First-Quantized Theory of Expanding Universe from Field Quantization in Mini-Superspace
aa r X i v : . [ g r- q c ] A ug FIRST–QUANTIZED THEORY OF EXPANDING UNIVERSEFROM FIELD QUANTIZATION IN MINI–SUPERSPACE
DAISUKE IDA AND MIYUKI SAITO
Department of Physics, Gakushuin University, Tokyo 171-8588
Abstract.
We propose an improved variant of the third–quantization scheme,for the spatially homogeneous and isotropic cosmological models in Einsteingravity coupled with a neutral massless scalar field. Our strategy is to spec-ify a semi–Riemannian structure on the mini–superspace and to consider thequantum Klein–Gordon field on the mini–superspace. Then, the Hilbert spaceof this quantum system becomes inseparable, which causes the creation ofinfinite number of universes. To overcome this issue, we introduce a vectorbundle structure on the Hilbert space and the connection of the vector bun-dle. Then, we can define a consistent unitary time evolution of the quantumuniverse in terms of the connection field on the vector bundle. By doing this,we are able to treat the quantum dynamics of a single–universe state. We alsofind an appropriate observable set constituting the CCR–algebra, and obtainthe Schr¨odinger equation for the wave function of the single–universe state.We show that the present quantum theory correctly reproduces the classicalsolution to the Einstein equation. Introduction
One fundamental problem in quantum cosmology is the issue on the observableset and the quantum state of the space-time. The purpose of this paper is toconstruct an appropriate set of observables in the mini–superspace model of quan-tum cosmology and quantum states of the universe with clear classical–quantumcorrespondence.While the perturbative quantization of the general relativity confronts the diffi-culty due to the non–renormalizability of the theory, alternative approaches such asstring theory, loop gravity, etc. have been widely discussed for a long time. Needlessto say, an important application of the quantum gravity would be in the quantumcosmology, which would give quantum descriptions of the expanding universe atearly times and the big-bang singularity. Although we don’t have a fully reliabletheory of quantum gravity, it will be worthwhile to discuss a quantum theory of theexpanding universe, such as the mini–superspace model, defined only on the finitedimensional reduced phase space of the gravitational field.In the Hamiltonian formulation of the general relativity, we are led, accordingto the Dirac’s prescription [1] for singular Lagrangian systems, to the constrainedHamiltonian system, in which the total Hamiltonian of the gravitational field H T = Z Σ t d x ( N Φ + X i =1 N i Φ i ) Date : Tuesday 12 August, 2014. consists of a linear combination of first–class constraint functions Φ µ ≈ µ =0 , , ,
3) [2, 3].The configuration space of the gravitational field is parametrized by the Rie-mannian 3-metric h ij on the Cauchy surface Σ t , and the canonical momentum isgiven by π ij = (16 πG ) − √ h ( K ij − Kh ij ) , where K ij = (1 / £ n h ij ( n is the future pointing unit normal vector field on Σ t ) isthe extrinsic curvature of Σ t . In terms of these canonical variables, the constraintsare written as Φ = G ij,kl π ij π kl − √ h R πG ≈ , Φ i = − D k π ki ≈ , where R and D k are the scalar curvature and the covariant derivative (for tensordensities) on (Σ t , h ij ), respectively, and G ij,kl := 8 πG √ h ( h ik h jl + h il h jk − h ij h kl )determines the semi–Riemannian structure of the configuration space of the 3-metrics.According to the Dirac’s algorithm of quantization, each first–class constraintfunction Φ µ is replaced with a corresponding linear operator b Φ µ in a Hilbert space H . Then, the observed physical state is required to be a ray in the subspace H phys = {| ψ i ∈ H ; b Φ µ | ψ i = 0 } determined by these constraint operators. Theserequirements: b Φ µ | ψ i = 0, for the physical state, which become functional differen-tial equations in the Schr¨odinger representation, are well known as the Wheeler–DeWitt equations [4, 5].This formulation of quantum gravity is purely kinematical, in that the Wheeler–DeWitt equations only make a restriction on the physical Hilbert space, but doesnot define the dynamics of the wave function, since the Hamiltonian itself becomesa constraint operator: b H T | ψ i = 0, which is called the problem of time in theliterature [6–8]. Furthermore, we don’t have a fully satisfactory interpretation ofthe quantum state subject to the Wheeler–DeWitt equations.In order to approach such conceptual issues, it will be worthwhile, as a first step,to discuss the mini–superspace model of the quantum cosmology. According tothe concept of the geometrodynamics by Wheeler [5], the solution to the Einsteinequation can be viewed as the motion of a test particle in the superspace [9, 10],which is the space of the 3–geometries on the Cauchy surface Σ t , more precisely, it isthe configuration space of 3-metrics above modulo the proper subgroup of the self–diffeomorphism group of Σ t . Since this viewpoint is very useful in our formulationof the quantum theory of cosmology, let us respect it in the mini–superspace modelin what follows.The organization of the paper is as follows. In section 2, we briefly review theHamiltonian formulation of the classical Einstein gravity applied to the Friedmann–Robertson–Walker (FRW) space–time, where we introduce the semi–Riemannianmetric g C on the mini–superspace. In section 3, we consider the quantum Klein–Gordon field in the mini–superspace with respect to the semi–Riemannian structuregiven by g C , where we are faced with the issue regarding the inseparability of the IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 3
Hilbert space in the cases of the spatially non–flat universe. In section 3.2 and 3.3,we introduce the vector bundle structure in the inseparable Hilbert space to definethe consistent quantum theory with the unitary time evolution described by theglobal section of the vector bundle, where we obtain the first–quantized theory ofthe expanding universe on the space of 1–particle states of the quantum Klein–Gordon field. In section 4, we show that our quantum theory correctly reproducesthe classical solution to the Einstein equation. In section 5, we give concludingremarks. 2.
