumyajit Seth, Srijita Sinha,viiushant Kumar Sinha, Dr. Swati Sen; they made my years at IISER Kolkata an unfor-gettable experience. I am thankful to my college and school friends Aastha, Abhishek,Akhileshwar, Amit, Anuj, Anurag, Arif, Atul, Gaurav, Jewel, Meenal, Vipul for theircontinuous support and motivation.I would also like to thank Ek Pehal kids with whom I have spent the memorable eveningsand have learned from them while trying to teach them basic math and science.Finally, I thank my parents and my sister for their unconditional love and support.At the end of this fairly long testimonial, I once again thank everyone from whom I havelearned something or the other...
Sachin PandeyIISER Kolkata,30 December, 2020 viii ist of Publications
Content of this thesis is based on the following publications:• Sachin Pandey,
Anisotropic n-dimensional quantum cosmological model with fluid ,Eur. Phys. J. C , 487 (2019).• Sachin Pandey, Sridip Pal and Narayan Banerjee, Equivalence of Einstein andJordan frames in quantized anisotropic cosmological models , Annals Phys. ,93-107 (2018).• Sachin Pandey and Narayan Banerjee,
Equivalence of Jordan and Einstein framesat the quantum level , Eur. Phys. J. Plus. , 107 (2017).• Sachin Pandey and Narayan Banerjee,
Unitary evolution for anisotropic quantumcosmologies: models with variable spatial curvature , Phys. Scripta , 115001(2016).• Sachin Pandey and Narayan Banerjee, Unitarity in quantum cosmology: symme-tries protected and violated , arXiv:1911.11839 (Submitted for the publication).ix bstract
In this thesis, we try to resolve the alleged problem of non-unitarity for variousanisotropic cosmological models. Using Wheeler-DeWitt formulation, we quantized theanisotropic models with variable spatial curvature, namely Bianchi II and Bianchi VI. Weused the dynamical variable related to the fluid as a time parameter following Schutz’sformalism. We showed that Hamiltonian of respective models admits self-adjoint ex-tension, thus unitary evolution. We further extended the unitary evolution for higherdimensional anisotropic cosmological models. We obtained finite normed and time inde-pendent wave packet showing the unitary evolution of the models. Expectation valuesof scale factors showed that the quantized model avoids the problem of singularity. Wealso showed that unitarity of the model preserves the Noether symmetry but loses thescale invariance.Using the Wheeler-DeWitt quantization scheme, we showed the equivalence of Jordanand Einstein frames at the quantum level. We quantized the Brans-Dicke theory in boththe frames for the flat FRW model using dynamical variable related to the scalar fieldas a time parameter. Obtained expressions for wave packet matched exactly in boththe frames indicating the equivalence of frames. We also showed that equivalence holdstrue for various anisotropic quantum cosmological models, i.e., Bianchi I, V, X, LRSBianchi-I and Kantowski-Sachs models. xi ontents
List of figures xvList of tables xvii1 Introduction 1 α = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1.2 General perfect fluid: α = 1 . . . . . . . . . . . . . . . . . . . . . 312.1.3 α = − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Bianchi II models: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3 A note on spatial curvature . . . . . . . . . . . . . . . . . . . . . . . . . 352.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 36 α = 1 and n > α = 1 and n = 5 . . . . . . . . . . . . . . . . . . . . 463.2.3 For fluid with α = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Discussion with Concluding Remarks . . . . . . . . . . . . . . . . . . . . 50 xv ist of Figures < V = a b n − > . . . . . . . . . . . . . . . . 483.5 Expectation value of scale factors for n=5. . . . . . . . . . . . . . . . . . 49xvi ist of Tables HAPTER Introduction
A quantum theory of gravity should prescribe the quantum description of the universe.However, in the absence of a more generally accepted quantum theory of gravity, quan-tum cosmology is a moderately ambitious attempt to apply quantum principles in grav-itational systems. Another strong motivation of quantum cosmology is the expectationthat it might resolve the beginning of the universe from a singular state which plaguesthe classical description of the universe. This issue is discussed extensively in work byHawking and Ellis[1], Penrose[2], and Belinsky-Khalatnikov-Lifshitz [3]. Dynamics nearthe initial singularity is not well known even today.Wheeler-DeWitt equation[4, 5] provides a procedure of quantization which uses theHamiltonian formulation in 3 + 1 decomposition of gravitational configuration. It givesthe evolution of the universe in infinite dimensional space of all possible three-geometriesknown as wheeler superspace. The problem to deal with infinite degrees of freedom issimplified in quantum cosmology, where the imposition of symmetries reduces infinitedimensional space to finite one, which is called minisuperspace. Quantum cosmology hasits own sets of conceptual problems. The first one is the identification of time as our wellknown time parameter is a part of dynamics as a coordinate in relativistic theory. Therehave been several approaches to address this issue like the use of internal coordinates,1hapter 1 Introductioni.e., volume element or scalar field or external factors like matters field leading to sometime parameter for the evolution of the universe. The method developed by Lapchinskiiand Rubakov [6] uses the monotonic evolution of fluid density as a time parameter basedon Schutz’s formalism, which is discussed in detail in this thesis.Another major problem of quantum cosmology is a non-unitary evolution of theanisotropic universe [7–9], which questions the Wheeler-DeWitt formulation. The pos-sibility to resolve this problem has been initiated for a few anisotropic models withthe help of the self-adjoint extension of the Hamiltonian for the anisotropic models likeBianchi-I, V, IX [10–13]. The first part of this thesis aims to deal with alleged non-unitarity in quantum cosmological models, especially anisotropic models with variablespatial curvature.The latter part of the thesis deals with the equivalence of Jordan and Einstein framesin non-minimally coupled scalar field theories, particularly in Brans-Dicke theories atthe quantum level.In this thesis following notations have been used: Greek indices range over 0,1,2,3,4...and Latin indices range over 1,2,3.... A comma as in A, k denotes derivative of A withrespect to x k . The signature convention of + - - - for the metric g µν is adopted. Curvaturetensors are used in following forms: R τµνσ = Γ τµσ,ν − Γ τµν,σ + Γ γµσ Γ τγν − Γ γµν Γ τγσ , (1.1) R µν = R τµτν , (1.2) R = g µν R µν , (1.3)(1.4)where Γ τµν = g τγ g νγ,µ + g µγ,ν − g µν,γ ) , (1.5) g µγ g γν = δ µν . (1.6)The choice of units, unless otherwise specified, will be (cid:126) = c = 16 πG = 1, where G is2hapter 1 Introductiongravitational constant, (cid:126) = h π ( P lanck sconstant ) and c is the speed of light in vacuum. The line element for the space-time defined on four dimensional manifold M character-ized with metric tensor g µν ( x γ ) assigned on coordinate x γ can be given as ds = g µν dx µ dx ν . (1.7)Einstein-Hilbert action which contains the gravitational contribution and matter contri-bution is A = Z M d x ( √− gR + L m ) . (1.8)We need to introduce the Hamiltonian for the gravitational field for a canonical quanti-zation given by DeWitt [4]. This requires the 3 + 1 split of the metric in which spatialcomponents of metric are dynamical degrees of freedom. In geometrical language, thiscan be considered as a decomposition of the four-dimensional manifold M as M → R × Σ,where Σ denotes a three-dimensional hypersurface. Arnowitt, Deser, Misner(ADM)[14]describe the usual 3 + 1 split through the decomposition of the space-time metric interms of a lapse function N , a shift vector N i and an induced spatial metric h ij . Onecan begin by constructing the hypersurfaces Σ t , which is parameterized by some globaltime-like variable t .In the 3+1 decomposition metric tensor is given as, g µν = N − N k N k − N j − N j − h ij . (1.9)The extrinsic curvature of a hypersurface Σ t is defined as K ij = 12 N ( N i | j + N j | i − h ij, ) , (1.10)3hapter 1 Introductionwhere vertical bar denotes the covariant derivative with respect to three dimensionalmetric h ij . Gauss-Codazzi relation relates the four dimensional Ricci scalar R to threedimensional one R with the help of the extrinsic curvature K ij , √− gR = N √ h ( R + K ij K ij − K ) − √ hK ) , + 2( √ hKN j √ h − h ij N ,i ) ,j , (1.11)where h = det ( h ij ), K ij = h ik h jl K kl and K = h ij K ij .The last two terms of Eq.(1.11) represent a derivative on the boundary in the action anddo not contribute to dynamics and thus may be dropped. Now the gravitational part ofthe action(1.8) can be written in 3 + 1 formalism as A g ( N, N i , h ij ) = Z M dtdx L g = Z M N √ h ( R + K ij K ij − K ) dtdx . (1.12)We can write conjugate momenta to the variables ( N, N i , h ij ) from the Lagrangiandensity in the following way, Π = δ L δ ˙ N = 0 , (1.13)Π i = δ L δ ˙ N i = 0 , (1.14)Π ij = δ L δ ˙ h ij = −√ h ( K ij − h ij K ) . (1.15)Equations (1.13) and (1.14) are known as primary constraints . They express the factthat the Lagrangian does not depend on the velocities ˙ N and ˙ N i .Hamiltonian can be written in terms of N, N i , h ij and their conjugate momenta usingthe Legendre transformation as H = Z d x (Π ˙ N + Π i ˙ N i + Π ij ˙ h ij − L ) , (1.16) H = Z d x (Π ˙ N + Π i ˙ N i + N i H i + N H ) , (1.17)4hapter 1 Introductionwhere H = G ijkl Π ij Π kl − √ h ( R ) , (1.18) H i = − h ik Π kj | j , (1.19) G ijkl = 12 √ h ( h ik h jl + h jk h il − h ij h kl ) . (1.20) H , H i and G ijkl are called as super Hamiltonian, super momentum and super-metric orDeWitt metric respectively [4].In terms of these variables, the action for gravity part(1.12) becomes A g = Z dtdx (Π ˙ N + Π i ˙ N i − N i H i − N H ) . (1.21)If we vary the above action with respect to Π ij , we get back the relation given in equa-tion(1.15). Variation of action with respect to lapse function N gives the Hamiltonianconstraint H = 0 , (1.22)while variation of the action with respect to shift vector N i gives the supermomentumconstraint H i = 0 (1.23)These constraints can also be obtained as˙Π = ∂ ( N H ) ∂N = H = 0 , (1.24)˙Π i = ∂ ( N i H i ) ∂N i = H i = 0 . (1.25)They are also known as secondary constraints in Dirac terminology.5hapter 1 Introduction
The super Hamiltonian as given in equation(1.24) provides the evolution of the system.The supermomentum constraint defines the configurational space of the canonical gravitythat it is the infinite dimensional space of all the possible geometries[15].
For an infinite-dimensional space x = { x i } , which specifies the point on the hypersurfaceΣ, there are finite number of degrees of freedom at each point, one considers the spaceof all Riemannian 3-metric configuration on the spatial hypersurfaces, Riem (Σ) = { h ij ( x ) , x ∈ Σ } . (1.26)Since we are interested in the geometry and configuration related to each other by adiffeomorphism, we identify the superspace [5] as Riem (Σ)
Dif f (Σ) , where zero subscript denotes the diffeomorphism connected to the identity. Then, theDeWitt metric (1.20) can be written as G AB ( x ) = G ( ij )( kl ) ( x ) , (1.27)where the indices A, B ∈ ( h , h , h , h , h , h ), run over all the independent com-ponents of intrinsic metric h ij . Signature of DeWitt metric at each point x is always(- + + + + +), regardless of the signature of the space-time metric g µν . We can alsoextend the range of indices to include the matter fields by defining appropriate G φφ ( x ).Classical Poisson brackets for the configurational variables can be written as { h αβ ( x, t ) , h γδ ( x , t ) } = 0 , (1.28)6hapter 1 Introduction { Π αβ ( x, t ) , Π γδ ( x , t ) } = 0 , (1.29) { h αβ ( x, t ) , Π γδ ( x , t ) } = 0 , (1.30)and the corresponding quantum Poisson brackets can be given as { ˆ h αβ ( x, t ) , ˆ h γδ ( x , t ) } = 0 , (1.31) { ˆΠ αβ ( x, t ) , ˆΠ γδ ( x , t ) } = 0 , (1.32) { ˆ h αβ ( x, t ) , ˆΠ γδ ( x , t ) } = 0 . (1.33)The quantum operators corresponding to the classical variables can be written as N → ˆ N , N i → ˆ N i , h ij → ˆ h ij , (1.34)Π → ˆΠ = − i δδN , Π i → ˆΠ i = − i δδN i , Π ij → ˆΠ ij = − i δδh ij . (1.35)Let us consider the wave function of the universe to be a functional state Ψ( N, N i , h ij ),which is annihilated by the quantum version of the primary constraint(1.13,1.14) asˆΠΨ = − i δ Ψ δN = 0 , ˆΠ i Ψ = − i δ Ψ δN i = 0 . (1.36)These relations establish the fact that the wave function of the universe is independentof ( N, N i ) and become a functional of three metric h ij only. Similarly, using supermo-mentum constraint, we get ˆ H i = 2 i δ Ψ δh ij ! | j = 0 . (1.37)The above relation implies the fact that the wave function of the universe does notdepend on a particular metric used to represent the geometry. Instead, it is defined onthe whole class of the three geometries. This can be expressed asΨ = Ψ { h ij } . (1.38)7hapter 1 IntroductionThis indicates that wave function exists in the same configurational space, i.e., super-space, as discussed in the beginning. Finally, the quantum counterpart of the superHamiltonian constraint read asˆ H Ψ = G ijkl δ Ψ δh ij δh kl − √ h R Ψ = 0 . (1.39)Equation (1.39) is known as Wheeler-DeWitt (WDW) equation[4, 5]. It is the second-order differential equation at each point x ∈ Σ on the superspace. We can see theambiguity of factor ordering in this WDW equation, but there exist natural choices ofordering for which derivative terms become a Laplacian in the supermetric.
