Quantum Gowdy T 3 Model: Schrodinger Representation with Unitary Dynamics
Alejandro Corichi, Jeronimo Cortez, Guillermo A. Mena Marugan, Jose M. Velhinho
aa r X i v : . [ g r- q c ] N ov Quantum Gowdy T Model: Schr¨odinger Representation withUnitary Dynamics
Alejandro Corichi,
1, 2, ∗ Jer´onimo Cortez, † GuillermoA. Mena Marug´an, ‡ and Jos´e M. Velhinho § Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico,UNAM-Campus Morelia, A. Postal 61-3,Morelia, Michoac´an 58090, Mexico. Institute for Gravitation and the Cosmos,Pennsylvania State University, University Park PA 16802, USA. Departamento de F´ısica, Facultad de Ciencias,Universidad Nacional Aut´onoma de M´exico,A. Postal 50-542, M´exico D.F. 04510, Mexico. Instituto de Estructura de la Materia,CSIC, Serrano 121, 28006 Madrid, Spain. Departamento de F´ısica, Universidade da Beira Interior,R. Marquˆes D’ ´Avila e Bolama, 6201-001 Covilh˜a, Portugal.
Abstract
The linearly polarized Gowdy T model is paradigmatic for studying technical and conceptualissues in the quest for a quantum theory of gravity since, after a suitable and almost completegauge fixing, it becomes an exactly soluble midisuperspace model. Recently, a new quantizationof the model, possessing desired features such as a unitary implementation of the gauge group andof the time evolution, has been put forward and proven to be essentially unique. An appropriatesetting for making contact with other approaches to canonical quantum gravity is provided by theSchr¨odinger representation, where states are functionals on the configuration space of the theory.Here we construct this functional description, analyze the time evolution in this context and showthat it is also unitary when restricted to physical states, i.e. states which are solutions to theremaining constraint of the theory. PACS numbers: 04.62.+v, 04.60.Ds, 98.80.Qc ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] . INTRODUCTION In the quest for a quantum theory of gravity, the use of simple models has proven to bevery effective. The simplest possible models, where the most symmetries are imposed fromthe outset [1], have become important for the study of Planck scale modifications to the BigBang scenario (see, e.g., Ref. [2]). However, these models suffer from an oversimplificationsince all inhomogeneous degrees of freedom are neglected. A natural question is how theinclusion of these inhomogeneous modes affects the qualitative picture near the singularitythat the homogeneous models possess. In this regard, the linearly polarized Gowdy T modelis a natural candidate for a detailed study. It is the simplest inhomogeneous, spatially closed,cosmological model in vacuo [3]. One important reason for the appeal of such a model isthat, after a convenient almost complete gauge fixing and the introduction of a geometricallymotivated internal time, the model becomes soluble. Any solution of the full set of Einsteinequations can be obtained from the solutions of an auxiliary scalar field in a fixed fiducialbackground. However, this auxiliary scalar field system is not unique. Different “fieldparametrizations” of the metric may give rise to different scalar field systems. Classicallythey are all equivalent, but in the quantum theory this may not be so. In addition, thequantization of field systems possesses an infinite degree of ambiguity, even if one restrictsall considerations to standard quantizations, e.g., of the Fock type. As a consequence, thereexist in principle infinitely many inequivalent quantizations of the Gowdy T midisuperspacemodel.Among the different possibilities available in this route to quantization, two fieldparametrizations have received special attention in past years. One of them can be con-sidered a somewhat conventional field parametrization from the viewpoint of a dimensionalreduction of the model [4]. However, this proposal for the choice of fundamental field has theundesirable property of not implementing the dynamics (generated by the internal notion oftime) unitarily. Actually, although this lack of unitarity was first proven [5] for a “naturalquantization” of the associated scalar field, introduced by Pierri [4], it has been recentlyshown that there exists no Fock quantization with a unitary dynamics, at least if one alsodemands an invariant unitary implementation of the gauge group that remains on the modelafter gauge fixing [6]. To solve this problem, a new