Preliminaries: Mini–Superspace Model
In the mini–superspace model of the isotropic and homogeneous universe, weconsider the class of space–times represented by the FRW metric g = − N ( t ) dt + a ( t ) γ K , where γ K ( K = 0 , ±
1) denotes the Riemannian metric of the 3–space of the constantsectional curvature K . Let us consider the neutral massless scalar field minimallycoupled with the Einstein–Hilbert action, which is provided by the classical action S [ g, X ] = 116 πG Z d x √− g (cid:18) R − X ,µ X ,µ (cid:19) . In the standard procedure, we are led to the total Hamiltonian H T = N Φ , (1) Φ = 8 πGv (cid:18) − p a a + p X a (cid:19) − Kva πG , where we consider the fixed spatial coordinate volume and set v = R d x √ γ K . Theconfiguration space M , which is called the mini–superspace, is parametrized by q m = ( a, X ), and their conjugate momenta are denoted as p m = ( p a , p X ). Themomentum constraints Φ i ≈ H T = N [( g S ) mn p m p n + u ] ,g S = v πG ( − ada + a dX ) ,u = − Kva πG , the dynamics of the universe is equivalent to that of a point particle in the curvedspace–time with the metric g S and the potential function u . In fact, the equationof motion is given by d q k dt + Γ[ g S ] kmn dq m dt dq n dt ≈ N − dNdt dq k dt − N ( g S ) kl dudq l , Φ = 14 N ( g S ) mn dq m dt dq n dt + u ≈ . The present system has an invariance under the coupled conformal transfor-mation of the Lorentzian metric g S and the pointwise scaling of the potential:( g S , u ) ( f g S , f − u ) for f ( q m ) = 0. Using this invariance, the system can alwaysbe reduced to that of the geodesic particle in M but with a conformally relatedmetric [9, 10]. DAISUKE IDA AND MIYUKI SAITO
In fact, assuming u = 0, under the conformal transformation( g S ) mn = C u ( g C ) mn (2)and the reparametrization of the time function t ( s ) = Z s ds C N u , the equations of motion become d q k ds + Γ[ g C ] kmn dq m ds dq n ds ≈ , (3) Φ = u (cid:18) ( g C ) mn dq m ds dq n ds + 1 (cid:19) ≈ , (4)where C is a constant with the mass dimension 1. The Eq. (3) is the geodesicequation with respect to the semi–Riemannian metric g C , and the Hamiltonianconstraint (4) requires that the new parameter s is the proper time. Hence, theequivalent system is given by the Hamiltonian of the geodesic particle in ( M , g C ), H ′ T = λ (( g C ) mn p m p n + 1) , where λ is the Lagrange multiplier.On the other hand, when u = 0, we set g C = g S . Then, the Hamiltonian issimply given by H ′ T = N ( g C ) mn p m p n . This describes the null geodesic particle in ( M , g C ).As a quantized system corresponds to the classical system defined by Eq. (1), weconsider the quantum fields in the mini–superspace ( M , g C ) as a semi–Riemannianmanifold. 3. Quantum Klein–Gordon field in mini–superspace
Here, we consider the Klein–Gordon field in ( M , g C ) and give a consistent quan-tized theory. Let us see K = 0 , ± Case of flat universe ( K = 0) : The spatially flat case turns out to be themost simple, so that it is appropriate to explain this case first. In this case, thedynamics of the universe is equivalent to that of a null geodesic particle in ( M , g C ),where the metric is given by g C = v πGC ( − ada + a dX ) . We construct the localized 1–particle states of the massless quantum Klein–Gordonfield to describe a quantum mechanical counterpart of this classical light–like par-ticle.Firstly, we introduce the new time function T by a = a e βT . Then, the metricon M becomes the conformally flat form g C = A e βT ( − dT + dX ) , IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 5 where we set β = (12) − / , A = va πGC . The classical action of the massless Klein–Gordon field in ( M , g C ) becomes thatin the flat space: S [ φ ] = 12 Z dT dX (cid:2) ( ∂ T φ ) − ( ∂ X φ ) (cid:3) . A solution to the Klein–Gordon equation can be decomposed into a linear combi-nation of the mode functions { f ( p ; T, X ) , f ∗ ( p ; T, X ) } ( p ∈ R ), where f ( p ; T, X ) = (4 π | p | ) − / e − i | p | T e ipX . The quantum Klein–Gordon field is expanded into the form φ ( T, X ) = Z ∞−∞ dp [ a ( p ) f ( p ; T, X ) + a ∗ ( p ) f ∗ ( p ; T, X )] , and the conjugate momentum operator is written as π ( T, X ) = ∂ T φ ( T, X ). Thecanonical commutation relations (CCRs)[ φ ( T, X ) , π ( T, X ′ )] − = iδ ( X − X ′ ) , [ φ ( T, X ) , φ ( T, X ′ )] − = 0 , [ π ( T, X ) , π ( T, X ′ )] − = 0 , are equivalent to[ a ( p ) , a ∗ ( p ′ )] − = δ ( p − p ′ ) , [ a ( p ) , a ( p ′ )] − = 0 , [ a ∗ ( p ) , a ∗ ( p ′ )] − = 0 . In terms of these, the quantum Hamiltonian operator is given by H = Z ∞−∞ dp | p | a ∗ ( p ) a ( p ) . As usual, the vacuum state | Ω i is defined by the requirement: a ( p ) | Ω i = 0 , for all p ∈ R , and the Fock space is constructed in the standard procedure. Let F (1)Ω denote theHilbert space of 1–particle states, which we are mainly concerned with.Define, for each p ∈ R , the momentum operator P := Z ∞−∞ dp pa ∗ ( p ) a ( p ) , which is a Hermitian operator on F (1)Ω . The eigenvectors of P are given by | p i := a ∗ ( p ) | Ω i , for p ∈ R , which satisfy P | p i = p | p i . These satisfy the orthogonality condition h p | p ′ i = δ ( p − p ′ )and the completeness condition Z ∞−∞ dp | p ih p | = , where denotes the identity operator on F (1)Ω . DAISUKE IDA AND MIYUKI SAITO
The position operator is defined by Q := i Z ∞−∞ dp a ∗ ( p ) ∂ p a ( p ) , which is a Hermitian operator on F (1)Ω . The eigenvector of Q is formally writtenas a state | X i Q := Z ∞−∞ dp e − ipX | p i , for X ∈ R , and it satisfies Q | X i Q = X | X i Q . The vector | X i Q is the localized state considered by Newton and Wigner [11] longtime ago.The operators P and Q constitute the CCR–algebra[ Q, P ] − = i . Thus, we obtain the canonical observable set (
P, Q ) and its Schr¨odinger represen-tation on F (1)Ω .We regard the quantum state of the universe as being described by a 1–particlestate of the Fock space. Since the Hamiltonian of the Klein–Gordon field actson F (1)Ω , any 1–particle state remains in F (1)Ω in the course of the unitary timeevolution. In fact, the Hamiltonian restricted on F (1)Ω can be written as H (1) = | P | . The expansion of the 1–particle state at the time T | ψ ( T ) i = Z ∞−∞ dp ψ ( p, T ) | p i defines the wave function ψ ( p, T ) of the universe in the momentum representation.Then, the time evolution of the 1–particle state is determined by the Schr¨odingerequation i∂ T ψ ( p, X ) = h p | H (1) | ψ ( T ) i = | p | ψ ( p, T ) , according to the unitary time evolution of the quantum Klein–Gordon field. In thisway, we obtain a quantum Hamiltonian for the wave function of the universe withoutoperator ordering ambiguity [12], which clearly describes the massless particles in( M , g C ), besides the well–defined canonical observable set ( P, Q ).3.2.