In section 1.2.1, the canonical quantization method considers the configurational space ofthe infinite dimensional space of all possible three-geometries. The presence of an infinitenumber of degrees of freedom makes the problem intractable with the techniques thathave been developed so far. It is necessary to truncate the infinite degrees of freedom tofinite numbers from a practical perspective. One can obtain some particular minisuper-space model through the imposition of symmetries in superspace [16–19]. Consideringhomogeneous metrics is an easy choice to achieve this. Let us consider a spatially ho-mogeneous system characterized by zero shift vector N i = 0, only time dependent lapsefunction N ( t ) given as ds = N ( t ) dt − h ij dx i dx j . (1.40)In this minisuperspace, the three metric h ij depends on the finite number of coordinates q A , unlike the superspace case where we have the infinite dimensional degree of freedom.The Einstein-Hilbert action for this minisuperspace can be written as A = Z dt [ 12 N G AB ( q ) ˙ q A ˙ q B − N U ( q )] , (1.41)where G AB is the minisupermetric, the reduced version of the entire supermetric G ijkl G AB dq A dq B = Z d x G ijkl δh ij δh kl , (1.42)and U ( q ) is the potential term given as U = Z d x √ h ( − R ) . (1.43)Since our configurational space is finite dimensional, quantization is simplified as we aredealing with the quantum mechanics of the constrained system.Now canonical momenta and Hamiltonian can be obtained asΠ A = ∂L∂ ˙ q A = G AB ˙ q B N , (1.44) H = Π A ˙ q A − L = N ( 12 G AB Π A Π B + U ( q )) . (1.45)We can obtain the super Hamiltonian constraint by varying the action with respect tothe lapse function as H = 1 N H = 12 G AB Π A Π B + U ( q ) = 0 . (1.46)With natural choices of ordering [20], canonical quantization of equation(1.46) leads tothe Wheeler-DeWitt equation asˆ H Ψ = [ − ∇ + U ( q )]Ψ = 0 , (1.47)where the symbol ∇ denotes the covariant derivative constructed from minisupermetricand the Laplacian ∇ for the minisupermetric is given as ∇ = 1 √−G ∂ A [ √−GG AB ∂ B ] , (1.48)where G = det ( G AB ). 9hapter 1 Introduction The quantum Hamiltonian constraint H (1.24) is that the operator ˆ H annihilates thephysical states. The well known Schrödinger equation for our Hamiltonian constraintequation can be written as ˆ H Ψ = i ∂ Ψ ∂t = 0 . (1.49)This equation denotes the time independence of the wave function Ψ, which apparentlydiscard the quantum evolution of the system. Such frozen formalism[21] seems to suggestthat the quantum theory of gravity does not evolve with the time. This issue is knownas the problem of time [22].Difference between the intrinsic nature of the two theory, general relativity, and quan-tum mechanics give rise to this issue. In quantum mechanics, the time is used for theevolution of the system, and events occur on it as time is an externally fixed scalarparameter.The absence of a suitable time parameter creates countless problems in quantumtheory. Any quantum theory is unable to explain the concepts of probability and mea-surement in the absence of proper time description unless one redefines these concepts.In order to avoid the problem of time, we must have a first order momenta term inHamiltonian. One can identify a time coordinate q A at the classical level. Then one canwrite scalar constraint with the help of the conjugate momenta p A to the time variableas p A + H A = 0 , (1.50)where H A is the physical reduced Hamiltonian which can evolve with respect to the timevariable q A and subsequently Schrödinger like equation can be written asˆ H A Ψ = i d Ψ dq A . (1.51)The selection of time variable q A can be made in two ways. First, one can select q A from gravitational configurational space. But with this selection, one can only quantize10hapter 1 Introductiona part of gravitational space. The quantization of the reduced ADM Hamiltonian usingthis scheme is shown by Arnowitt, Deser, and Misner [14].The second possibility can be a selection of the time coordinate q A from the externalmatter field and evolve the gravity part of the Hamiltonian with respect to time, whichis picked up from the evolution of the matter distribution. One such procedure is theuse of the fluid so that the monotonic evolution of the fluid density can be identified asa time parameter. Lapchinskii and Rubakov [6] uses this procedure with the help of the Schutz’s formalism , and it is discussed in detail in the following section.
In hydrodynamics, a perfect fluid can be described with the help of velocity potentials[23]. Schutz [24] generalized it as a nonlinear relativistic field theory for five coupledscalar fields, whose Lagrangian density is simply the pressure of the fluid. A similargeneralization was also independently suggested by Schmid [25]. In this section, wediscuss the conversion of perfect-fluid hydrodynamics into the Hamiltonian form, asgiven by Schutz [26].Action for fluid part can be written as A f = Z dtd x √− g P = Z dtd xN √ h P , (1.52)where P is pressure of fluid related to density ρ by an equation of state ( P = αρ ) . Nowlet us express the four velocity U ν as U ν = 1 µ ( (cid:15) ,ν + ζβ ,ν + θS ,ν ) , (1.53)where S is specific entropy, ζ and β are potentials connected with rotation, and (cid:15) and θ are potentials with no clear physical meaning and µ is specific enthalpy given as µ = P + ρρ = 1 + u + P ρ . (1.54)11hapter 1 IntroductionHere ρ is rest mass density and u is specific internal energy. The four velocity isnormalized as U ν U ν = 1 . (1.55)From the laws of thermodynamics, one can write τ ( ρ , u ) dS = d u + P d (1 /ρ ) = (1 + u ) d ( ln (1 + u ) − α lnρ ) . (1.56)From above relation, expression for entropy S can be given as S = ln (1 + u ) − α lnρ . (1.57)Using equations (1.54) and (1.57), the fluid pressure can be given as P = α µ /α (1 + α ) /α e − Sα . (1.58)In co-moving system, the four-velocity U ν = ( N, , , µ = 1 N ( ˙ (cid:15) + ζ ˙ β + θ ˙ S ) . (1.59)Using equations (1.58) and (1.59), action for the fluid part(1.52) can be given as A f = Z d xdt L f = Z dtd x N − α √ h α (1 + α ) α (cid:16) ˙ (cid:15) + ζ ˙ β + θ ˙ S (cid:17) α e − Sα . (1.60)If q a stand for the five potential fields (cid:15), ζ, β, θ and S , then their conjugate momenta canbe given as p a = ∂ L f ∂ ˙ q a . (1.61)12hapter 1 IntroductionThis gives p (cid:15) = N − α √ h α ) α (cid:16) ˙ (cid:15) + ζ ˙ β + θ ˙ S (cid:17) α e − Sα , (1.62) p ζ = p θ = 0 , (1.63) p β = ζp (cid:15) , (1.64) p S = θp (cid:15) . (1.65)Potential fields ζ and β are related to the vorticity of the system. They do not contributeto any space-time without rotation. Hence they and their respective conjugate momentacan be dropped. This shows that we only have one independent momentum.The Lagrangian density for the fluid part read as L f = N − α √ h α (1 + α ) α (cid:16) ˙ (cid:15) + θ ˙ S (cid:17) α e − Sα . (1.66)Since p (cid:15) is only independent momentum, we can introduce the following canonical trans-formation T = − p S e − S p − α − (cid:15) , (1.67) p T = p α +1 (cid:15) e S , (1.68) (cid:15) = (cid:15) + ( α + 1) p S p (cid:15) , (1.69) p (cid:15) = p (cid:15) . (1.70)The corresponding Hamiltonian for the fluid part can be obtained from (1.66) usingabove canonical transformed variable as H f = N √ h α p T . (1.71)The Poisson bracket { T, p T } corresponds to the quantum commutator in a canonicalquantization, p T = − i ∂∂T as [ T, p T ] = i. (1.72)13hapter 1 IntroductionThe classical Poisson bracket { T, H f } gives { T, H f } = 1 N dTdt = 1 √ h α . (1.73)Since √ h α is a positive quantity, parameter T can have the same orientation as thecosmic time everywhere in the system. Hence, parameter T can be chosen as a timeparameter, as discussed in the previous section. By varying the action with respect to N , one can get the super Hamiltonian H = √ h α N ( H g + H f ) = 0 which reads as H g + p T = 0 . (1.74)Now the Wheeler-DeWitt equation ( ˆ H Ψ = 0) reads asˆ H g Ψ = i ∂ Ψ ∂T . (1.75)Equation (1.75) looks like the Schrödinger equation. Also T, which we identify as time,is indeed a scalar parameter and not a coordinate, unlike the cosmic time t. Thus the problem of time , as discussed in section 1.3 is now resolved. Schutz’s formalism notjust solves the problem of time, but also help to quantize the full part of gravitationalspace as mentioned in previous section 1.3. As long as we can have 3+1 decompositionof metric of homogeneous system, this formalism should give us time parameter havingsame orientation as cosmic time(1.73). We notice regularity in the distribution of matter and radiation in our universe if welook at a large scale. An observable universe at a large scale suggests that our universeis homogeneous and isotropic everywhere. This phenomenon supported by observationis also known as
Cosmological principle . Isotropy means that there are no particulardirections in the universe, i.e., all direction looks similar. Homogeneity means that14hapter 1 Introductionthere exist no specific or preferred places in our universe.Roberston-Walker (RW) metric given as ds = dt − a ( t ) dr − kr + r dθ + r sin θdφ ! (1.76)is widely used isotropic and homogeneous space-time. Here a ( t ) is the scale factor,( r, θ, φ ) are spherical polar coordinates and k is a constant indicating the spatial cur-vature of the three space, which can take normalized values +1 , , −
1. When k = 0,the three-space is flat. When k = +1 and k = −
1, the three-space is of positive andnegative constant curvature; they are known as the closed and open Friedmann models,respectively.From the general point of view, the cosmological model can be divided into threeparts based on isotropy. The first one is the isotropic model. The second one is knownas Locally Rotational Symmetry(LRS) models in which kinematical quantities are rota-tionally symmetric about a preferred spatial direction. All observations are rotationallysymmetric about this direction at every general point. Third, the obvious one in whichobservation in a spatial direction differs from observations in other directions is knownas anisotropic models.With the help of three independent differential forms defined as ω a = e aα dx α where e aα are a set of four linearly independent vectors, we can express more general homogeneousspace-time metric as ds = N ( t ) dt − η ab ω a ω b , (1.77)where η ab = e ia e ib is a symmetric tensor depending on time only. Now homogeneityconditions can be expressed as C cab = ∂e cα ∂x β − ∂e cβ ∂x α ! e αa e βb , (1.78)where C cab are known as structure constants. They are antisymmetric in the lowerindices. All possible homogeneous models can be expressed through the dual of these15hapter 1 Introduction Type of Bianchi Model a n n n I 0 0 0 0II 0 1 0 0VI 0 1 − − − V II a a V I a ( a = 1) a − Table 1.1: Classification of Bianchi Models. constants with the help of antisymmetric Levi-Civita tensor (cid:15) abc , i.e., C ab = (cid:15) cda C bcd .Then expression given by (1.78) can be written as, (cid:15) bcd C cd C ba = 0 . (1.79)We can decompose the tensor C ab into symmetric and antisymmetric part as C ab = n ab + (cid:15) abc a c , (1.80)where (cid:15) abc a c is antisymmetric part and n ab is symmetric part. We can redefine symmetrictensor n ab to a diagonal matrix n ab = diag ( n , n , n ) and similarly a c = ( a, ,
0) withoutlosing the generality. Then condition given by (1.79) takes the form as an = 0 . (1.81) Bianchi classified various geometric model based on this homogeneity condition. For allpossible combinations, n , n , n can assume values 0 , , − a ≥
0. Brief classifica-tion of all possible models is given in the table 1.1. One can look at this classification indetail in work done by Bianchi [27]. These models are known as
Bianchi cosmologicalmodels. 16hapter 1 Introduction
Resolving the problem of time, as discussed in section 1.3, does not solve all the concep-tual problems of quantum cosmology. In the standard quantum mechanics, the evolutionof wave function, inner product, and conservation of probability is well defined. We needto see whether the Wheeler-DeWitt method of quantization help in defining these forquantum cosmology or not. Let’s consider the Bianchi-I cosmological model given as ds = N ( t ) dt − a ( t ) dx − b ( t ) dy − c ( t ) dz , (1.82)where a ( t ) , b ( t ) , c ( t ) are respective scale factors along the 3-space directions indicatingthe anisotropic nature of the model. The Lagrangian for the gravity section of theaction(1.12) can be given as L g = − e β N ( ˙ β − ˙ β − ˙ β − ) . (1.83)The corresponding Hamiltonian for the gravity sector read as H g = − N e − β
24 ( p − p − p − ) , (1.84)where p i (for i =0,+,-) are canonical momentas, p i = ∂ L g ∂ ˙ β i . With the help of Schutz’sformalism, as discussed in section 1.5, one can find the Hamiltonian for the fluid part, H f from (1.71). Then the total Hamiltonian( H g + H f ) for Bianchi I Model takes theform as H = N e − β (cid:18) −
124 ( p − p − p − ) + e − α ) β p T (cid:19) . (1.85)It is quite apparent from equation (1.85) that the signature of the kinetic term of Hamil-tonian is hyperbolic. From equation (1.85) Wheeler-DeWitt equation for Bianchi I Uni-verse can be written as ∂ ∂β − ∂ ∂β − ∂ ∂β − ! ψ = − ie − α ) β ∂ψ∂T . (1.86)17hapter 1 IntroductionThe hermiticity condition of the Hamiltonian requires the wave function ψ of equation(1.86) must follow the boundary condition given as ∂ψ∂β i ! β i →±∞ = K ( ψ ) β i →±∞ = 0 . (1.87)To ascertain whether the wave function for Bianchi I universe obtained from equation(1.86) follow the above condition or not, one needs to calculate the expression of the wavefunction. Alvarenga et al have obtained the explicit expression for the wave function forequation (1.86) in work [8]. The expression for the wave function read as ψ ( β i , T ) = e i ( k + β + + k − β − ) " c J v √ E − α ) / e − α ) β / ! + c J − v √ E − α ) / e − α ) β / ! e − iET , (1.88)where c , c are constant of integration and J v is Bessel function of order v . By integrat-ing out parameters k + , E and fixing k − = 0, wave packet read asΨ = 1 B s πγ exp " − e − α ) β B − ( β + + C ( β , B )) γ , (1.89)where B = λ − i − α ) T , C ( β , B ) = e β − − α ) ln (2 B ) and γ and λ are constant. Itis quite evident that the wave packet given by equation (1.89) is square integrable andalso follows the boundary conditions described by equation (1.87).We can obtain the norm || Ψ || from equation (1.89) as || Ψ || = Z ∞−∞ Z ∞−∞ e − α ) β ψ ∗ ψdβ dβ + = 2 √ γπ − α ) λ F ( T ) , (1.90)where F ( T ) = exp (cid:16) [ Img ( C ( β ,B ))] γ (cid:17) , Img ( C ( β , B )) = − − α ) arctan ( − − α ) T λ ).From equation (1.90), it is quite clear that norm || Ψ || is time dependent; thus, modeldoes not have unitary evolution.Time dependent norm also leads to the inequivalence between the Copenhagen interpretation[28]18hapter 1 Introductionand the de Broglie-Bohm interpretation [29]. Even if the Hamiltonian is hermitian, i.e.,the eigenvalues are real, the evolution may not be unitary. For unitarity, the Hamilto-nian has to be self-adjoint, i.e., H † = H , which means that they act on the same Hilbertspace. For a comprehensive discussion on this issue, we refer to the monograph by Reedand Simon[30].An operator A can be self adjoint only if the domain of A † is the same as the domainof A . One can calculate the deficiency indices n ± in order to verify the self-adjointness ofthe operator A . Here n ± are the dimensions of the linear independent square integrablesolutions of the indicial equation given as Aφ = ± iφ. (1.91)If n + = n − = 0 then operator A is self-adjoint. Even if n ± = 0, a self-adjoint ’extension’of A is possible if n + = n − [30].Alvarenga et al [8] showed that, for Hamiltonian H given in the form as equation(1.85) which leads to time dependent norm (1.90) and the probability is not conserved.Incidentally values of deficiency indices come out as n + = 0 and n − = 1. Thus H isneither self adjoint, nor it admits any self adjoint extension.We now discuss the reason for this problem of non-unitarity in the model. Apparently,the alleged non-unitarity stems from the hyperbolicity of the Hamiltonian (equation1.84). The momentum squared terms contribute with both positive and negative signs.FRW models have only one scale factor, so this problem does not arise at all.However, it was shown that the clue lies in the operator ordering rather than anythingelse[10].In the case of the Bianchi I model, Wheeler-DeWitt equation (1.86), a particular opera-tor ordering e α − β ∂ ∂β is used. If one uses the operator ordering as e α − β / ∂∂β ( e α − β / ∂∂β ),then Pal and Banerjee [10] show that the Hamiltonian can be written as H g = d dχ + σχ , (1.92)19hapter 1 Introductionwith the help of a new variable χ = e − ( α − β . This Hamiltonian is similar to thatof the inverse square potential, which is a well studied problem in physics [31]. Paland Banerjee show that deficiency indices for H g , n ± = 1; thus, it admits self adjointextension. Hence the alleged problem of the non-unitarity may be avoided. A similarself-adjoint extension is shown for anisotropic models like Bianchi-I, III, IX, Kantowski-Sachs (KS) models in the subsequent work [11–13].Very recently, Pal and Banerjee [32] show that a proper ordering can in fact resolvethe issue of the alleged non-unitarity. Anyway, operator ordering has a one parameterof U(1) family of self-adjoint extensions. We can have various operator ordering, whichleads to a U(1) group of self-adjoint extension; thus, unitarity can be preserved for abunch of the orderings. Hence the choice of operator ordering is not unique for theunitary evolution of the model. However, the above mentioned particular ordering is agood choice that can preserve the ground state energy, unlike other extensions. Theyalso emphasize that explicit evaluation of deficiency index and construction of boundaryconditions for self-adjoint extension may not be analytically possible in all cases.The alleged discrepancy between the Copenhagen interpretation and the Bohm-deBroglie interpretation can also be resolved by the proper choice of boundary conditions.However, it may be a non-trivial task to prove it due to the presence of hyperbolicityin the Hamiltonian. Incidentally, this discrepancy is also shown for the isotropic casewhere the hyperbolicity is absent by Falciano, Pinto-Neto, and Struyve [33]. The theory of general relativity is based on the equivalence principle. However, Mach’sprinciple [34] states that inertial mass is affected by the global distribution of matter;thus, it creates the compatibility issue with general relativity. Brans and Dicke [35, 36]made an attempt to incorporate the Mach’s principle in a relativistic theory of thegravity. They introduced a scalar field φ = φ ( x i , t ), which effectively makes G , theNewtonian constant of gravity, vary with the spacetime coordinates.20hapter 1 IntroductionBrans-Dicke theory of gravity, which is among the widely used modified theory of thegravity. It was believed that the theory reduces to standard general relativity when thecoupling constant ω → ∞ . It was later proved that this was not in general correct[37, 38].Still, Brans-Dicke theory and some generalization of that find application, particularlyin cosmological scenario, such as for resolving the general exit issue of inflation [39, 40]or deriving a late time acceleration even without a dark energy[41]. With the assumption that only that Gravitational constant G varies with space-time,action for Brans-Dicke theory can be written as A J = Z d x " √− g ( φR + ωφ ∂ µ φ∂ µ φ ) + 16 πc L m (1.93)where φ is the scalar field which plays the role analogous to the G − and ω is a dimen-sionless parameter. The second term is the contribution of the Lagrangian density of ascalar field. L m is the Lagrangian density for matter field.This theory, given by equation (1.93), is also known as the Brans-Dicke theory in Jor-dan frame for having a formal connection with Jordan’s theory [42]. The field equationfor the theory can be written as R µν − g µν R = 8 πφ − c T µν + ωφ ( φ ,µ φ ,ν − g µν φ ,δ φ ,δ ) + 1 φ ( φ ,µ ; ν − g µν (cid:3) φ ) , (1.94)where T µν is energy momentum tensor coupled with the variable gravitational param-eter φ − unlike constant G in the usual Einstein field equation and (cid:3) is covariantd’Alembertian operator. The wave equation for the scalar field φ is given as (cid:3) φ = 8 π (2 ω + 3) c T. (1.95)21hapter 1 Introduction Later Dicke[36] showed the theory in the conventional form, in which the Einstein fieldequation holds true, with the help of the coordinate dependent transformation of theunits of measure. Metric tensor transforms as g µν → Ω g µν and mass transforms as m → Ω m . Then after affecting the transformation as stated, which is also a conformaltransformation, action given by equation (1.93) takes the form as A E = Z d x √− ¯ g (cid:20) ¯ R + 2 ω + 32 ∂ µ ¯ φ∂ ν ¯ φ + + 16 πc ¯ L m (cid:21) , (1.96)where ¯ R is the Ricci scalar in transformed coordinates, ¯ φ = ln φ and ¯ L m = L m / Ω isthe contribution from matter field. Here we can notice that the scalar field is coupledto the gravity part only in a minimal way, and there is no "interference term" like φR asin equation (1.93). This representation is called as Brans-Dicke Theory in the Einsteinframe. If Ω = φ , this frame restores the constancy of the gravitational constant butit effects the rest mass of the particles as it is now a function of the scalar field. Thisfurther leads to the loss of the equivalence principle as rest mass is no longer a constantand geodesic equations are no longer valid. In this frame, the field equation can bewritten as, ¯ R µν −
12 ¯ g µν ¯ R = 8 πG c T µν + (2 ω + 3)2 ( ¯ φ ,µ ¯ φ ,ν −
12 ¯ g µν ¯ φ ,δ ¯ φ ,δ ) , (1.97)and wave equation for the scalar filed is (cid:3) ¯ φ = 8 π (2 ω + 3) c T. (1.98)Now the bigger question arises: which frame should be used? Physicists are dividedin order to answer this question. This topic is widely debated, and it can be categorized.There are authors[43–47] argue that the two frames are physically equivalent. In works[48–52], physicists consider these frames are physically non-equivalent for various rea-sons. The other group of authors [53–57] regards only Jordan frame as a physical frame,22hapter 1 Introductionbut they show that the Einstein frame can also be used for mathematical convenience.Many physicists also believe that the Einstein frame is the only physical frame[58–63].Detail summary can be found in review work by Faraoni, Gunzig, and Nardone[64]. Forquantum aspects of the equivalence, we refer to the discussion by Almedia et al[65, 66]. The purpose of the thesis is two fold. One is to look at some anisotropic models inconnection with the possibility of unitary evolutions of quantized models in the Wheeler-DeWitt formulation. Chapters 2, 3 and 4 are devoted to that. The second purpose is touse Wheeler-DeWitt formulation to find the resolution of the equivalence of Jordan andEinstein frames at the quantum level. Chapters 5 and 6 deal with that.In chapter 2, we quantize the Bianchi II and Bianchi VI cosmological models, which areanisotropic models with variable spatial curvature, using the Wheeler-DeWitt method ofquantization. As time is itself a coordinate in a relativistic theory, dynamical variablesrelated to fluid have been used as “time” following Schutz’s formalism. We show unitaryevolution is indeed possible for these models as their respective Hamiltonians admit self-adjoint extensions.In chapter 3, we work on extending the unitary evolution in higher dimensionalanisotropic quantum cosmological models. We discuss the Wheeler-DeWitt quantiza-tion scheme for the model with a perfect fluid in the presence of a massless scalar field.We identify the time parameter using a generalization of Schutz’s formalism and find thewave packet of the universe following standard Wheeler-DeWitt methodology of quan-tum cosmology. We establish the unitary evolution for the model. We also calculatethe expectation values of scale factors and volume elements for different dimensions,which show that the quantized model escapes the singularity and supports the bouncinguniverse solutions.In order to achieve the unitary evolution of cosmological models, what are the price onehas to pay for the self-adjoint extension, do we lose symmetries like Noether symmetry23hapter 1 Introductionand scale-invariance? This has been discussed for the Bianchi-I cosmological model inchapter 4.There has been a long standing debate regarding the equivalence of Jordan and Ein-stein frames in literature. We quantize the Brans-Dicke theory in both the frame usingdynamic variable related to the scalar field as time parameter and then address thequestion of equivalence of these two frames at the quantum level for the isotropic cos-mological model, i.e., FRW in chapter 5. The obtained expressions for wave packets inboth the frames and show the equivalence of the frames at the quantum level.In chapter 6, we use a similar method of the quantization, i.e., Wheeler-DeWitt,with dynamic variables related to the scalar field as the time for the various anisotropicmodel, i.e., Bianchi I, V, IX and LRS Bianchi -I and Kantowski-Sachs models in boththe frames. We try to generalize the equivalence of both frames at the quantum levelfor all these models. 24
HAPTER Anisotropic cosmological models with variable spatial curvature Quantum cosmology has its own motivation, such as looking for a resolution of theproblem of singularity at the birth of the universe. The basic framework for quantumcosmology is provided by the Wheeler-DeWitt equation[4, 5, 67]. The Wheeler-DeWittformulation actually has a very general appeal, the approach is very similar to the usualpractice in standard quantum physics. From the classical Lagrangian, the momenta cor-responding to the identified coordinates are found out so as to write the Hamiltonian,the variables are then promoted to operators (usually in the coordinate representation),and the relevant Schrodinger like equations are found out which govern the system. Insome cases, the Wheeler-DeWitt equation has a straightforward analogue with somesituations in other branches of physics. For instance, one anisotropic quantum cosmo-logical model, namely the Bianchi-I model eventually reduces to the standard quantummechanical problem with an inverse square potential[10] which has many applicationsin other branches of physics[68].One major problem of quantum cosmology is that the quantization of anisotropic The work illustrated in this chapter is published;
S. Pandey and N. Banerjee, Phys. Scr. ,115001(2016). We start with the standard Einstein-Hilbert action for gravity along with a perfect fluidgiven by A = Z M d x √− gR + Z M d x √− gP, (2.1)where R is the Ricci Scalar, g is the determinant of the metric and P is the pressureof the ideal fluid related to density ρ by an equation of state ( P = αρ ). The first termcorresponds to the gravity sector and the second term is due to the matter sector. Herewe have ignored the contributions from boundary as it would not contribute to the vari-ation. The units are so chosen that 16 πG = 1.A Bianchi VI model is given by the metric ds = n ( t ) dt − a ( t ) dx − e − mx b ( t ) dy − e x c ( t ) dz , (2.2)where the lapse function n and a, b, c are functions of time t and m is a constant.27hapter 2 Anisotropic cosmological models with variable spatial curvatureFrom the metric given above, we can write the Ricci Scalar as √− gR = e (1 − m ) x " ddt [ 2 n ( ˙ abc + ˙ bca + a ˙ cb )] − n [ ˙ a ˙ bc + ˙ b ˙ ca + ˙ c ˙ ab + n bc a ( m − m + 1)] . (2.3)Using this, we can find the action for the gravity sector from equation (2.1) which isgiven as A g = Z dt " − n [ ˙ a ˙ bc + ˙ b ˙ ca + ˙ c ˙ ab + n bc a ( m − m + 1)] , (2.4)where an overhead dot indicates a derivative with respect to time.Now we make a set of transformation of variables as a ( t ) = e β , (2.5) b ( t ) = e β + √ β + − β − ) , (2.6) c ( t ) = e β −√ β + − β − ) . (2.7)This introduces a constraint a = bc , but the model still remains Bianchi Type VIwithout any loss of the typical characteristics of the model. Such type of transformationof variables has been extensively used in the literature[8–10]. One can now write theLagrangian density of the gravity sector as L g = − e β n [ ˙ β − ( ˙ β + − ˙ β − ) + e − β n ( m − m + 1)12 ] . (2.8)Here β , β + and β − has been treated as coordinates. So corresponding Canonicalmomentum will be p , p + and p − where p i = ∂ L g ∂ ˙ β i . It is easy to check that one has p + = − p − . Hence we can write the corresponding Hamiltonian as H g = − ne − β [ 124 ( p − p − m − m + 1) e β )] . (2.9)28hapter 2 Anisotropic cosmological models with variable spatial curvatureWith the widely used technique, developed by Lapchinskii and Rubakov[6] by usingthe Schutz’s formalism as discussed in section 1.4, the action the fluid sector can bewritten as A f = Z dt L f = Z dt n − α e β α (1 + α ) α (cid:16) ˙ (cid:15) + θ ˙ S (cid:17) α e − Sα . (2.10)Here (cid:15), θ, S are thermodynamic potentials. A constant spatial volume factor V comesout of the integral in both of (2.4) and (2.10). This V is inconsequential and can beabsorbed in the subsequent variational principle. With a canonically transformed set ofvariables T, (cid:15) in place of S, (cid:15) , one can finally write down the Hamiltonian for the fluidsector as H f = ne − β e − α ) β p T . (2.11)The canonical transformation is given by the set of equations T = − p S exp( − S ) p − α − (cid:15) , (2.12) p T = p α +1 (cid:15) exp( S ) , (2.13) (cid:15) = (cid:15) + ( α + 1) p S p (cid:15) , (2.14) p (cid:15) = p (cid:15) , (2.15)This method and the canonical nature of the transformation are comprehensively dis-cussed in reference [10].The net or the super Hamiltonian is H = H g + H f = − ne − β
24 [ p − p − m − m + 1) e β − e − α ) β p T ] . (2.16)Using the Hamiltonian constraint H = 0, which can be obtained by varying the action29hapter 2 Anisotropic cosmological models with variable spatial curvature A g + A f with respect to the lapse function n , one can write the Wheeler-DeWitt equationas [ e α − β ∂ ∂β − e α − β ∂ ∂β + 12( m − m + 1) e (3 α +1) β ] ψ = 24 i ∂∂T ψ. (2.17)This equation is obtained after we promote the momenta to the corresponding operatorsgiven by p i = − i ∂∂β i in the units of (cid:126) = 1.It is interesting to note that for a particular value of m = m where m is a root ofequation m − m + 1 = 0, the spatial curvature vanishes and the equation (2.17) re-duces to the corresponding equation for a Bianchi Type I model[10]. We shall discuss thesolution of the Wheeler-DeWitt equation in two different cases, namely α = 1 and α = 1. α = 1 For a stiff fluid ( P = ρ ), the equation (2.17) becomes simple and easily separable. Itlooks like " ∂ ∂β − ∂ ∂β + 12( m − m + 1) e β ψ = 24 i ∂∂T ψ. (2.18)With the separation ansatz ψ = e i k + β + φ ( β ) e − iET , (2.19)one can write ∂ φ∂β + (4 k − E + 4 N e β ) φ = 0 , (2.20)where N = 3( m − m + 1). After making the change in variable as q = N e β , aboveequation can be written as q ∂ φ∂q + q ∂φ∂q + [ q − (6 E − k )] φ = 0 . (2.21)30hapter 2 Anisotropic cosmological models with variable spatial curvatureSolution of this equation can be written in terms of Bessel’s functions as φ ( q ) = J ν ( q ) , (2.22)where ν = q E − k . Now for the construction of the wave packet, we need to fix ν .If we take (cid:15) = − ν = k − E then wave packet can have the following expressionΨ = Φ( q ) ζ ( β + ) e i(cid:15)T/ . (2.23)where ζ ( β + ) = Z dk + e − ( k + − k +0 ) e i (2 k + β + − k T ) (2.24)The norm indeed comes out to be positive and finite (for the details of the calculations,we refer to work of Pal and Banerjee [12]). Thus one indeed has a unitary time evolution. α = 1 Now we shall take the more complicated case of α = 1 and try to solve the Wheeler-DeWitt equation (2.17). We use a specific type of operator ordering with which equation(2.17) takes the form " e ( α − β ∂∂β e ( α − β ∂∂β − e α − β ∂ ∂β + 12( m − m + 1) e (3 α +1) β Ψ = 24 i ∂∂T Ψ . (2.25)Now with the standard separation of variable as,Ψ( β , β + , T ) = φ ( β ) e ik + β + e − iET , (2.26)the equation for φ becomes " e ( α − β ∂∂β e ( α − β ∂∂β + e α − β k + 12( m − m + 1) e (3 α +1) β − E φ = 0 . (2.27)31hapter 2 Anisotropic cosmological models with variable spatial curvatureFor α = 1 we make a transformation of variable as χ = e − ( α − β , (2.28)and write equation (2.27) as94 (1 − α ) ∂ φ∂χ + k χ φ + 12( m − m + 1) χ α +1)3(1 − α ) φ − Eφ = 0 . (2.29)We define some parameters as σ = 4 k − α ) , (2.30) E = 323(1 − α ) E, (2.31) M = 16( m − m + 1)3(1 − α ) . (2.32)Equation (2.29) can now be written as − ∂ φ∂χ − σ χ φ − M χ α +1)3(1 − α ) φ = − E φ. (2.33)Above equation can be compared to −H g = − d dχ + V ( χ ) with V ( χ ) = − σ χ − M χ α +1)3(1 − α ) which is a continuous and real valued function on the half line. One can show that theHamiltonian H g admits self-adjoint extension as H g has equal deficiency indices. Wecan refer to the text of Reed and Simon[30] for a systematic and detailed description ofthe self-adjoint extension.So it can be said that for perfect fluid with α = 1, Bianchi VI quantum models do admita unitary evolution. 32hapter 2 Anisotropic cosmological models with variable spatial curvature α = − We take a specific choice, where ρ + 3 P = 0, as an example. This equation of state willmake equation (2.33) much simpler. With α = − /