Case of hyperbolic universe ( K = − : In this case, we are led to themassive field in the expanding chart of the Milne space. Then, according to thestandard approach, the quantum field theory in this background geometry describesthe continuous pair creations of scalar particles, which is hard to interpret in thecontext of the quantum cosmology. This occurs due to the inseparability of theHilbert space involved. Nevertheless, we show that it is possible to construct aconsistent quantum theory which describes the dynamics of a 1–particle state witha correct classical interpretation.The semi–Riemannian metric on M in this case is given by [See Eq. (2)] g C = 3 v (8 πG ) C ( − a da + a dX ) . IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 7
Introducing the new time function T by a = a e βT , the metric is written as g C = A e βT ( − dT + dX ) , where we set β = (12) − / , A = (cid:12)(cid:12)(cid:12)(cid:12) √ va πGC (cid:12)(cid:12)(cid:12)(cid:12) . Now, we start with the action of the massive Klein–Gordon field S [ φ ] = 12 Z dT dX [( ∂ T φ ) − ( ∂ X φ ) − A m e βT φ ] , of the mass m . The complete set of the solutions to the Klein–Gordon equation isgiven by { f ( p ; T, X ) , f ∗ ( p ; T, X ) } ( p ∈ R ), where f ( p ; T, X ) := (4 π | p | ) − / F ( | p | ; T ) e ipX ,F ( | p | ; T ) := Γ(1 − i (2 β ) − | p | ) (cid:18) Am β (cid:19) i (2 β ) − | p | J − i (2 β ) − | p | (cid:18) Ame βT β (cid:19) , and Γ( x ) and J α ( x ) denote the Gamma and Bessel functions, respectively. Themode functions have been normalized in terms of the Klein–Gordon product:( f, g ) KG := i Z ∞−∞ dX [ f ∗ ∂ T g − ( ∂ T f ∗ ) g ] , such that( f ( p ; T, X ) , f ( p ′ ; T, X )) KG = − ( f ∗ ( p ; T, X ) , f ∗ ( p ′ ; T, X )) KG = δ ( p − p ′ ) , ( f ( p ; T, X ) , f ∗ ( p ′ ; T, X )) KG = 0hold. Since the mode function f ( p ; T, X ) approximates the plane wave in theMinkowski space–time in the limit T → −∞ : f ( p ; T, X ) = (4 π | p | ) − / e − i | p | T e ipX + O ( e βT ) , we take { f ( p ; T, X ) } as the positive frequency solutions and expand the quantumKlein–Gordon field in the form φ = Z ∞−∞ dp [ a ( p ) f ( p ; T, X ) + a ∗ ( p ) f ∗ ( p ; T, X )] . The Fock space F Ω is constructed from the vacuum state | Ω i defined by the re-quirement: a ( p ) | Ω i = 0 , (for all p ∈ R ) , according to the standard procedure.The computation of the quantum Hamiltonian operator for the Klein–Gordonfield in this case leads to H ( T ) = Z dp (cid:20) σ ( | p | ; T ) a ∗ ( p ) a ( p ) + τ ( | p | ; T )2 a ( p ) a ( − p ) + τ ∗ ( | p | ; T )2 a ∗ ( p ) a ∗ ( − p ) (cid:21) , DAISUKE IDA AND MIYUKI SAITO where σ ( | p | ; T ) := | Γ(1 − i (2 β ) − | p | ) | × (cid:20) A m e βT | p | (cid:12)(cid:12)(cid:12)(cid:12) J i (2 β ) − | p |− (cid:18) Ame βT β (cid:19) − J i (2 β ) − | p | +1 (cid:18) Ame βT β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:18) | p | + A m e βT | p | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) J i (2 β ) − | p | (cid:18) Ame βT β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ,τ ( | p | ; T ) := Γ(1 − i (2 β ) − | p | ) (cid:18) Am β (cid:19) iβ − | p | × (cid:26) A m e βT | p | (cid:20) J − i (2 β ) − | p |− (cid:18) Ame βT β (cid:19) − J − i (2 β ) − | p | +1 (cid:18) Ame βT β (cid:19)(cid:21) + 12 (cid:18) | p | + A m e βT | p | (cid:19) (cid:20) J − i (2 β ) − | p | (cid:18) Ame βT β (cid:19)(cid:21) (cid:27) . The Hamiltonian can be put into the diagonal form by the transformation [13,14] a ( p ; T ) = a ( p ) cosh θ ( | p | ; T ) + a ∗ ( − p ) e iγ ( | p | ; T ) sinh θ ( | p | ; T ) , (5)where the real functions θ ( | p | ; T ) and γ ( | p | ; T ) are determined by e θ ( | p | ; T ) = s σ ( | p | ; T ) + | τ ( | p | ; T ) | σ ( | p | ; T ) − | τ ( | p | ; T ) | ,e − iγ ( | p | ; T ) = τ ( | p | ; T ) | τ ( | p | ; T ) | . Then, the Hamiltonian becomes H = Z dp ω ( | p | ; T ) a ∗ ( p ; T ) a ( p ; T ) , where ω ( | p | ; T ) := p σ ( | p | ; T ) − | τ ( | p | ; T ) | . Taking the number operators N ( p ; T ) := a ∗ ( p ; T ) a ( p ; T ) as the observables atthe time T , the evaluation of h Ω | N ( p ; T ) | Ω i gives a non–zero value. Then, it isargued that it describes the creation of scalar particles in the expanding universe,which is the approach frequently taken in the context of the quantum field theoryin the FRW universe. Here, we seek for an appropriate formulation of the quantumcosmology assuming that we can observe only a 1–particle state of the universe asa guiding principle.Since the defining equation (5, section 3) of a ( p ; T ) has the form of the Bogoli-ubov transformation, it can be also written as a ( p ; T ) = U ( T ) a ( p ) U ∗ ( T ) ,U ( T ) := exp (cid:26) Z dp θ ( | p | ; T )[ e − iγ ( | p | ; T ) a ( p ) a ( − p ) − e iγ ( | p | ; T ) a ∗ ( p ) a ∗ ( − p )] (cid:27) . The unitary operator U ( T ) is improper [15] in the sense that it is not a unitaryoperator on the Fock space F Ω . The Fock space F T ( T ∈ R ) can be built fromthe “ T –vacuum” | Ω; T i := U ( T ) | Ω i , IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 9 subject to a ( p ; T ) | Ω; T i = 0 ( p ∈ R ), by applying all polynomials of a ∗ ( p ; T )according to the standard procedure for the Fock representation. In this way, the T –vacuum | Ω; T i is regarded as the instantaneous vacuum of the Klein–Gordon fieldat the finite time T , and we have a continuum of mutually improperly equivalentFock spaces parametrized by T .Since the Hamiltonian H ( T ) is a Hermitian operator in F T , a finite particle statein F T remains in a finite particle state under the action of the unitary operatorexp( − iH ( T ) dT ) in F T . This is of course not the state in F T + dT . Instead, it isphysically interpreted as the