3, the term − M χ α +1)3(1 − α ) becomes aconstant ( M ). Equation (2.33) becomes − ∂ φ∂χ − σ χ φ = − ( E − M ) φ, (2.34)which is in fact a well known Schrodinger equation of a particle with mass m = 1 / φ a ( χ ) = √ χ [ AH (2) iβ ( λχ ) + BH (1) iβ ( λχ )] , (2.35) φ b ( χ ) = √ χ [ AH (2) α ( λχ ) + BH (1) α ( λχ )] , (2.36)for σ > / σ < / β = q σ − / β = q / − σ respectively. Here both α and β are real numbers and in both cases the energy spectra is given as E = M − λ . (2.37)For H g = d dχ + σ χ , the value of deficiency index n ± , which is the number of linearlyindependent solutions for equation H g φ ± = ± iφ ∓ , comes out to be n + = n − = 1. It isalways possible to have self adjoint extension of the Hamiltonian having equal deficiencyi.e. indices n + = n − . For an inverse square potential, the method is described in detailby Essin and Griffiths [31]. Using the asymptotic expression for φ a and φ b , self-adjointextension guarantees that | B/A | takes a value so as to conserve probability and makethe model unitarity. 33hapter 2 Anisotropic cosmological models with variable spatial curvature Bianchi Type II model is given by the line element ds = dt − a ( t ) dr − b ( t ) dθ − [ a ( t ) θ + b ( t )] dφ + 2 a ( t ) θdrdφ. (2.38)The calculation in this case is a bit more involved for the presence of the non-diagonalterms in the metric.The Ricci scalar R in this case is given by R = − a b − a ˙ bab − b b − aa − bb . (2.39)If we define a new variable β = ab as prescribed in [8], then Lagrangian density forgravity sector looks like L g = 2 β ˙ a a − β a − a β , (2.40)and the corresponding Hamiltonian density for gravity sector can be written as H g = a p a β − a p β + a β . (2.41)Using Schutz’s formalism and proper identification of time as we did before, the Hamil-tonian density for fluid sector can be written as H f = a α β − α p T . (2.42)The super Hamiltonian can now be written in following form H = H g + H f = a p a β − a p β + a β + a α β − α p T . (2.43)As an example we take up the case of a stiff fluid given by α = 1.34hapter 2 Anisotropic cosmological models with variable spatial curvatureAfter promoting the momenta by operators as usual, the Wheeler-DeWitt equation H Ψ = 0 takes following form − a ∂ ψ∂a + β ∂ Ψ ∂β + a i ∂ Ψ ∂T . (2.44)Using a separation of variables Ψ = e − iET φ ( a ) ψ ( β ) , (2.45)we get following equations for ψ and φ respectively − d ψdβ + 8 kβ ψ = 0 , (2.46) a d φda − a φ − k − E ) φ = 0 . (2.47)With φ = φ √ a and χ = a , last equation can be written as − d φ dχ − σχ φ = − φ , (2.48)where σ = [ − k − E )] . Equations (2.46) and (2.48) are the governing equations forBianchi Type II with a stiff fluid.Equations for both ψ and φ can be mapped to a Schrodinger equation for a particlein an inverse square potential. In order to get a solution we actually have ensure anattractive regime, which requires k ≤ E ≤ k − /
32. We see that both the equationsare that for inverse square potentials, and thus a self-adjoint extension is possible.
It has already been mentioned that the difference between the present examples ofBianchi VI and Bianchi II models on one hand, and most of the models discussed ear-35hapter 2 Anisotropic cosmological models with variable spatial curvaturelier on the other, is the fact that the present models have variable spatial curvature asopposed to most the models discussed in connection with the quantization of cosmolog-ical models according to the Wheeler-DeWitt scheme. In a (3+1) decomposition of thespace-time metric, one can calculate the Ricci curvature ( R ) of the three dimensionalspace section, embedded in a four dimensional space-time. Both Bianchi VI and II willhave R which vary with time. For example, if we take the Bianchi VI metric as anexample in a more general form, than that used here (equation (2.2)), given by ds = n ( t ) dt − a ( t ) dx − e − mx b ( t ) dy − e lx c ( t ) dz , (2.49)the 3-space Ricci curvature looks like R = m − ml + l a ( t ) , (2.50)which indeed is a function of the cosmic time t through a . In the case of the metric(2.2), this becomes R = m − m + 12 a ( t ) . (2.51)The most talked about anisotropic models, like Bianchi I, V and IX all have costant R . For example, if we put m = l = 0 in the metric (2.49), we get a Bianchi I metric,and for this choice, equation (2.50) clearly shows that R = 0. For some more informa-tion regarding the spatial curvature, we refer to the recent work by Akarsu and Kilinc[73]. The work discussed in this chapter deals with two examples of anisotropic quantumcosmological models with varying spatial curvature. We show that there is indeed apossibility of finding unitary evolution of the system. The earlier work on anisotropicmodels with constant spatial curvature[10, 11] disproved the belief that anisotropic quan-36hapter 2 Anisotropic cosmological models with variable spatial curvaturetum cosmologies generically suffer from a pathology of non-unitarity. The present worknow strongly drives home the fact that this feature is not at all a charactristic of mod-els with constant spatial curvature. It was also shown before that the unitarity is notachieved at the cost of anisotropy itself[13]. One can now indeed work with quantumcosmologies far more confidently, as there is actually no built-in generic non-conservationof probability in the models.Very recently it has been shown that in fact all homogeneous models, isotropic oranisotropic, quite generally have a self-adjoint extension[32], although the extension isnot unique. The present work gives two more examples, and consolidates the resultproved in reference [32]. The examples chosen indeed have physical implications. It hasbeen shown very recently that a Bianchi VI model plays an important role in producinganisotropic inflation[74]. We also refer to the work of Barrow[75] for various cosmo-logical implications of Bianchi type VI models. Bianchi II models, on the other hand,are instrumental in understanding the Belinskii, Khalatnikov, Lifshitz conjecture in thediscussion of spacelike singularities[3, 76].Thus the standard canonical quantization of cosmological models via Wheeler-DeWittequation still proves to be useful in the absence of a more general quantum theory ofgravity. 37
HAPTER Quantization of n-dimensional cosmological models Higher dimensional models had been investigated widely in the past in order to find a the-ory to unify gravity with other fundamental forces of physics. It started with Kaluza andKlein’s assertion that a fifth dimension in general relativity will unify gravity with theelectromagnetic field[77–79]. Further motivation of resorting to higher dimensional mod-els came from the expectation of unifying gravity with non-Abelian gauge fields[80, 81].Later, spacetime having more than four dimensions were motivated by 10-dimensionalsuperstring theory and 11-dimensional supergravity theory[82, 83].In Chapter 2, we use the Wheeler-DeWitt quantization method to quantize anisotropicmodels, where the evolution of the cosmic fluid is identified as a time parameter usingSchutz formalism. In cosmological models related to Brans-Dicke theory[35, 36], theevolution of the scalar field have been used as a time parameter in order to quantize thesame without adding any additional matter[84–86]. Brans-Dicke theory of gravity hasalso been quantized using Schutz’s formalism[65, 87]. Similar formalism have been usedby Khodadi e t al in scalar-energy dependent metric cosmology[88]. So long as a method The work illustrated in this chapter is published;
S. Pandey , European Physical Journal C, ,487(2019). e t al[89] have shown the canonical quantizationof n-dimensional anisotropic model coupled with a massless scalar field in the absenceof any fluid where they have emphasized on natural identification of scalar field as timeparameter for the evolution of quantum variables.In this chapter, we try to investigate the quantum cosmological solutions of an n-dimensional anisotropic model in which massless scalar field is minimally coupled withgravity in presence of a barotropic perfect fluid( P = αρ ). The idea is to generalize thework of Alves-Junior et al to include a fluid, as well as to generalize the recent work onanisotropic cosmologies by Pal and Banerjee[10, 11] to higher dimensions. Keeping inmind the importance of various aspects of higher dimensions, it is quite relevant to askthe questions like whether singularity-free n-dimensional quantum cosmological modelsmay be realized. Also, this investigation should confirm if the general result, givenby Pal and Banerjee[32], that it is always possible to find a self-adjoint extension ofhomogeneous quantum cosmologies in four dimensions, is applicable to higher dimensionsalso. We start with the Hilbert-Einstein action in n-dimension with a minimally coupled scalarfield in the presence of a fluid, A = Z d n x √− g ( R + ωg µν φ ,µ φ ,ν ) + Z d n x √− gP, (3.1)where ω is a dimensionless parameter and φ is the massless scalar field. The last term ofequation (3.1) represents the matter contribution coming from fluid pressure P relatedwith density ρ by the equation of state P = αρ . The metric for an n dimensional ( n > ds = N ( t ) dt − a ( t ) ( dx + dy + dz ) − b ( t ) n − X i =1 dl i , (3.2)where N ( t ) is the lapse function, a ( t ) is our good old scale factor used in flat FLRWcosmology and b ( t ) is scale factor coming from remaining ( n −
4) dimensions. So the usual3-space is isotropic, but the extra dimensions, though isotropic in itself, is anisotropicwith respect to the usual 3-space section.The action for scalar gravity part can be written in a reduced form for metric given by(3.2) as A g = V Z dt N " − ˙ a ab n − − ( n − b ˙ aa b n − − ( n − n − b a b n − + ω a b n − ˙ φ , (3.3)where an over-dot represents a derivative with respect to coordinate t and V denotes( n −
1) dimensional volume. We have ignored the surface terms as they do not contributeto the field equations.From (3.3), we can write the Lagrangian for the gravity sector as, L g = − N ˙ a ab n − − n − N ˙ b ˙ aa b n − − ( n − n − N ˙ b a b n − + ωN a b n − ˙ φ . (3.4)Schutz [24, 26] showed that four dimensional velocity vector of a perfect fluid can bewritten using six thermodynamic quantities as, U ν = 1 µ ( (cid:15) ,ν + ζβ ,ν + θS ,ν ) , (3.5)where µ is specific enthalpy, S is specific entropy, ζ and β are potentials connectedwith rotation and (cid:15) and θ are potentials with no clear physical meaning. For n dimen-41hapter 3 Quantization of n-dimensional cosmological modelssional velcoity vector, we can consider additional potential terms[90] like ζ and β inequaution(3.5) as U ν = 1 µ (cid:15) ,ν + ζβ ,ν + θS ,ν + n − X i =1 ζ i β i,ν ! . (3.6)Since the potentials ζ , ζ i , β and β i are related to rotation, we can neglect them in thepresent case as there is no vorticity in the spacetime given by equation (3.2). Followingthe widely used technique, developed by Lapchinskii and Rubakov [6] by writing thefluid pressure in terms of potentials given in equation (3.6), the fluid sector action givenin equation (3.1) can be written as A f = V Z dt " N − /α a b n − α ( α + 1) /α +1 ( ˙ (cid:15) + θ ˙ S ) /α +1 exp ( − S/α ) . (3.7)From above equation, Lagrangian for fluid sector can be written as L f = N − /α a b n − α ( α + 1) /α +1 ( ˙ (cid:15) + θ ˙ S ) /α +1 exp ( − S/α ) . (3.8) Following standard procedure [91], Hamiltonian for gravity sector from correspondingLagrangian (3.4) can be written as H g = N ( n − ab n − " ( n − p a b + 12( n − p b a − p a p b ab + ( n − ωa b p φ . (3.9)For fluid sector, Hamiltonian corresponding to Lagrangian (3.8) can be written as H f = Na α b ( n − α p T (3.10)where T and p T are related by following canonical transformations T = p S e − S p − ( α +1) (cid:15) , p T = p α +1 (cid:15) e S , ¯ (cid:15) = (cid:15) − ( α + 1) p s p (cid:15) , ¯ p (cid:15) = p (cid:15) . (3.11)42hapter 3 Quantization of n-dimensional cosmological modelsUsing (3.9) and (3.10), the net Hamiltonian can be written as H = Na b ( n − " ( n − a n − p a + b n − n − p b − ab n − p a p b + 14 ω p φ + a − α b ( n − − ( n − α p T . (3.12)The above expression for H may look formidable, but can be written in a simpler formwith the help of following canonical transformations, A = lna, p A = ap a , B = lnb, p B = bp b , (3.13)as H = ¯ N " ( n − n − p A − n − p A p B + 12( n − n − p B + 14 ω p φ + e (3 α − A +(( n − α − ( n − B p T , (3.14)where ¯ N = Na b n − . By varying the action with respect to ¯ N , Hamiltonian constraint can be written asˆ H = H ¯ N = 0. Now the coordinates ( A, B, φ and their corresponding conjugate momentaare promoted to operators using the standard commutation relations ([ q i , p i ] = i in theunits h π = 1). ˆ H can be written in terms of these operators using eq (3.14) and theWheeler-DeWitt equation ( ˆ Hψ = 0) looks like, " − ( n − n − ∂ ∂A + 12( n − ∂ ∂A∂B − n − n − ∂ ∂B − ω ∂ ∂φ − ie (3 α − A +(( n − α − ( n − B ∂∂T ψ ( A, B, φ, T ) = 0 . (3.15)43hapter 3 Quantization of n-dimensional cosmological modelsFor n = 5, the first term of equation (3.15) will vanish and Wheeler-DeWitt equationbecomes simpler. We shall be considering it later.With ansatz, ψ ( A, B, φ, T ) = ξ ( A, B, φ ) e − iET , equation (3.15) takes following form, " − vuut n − n − ∂ ∂u + vuut n − n − ∂ ∂v + 14 ω ∂ ∂φ + exp ( (3 α − s n − n − u + v ! + (( n − α − ( n − s n − n − u − v !) E ξ ( u, v, φ ) = 0 , (3.16)where u and v are given as, u = vuut n − n − A + q n − n − B, (3.17) v = vuut n − n − A − q n − n − B. (3.18) α = 1 and n > For α = 1 i.e. P = ρ , equation (3.16) can be rewritten as " − vuut n − n − ∂ ∂u + vuut n − n − ∂ ∂v + 14 ω ∂ ∂φ + E ξ ( u, v, φ ) = 0 . (3.19)Solution for equation (3.19) can be written as ψ λ,E,k ( u, v, φ, T ) = K sin( u s λ | − ¯ n | ) × sin( v s E − k − λ n ) sin( φ √ ωk ) e − iET , (3.20)44hapter 3 Quantization of n-dimensional cosmological modelswhere ¯ n = q n − / ( n −
5) and λ , k and K are constant. Here we take E > k + λ .By superposing the function ψ λ,E,k ( u, v, φ, T ), the wave packet can be formed as;Ψ( u, v, φ, T ) = Z ∞ dλ Z ∞ dk Z ∞ dE sin( u s λ − ¯ n ) × sin( v s E n ) sin( φ √ ωk ) e − ( λ + E + k )( γ + iT ) , (3.21)where E = E − k − λ and γ is a positive constant. We are choosing e − γ ( λ + E + k ) as weightfactor, which may not be unique for superposition, which gives us the normalized wavepacket.Norm k Ψ k can be calculated as, k Ψ k = Z ΨΨ ∗ d ¯ ud ¯ vdφ = 132 √ ω πγ ! / , (3.22)where ¯ u = u q | − √ n − / ( n − | and ¯ v = v q √ n − / ( n − .This clearly shows that we have a finite and time-independent norm. From equation(3.21), the normalized wave packet can be written as,Ψ(¯ u, ¯ v, φ, T ) = ωπ ! / √ γγ + iT ! / ¯ u ¯ vφ × exp " − ωφ + ¯ u + ¯ v γ + iT ! . (3.23)Now we can calculate the expectation values of a , b and the proper volume measure V = ( − g ) = a b n − in terms of ¯ u and ¯ v as, h a i = Z ∞−∞ d ¯ u Z ∞−∞ d ¯ v Z ∞−∞ dφ | Ψ(¯ u, ¯ v, φ, T ) | exp " s n − n − × vuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − vuut n − n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ u + vuuut vuut n − n −
5) ¯ v ! , (3.24)45hapter 3 Quantization of n-dimensional cosmological models Figure 3.1: Expectation value of scale factors for n=6. h b i = Z ∞−∞ d ¯ u Z ∞−∞ d ¯ v Z ∞−∞ dφ | Ψ(¯ u, ¯ v, φ, T ) | exp " s n − n − × vuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − vuut n − n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ u − vuuut vuut n − n −
5) ¯ v ! , (3.25) h V i = Z ∞−∞ d ¯ u Z ∞−∞ d ¯ v Z ∞−∞ dφ | Ψ(¯ u, ¯ v, φ, T ) | a (¯ u, ¯ v ) b n − (¯ u, ¯ v ) . (3.26)The plots for the expectation values are shown in the figures taking the value of positiveconstant γ = 1. The figures clearly indicate that there is no singularity of a zero propervolume. In fact both the scale factors a and b are quite well behaved individually. α = 1 and n = 5 For n = 5 with α = 1,equation(3.15) will become " ∂ ∂A∂B − ∂ ∂B − ω ∂ ∂φ − i ∂∂T ψ ( A, B, φ, T ) = 0 . (3.27)46hapter 3 Quantization of n-dimensional cosmological models Figure 3.2: Expectation value of scale factors for n=8 .Figure 3.3: Expectation value of scale factors for n=7 is given above and as it is shown, theycoincide.