infinite particle state in F T + dT . This implies thatthe Klein–Gordon Hamiltonian H ( T ) does not define a unitary time evolution in aseparable Hilbert space. In this sense, this theory deviates from the framework ofthe standard quantum theory.This leads to a paradoxical conclusion that an infinitely many universes arecontinuously created. While in the context of the quantum field theory in the FRWbackground, such divergence of the particle number might not be regarded as soproblematic, since its physical meaning is clearly interpretable. In fact, this kind ofdivergence comes from the infinite spatial volume of the background geometry, andthe expectation value of the appropriate number density operator remains finite.Nevertheless, in the present context of the quantum cosmology, it is hard to seethe correspondence to the classical solution to the Einstein equation with such aninterpretation, because we always observe only a single–universe state in any case.Hence, an alternative framework for the quantum dynamics is required.In order to provide a natural notion of the unitary time evolution, we need todefine a time derivative between states belonging to mutually different Fock spaces.A natural way to realize this would be provided by considering the fibre bundlestructure [16] in the continuum of the Fock spaces by π : [ T ∈ R F T =: F → R ; | ψ ; T i 7→ T, and the local trivialization of this vector bundle is supposed to be given by ϕ : R × F Ω → F ; ( T, | ψ ; Ω i )
7→ | ψ ; T i := U ( T ) | ψ ; Ω i . In this setting, the dynamics of the quantum state can be described by a globalsection ψ : R → F ; T
7→ | ψ ( T ); T i of F (See Fig. 1).In order to formulate the quantum dynamics, we have to define the time deriva-tive of the motion | ψ ( T ); T i , where a problem is that d/dT is not an anti–Hermitianoperator on the Fock space F T . To make the time derivative of the quantum statea meaningful operation, we need to introduce the notion of parallel transport of thequantum state from F T to F T + dT . Then, our vector bundle structure provides anatural framework to define the parallel transport in terms of the connection of thevector bundle. The most natural choice of the parallel transport would be given by u ( T ′ , T ) := U ( T ′ ) U ∗ ( T ) : F T → F T ′ . Figure 1.
Schematic picture of the vector bundle structure in thecontinuum of Fock spaces. The dynamics of the quantum state isgiven by the unitary transformation exp[ − iH ( T ) dt ] in F T followedby the parallel transport defined by U ( T + dT ) U ∗ ( T ).Now, we define the covariant time derivative of the quantum state with respect tothis parallel transport by D T := ddT + U ( T )( ∂ T U ∗ ( T )) , which is an anti–Hermitian operator on F T . The second term appeared in D T is, in a sense, the gauge field, which belongs to the Lie algebra of the structuregroup: the group of all unitary operators on the Fock space F Ω . In terms of thiscovariant time derivative operator, we postulate that the unitary time evolution ofthe quantum state is determined by the covariant Schr¨odinger equation iD T | ψ ( T ); T i = H ( T ) | ψ ( T ); T i . In terms of the local coordinates in F , this is equivalent to the Schr¨odinger equation i∂ T | ψ ( T ); Ω i = H Ω ( T ) | ψ ( T ); Ω i ,H Ω ( T ) : = Z dp ω ( | p | ; T ) a ∗ ( p ) a ( p ) , in F Ω . Thus, in this formulation, everything is described in the language of a singleseparable Hilbert space F Ω .Our quantum dynamics has an advantage that any 1–particle state remains in a1–particle state, which would be particularly suitable for the quantum cosmology.So, let us introduce the canonical observable set on the 1–particle Fock spaces F (1) T ,along the line of the case of the flat universe. The general form of the covariant Schr¨odinger equation is possibly given by ( iD T + A ( T )) | ψ ( T ); T i = H ( T ) | ψ ( T ); T i in terms of a gauge field A ( T ). Here we simply consider thecase: A ( T ) = 0, because an additional structure to determine A ( T ) = 0 is not equipped with thepresent settings. IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 11
Firstly, define the Hermitian operators P ( T ) := Z ∞−∞ dp pa ∗ ( p ; T ) a ( p ; T ) ,Q ( T ) := i Z ∞−∞ dp a ∗ ( p ; T ) ∂ p a ( p ; T ) , on F (1) T are regarded as the momentum and position operators, respectively. Infact, these realize the representation of the CCR–algebra[ Q ( T ) , P ( T )] − = i , on the 1–particle Fock space F (1) T .The eigenvector of P ( T ) is written as | p ; T i := a ∗ ( p ; T ) | Ω; T i , corresponding to the eigenvalue p ∈ R , and the eigenvector of Q ( T ) is formallywritten as | X ; T i Q := (2 π ) − / Z ∞−∞ dp e − ipX | p ; T i , which corresponds to the eigenvalue X ∈ R . Hence, | X ; T i Q can be regarded asthe 1–particle state localized at the position X at time T .Since | X ; T ′ i Q = u ( T ′ , T ) | X ; T i Q holds, the localized 1–particle state | X ; T ′ i Q ∈ F (1) T ′ is the parallel transport of | X ; T i Q ∈ F (1) T with the same position. It givesjustification for our choice of the connection of the vector bundle.From the completeness of the 1–particle states Z ∞−∞ dp | p ; T ih p ; T | = , in F (1) T , we can expand the 1–particle state | ψ ( T ); T i as | ψ ( T ); T i = Z ∞−∞ dp ψ ( p, T ) | p ; T i , and the coefficient ψ ( p, T ) is the wave function in the momentum representation.The Hamiltonian operator H ( T ) restricted on the 1–particle Fock space F (1) T canbe written as H (1) ( T ) = Z ∞−∞ dp ω ( | p | , T ) a ∗ ( p ; T ) a ( p ; T ) . By calculating its matrix element, we obtain the Schr¨odinger equation i∂ T ψ ( p, T ) = Z ∞−∞ dp ′ h p ; T | H (1) ( T ) | p ′ ; T i ψ ( p ′ , T )= ω ( | p | , T ) ψ ( p, T ) . (6)in the momentum representation on F (1) T . We show in section 4 that this reproducesa correct classical dynamics. Case of elliptic universe ( K = 1) : The semi–Riemannian metric on M inthis case is given by [See Eq. (2)] g C = 3 v (8 πG ) C (12 a da − a dX ) . Introducing T by a = a e βT , this becomes g C = A e βT ( dT − dX ) , where β = (12) − / , A = (cid:12)(cid:12)(cid:12)(cid:12) √ va πGC (cid:12)(cid:12)(cid:12)(cid:12) . Since the classical Hamiltonian is given by H ′ T = λ (( g C ) mn p m p n + 1) , a classical motion corresponds to a space–like geodesic in M , if we take T as thetime function.As a quantum version of this classical system, we take the tachyonic quantumKlein–Gordon field [17] described by the action S [ φ ] = 12 Z dT dX [( ∂ T φ ) − ( ∂ X φ ) + A m e βT φ ] . The mode functions { f ( p ; T, X ) , f ∗ ( p ; T, X ) } for the Klein–Gordon equation inthis case can be written as f ( p ; T, X ) := (4 π | p | ) − / F ( | p | ; T ) e ipX ,F ( | p | ; T ) := Γ(1 − i (2 β ) − | p | ) (cid:18) Am β (cid:19) i (2 β ) − | p | I − i (2 β ) − | p | (cid:18) Ame βT β (cid:19) , where I α ( x ) denotes the modified Bessel function of the first kind. Since the modefunction f ( p ; T, X ) becomes the plane wave f ( p ; T, X ) = (4 π | p | ) − / e − i | p | T e ipX + O ( e βT ) , as T → −∞ , we regard { f ( p ; T, X ); p ∈ R } as the positive frequency solutions.The subsequent procedure to obtain the first–quantized theory is parallel tothe K = − φ ( T, X ) = Z dp [ a ( p ) f ( p ; T, X ) + a ∗ ( p ) f ∗ ( p ; T, X )] , with the operators { a ( p ) , a ∗ ( p ); p ∈ R } . In the case of the tachyonic scalar fieldin the Minkowski background, it is known that the CCRs for the scalar field isnot compatible with the harmonic–oscillator commutation relations for the cre-ation and annihilation operators [18]. It however turns out that we don’t encounterthis kind of difficulties in the present background geometry. Here, the CCRs for { φ ( T, X ) , ∂ T φ ( T, X ); X ∈ R } are equivalent to the harmonic–oscillator commuta-tion relations[ a ( p ) , a ∗ ( p ′ )] − = δ ( p − p ′ ) , [ a ( p ) , a ( p ′ )] − = 0 , [ a ∗ ( p ) , a ∗ ( p ′ )] − = 0 , IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 13 for the operators { a ( p ) , a ∗ ( p ); p ∈ R } . The vacuum state is defined by a ( p ) | Ω i = 0( p ∈ R ) and we obtain the Fock representation on F Ω in the standard procedure.Then, the Hamiltonian operator becomes H ( T ) = Z dp h ( p ; T ) , where h ( p ; T ) := σ ( | p | ; T ) a ∗ ( p ) a ( p ) + τ ( | p | ; T )2 a ( p ) a ( − p ) + τ ∗ ( | p | ; T )2 a ∗ ( p ) a ∗ ( − p ) ,σ ( | p | ; T ) := | Γ(1 − i (2 β ) − | p | ) | × (cid:20) A m e βT | p | (cid:12)(cid:12)(cid:12)(cid:12) I i (2 β ) − | p |− (cid:18) Ame βT β (cid:19) + I i (2 β ) − | p | +1 (cid:18) Ame βT β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:18) | p | − A m e βT | p | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) I i (2 β ) − | p | (cid:18) Ame βT β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ,τ ( | p | ; T ) := Γ(1 − i (2 β ) − | p | ) (cid:18) Am β (cid:19) iβ − | p | × (cid:26) A m e βT | p | (cid:20) I − i (2 β ) − | p |− (cid:18) Ame βT β (cid:19) + I − i (2 β ) − | p | +1 (cid:18) Ame βT β (cid:19)(cid:21) + 12 (cid:18) | p | − A m e βT | p | (cid:19) (cid:20) I − i (2 β ) − | p | (cid:18) Ame βT β (cid:19)(cid:21) (cid:27) . The Hamiltonian at the time T can be made in a diagonalized form in the integra-tion range | p | > p ( T ), where the function p ( T ) is the smallest positive solution ofthe equation σ ( p ( T ); T ) = | τ ( p ( T ); T ) | , that is given by p ( T ) = Ame βT . Keeping this in mind, we separate the Hamiltonian into two parts as H ( T ) = Z | p | >ξ + Z | p |≤ ξ ! dp h ( p ; T ) , in terms of an arbitrarily fixed positive parameter ξ . The first part of the Hamil-tonian can be diagonalized via the Bogoliubov transformation a ( p ; T ) = U ( T ) a ( p ) U ∗ ( T ) ,U ( T ) := exp (cid:26) Z | p | >ξ dp θ ( | p | ; T )[ e − iγ ( | p | ; T ) a ( p ) a ( − p ) − e iγ ( | p | ; T ) a ∗ ( p ) a ∗ ( − p )] (cid:27) ,e θ ( | p | ; T ) := s σ ( | p | ; T ) + | τ ( | p | ; T ) | σ ( | p | ; T ) − | τ ( | p | ; T ) | ,e − iγ ( | p | ; T ) := τ ( | p | ; T ) | τ ( | p | ; T ) | . The above expressions are valid for the range T ∈ ( −∞ , T ), where T is determinedby ξ = p ( T ), i.e. T = (2 β ) − log( ξ/Am ).In this way, we have a partly diagonalized form of the Hamiltonian H ( T ) = Z | p | >ξ dp ω ( | p | ; T ) a ∗ ( p ; T ) a ( p ; T ) + Z | p |≤ ξ dp h ( p ; T ) ,ω ( | p | ; T ) := p σ ( | p | ; T ) − | τ ( | p | ; T ) | . where ω ( | p | ; T ) is a real function, hence the first diagonal part is a Hermitianoperator on F T for T < T .Accordingly, as shown in the following, we can at best obtain the unitary theoryof the 1–particle states within the time duration T ∈ ( −∞ , T ) for K = 1 uni-verse. This would correspond to the fact that each classical solution describes arecollapsing universe, and hence the scale factor a = a e βT always has an upperbound.To describe this system, we regard the Fock space F Ω as the tensor productspace A ⊗ B of the subsystems A and B , where A (resp. B ) is the Fock spacewith respect to the creation and annihilation operators { a ( p ) , a ∗ ( p ); | p | > ξ } (resp. { a ( q ) , a ∗ ( q ); | q | ≤ ξ } ). The T –vacuum is defined by a ( p ; T ) | Ω; T i = 0 , for | p | > ξa ( q ) | Ω; T i = 0 , for | q | ≤ ξ and the Fock space F T is constructed with respect to this vacuum by applying poly-nomials of { a ∗ ( p ; T ) , a ∗ ( q ); | p | > ξ, | q | ≤ ξ } . The Fock space F T can be regardedas the tensor product of a pair of Fock spaces A T and B , A T (resp. B ) being theFock representation