With the change of variable as x = 2 A + B and y = B , it can be re-written as, " − ∂ ∂x + 16 ∂ ∂y + 14 ω ∂ ∂φ + E ξ ( x, y, φ ) = 0 . (3.28)47hapter 3 Quantization of n-dimensional cosmological models Figure 3.4: Expectation value of Volume < V = a b n − > . Solution of above equation can be found using similar method what was done in theprevious section. Normalized wave packet is given as,Ψ(¯ x, ¯ y, φ, T ) = ωπ ! / γ / ( γ + T )( γ + iT ) ! / × ¯ x ¯ yφexp " − ¯ x γ − iT + 4 ωφ + ¯ y γ + iT ! , (3.29)where ¯ x = √ x and ¯ y = √ y .The relevant expectation values are given as h a i = e γ T γ ( γ ( γ + 24) + T ) γ , (3.30) h b i = e γ T γ ( γ ( γ + 6) + T )6 γ . (3.31)48hapter 3 Quantization of n-dimensional cosmological models Figure 3.5: Expectation value of scale factors for n=5. α = 1 We will be considering only n = 7 as an example for a general fluid with α = 1, in whichcase the equation is tractable. Now Wheeler-DeWitt equation (3.16) can be written as, " − ∂ ∂u + 5 ∂ ∂v + 14 ω ∂ ∂φ + e α − √ u E ξ ( u, v, φ ) = 0 . (3.32)With suitable operator ordering as done in work by Pal and Banerjee[10, 11], equation(3.32) takes following form, " e (3 − α )2 √ u ∂∂u e (3 − α )2 √ u ∂∂u − e (3 − α ) √ u ∂ ∂v − e (3 − α ) √ u ω ∂ ∂φ − E ξ ( u, v, φ ) = 0 . (3.33)Now with ansatz ξ ( u, v, φ ) = U ( u ) e iV v/ √ e i √ ωφ , above equation can be re-written as, " e (3 − α )2 √ u ∂∂u e (3 − α )2 √ u ∂∂u + e (3 − α ) √ u ( V + k ) − E U ( u ) = 0 . (3.34)49hapter 3 Quantization of n-dimensional cosmological modelsWith change of variable like χ = e (3 α − √ u , this equation can be written as, " α − ∂ ∂χ + V + k χ − E U ( χ ) = 0 . (3.35)Above equation is similar to the inverse square potential problem of physics which admitsthe self-adjoint extensions. The solution to equation(3.35) can be given by Hankelfunctions. It has been extensively shown in the work[10]. In this work, using the n-dimensional generalization of Schutz formalism[90], we formu-lated the Lagrangian and Hamiltonian for our model. Followed by multiple canonicaltransformations, we obtained the complicated looking Wheeler-DeWitt equation(3.16)for the wave function of the n-dimensional anisotropic universe. We obtained the generalwave packet of Wheeler-DeWitt equation for stiff fluid α = 1. We are able to find a timeindependent and finite normed wave packet which establishes the unitary evolution ofthe model.We have found the non-singular expectation values of the scale factors and these areshown in Fig. 3.1, 3.2, 3.3 and 3.5 for different n. Expectation values of scale factors a and b coincide for n = 7. This is not surprising as both usual space-section and theextra 3-dimensional space, although anisotropic between each other to start with, are3-dimensional isotropic space like sections in themselves.Non-zero minima of the expectation values of volume element(Fig. 3.4) clearly showthe contraction of the universe followed by expansion avoiding any singularity indicatinga bouncing universe. Similar results were obtained in quantization of anisotropic modelsin absence of fluid in the past [89]. This result is obtained with quite reasonable boundaryconditions, Ψ → u and ¯ v . For a discussion, werefer to the work of Vilenkin and Yamada [92] and Tuccii and Lehners [93].For a general fluid with α = 1, we are able to show the self-adjoint extension andthus unitary evolution of the model for n = 7 despite the computational difficulty of50hapter 3 Quantization of n-dimensional cosmological modelsthe model. However, the result of it is in accordance with four-dimensional anisotropicmodel Bianchi-I as expected [10].Our results may also be considered as the higher dimensional generalization of unitaryevolution of the anisotropic models as shown in [10–12, 94].51 HAPTER Unitarity in quantum cosmology: symmetries protected and violated The Wheeler-DeWitt scheme of quantization[4, 5] of cosmological models was believed tohave been plagued with the non-unitartity in anisotropic cosmological models[8, 96]. Inthe first chapter, we have seen that there are ample examples [9–13, 65] of the possibilityof a unitary evolution achieved by an operator ordering.. This unitarity is achieved by asuitable operator ordering. In fact, a theorem has recently been proved[32] to show thatat least for homogeneous models, it is always possible a have a self-adjoint extensionfor the Hamiltonian, and thus to have a unitary evolution for the system. The last twochapters, some different kinds of anisotropic models were shown to have similar features.The purpose of this chapter is to look at the required price for the self-adjoint exten-sion, in terms of symmetry. We shall look at two aspects, one is the Noether symmetryand the other being the scale invariance. We shall deal with one example, the BianchiI cosmological model, which is the simplest anisotropic cosmological model, but bringsout the associated physical content quite comprehensively. The work illustrated in this chapter is under consideration for publication; [95].
This example has already been worked out in an earlier work[10]. We narrate theimportant steps here so as to make the presentation self sufficient.We start with the action A = Z M d x √− gR + 2 Z ∂M d x √ hh ab K ab + Z M d x √− gP, (4.1)in a four dimensional space-time manifold M. R is the Ricci Scalar, K ab is extrinsic cur-vature and h ab is induced metric on the boundary ∂M . The first two terms correspondto the gravity sector and third term is due to a perfect fluid which is taken as the matterconstituent of the universe, P is the pressure of the fluid. We have chosen our units, suchthat 16 πG = 1. The second term will not contribute in the Euler-Lagrange equationsas there is no variation in the boundary.Bianchi I metric is given as ds = n ( t ) dt − a ( t ) dx − b ( t ) dy − c ( t ) dz . (4.2)With this metric, the gravity sector of the action can be written as A g = Z dt " − n [ ˙ a ˙ bc + ˙ b ˙ ca + ˙ c ˙ ab ] . (4.3)A transformation of variables as a ( t ) = e β + β + + √ β − , (4.4) b ( t ) = e β + β + −√ β − , (4.5) c ( t ) = e β − β + , (4.6)54hapter 4 Unitarity in quantum cosmology: symmetries protected and violatedwill make the Lagrangian in equation (4.3) look like, L g = − e β n [ ˙ β − ˙ β − ˙ β − ] . (4.7)The canonically conjugate momenta are defined as usual as p i = ∂ L g ∂x i . The Hamiltonianfor the gravity sector looks like H g = − n exp( − β ) (cid:26) (cid:16) p − p − p − (cid:17)(cid:27) . (4.8)For the fluid sector, the practice is to adopt the Schutz formalism of expressing thefluid properties like density and pressure in terms of some thermodynamic variables andthen effect a set of canonical transformations. The method is described elaborately inthe introduction. The relevant action is A f = Z dt L f = Z dt n − α e β α (1 + α ) α (cid:16) ˙ (cid:15) + θ ˙ S (cid:17) α e − Sα , (4.9)where (cid:15), θ and S are the thermodynamic quantities (Schutz variables), and α is a con-stant that connects the density ( ρ ) and pressure ( P ) as P = αρ .As the metric components do not depend on spatial coordinates, the spatial volumeintegrates out as a constant and will not participate in the subsequent calculations. Also,the boundary term is ignored as that does not contribute to the variation of the action.The Canonical momenta are defined as p (cid:15) = ∂ L f ∂ ˙ (cid:15) and p S = ∂ L f ∂ ˙ S and execute a set ofcanonical transformations, T = − p S exp( − S ) p − α − (cid:15) , (4.10) p T = p α +1 (cid:15) exp( S ) , (4.11) (cid:15) = (cid:15) + ( α + 1) p S p (cid:15) , (4.12) p (cid:15) = p (cid:15) . (4.13)55hapter 4 Unitarity in quantum cosmology: symmetries protected and violatedThe Hamiltonian for the fluid becomes H f = ne − β e − α ) β p T . (4.14)One can now write the net Hamiltonian as H = n exp( − β ) (cid:26) − (cid:16) p − p − p − (cid:17) + e − α ) β p T (cid:27) , (4.15)and proceed to quantize the system by using the Hamiltonian constraint H = 0 (obtainedby varying the action with respect to the lapse function n ) and raising the variables asoperators, which yields the Wheeler-DeWitt equation as ∂ ∂β − ∂ ∂β − ∂ ∂β − ! ψ = 24 ıe − α ) β ∂ψ∂T . (4.16)In writing this equation, a choice of gauge has been made ( n = e αβ ). With theusual separability ansatz ψ ( β , β + , β − , T ) = φ ( β , β + , β − ) e − ıET where E is a constant,the equation (4.16) becomes ∂ ∂β − ∂ ∂β − ∂ ∂β − ! φ = 24 Eφe − α ) β . (4.17)For the general case 0 ≤ α ≤
1, choose another separation of variables as φ = ξ ( β ) η ( β + , β − ) and a coordinate transformation as χ = e − ( α − β , a long but straight-forward calculation will yield an equation for ξ as − d ξdχ − σχ ξ = − E ξ, (4.18)where E = − α ) E and σ is composed of the separation constants coming in the processof separation of the functions of β + and β − . This equation indeed has a favourabledeficiency index, which guarantees the existence of a self-adjoint extension[97] and thus,the evolution of the system is unitary.The work of Pal and Banerjee[10] shows that this transformation, even at the classical56hapter 4 Unitarity in quantum cosmology: symmetries protected and violatedlevel, gives rise to a Hamiltonian which, when raised to operators, is self-adjoint. Itdeserves mention that often the unitarity is achieved by means of an operator ordering,which is not unique[32]. Thus the most unambiguous example would be the one if onecan effect a coordinate transformation at the classical level so that the operator orderingis irrelevant at the quantum level. The present example is exactly that and this is onegood reason for choosing Bianchi I at the outset. For a very brief review of variousaspects of factor ordering, we refer to [98]. The generator for Lagrangian (4.7) can be written as: X = b ∂∂β + b + ∂∂β + + b − ∂∂β − + ˙ b ∂∂ ˙ β + ˙ b + ∂∂ ˙ β + + ˙ b − ∂∂ ˙ β − , (4.19)where b i ( β j )s ( i, j = 0 , + , − ) are to be determined from the Noether symmetry condition £ X L g = 0 , (4.20)meaning the Lie derivative of the Lagrangian with respect to X is zero.This condition yields the set of equations32 b + ∂b ∂β = 0 , (4.21)32 b + ∂b + ∂β + = 0 , (4.22)32 b + ∂b − ∂β − = 0 . (4.23)57hapter 4 Unitarity in quantum cosmology: symmetries protected and violatedThe solution for this set of equations can be written as e β b = Q = constant, (4.24)32 β + b + b + = Q = constant, (4.25)32 β − b + b − = Q = constant. (4.26)A coordinate transformation of the form χ = e − ( α − β transforms the Lagrangiandensity given in (4.7) to the form L g T = − n [ 4 χ α/ (1 − α ) − α ) ˙ χ − χ / (1 − α ) ( ˙ β + ˙ β − )] . (4.27)It is now required to check whether the Noether symmetry corresponding to L g isretained in L g T . The corresponding generator for the Lagrangian as in equation (4.27) can be writtenas, X = q ∂∂χ + q + ∂∂β + + q − ∂∂β − + ˙ q ∂∂ ˙ χ + ˙ q + ∂∂ ˙ β + + ˙ q − ∂∂ ˙ β − , (4.28)where q i ( χ, β j ) ( i, j = 0 , + , − ) are to be determined from the Noether symmetry condi-tion £ X L g T = 0, which in this case gives following partial differential equations, α − α q χ + ∂q ∂χ = 0 , (4.29) α − α q χ + ∂q + ∂β + = 0 , (4.30) α − α q χ + ∂q − ∂β − = 0 . (4.31)Solution to above three equations can be given as, q /α χ / (1 − α ) = Q A = constant, (4.32) q β + (1 − α ) χ + q + = Q B = constant, (4.33) q β − (1 − α ) χ + q − = Q C = constant. (4.34)58hapter 4 Unitarity in quantum cosmology: symmetries protected and violatedIt is easy to check that the solution to Noether symmetry conditions for both La-grangian match exactly with the identification b = q /α = [ (1 − α )] / ( α − e − β . It should be noted that a self-adjoint extension actually involves working on the Hilbertspace[97]. But it has been brought about by ordering of operators[10]. As there isindication of the uniqueness of the operator ordering to achieve this[32], a safe way isto get a coordinate at the classical level, such that while promoting the variables tooperators for the quantization, the ordering is at least unambiguous. The coordinate χ exactly serves this purpose. We shall now look at the issue of scale invariance. Under a scale transformation, thereis a rule of transformation of the coordinate and time, e.g., if x = x ( t ) is transformedlike ¯ x = λ − / x , t should go like ¯ t = λt , where λ is the scale.We now look back at the equation (4.18), where we easily identify χ and T as thecoordinate and time respectively. So the relevant transformations will be ¯ T = λT ,¯ χ ( ¯ T ) = λ − / χ ( λT ). Energy will transform ¯ E = E /λ and ∂∂ ¯ χ = λ / ∂∂χ . For a compre-hensive description of this, we refer to references [99] and [100].If we effect this scale transformation, the equation (4.18)takes the form as − λ d ξdχ − λ σχ ξ − = − E λ ξ. (4.35)This clearly does not preserve the scale invariance! However, this should not perhapsbe considered too costly, as there is indeed an incompatibility between hermiticity andscale invariance in general. This has been proved very elegantly by Pal[101]. So this isnot at all an artefact of anisotropic quantum cosmology.59hapter 4 Unitarity in quantum cosmology: symmetries protected and violated It is now known that the allaged non-unitarity of the anisotropic quantum cosmologicalmodels is not true. Anistropic models, at least if they are spatially homogeneous, areshown to have self-adjoint extension[32]. The present work looks at the cost of thisextension in terms of symmetry.With the example of the Bianchi I metric, it is quite clearly shown that the self-adjointextension indeed retains the Noether symmetry. However, the scale invariance is clearlylost. So apparently, this is the cost of a self-adjoint extension. As already mentioned,this does not appear to be too much of a price for securing unitarity as this feature isquite generic[101]. 60