space of the algebra generated by { a ( p ; T ) , a ∗ ( p ; T ); | p | > ξ } (resp. { a ( q ) , a ∗ ( q ); | q | ≤ ξ } ). In other words, F T is spanned by the vectors in theform | n p , n p , · · · , n p r ; T i ⊗ | n q , n q , · · · , n q s i , where | n p , n p , · · · , n p r ; T i ∈ A T (resp. | n q , n q , · · · , n q s i ∈ B ) denotes the si-multaneous eigenvector of the number operators a ∗ ( p ; T ) a ( p ; T ) ( | p | > ξ ) [resp. a ∗ ( q ) a ( q ) ( | q | ≤ ξ )]. Accordingly, the Hamiltonian H ( T ) is regarded as the sum H ( T ) = H ξ ( T ) ⊗ + ⊗ Z | p |≤ ξ dp h ( p ; T ) ,H ξ ( T ) := Z | p | >ξ dp ω ( | p | ; T ) a ∗ ( p ; T ) a ( p ; T ) , of the operator on A T and the operator on B . Hence, for a vector state | ψ i ⊗ | ψ ′ i ∈ A T ⊗ B , each vector | ψ i (resp. | ψ ′ i ) independently undergoes the unitary timeevolution in the inseparable Fock space F ξ := S T 1, the Hilbert spaces A T and A T ′ are improperly unitaryequivalent, when T = T ′ . In a similar procedure to the K = − π : [ T 7→ | ψ ; T i := U ( T ) | ψ ; Ω i . Our Hilbert space describing the 1–particle state of the universe at the time T isassumed to be the subspace A (1) T ⊗ B , where A (1) T denotes the space of 1–particlestates in A T . Each such state is projected onto the space of quantum state on A T via the partial trace over the subsystem B , which may produce a mixed state on A T . Since the structure of the Hamiltonian ensures that the operation of the partialtrace and the unitary time evolution by the Hamiltonian commute, the quantumdynamics is reduced to that of a separable state, which can be represented as avector state consisting of a single term | ψ ; T i ⊗ | ψ ′ i . Such a separable state simplycorresponds to the vector state | ψ ; T i ∈ A T , and its time evolution is assumed todescribed by the Schr¨odinger equation iD T | ψ ( T ); T i = H ξ ( T ) | ψ ( T ); T i ,D T := ddT + U ( T )( ∂ T U ( T ) ∗ ) , for T ∈ ( −∞ , T ).The canonical observable set { Q ( T ) , P ( T ) } on A (1) T are also constructed in asimilar manner. Firstly, we define the momentum operator by P ( T ) := Z | p | >ξ dp p a ∗ ( p ; T ) a ( p ; T ) , which is a Hermitian operator on A (1) T . The eigenstate of P ( T ) is given by | p ; T i := a ∗ ( p ; T ) | Ω A ; T i , which satisfies P ( T ) | p ; T i = p | p ; T i , where | Ω A ; T i denotes the vacuum state in A T . Next, the position operator isdefined by Q ( T ) := i Z | p | >ξ dp a ∗ ( p ; T ) ∂ p a ( p ; T ) , which is a Hermitian operator on A (1) T . These constitute the CCR–algebra[ Q ( T ) , P ( T )] − = i , where we abbreviate the identity operator on A (1) T as := Z | p | >ξ dp a ∗ ( p ; T ) a ( p ; T ) . The eigenvector of Q ( T ) can be written as | X ; T i Q := (2 π ) − / Z | p | >ξ dp e − ipX | p ; T i , for X ∈ R , and it satisfies Q ( T ) | X ; T i Q = X | X ; T i Q . Unlike the K = − {| X ; T i Q ; X ∈ R } does not satisfy the orthogonalitycondition, i.e. it holds Q h X ; T | X ′ ; T i Q = δ ( X − X ′ ) , in the present case. However, since for the operator F X ( T ) := | X ; T i QQ h X ; T | , ithold h ψ ; T | F X ( T ) | ψ ; T i > , for all | ψ ; T i ∈ A (1) T Z ∞−∞ dXF X ( T ) = , the measurement of the position operator Q ( T ) = Z dX XF X ( T )can be regarded as a POVM measurement, though not a von Neumann measure-ment.In the momentum representation on the 1–particle Fock space A (1) T , the 1–particle state is described by the wave function ψ ( p, T ) := h p ; T | ψ ( T ); T i , and it issubject to the Schr¨odinger equation i∂ T ψ ( p, T ) = ω ( | p | , T ) ψ ( p, T ) . We see in the next section that the quantum system described here reproduces thecorrect classical theory.4. Classical–quantum correspondence in the first–quantized theory Here, we show that the first–quantized theory obtained in the previous sectioncorrectly reproduces the classical Einstein gravity.In the previous section, we have obtained a quantum theory of the expand-ing universe described by the wave function ψ ( p, T ) of the universe subject to theSchr¨odinger equation The expression of ω ( | p | ; T ) includes the Bessel functions when K = ± 1, which makes it difficult to solve the Schr¨odinger equation strictly. Nev-ertheless we can solve it approximately in terms of asymptotic forms of the Besselfunctions. The Bessel function and the modified Bessel function in the mode func-tions are expanded around their argument Ame βT = 0 as J − i (2 β ) − | p | (cid:18) Ame βT β (cid:19) = ∞ X n =0 ( − n n !Γ( n − i (2 β ) − | p | + 1) (cid:18) Ame βT β (cid:19) n − i (2 β ) − | p | ,I − i (2 β ) − | p | (cid:18) Ame βT β (cid:19) = ∞ X n =0 n !Γ( n − i (2 β ) − | p | + 1) (cid:18) Ame βT β (cid:19) n − i (2 β ) − | p | . From these, it can be seen that the series expansion of ω ( | p | ; T ) becomes ω ( | p | ; T ) = p p − KA m e βT + K p − p + 18 p )32 | p | (1 + 3 p ) (4 + 3 p ) A m e βT + O ( e βT ) . For K = − 1, this expression is valid for p = 0, and for K = 1, it is valid for | p | > Ame βT . Thus, our Hamiltonian operator ω ( | p | , T ) well approximates the IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 17 naive Hamiltonian H c := p p − KA m e βT , for e βT ≪ ( Am ) − . The classical counterpart of this naive Hamiltonian exactlygives the equation of motion for the classical trajectory.This correspondence between our Hamiltonian ω ( | p | , T ) and the naive Hamil-tonian H c holds wider range of T . To see this, we next consider the late timebehavior of ω ( | p | ; T ). For e βT ≫ ( Am ) − and p = 0, the Bessel function in themode function behaves as J − i (2 β ) − | p | (cid:18) Ame βT β (cid:19) = r βπAm e − βT hn − p )(3 + 4 p )128 A m e − βT o × cos (cid:18) Am β e βT − π iπ | p | β (cid:19) + β (1 + 12 p )4 Am e − βT sin (cid:18) Am β e βT − π iπ | p | β (cid:19) i + O ( e − βT ) , which yields the behavior of ω ( | p | ; T ) for K = − ω ( | p | ; T ) = Ame βT + 12 e − βT Am sinh ( π | p | / (2 β )) h (1 − p )12+ 5 + 48 p n cos (cid:18) Amβ e βT − π (cid:19) cosh π | p | β − o + 2 p cos (cid:18) Amβ e βT − π (cid:19) cosh π | p | β i + O ( e − βT ) . On the other hand, the naive Hamiltonian H c for K = − e βT ≫ ( Am ) − as H c = p p + A m e βT = Ame βT + p A m e − βT + O ( e − βT ) . This shows ω ( | p | ; T ) approximates H c in the limit T → ∞ , ω ( | p | ; T ) − H c H c = O (cid:0) e − βT (cid:1) , for p = 0.For the intermediate range of T , i.e. for Ame βT = O (1), we can see the goodnumerical coincidence between ω ( | p | , T ) and H c . A similar argument also holds forthe closeness between ∂ p ω ( | p | , T ) and ∂ p H c . Thus, we conclude that ω ( | p | ; T ) ≈ p p − KA m e βT ,∂ p ω ( | p | ; T ) ≈ p p p − KA m e βT , holds for K = 0 , ± 1, and for all ranges of T in consideration.Next, we show that the approximate Schr¨odinger equation i∂ T ψ ( p ; T ) = p p − KA m e βT ψ ( p ; T )(7)applied to the wave packet state, reproduces the geodesic motion in the mini–superspace, which corresponds to the classical solution to the Einstein equation. Firstly, we note that the Ehlenfest–type theorem can be applied in the presentcase. The expectation values of the momentum and position operators for thenormalized state ψ ( p, T ) are given by h P i = Z dp pψ ∗ ( p, T ) ψ ( p, T ) , h Q i = i Z dp ψ ∗ ( p, T ) ∂ p ψ ( p, T ) , where the range of integration is p ∈ R for K = 0 , − 1, and | p | > ξ for K = 1. Thetime derivative of these expectation values are readily obtained as ddT h P i = 0 ,ddT h Q i = (cid:28) ∂H c ∂p (cid:29) := Z dp ∂H c ∂p ψ ∗ ( p, T ) ψ ( p, T ) . This shows formal, though not strict, correspondence between quantum and clas-sical dynamics.Next, we consider the dynamics of the wave packet state. The solution of Eq. (7)can be written as ψ k ( p, T ) = exp − i Z T ds p k − KA m e βs ! δ ( p − k ) , where the solutions are labeled by the simultaneous eigenvalue k of the momentumoperator subject to k ∈ R for K = 0 , − 1, and | k | > ξ for K = 1. The generalsolution is written as ψ ( p, T ) = Z dk c ( k ) ψ k ( p, T ) , in terms of a coefficient c ( k ) subject to the normalization condition Z dk | c ( k ) | = 1 . Now, we consider the position representation of the wave function. This is given by e ψ ( X, T ) := Q h X ; T | ψ ( T ); T i = (2 π ) − / Z dp e ipX ψ ( p, T ) , so that it is generally written as e ψ ( X, T ) = (2 π ) − / Z dk c ( k ) exp − i Z T ds p k − KA m e βs ! e ikX . (8)The Born rule here could be stated as that | e ψ ( X, T ) | gives the probability dis-tribution that a measurement of the position operator at time T yields the value X ∈ R .A wave packet state corresponding to a geodesic particle in mini–superspace isobtained if we take c ( k ) as a bell curve, say a Gaussian–like function, centered at k = k = 0, with the appropriately broad width σ − . Then, e ψ ( X, T ) describes awave packet with the width σ . IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 19 From Eq. (8), we can read off the time dependence of the dispersion relation ω ′ ( k, T ) = p k − KA m e βT , between the wave–number k and the instantaneous angular frequency ω ′ . Then,we readily find that the group velocity v q ( k , T ) of a wave packet with the centralwave–number k = 0 is given by v q ( k , T ) = ∂ω ′ ( k, T ) ∂k (cid:12)(cid:12)(cid:12)(cid:12) k = k = k p k − KA m e βT . The corresponding quantity can be obtained from the classical theory. From theHamiltonian constraint p T − p X + KA e βT ≈ , we get the coordinate velocity of the geodesic particle as v c = dX/dsdT /ds = − p X p T ≈ p X p p X − KA e βT . This shows good agreement between the classical and quantum predictions if weidentify the dimensionless momentum mp X of the geodesic motion with the centralwave–number k of the wave packet state.5. Concluding remarks The quantum mechanical model of the cosmology studied here is conceptuallysimilar to the third–quantization model, but technically different in that we re-spect the specific semi–Riemannian structure ( M , g C ) on the mini–superspace M ,where the classical solution to the Einstein equation is given by the geodesic mo-tion. In the case of the Einstein gravity coupled with a massless scalar field inthe FRW background, the mini–superspace as a semi–Riemannian manifold be-comes a two–dimensional expanding universe. Then, the classical solution to theEinstein equation is given by a null (resp. time–like, space–like) geodesic in themini–superspace for the flat (resp. hyperbolic, elliptic) FRW space–time. Hence,we consider the quantum Klein–Gordon field in the mini–superspace, which be-comes massless (resp. massive, tachyonic) for the flat (resp. hyperbolic, elliptic)universe, as a quantized model of the geodesic motion.In the case of the massive or tachyonic field, we are faced with a known prob-lem in the quantum field theory in a dynamical space–time that an inseparableHilbert space is required to describe the quantum states of the field, where theKlein–Gordon Hamiltonian at different time belongs to a different Fock space. Thiscontinuum of the Fock spaces F T parametrized by T ∈ R constitutes the insepara-ble Hilbert space, which is too big. By introducing the vector bundle structure inthe continuum of the Fock space and the connection of the vector bundle, we definethe covariant time derivative for the quantum states, in which the Klein–GordonHamiltonian can be regarded as a Hermitian operator in a separable Hilbert space.Furthermore, this framework gives a unitary time evolution