HAPTER Jordan and Einstein frame: Are they same at quantum level? A non-minimally coupled theory of gravity, where a field interferes with the curvaturescalar, has two popular frameworks for its description. One is called the Jordan frame,where the theory is manifestly non-minimal in the sense that the interference term isvisible in the action and also in the field equations derived from the action by meansof a variational principle. In the second framework, known as the Einstein frame, thenon-minimal coupling is broken by means of a conformal transformation of the form¯ g µν = Ω g µν , the theory appears to be simpler and looks similar to General Relativ-ity where the non-minimally coupled field appears as an additional term in the mattersector. In Jordan frame, the Newtonian constant of gravity G becomes a variable. Ein-stein’s frame has a restored constancy of G but the rest mass of the test particle becomesa function and thus, one has to pay a bigger price, the validity of equivalence principle,and hence the significance of the geodesic equation is lost. In fact, this loss of the prin-ciple of equivalence is the key to understand the nature of the non-minimal coupling inspite of the apparent resemblance with general relativity. The work illustrated in this chapter is published;
S. Pandey and N. Banerjee , European PhysicalJournal Plus, , 107(2017). φ is coupled with the Ricci scalarR in the action. We work in a spatially flat, homogeneous and isotropic cosmological62hapter 5 Jordan and Einstein frame: Are they same at quantum level?model in vacuum, and quantize the model following the standard canonical Wheeler-deWitt quantization scheme[4, 5], and form the relevant wave packet Ψ Jordan in Jordanframe. The action is then written in the Einstein frame via the conformal transformation¯ g µν = φg µν suggested by Dicke[36]. We pretend that this is a completely different theoryand quantize a same cosmological model following the same Wheeler-DeWitt scheme.The wave packet Ψ Einstein is formed. Naturally, it looks different from the wave packetin the Jordan frame. We now effect the inverse transformation in ¯ g µν in the wave packetΨ Einstein , and see that it is exactly the same as Ψ
Jordan . The result is quite general,inthe sense that this does not depend upon the parameter of the theory ω .In the next two sections, the quantization of the cosmological model in vacuum is dis-cussed in Jordan and Einstein frames respectively. Finally, the result is critically ana-lyzed in the section 5.3. The relevant action in Brans Dicke theory without any contribution from the mattersector, in the so-called Jordan frame, is written as A = Z d x √− g " φR + ωφ ∂ µ φ∂ ν φ , (5.1)where R is the Ricci scalar, φ is the scalar field and ω is a dimensionless parameter. Itis generally believed that the higher the value of ω , the closer the theory is to generalrelativity, and for ω → ∞ , the two theories (GR and BD) are identical. However, it hasbeen proved that this equivalence of the two theories is not at all generic[37, 38].A spatially homogeneous and isotropic space-time with a flat spatial section is givenas ds = n ( t ) dt − a ( t ) dl , (5.2)where the lapse function n and the scale factor a are functions of the time alone. With63hapter 5 Jordan and Einstein frame: Are they same at quantum level?this metric, the Lagrangian can be extracted from the action (5.1) as L = − φa ˙ a n − a ˙ φa n + ωnφ ˙ φ a . (5.3)With the change of variable as a ( t ) = e − α/ β , (5.4) φ ( t ) = e α (5.5)Lagrangian can be written as L = e − α/ β n " − β + 2 ω + 32 ˙ α . (5.6)The corresponding Hamiltonian comes out to be H = ne α/ − β " − p β + 122 ω + 3 p α . (5.7)By a variation of the action in the first order with respect to the lapse function n , onehas the Hamiltonian constraint as H = He − α/ β = 0. Now we consider a canonicaltransformation of variables as ( α, p α ) to ( T, p T ) given by T = αp α , (5.8) p T = p α . (5.9)It is easily verified that T and p T are canonically conjugate variables.One can now write H as H = − p β + 242 ω + 3 p T . (5.10)Here β , T are the coordinates and p β , p T are the corresponding canonically conjugatemomenta. The canonical structure can be verified from the relevant Poisson brackets.64hapter 5 Jordan and Einstein frame: Are they same at quantum level?The Wheeler-DeWitt (WDW) equation, H ψ = 0, can be written as " ∂ ∂β − i ω + 3 ∂∂T ψ = 0 . (5.11)Then solution for above equation is obtained as ψ E ( β, T ) = e iET sin[ q E/ (2 ω + 3) β ] , (5.12)or ψ E ( a, φ, T ) = e iET sin[ q E/ (2 ω + 3) ln ( q φa )] , (5.13)where E is a constant of separation.Using R ∞ e − γx sin √ mxdx = √ πm γ / e − m/ γ , wave packet can be written asΨ( a, φ, T ) = s π ω + 3 ln ( √ φa )( γ − iT ) / exp " − ln ( √ φa )(2 ω + 3)( γ − iT ) . (5.14) If one effects a conformal transformation given by¯ g µν = Ω g µν , (5.15)the action will look like A = Z d x √− ¯ g " ¯ R + 2 ω + 32 ∂ µ ξ∂ ν ξ , (5.16)where ξ = ln φ [36]. The Lagrangian in this case can be written as L = − a ¯ a ¯ n + 2 ω + 32¯ n ˙ ξ ¯ a , (5.17)65hapter 5 Jordan and Einstein frame: Are they same at quantum level?and the corresponding Hamiltonian becomes H = (¯ n/ ¯ a ) " − (¯ a / p a + 12(2 ω + 3) p ξ . (5.18)The Hamiltonian constraint, as usual, can be obtained by varying the action with respectto the lapse function ¯ n as, H = ¯ a H = 0.Again with a similar canonical transformation as¯ T = ξp ξ , (5.19) p ¯ T = p ξ , (5.20)then WDW equation can be written as " ¯ a ∂ ∂ ¯ a − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (5.21)One can easily see from the transformation equations that the scalar time parameters inthe two frames, T and ¯ T are actually equal. With an operator ordering of the first termon the left-hand side ¯ a ∂∂ ¯ a ¯ a ∂∂ ¯ a and taking χ = ln ¯ a , eq. (5.22) can be written as " ∂ ∂χ − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (5.22)Then solution for above equation can be given as¯ ψ E (¯ a, ¯ T ) = e iE ¯ T sin [ s E (2 ω + 3) ln (¯ a )] . (5.23)The corresponding wave packet isΨ(¯ a, ¯ T ) = s π ω + 3 ln (¯ a )( γ − i ¯ T ) / exp " − ln (¯ a )(2 ω + 3)( γ − i ¯ T ) . (5.24)66hapter 5 Jordan and Einstein frame: Are they same at quantum level?If one now reverts the conformal transformation and go back to the Jordan frame, byusing ¯ a = a φ and ξ = lnφ , it is quite easy to see that the wave packet given in equation(5.24) in the Einstein frame is exactly same as that in the Jordan frame given in equation(5.14).The criticism of this proof could well be that in order to separate the variables inthe Jordan frame, a transformation has been used which is connected to the conformaltransformation that takes the action to the Einstein frame! Thus the calculations areactually in the same Einstein frame. But a closer scrutiny will reveal that the transfor-mation, being a point transformation, is a canonical transformation, so the equivalenceis actually built in. The result obtained carries a clear message. If the action is not contaminated with otherfields, such as a fluid, the Jordan and Einstein frames are completely equivalent in thesense that one can go from one description to the other at the final stage, i.e., at the levelof the solution via the conformal transformation. This result is completely independentof the choice of the coupling constant ω , which actually determines the deviation of thetheory from general relativity. The work is carried out in Brans-Dicke theory. Of course,there are other more complicated non-minimally coupled theories where this has to beverified, but the message is significant.Very recently a result contrary to this has been given[108], where it was shown that thewave packets in the two frames behave in different ways even after Ψ einstein is trans-formed back to the Jordan frame. The solutions were obtained for particular values ofthe Brans-Dicke parameter ω , but that should not infringe upon the result. Perhaps theaddition of a contribution of a fluid in the action results in the requirement of an or-dering of operators, as the fluid variable and the geometry cannot be separated efficiently.Furthermore, the conformal transformation, ¯ g µν = Ω g µν , inflicts a change of units in67hapter 5 Jordan and Einstein frame: Are they same at quantum level?the variables as indicated by Dicke[36], so one has to be careful about the interpretationof the results as shown by Faraoni and Nadeau[105]. For various choices of units andtheir significance, we also refer to the early work by Morganstern[109].It deserves mention that as the cosmic time t is a coordinate and not a scalar pa-rameter, the evolution of the quantum system requires a properly oriented scalar timeparameter in the scheme, which is very efficiently constructed out of the fluid parametersas shown by Lapchinski and Rubakov[6]. In the present work, as no fluid is considered,the scalar time parameter ( T and ¯ T respectively in the two frames) is constructed fromthe scalar field and the scale factor following the work of Vakili[84]. The derivativeswith respect to T and ¯ T appear in the first order in the Hamiltonian (equations (5.11)and (5.22) ), indicating their role in the scheme as time. One can easily check that thePoisson brackets { T, H } and { ¯ T , ¯ H } have the correct signatures which ensure the properorientation of the time parameter. For a detailed description of this issue, we refer tothe recent work of Pal and Banerjee[10].The present example discussed in this chapter is definitely a particular theory, namelythe Brans-Dicke theory. But the calculations are clean and the results are so unam-biguous and general (independent of the Brans-Dicke coupling parameter ω ), that onecan claim with confidence that the equivalence of the two frames are established, atthe quantum level, at least when the action is taken in the pure form, i.e., without anymatter field. 68 HAPTER Equivalence of Jordan and Einstein frames in anisotropic quantum cosmological models In the previous chapter, we established the mathematical equivalence of Jordan and Ein-stein frames at a quantum level using an isotropic cosmological model. In this chapter,we shall discuss this equivalence using more involved anisotropic models.Although the physical inequivalence and henceforth the question of which frame isthe physical one is a moot point for decades[64, 109–113], mathematical equivalence ofthese two frames at a classical level has almost been taken for granted. By mathematicalequivalence, we mean the following: we evaluate a quantity T J in the Jordan frame, eval-uate the same quantity in the Einstein frame and obtain T E , mathematical equivalenceis defined to be the statement that T E is just a transformed version of T J . Given thisscenario, it is meaningful to explore whether this mathematical equivalence survives aquantization process.In the light of recent resurgence in the Wheeler-DeWitt (WDW) quantization [10–13, 32, 94, 114] scheme, we have a natural framework to answer the question of equiva- The work illustrated in this chapter is published;