of 1–particle states,which would be preferable in the context of the quantum cosmology. However, inthe case of elliptic universe, the momentum space for the Hilbert space of 1–particlestates has to be restricted. Then, we obtain the unitary theory for a restricted range of the time parameter T ∈ ( −∞ , T ). This time parameter corresponds to the scalefactor of the elliptic universe, which represents the recollapsing universe in the clas-sical theory, so that this limitation of the time parameter would correspond to theexistence of the upper bound for the scale factor. It would not imply that the closeduniverse does not recollapse, but rather it should be taken as an indication of thelimit of applicability of our model. This might be a common issue of the quantumcosmological model based on the minisuperspace, where the time function is givenby the scale factor of the universe.We construct the Hilbert space for the 1–particle states and the canonical ob-servable set constituting the CCR–algebra on the space of 1–particle states. Ac-cordingly, we obtain the Schr¨odinger equation for the wave function, which is theSchr¨odinger representation of a 1–particle state. The present quantization schemeis free from the operator ordering ambiguities and the problem of time unlike theWheeler–DeWitt quantization scheme. We find that the Hamiltonian is close tothe naive Hamiltonian predicted from the Klein–Gordon equation but with a smallcorrection in the non–flat background cases. We see that this ensures that ourquantum theory correctly reproduces the Einstein equation.A possible advantage of the present formalism over other third–quantized modelsis that it is applicable to the fermionic field by considering the quantum Dirac fieldon ( M , g C ). Another good point is that we can treat the dynamics of the single–universe state, while in typical third–quantization models [19–24], we always sufferfrom creation of infinite number of universes [25–27]. However, we have to introducea mass scale of the quantum field, which is a disadvantage of the theory. In fact,we have left an unknown dimensionless parameter Am in our formulation, whichshould be determined by a more fundamental theory. Acknowledgments We would like to thank Takahiro Okamoto for useful discussions. References [1] P. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva Univ., New York, 1964).[2] P. A. M. Dirac, “The Theory of Gravitation in Hamiltonian Form” Proc. R. Soc. Lond. A (1958) 333.[3] R. Arnowitt, S. Deser and C. W. Misner, “The Dynamics of General Relativity,” in Gravita-tion: an Introduction to Current Research , ed. L. Witten (Wiley, New York, 1962).[4] B. S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory” Phys. Rev. (1967)1113.[5] J. A. Wheeler, “Superspace and the nature of quantum geometrodynamics” in Battelle Ren-contres , ed. C. M. DeWitt and J. A. Wheeler (Benjamin, New York, 1968).[6] C. Rovelli, “Time in quantum gravity: An hypothesis” Phys. Rev. D (1991) 442.[7] K. V. Kuchaˇr, “Time and Interpretations of Quantum Gravity” in Proceedings of the FourthCanadian Conference on General Relativity and Relativistic Astrophysics ed. G. Kunstatter,D. E. Vincent and J. G. Williams (World Scientific, Singapore, 1992).[8] C. J. Isham, “Canonical Quantum Gravity and the Problem of Time” in Integrable Systems,Quantum Groups, and Quantum Field Theories ed. L. A. Ibort and M. A. Rodriguez (Kluwer,Dordrecht, 1993).[9] B. S. DeWitt, “Spacetime as a sheaf of geodesics in superspace” in Relativity , ed. M. Carmeli,S. I. Fickler and L. Witten (Plenum Press, New York–London, 1970).[10] C. W. Misner, “Minisuperspace”, in Magic Without Magic , ed. J. R. Klauder (Freeman, SanFrancisco, 1972).[11] T. D. Newton and E. P. Wigner, “Localized states for elementary systems” Rev. Mod. Phys. (1949) 400. IRST–QUANTIZED THEORY OF EXPANDING UNIVERSE 21 [12] S. W. Hawking and D. N. Page, “Operator ordering and the flatness of the universe” Nucl.Phys. B264 (1986) 185.[13] L. Parker, “Quantized Fields and Particle Creation in Expanding Universes. I” Phys. Rev. (1969) 1057.[14] T. S. Bunchi, S. M. Christensen and S. A. Fulling, “Massive quantum field theory in two-dimensional Robertson-Walker space-time” Phys. Rev. D (1978) 4435.[15] P. Roman, Introduction to Quantum Field Theory (Wiley, New York, 1969).[16] N. Steenrod, The Topology of Fibre Bundles (Princeton Univ. Press, Princeton New Jersey,1951).[17] S. Tanaka, “Theory of Matter with Super Light Velocity” Prog. Theor. Phys. (1960) 171.[18] G. Feinberg, “Possibility of Faster-Than-Light Particles” Phys. Rev. (1967) 1089.[19] M. McGuigan, “Third quantization and the Wheeler–DeWitt equation” Phys. Rev. D (1988) 3031.[20] T. Banks, “Prolegomena to a theory of bifurcating universes: a nonlocal solution to thecosmological constant problem or little lambda goes back to the future” Nucl. Phys. B309 (1988) 493.[21] S. B. Giddings and A. Strominger, “Baby universes, third quantization and the cosmologicalconstant” Nucl. Phys. B (1989) 481.[22] R. M. Wald, “Proposal for solving the “problem of time” in canonical quantum gravity” Phys.Rev. D (1993) R2377.[23] A. Higuchi and R. M. Wald, “Application of a new proposal for solving the “problem of time”to some simple quantum cosmological models” Phys. Rev. D (1995) 544.[24] S. J. Robles–P´erez, “Third quantization: modeling the universe as a ‘particle’ in a quantumfield theory of the minisuperspace” J. Phys.: Conf. Ser. (2012) 012133.[25] V. A. Rubakov, “On third quantization and the cosmological constant” Phys. Lett. B (1988) 503.[26] XA. Hosoya and M. Morikawa, “Quantum field theory of the Universe” Phys. Rev. D (1989) 1123.[27] A. Vilenkin, “Approaches to quantum cosmology” Phys. Rev. D50