S. Pandey , S. Pal and N. Banerjee, Annals ofPhysics, , 93(2018). , there hasbeen claim of non-equivalence at quantum level in literature [108], albeit for cosmologicalmodels with a matter content. Very recently Kamenshchik and Steinwachs[115], withan estimate of the one loop divergence in the two frames, showed that the frames arenot equivalent. Nonetheless, the debate seems to be open enough. The hint of quan-tum equivalence is also speculated in [87] and further claimed to be true for isotropichomogeneous model with scale invariant matter content in [65] and quite generically in[89]. The purpose of this work is to settle the issue and show that a consistent operatorordering can be chosen in the two frames so that the frames become equivalent. Thisordering has to do with the ordering of generalized position and conjugate momentain the Hamiltonian, once we promote the classical Hamiltonian to a quantum one andposition, momenta become non-commuting operators. For every choice of an orderingin one frame, there is a particular choice of ordering in another frame, such that quan-tum Hamiltonian written in terms of operators become equivalent. In particular, weexplicitly choose a consistent set of parametrization and operator ordering to show theequivalence for homogeneous models without matter content.The rest of the chapter is organized as follows. In section 6.1, we analyze the modelwith zero spatial curvature, namely a Bianchi I model in detail. The techniques andmethod of the proof to study Bianchi-I is further generalized in the concluding section.We conclude our work with a general discussion along with a sketch of a general proofof equivalence between the two frames for all homogeneous models. It deserves mentionthat our result agrees with the work in references [89] and [116]. Models which areanisotropic generalization of constant non-zero spatial curvature isotropic models, i.e.,Bianchi-V and Bianchi-IX are discussed in section 6.2. Section 6.3 contains models withno isotropic analogue and discussion on Locally rotationally symmetric Bianchi-I andKantowski-Sachs models, respectively. The following table 6.1 shows the curvature ofthe spatial slice, for the various models discussed in this chapter. Similar non-equivalence is shown in the framework of loop quantum cosmology as well[107]
Anisotropic Models Spatial Curvature
Bianchi-I 0Bianchi-V m a ( t ) Bianchi-VI m − m +1) a ( t ) Bianchi-I(LRS) h a ( t ) Kantowski-Sachs − b ( t ) Table 6.1: Spatial curvature of various models
In the absence of any matter field, action for Brans Dicke theory in the Jordan frame isgiven by A J = Z d x L = Z d x √− g " φR + ωφ ∂ µ φ∂ ν φ , (6.1)where ω is the dimensionless Brans-Dicke coupling parameter and φ is the scalar fieldwhich depends only on time t . The Bianchi-I metric is described by ds = n dt − a ( t ) dx − b ( t ) dy − c ( t ) dz , (6.2)where a, b, c are the scale factors along three spatial direction. They encode the anisotropypresent in the metric.The Lagrangian arising out of Eq. (6.1) can be explicitly written as: L J = − φabcn " ˙ aa ˙ bb + ˙ aa ˙ cc + ˙ bb ˙ cc + ˙ aa ˙ φφ + ˙ bb ˙ φφ + ˙ cc ˙ φφ − ω φ φ , (6.3)where an overhead dot signifies a differentiation with respect to t . We reparametrize the71hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsscale factors as, a ( t ) = e σ + σ + + √ σ − , (6.4) b ( t ) = e σ + σ + −√ σ − , (6.5) c ( t ) = e σ − σ + , (6.6) φ ( t ) = e α , (6.7)which recast the Lagrangian in following form: L J = e α +3 σ n h −
6( ˙ σ − ˙ σ − ˙ σ − ) − α ˙ σ + ω ˙ α i . (6.8)Now to kill the cross-term ˙ α ˙ σ , we do a change of variables as, (which is in fact acanonical one) β = σ + α , (6.9) β ± = σ ± . (6.10)which yield following Lagrangian L J = e − α +3 β n (cid:20) − β + 6 ˙ β + 6 ˙ β − + 2 ω + 32 ˙ α (cid:21) , (6.11)and the corresponding Hamiltonian is given by H J = ne α − β (cid:20) −
16 ( p − p − p − ) + 22 ω + 3 p α (cid:21) , (6.12)where p and p ± are momenta conjugate to β and β ± respectively while p α is momentumconjugate to α .The transformation (6.9) looks like the one which takes us from the Jordan to the We thank Ott Vilson for pointing out to us that this is a canonical transformation even at quantumlevel. n , we obtain Hamiltonian constraint, H = (cid:20) −
16 ( p − p − p − ) + 22 ω + 3 p α (cid:21) = 0 . (6.13)If we now consider a canonical transformation ( α, p α ) to ( T, p T ) as T = αp α , (6.14) p T = p α , (6.15)then Eq. (6.13) can be rewritten as H = − (cid:18) p − p − p − − ω + 3 p T (cid:19) . (6.16)In the absence of a properly oriented scalar time parameter in the theory, we have usedthe evolution of the scalar field as the relevant time parameter. The method is the sameas that suggested by Vakili[84]. That T has the proper orientation can be ascertainedfrom its dependence on the cosmic time in the right direction. For a summary of themethod, we refer to [10].Upon quantization, we obtain the Wheeler-DeWitt equation as follows, " ∂ ∂β − ∂ ∂β − ∂ ∂β − − i ω + 3 ∂∂T ψ = 0 . (6.17)In the Einstein frame, the transformed metric components are ¯ g µν = φg µν , and theaction is given by A E = Z d x √− ¯ g (cid:20) ¯ R + 2 ω + 32 ∂ µ ξ∂ ν ξ (cid:21) , (6.18)73hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelswhere ξ = ln φ . Again, Bianchi I metric is given by, ds = ¯ n ( t ) dt − ¯ a ( t ) dx − ¯ b ( t ) dy − ¯ c ( t ) dz . (6.19)The Lagrangian can be written as L E = e r ¯ n (cid:20) −
6( ˙ r − ˙ r − ˙ r − ) + 2 ω + 32 ˙ ξ (cid:21) , (6.20)where ¯ a ( t ) = e r + r + + √ r − , (6.21)¯ b ( t ) = e r + r + −√ r − , (6.22)¯ c ( t ) = e r − r + , (6.23)and we obtain the following Hamiltonian H E = ¯ ne − r (cid:20) −
16 (¯ p − ¯ p − ¯ p − ) + 22 ω + 3 p ξ (cid:21) . (6.24)Now varying the action with respect to ¯ n , one obtains the Hamiltonian constraint H E = e r ¯ n H E = 0. Again with similar canonical transformations as¯ T = ξp ξ , (6.25) p ¯ T = p ξ , (6.26)the WDW equation can be written as " ∂ ∂r − ∂ ∂r − ∂ ∂r − − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (6.27)One can easily show that r i = β i and T = ¯ T from canonical transformations donein both the frame work. Hence it is quite obvious to see that the WDW equation in74hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsboth the frame is exactly same. It is a trivial exercise to check that the wave packet,obtained solving Eq. (6.27), will be the same as that formed out of the solution of theEq. (6.17), the WDW equation in the Jordan frame. To be mathematically precise,in Einstein frame, the wave function ψ E is a mapping from the configuration space˜ C , parameterized by r i to the real line, in Jordan frame we have wave function ψ J ,a mapping from the configuration space C , parameterized by β i to the real line. Theequality r i = β i implies that C = ˜ C and the same form of WDW equation tells us thatthe functional form of ψ E is same with that of ψ J ; in short, wave functions ψ E : ˜ C 7→ R and ψ J : C 7→ R are precisely same functions, living in the same Hilbert space.It further implies that not only the two formulations are equivalent in the formalstructure of the WDW equation, but also the probabilities and expectation valuesof observables will be same. The expectation value for operator A can be found as R dr i ψ ∗ J Aψ J = h A J i and R dβ i ψ ∗ E Aψ E = h A E i , where A should be expressed as a func-tion of r i or β i respectively, but they are same anyway. So the equivalence is not merelyin the classical limit, but rather in a real quantum picture. It also deserves mentionthat the quantities r i and β i are unitarily related, one can get one set from the otheran identity transformation. Moreover, Should one treat σ i as fundamental variable inJordan frame, the unitary equivalence is preserved as σ i is also related to β i by a unitarytransformation, to be precise exp (cid:16) ı α P σ (cid:17) σ exp (cid:16) − ı α P σ (cid:17) = β and β ± is related to σ ± by identity transformation where P σ is the momentum conjugate to σ .We further make a remark that if in Eq. (6.2) we put a = b = c , then in Eq. (6.4),both σ + and σ − become zero and we recover the same result for a spatially flat isotropiccosmological model in chapter 5. Bianchi V and Bianchi IX models have constant spatial curvature, with a negative anda positive signature respectively and are thus anisotropic generalizations of open and75hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsclosed isotropic models.
The Bianchi-V metric is given by ds = n ( t ) dt − a ( t ) dx − e mx [ b ( t ) dy + c ( t ) dz ] , (6.28)where m is a constant. We parametrize the scale factors in following manner, a ( t ) = e σ , (6.29) b ( t ) = e σ + √ β + − β − ) , (6.30) c ( t ) = e σ −√ β + − β − ) , (6.31) φ ( t ) = e α (6.32)and write down the Lagrangian, which will have some cross-term like ˙ α ˙ σ . Now to killthe cross-term, we define β = σ + α , which is again a canonical transformation. Thisimmediately recasts the Lagrangian in following form, L J = e − α +3 β n (cid:20) − β + 6( ˙ β + − ˙ β − ) − e − β n m + 2 ω + 32 ˙ α (cid:21) . (6.33)The non-trivial part of this parametrization is that the canonical transformation re-quired to make the Lagrangian in a diagonal form is essentially same as conformal trans-formation to the Einstein frame. There is no a priori reason for them to be the same. Weemphasize that the conformal transformation is indeed canonical even, at quantum level.The corresponding Hamiltonian in the Jordan frame can be written as H J = ne α/ − β (cid:20) − ( p − p − m e β ) + 122 ω + 3 p α (cid:21) . (6.34)A variation of the action with respect to n yields H = e − α/ β Hn = 0.76hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsNow we consider a canonical transformation ( α, p α ) to ( T, p T ), as before, given byfollowing, T = αp α , (6.35) p T = p α , (6.36)so that Wheeler de Witt equation becomes " ∂ ∂β − ∂ ∂β − m e β − i ω + 3 ∂∂T ψ = 0 . (6.37)In the Einstein frame the action is given by A E = Z d x √− ¯ g (cid:20) ¯ R + 2 ω + 32 ∂ µ ξ∂ ν ξ (cid:21) , (6.38)where ξ = ln φ . The Bianchi V metric is given by ds = ¯ n ( t ) dt − ¯ a ( t ) dx − e mx [¯ b ( t ) dy + ¯ c ( t ) dz ] . (6.39)Now the Lagrangian can be written as L E = e r ¯ n (cid:20) − r + 6( ˙ r + − ˙ r − ) − e − r n m + 2 ω + 32 ˙ ξ (cid:21) , (6.40)where ¯ a ( t ) = e r , (6.41)¯ b ( t ) = e r + √ r + − r − ) , (6.42)¯ c ( t ) = e r −√ r + − r − ) , (6.43)and the corresponding Hamiltonian is now of the form, H E = ¯ ne − r (cid:20) − (¯ p − ¯ p − m e r ) + 122 ω + 3 p ξ (cid:21) . (6.44)77hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsA variation of the action with respect to n yields H E = e r ¯ n H E = 0. Again with similarcanonical transformation, ¯ T = ξp ξ , (6.45) p ¯ T = p ξ , (6.46)Wheeler-DeWitt equation can be recast in following form, " ∂ ∂r − ∂ ∂r − m e r − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (6.47)It is evident from the canonical transformation done in both the framework that r i = β i and T = ¯ T , hence the WDW equation in both the frame comes out to beexactly same and leads to similar wave packet if we transform ¯ a, ¯ b, ¯ c back to the original √ φa, √ φb, √ φc respectively. A Bianchi-IX metric given as ds = n ( t ) dt − a ( t ) dr − b ( t ) dθ − [ a ( t ) cos θ + b ( t ) sin θ ] dφ . (6.48)In the Einstein frame, the metric components are indicated with an overhead bar.With the following change of variables b ( t ) = e − α βa , (6.49) a ( t ) = e − α/ a , (6.50) φ ( t ) = e α , (6.51)78hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsin the Jordan frame and following change¯ β = ¯ a ¯ b , (6.52)in the Einstein frame coupled with a canonical transformation as done in the previoustwo cases, one can write down the WDW equations in the two frames. The result is thesame, i.e., similar wave packet will be obtained if we revert the ¯ a, ¯ b to original √ φa, √ φb respectively. The equivalence of the two frames can also be shown in other Bianchi models as well.Bianchi I, V and IX are in fact examples of anisotropic models which reduce to isotropicFriedmann models under given condition. But the conclusion remains the same for othermodels which do not have this property.
The example of a Bianchi type VI model can be taken up. Such a model is given by themetric ds = n ( t ) dt − a ( t ) dx − e − mx b ( t ) dy − e x c ( t ) dz . (6.53)With the change of variables as a ( t ) = e − α/ β , (6.54) b ( t ) = e − α/ β + √ β + − β − ) , (6.55) c ( t ) = e − α/ β −√ β + − β − ) , (6.56) φ ( t ) = e α , (6.57)79hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsthe Lagrangian and the Hamiltonian in the Jordan frame can be written respectivelyas L J = e − α/ β n " − β + 6( ˙ β + − ˙ β − ) − e − β n ( m − m + 1)2 + 2 ω + 32 ˙ α , (6.58)and H J = ne α/ − β (cid:20) − [ p − p − m − m + 1) e β ] + 122 ω + 3 p α (cid:21) . (6.59)If we consider a canonical transformation ( α, p α ) to ( T, p T ) as T = αp α , (6.60) p T = p α , (6.61)then with the Hamiltonian constraint, H = e − α/ β Hn = 0, the WDW equation can bewritten as " ∂ ∂β − ∂ ∂β − m − m + 1) e β − i ω + 3 ∂∂T ψ = 0 . (6.62)In the Einstein frame, the metric is written as ds = ¯ n ( t ) dt − ¯ a ( t ) dx − e − mx ¯ b ( t ) dy − e x ¯ c ( t ) dz , (6.63)where the barred metric components are related to the unbarred components in theJordan frame as ¯ g µν = φg µν . With the transformation¯ a ( t ) = e r , (6.64)¯ b ( t ) = e r + √ r + − r − ) , (6.65)¯ c ( t ) = e r −√ r + − r − ) , (6.66)80hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsThe Lagrangian and the Hamiltonian in the Einstein frame look respectively as L E = e r ¯ n " − r + 6( ˙ r + − ˙ r − ) − e − r n ( m − m + 1)2 + 2 ω + 32 ˙ ξ , (6.67)and H E = ¯ ne − r (cid:20) − [ ¯ p − ¯ p − m − m + 1) e r ] + 122 ω + 3 p ξ (cid:21) . (6.68)Again with similar canonical transformation as¯ T = ξp ξ , (6.69) p ¯ T = p ξ , (6.70)and the Hamiltonian constraint H E = e r ¯ n H E = 0, the WDW equation can be writtenas " ∂ ∂r − ∂ ∂r − m − m + 1) e r − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (6.71)One can easily show that r i = β i and T = ¯ T from canonical transformation effectedin both the framework, thus the WDW equation in both the frame is exactly same,henceforth leads to similar the wave packet upon transforming ¯ a, ¯ b, ¯ c back to original a, b, c respectively as usual. A locally rotationally symmetric (LRS) Bianchi -I model can be written in terms of themetric ds = n dt − a ( t ) dx − b ( t ) e x/h ( dy + dz ) , (6.72)81hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelswhere n ( t ) is the lapse function, a ( t ) , b ( t ) are functions of time and h is a constant. TheRicci scalar can be written for this metric as √− g φR = − e xh b φnh a − bφ ˙ a ˙ bn − aφ ˙ b n + 2 b φ ˙ a ˙ nn + 4 abφ ˙ b ˙ nn − b φ ¨ an − ab ˙ b ˙ φn . (6.73)Lagrangian in the Jordan frame can be written as L J = b φan " − n h a − a ˙ bab − b b − a ˙ φaφ − b ˙ φbφ + ω ˙ φ φ . (6.74)With change of variable given as a ( t ) = e − α/ − β + , (6.75) b ( t ) = e − α/ β + + β − , (6.76) φ ( t ) = e α , (6.77)the Lagrangian assumes the form L J = e β + +2 β − − α/ n (cid:20) β − β − + 2 ω + 32 ˙ α (cid:21) − e α +3 β + +2 β − nh , (6.78)and corresponding Hamiltonian becomes H J = ne α/ − β + − β − " p β + − p β − + 42 ω + 3 p α + 48 e β + + β − ) h . (6.79)On varying of the action with respect to n , we get the Hamiltonian constraint H = e β ++2 β −− α/ Hn = 0.Now we consider a canonical transformation ( α, p α ) to ( T, p T ), as before, given by82hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsfollowing relations, T = αp α , (6.80) p T = p α . (6.81)so that Wheeler de Witt equation H ψ = 0 becomes " − ∂ ∂β + ∂ ∂β − + 48 e β + + β − ) h − i ω + 3 ∂∂T ψ = 0 . (6.82)If we now transform the metric to the Einstein frame via a conformal transformation¯ g µν = φg µν , the metric looks like ds = ¯ n dt − ¯ a ( t ) dx − ¯ b ( t ) e x/h ( dy + dz ) , (6.83)we can write the action in the Einstein frame as A E = Z d x √− ¯ g (cid:20) ¯ R + 2 ω + 32 ∂ µ ξ∂ ν ξ (cid:21) , (6.84)where ξ = ln ( φ ) and the barred quantities indicate that they are in the Einstein frame.With the transformation ¯ a ( t ) = e − r + , (6.85)¯ b ( t ) = e r + + r − . (6.86)the Lagrangian and the Hamiltonian in the Einstein frame look respectively as L E = e r + +2 r − n (cid:20) − r − + 2 ˙ r + 2 ω + 32 ˙ ξ (cid:21) − e r − +3 r + ¯ nh , (6.87) H E = ¯ ne − r + − r − " ¯ p r + − ¯ p r − + 42 ω + 3 p ξ + 48 e r + + r − ) h . (6.88)A variation of the action with respect to ¯ n yields H = e r ++2 r − H ¯ n = 0.83hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsNow we consider a canonical transformation of variables ( ξ, p ξ ) to ( ¯ T , p T ), as before,given by following, T = ξp ξ , (6.89) p T = p ξ , (6.90)so that Wheeler-DeWitt equation H ¯ ψ = 0 becomes " − ∂ ∂r + ∂ ∂r − + 48 e r + + r − ) h − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (6.91)It is evident from the canonical transformation done in both the framework that r + = β + , r − = β − and T = ¯ T , hence the WDW equations in the two frames comes out to beexactly same and leads to similar wave packet if we transform ¯ a, ¯ b back to the original √ φa, √ φb , respectively. The Kantowski-Sachs (KS) metric is written as ds = n dt − a ( t ) dx − b ( t )( dθ + sin θd Φ ) . (6.92)The Lagrangian in the Jordan frame is L J = b φan " n b − a ˙ bab − b b − a ˙ φaφ − b ˙ φbφ + ω ˙ φ φ . (6.93)With a change of the set of variables given as a ( t ) = e − α/ − β + , (6.94) b ( t ) = e − α/ β + + β − , (6.95) φ ( t ) = e α , (6.96)84hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsthe Lagrangian can be re-written as L J = e β + +2 β − − α/ n (cid:20) β − β − + 2 ω + 32 ˙ α (cid:21) + 2 ne α − β + (6.97)and the corresponding Hamiltonian looks like H J = ne α/ − β + − β − (cid:20) p β + − p β − + 42 ω + 3 p α − e β − (cid:21) . (6.98)A variation of the action with respect to n yields H = e β ++2 β −− α/ Hn = 0.Now we consider a canonical transformation ( α, p α ) to ( T, p T ), as before, given by fol-lowing, T = αp α , (6.99) p T = p α , (6.100)so that Wheeler-DeWitt equation H ψ = 0 becomes " − ∂ ∂β + ∂ ∂β − − e β − − i ω + 3 ∂∂T ψ = 0 . (6.101)In the Einstein frame, the transformed metric is ds = ¯ n dt − ¯ a ( t ) dx − ¯ b ( t )( dθ + sin θd Φ ) , (6.102)where the barred metric components are related to the unbarred components in theJordan frame as ¯ g µν = φg µν .With the transformations ¯ a ( t ) = e − r + , (6.103)¯ b ( t ) = e r + + r − , (6.104)85hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelsthe Lagrangian and the Hamiltonian in the Einstein frame look respectively as L E = e r + +2 r − n (cid:20) r − r − + 2 ω + 32 ˙ ξ (cid:21) + 2¯ ne − r + , (6.105) H E = ¯ ne − r + − r − (cid:20) ¯ p r + − ¯ p r − + 42 ω + 3 p ξ − e r − (cid:21) . (6.106)A variation of the action with respect to ¯ n yields H = e r ++2 r − ¯ n H = 0.Now we consider a canonical transformation ( ξ, p ξ ) to ( ¯ T , p T ), as before, given by fol-lowing, ¯ T = ξp ξ , (6.107) p ¯ T = p ξ , (6.108)so that Wheeler-DeWitt equation H ¯ ψ = 0 becomes " − ∂ ∂r + ∂ ∂r − − e r − − i ω + 3 ∂∂ ¯ T ¯ ψ = 0 . (6.109)It is evident from the canonical transformation done in both the framework that r + = β + , r − = β − and T = ¯ T , hence the WDW equation in both the frame comes out to beexactly same and leads to similar wave packet if we transform ¯ a, ¯ b back to the original √ φa, √ φb respectively. By a mathematical equivalence, we mean any physical quantity obtained in the Jordanframe can be mathematically transformed into an expression in the Einstein frame suchthat the expression is exactly the same in both the frames.Quantum mechanically, this mathematical equivalence seems to be broken as reportedin the literature several times[61, 108]. Here we show that the reported nonequivalence86hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelscan be restored with a consistent operator ordering. Given a frame, be it the Einsteinor the Jordan, we always have to choose a particular ordering of operators so as to goover to quantum theory. Now, choosing one particular operator ordering in the Jordanframe fixes the operator ordering in the Einstein frame and vice versa. This strips usoff the freedom of choosing operator ordering in both the frames independently. Shouldwe do the ordering in both the frames arbitrarily, it might become inconsistent witheach other, leading to a different quantum Hamiltonian and hence different behavior ofthe wave packets. Thus, the apparent discrepancy should not be attributed to quantumeffects, rather they are the artifacts of an inconsistent operator ordering. In this work,we basically have picked up a consistent choice of operator ordering in both the frames.We do the parametrization in a suitable way, which renders the most natural choice ofoperator ordering to be the consistent ones . It worths mentioning though that, [108]deals with a cosmological model with a matter content. Unless the matter content is aconformal one, i.e, radiation, the equivalence is broken explicitly. It would be interest-ing to show that with a consistent operator ordering at least for a conformal fluid, i.e.,radiation, equivalence can be restored.It deserves mention that the parametrization in this particular way and the naturaloperator ordering in the new variables β , β ± indicate a particular choice of operatorordering in terms of old variables a, b, c . This choice is by no means a unique one and itis not required to prove the uniqueness either. For homogeneous models, however, thereis a natural unique choice of the parametrization as shown in what follows.Here we explicitly show that for every choice of co-ordinates and corresponding quan-tization in the Jordan frame with a particular operator ordering, we have a particularchoice of co-ordinates and an operator ordering in the Einstein frame so that both the To avoid any potential confusion, the operator ordering we talk about is not the normal orderingin quantum field theory, and this also does not have anything to do with the field redifinitionor frame transformation. As we mentioned in the introduction, the ordering issue comes as wepromote the classical Himaltonian to a quantum one, position and momenta become non-commutingoperators. The important thing is that we choose an ordering in one frame, and the ordering shouldbe consistent with that in the other frame. The initial choice could be one in which the theorypreserves unitarity[32]. g ij = e − α h ij , φ = e α . (6.110)As we are dealing with homogeneous models, we parametrize h ij by some function oftime in some particular way. Thus we have L J = ne − α √ h (cid:20) R h ij + 2 ω + 32 ˙ α (cid:21) . (6.111)From this, we write down the Hamiltonian. Subsequently the Hamiltonian constraintis obtained and the theory is quantized in a canonical way.Now, the Einstein frame is a conformal transformation of the Jordan frame. Thehomogeneity guarantees that φ is a function of t alone, hence the conformal factor isa function of t alone. This implies that going from the Jordan to the Einstein framedoes not change the Bianchi class of the model considered. Had it been the case that φ is a function of space as well as time, things would have been more intricate and thismethod would have broken down, i.e., the proof would not have gone through.In the Einstein frame, we parametrize the metric ¯ g ij in same way as we have parametrizedthe metric h ij . This will lead to L E = ¯ n √− ¯ g (cid:20) ¯ R + 2 ω + 32 ˙ α (cid:21) (6.112)Since, ¯ g ij = φg ij = h ij , we have ¯ R = R h ij , and the parametrization in a similar wayguarantees that a consistent operator ordering is being chosen automatically, hence theequivalence becomes an obvious one. This is to be emphasized that we have shownequivalence for every choice of parametrization of h ij in this work.It deserves mention that this proof is really independent of how we are defining the88hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelstime variable for quantum theory, only a properly oriented time parameter is required.Nonetheless, one might wonder that the choice of parametrization or the transforma-tion, given by Eq. (6.9) to make the Lagrangian diagonal is actually effecting a confor-mal transformation to Einstein frame in disguise. However, this transformation can bethought of as a canonical one. Since, a canonical transformation preserves the quantumstructure, the two frames are indeed equivalent.It is a straight forward exercise to extend the prescription for a cosmological modelwith fluid. It deserves mention that the application of S-theorem [101] and result-ing anomalous symmetry breaking in quantum FRW cosmological model with radiationmatter content [87] is technically done using the same transformation. Thus, it is worth-while to check whether such symmetry breaking can happen in the Jordan frame as wellbefore effecting the canonical transformation and confirm the quantum equivalence.We point out that we have used the so called Misner variables[67] in writing the met-ric. In these coordinates, a lot of simplification is achieved for anisotropic but spatiallyhomogeneous spaces, namely the spacetimes that fall under the Bianchi classification.As general relativity is diffeomorphism invariant, this simplification does not lead to anychange in physical properties. In fact, it appears that Misner coordinates are arguablythe best set of coordinates for the purpose of quantization. We refer to the work ofAgostini, Cianfrani and Montani for a very recent application of Misner variables[117],which indicates that Misner variables and other approaches do yield equivalent results forisotropic matter distribution. In the present case, although the geometry is anisotropic,the matter distribution is indeed isotropic.It also deserves mention that we have tried other transformations involving the Brans-Dicke parameter ω , and at least for some particular vaules of ω could write down sep-arable expressions. One difficulty is that of picking up a nice time variable, whosecorresponding momentum appears only in the first order. But the most important prob-89hapter 6 Equivalence of Jordan and Einstein frames in anisotropic quantumcosmological modelslem is that it is impossible to pick the corresponding operator ordering in the two frames,which is irrecoverably lost in the transformation.Indeed this equivalence has been already shown to be true in isotropic models as inthe previous chapter. But isotropic models are special, the self-adjoint extension is quiteunique in such models[32]. But anisotropic models are not as simple, the uniqueness ofself-adjoint extension is not guaranteed, and the proper operator ordering is quite non-trivial. So the present work strengthens the present knowledge regarding the equivalencebetween conformally transformed versions of the same theory.We hope that the work will clarify the issue regarding equivalence prevailing in theliterature. The question of mathematical equivalence, thus being clarified, one could askfor more sophisticated questions like quantum mechanically which frame is more usefuland how the quantum behavior changes once we go from one frame to another frame.There is already some work in this connection[61].90 HAPTER Conclusion
Quantum cosmology makes an attempt to apply the principles of quantum mechanics tocosmological systems. The quantum description of the universe is required for the earlystage of the evolution at an energy scale where classical gravity loses its viability. Thequantum prescription requires a suitable time parameter for the evolution of the systemas cosmic time t is a coordinate itself in the general theory of relativity. Schutz’s formal-ism help us in this regard as discussed in Section 1.4. It is also believed that anisotropicquantum cosmological models suffer from the problem of a non-unitary evolution. Thisalleged problem of non-unitarity has recently been resolved for a few anisotropic modelswith constant spatial curvature by showing that the WDW quantization scheme canlead to either the self-adjoint Hamiltonian or the Hamiltonian admitting self-adjointextension by a suitable operator ordering [10, 11]. In chapter 2, we used the WDWquantization scheme to quantize the two examples of anisotropic cosmological modelswith varying spatial curvature, namely Bianchi II and VI. We showed the self-adjoint ex-tension of Hamiltonian for these models is quite possible; thus, they do admit a unitaryevolution.In chapter 3, we tried to generalize the work of unitary evolution to higher dimensionalanisotropic models. We quantize the n-dimensional anisotropic model with fluid. Using91hapter 7 Conclusionthe WDW quantization method, we obtained the finite normed and time independentwave packet for stiff fluid α = 1. Thus, the unitary evolution of the model is established.We also computed the expectation values of scale factors and were able to show that theywere non-singular. The expectation value of the volume element has non-zero minima,which indicates a bouncing universe. Thus, the model avoids the problem of singularitysimilar to earlier work done for anisotropic models in the absence of fluid [89]. Despitethe mathematical difficulty of the model with general fluid α = 1, we were able to exhibitthe unitary evolution for n=7.The unitary evolution of anisotropic models is achieved mainly through self-adjointextension. In chapter 4, we tried to investigate the cost of doing such extension eitherby operator ordering or transformation of variables at the classical level. We consideredthe Bianchi-I cosmological model as an example. We showed that Noether symmetry ispreserved in the process of self-adjoint extension. We also showed that scale invarianceis lost while achieving the unitarity of the model. It has already been shown that theunitary evolution is not achieved at the cost of anisotropicity itself [13].Since unitary evolution has been extensively shown in this thesis for anisotropic buthomogeneous models, the more difficult task would be to quantize the inhomogeneouscosmological models as it becomes challenging to do a 3+1 decomposition and applythe WDW quantization scheme in inhomogeneous models, even in the simpler modellike Lemaitre-Tolman-Bondi (LTB) model. One may attempt to quantize such modelsin the future.In the latter part of the thesis, we tried to quantize the Brans-Dicke theory, whichhas been studied in two frames, the Jordan Frame and The Einstein frame. We triedto address the question of the equivalence of these two frames at the quantum levelin chapter 5. We quantized the spatially homogeneous and isotropic spacetime. Weconstructed the time parameter from the scalar field in the absence of the fluid in themodel and obtained the wave packet in both the frames. Then we reverted the conformaltransformation, ¯ g µν = Ω g µν through which these two frames are related to each other.The wave packet obtained in the Einstein matched precisely with the one in the Jordan92hapter 7 Conclusionframe. Thus, we were able to show the equivalence of these two frames very clearly.After showing the equivalence of the Jordan and the Einstein frame for the isotropicquantum model, we tried to explore the equivalence of these two frames in anisotropicquantum cosmological models. In chapter 6, we first take the example of Bianchi Ispace time and derive the WDW equation in both the frames using various canonicaltransformation. The same form of the WDW equation in both the frames tells thefunctional form of the wave packet in the Einstein frame Ψ E is the same to one in theJordan frame Ψ J . We also remark that this equivalence is not only a mathematicalconstruct, rather a real physical one, as the expectation value of an operator A wouldbe same in these two frames.Then we tried to extend this equivalence to various other anisotropic models, modelswith constant but non-zero spatial curvature namely Bianchi V and Bianchi IX modeland models with varying spatial curvature, namely Bianchi VI, Bianchi I with localrotational symmetry and Kantawski-Sachs model.At the end of chapter 6, we show that we have a particular choice of coordinates trans-formations and an operator ordering in the Einstein frame for every such choice in theJordan frame, which leads to the equivalence of the frames for any homogeneous model.We also emphasize on the fact that transformations are canonical, which preserves thequantum structure. Thus, two frames are equivalent at quantum level.As a future work, one can look at this equivalence in the presence of matter, suchas a fluid. 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