field parametrization, together with anessentially unique quantum representation, was recently introduced. In this case, not onlythe evolution is unitary and the gauge group is naturally implemented, but it has been shownthat any other Fock quantization of the new field with such properties is unitarily equivalentto the constructed one [6–9]. Furthermore, the adopted field parametrization turns out tobe unique in a precise sense under the condition of the existence of a Fock representation(FR) with an invariant unitary action of the gauge group and a unitary dynamics [6]. Theseresults were mainly formulated in the language of Fock space, which is natural from theperspective of a scalar field in a fixed background.On the other hand, quantum gravity in its canonical formulation is commonly defined inthe Schr¨odinger functional picture, where states are functionals on the configuration spaceof the theory. Therefore, it is important to have a Schr¨odinger functional description of anysymmetry reduced model, such as the Gowdy T model. The purpose of this paper is topresent this description for the quantization which admits a unitary time evolution [7, 8],and analyze the implementation of such a unitary evolution in this framework, both beforeand after imposing the remaining constraint of the theory.We will adopt here the same viewpoint as in Refs. [7, 8]: instead of working with a fixed2uantum representation and considering the unitary implementability of the family of sym-plectic transformations defined by the evolution (together with the corresponding unitaryevolution operator), we will construct the associated 1-parameter family of representations.Notice that this is precisely the family of representations which is obtained by “evolvingin time” a fixed GNS state, and hence the complex structure defining the FR. The equiv-alence between the two viewpoints is then established by the fact that evolution betweenany two given times admits a unitary implementation if, and only if, the correspondingrepresentations are unitarily equivalent.Finally, we note that the 1-parameter family of complex structures that gives rise tothe 1-parameter family of unitarily equivalent representations can be obtained both on thecanonical phase space (the space of Cauchy data for the auxiliary scalar field) or on thecovariant phase space (the space of solutions). Since we are interested in the canonical func-tional description, we will obtain the family of complex structures directly on the canonicalphase space. As a particular consequence of unitarity, we will obtain a family of mutuallyequivalent Gaussian measures in the (quantum) configuration space.The structure of the paper is the following. In Sec. II we recall the quantization ofthe linearly polarized Gowdy T model constructed by Corichi, Cortez and Mena Marug´an,in which the time evolution is implemented unitarily [7, 8]. In Sec. III we construct theSchr¨odinger representation (SR) corresponding to this (unique) Fock quantization. In Sec.IV, we implement the canonical notion of time evolution within the Schr¨odinger description,showing explicitly the equivalence of the family of representations at different times. Theconclusions are presented in Sec. V. II. THE QUANTUM GOWDY MODEL
In this section we will review the quantization of the Gowdy T cosmological model asperformed in Refs. [7, 8]. We will start with a description of the classical model and itsdynamics. A. The classical model
The linearly polarized Gowdy T model describes globally hyperbolic four-dimensionalvacuum spacetimes, with two commuting hypersurface orthogonal spacelike Killing fieldsand compact spacelike hypersurfaces homeomorphic to a three-torus. In a coordinate system { ( t, θ, ν, δ ) , t ∈ R + ; θ, ν, δ ∈ S } with ( ∂ ν ) A and ( ∂ δ ) A being the hypersurface orthogonalKilling fields, the line element can be expressed asd s = e γ − ( ξ/ √ pt ) − ξ / (4 pt ) (cid:0) − d t + d θ (cid:1) + e − ξ/ √ pt t p d ν + e ξ/ √ pt d δ (1)after a gauge fixing procedure which removes all the gauge degrees of freedom except for ahomogeneous one [8]. The spatially homogeneous variable p is a positive constant of motion.On the other hand, the fields ξ and γ depend only on the time coordinate t and the spatialcoordinate θ . The field γ is completely determined by ξ , by P := ln p and by their respectivemomentum and configuration (canonically) conjugate variables, P ξ and Q (see Ref. [6] fordetails). Therefore, all local degrees of freedom reside in the field ξ .3s we have mentioned, the model is just partially gauge fixed: there is still a globalconstraint, C = 1 √ π I d θP ξ ξ ′ = 0 , (2)which comes from the homogeneous part of the θ -momentum constraint. Here, the primedenotes the derivative with respect to θ .After the reduction process, the Hamiltonian becomes H = 12 I d θ (cid:18) P ξ + ( ξ ′ ) + 14 t ξ (cid:19) . (3)Note that, since the reduced Hamiltonian does not depend on the degrees of freedom Q and P , these are constants of motion, and will be obviated in our subsequent discussion.Thus, the resulting system consists of a real scalar field ξ subject to the constraint (2).Its Hamiltonian (3) is that of a massless field with a quadratic time dependent potential V ( ξ ) = ξ / (4 t ) propagating in a (fictitious) background ( M ( f ) , g AB ), where M ( f ) ≃ S × R + and g AB = − (d t ) A (d t ) B + (d θ ) A (d θ ) B .We will now describe the (linear) dynamics of this field system, starting with the covariantdescription. The reduced Hamiltonian (3) leads to the field equations˙ ξ = P ξ , ˙ P ξ = ξ ′′ − ξ t , (4)where the dot denotes the derivative with respect to t . Hence, the field ξ satisfies the secondorder differential equation ¨ ξ − ξ ′′ + ξ t = 0 . (5)Since the general solution is most conveniently expressed in Fourier series, let us introducethe notation e k := e − ikθ √ π ∀ k ∈ Z . (6)With respect to some reference (“initial”) time t = t , all smooth solutions can then bewritten as [8] ξ ( t, θ ) = √ t [ q + p ln( t )] + X k ∈ Z −{ } h b k ( t ) G ( t ) k ( t, θ ) + b ∗ k ( t ) G ( t ) ∗ k ( t, θ ) i , (7)where we have singled out the homogeneous mode k = 0 and used the symbol ∗ to representcomplex conjugation. The constants b k ( t ) are complex coefficients, q and p are canonicallyconjugate variables and the mode solutions G ( t ) k ( t, θ ) are given by G ( t ) k ( t, θ ) = r πt c ∗ ( | k | t ) H ( | k | t ) − d ∗ ( | k | t ) H ∗ ( | k | t )] e ∗ k , (8)where d ( x ) = r πx (cid:20)(cid:18) i x (cid:19) H ∗ ( x ) − iH ∗ ( x ) (cid:21) , c ( x ) = r πx H ( x ) − d ∗ ( x ) , (9) We set 4
G/π = c = 1, G and c being Newton’s constant and the speed of light, respectively. H n ( n = 1 ,
2) is the n -th order Hankel function of the second kind [10]. Note that themode solutions satisfy G ( t ) k ( t, θ ) | t = t = e ∗ k p | k | , ∂ t G ( t ) k ( t, θ ) | t = t = − i r | k | e ∗ k . (10)We will refer to the linear space of solutions (7), equipped with the symplectic structureΩ ( ξ , ξ ) = H d θ ( ξ ∂ t ξ − ξ ∂ t ξ ), as the covariant phase space S .Alternatively, instead of S , we can consider the canonical phase space. This is the linearspace Γ coordinatized by the canonical pair which is formed by the configuration ϕ and themomentum P ϕ of the field ξ on a given section of constant time. We take this time to besome fixed reference time t . As we have seen above, the section of constant time t = t can be identified with the compact space S . In the following, we will refer to this sectionas the reference Cauchy surface (RCS). Let us also point out that, via Eq. (10), one canunderstand the way in which the solutions (7) are expressed as being specially adapted tothe choice of RCS, or vice-versa (given the RCS, such an adapted expression of the solutionsobviously simplifies the explicit form of the map between Γ and S ). On the other hand, thesymplectic structure on Γ is, of course, σ [( ϕ , P ϕ ) , ( ϕ , P ϕ )] = I d θ ( P ϕ ϕ − P ϕ ϕ ) . (11)The evolution generated by the Hamiltonian (3) in the canonical phase space gives riseto a 1-parameter family of symplectic linear transformations τ ( t f ,t ) : Γ → Γ (with t fixed)as follows. An initial state ( ϕ, P ϕ ) at t = t determines a solution ξ ∈ S , which in turndetermines a canonical pair of fields ( ξ | t = t f , ∂ t ξ | t = t f ) for any value of t f . This pair is thennaturally interpreted as new initial data at t = t . More rigorously, we have a natural1-parameter family of embeddings E t : S → M ( f ) , together with a 1-parameter familyof isomorphisms I E t , mapping Cauchy data at E t ( S ) into solutions. Then, the classicalevolution operator is τ ( t f ,t ) = (cid:0) E ∗ t (cid:1) − E ∗ t f I − E tf I E t , (12)with E ∗ t denoting the pull-back of the map E t . In this work we mostly ignore the distinctionbetween S and our RCS, E t ( S ), so that E t is trivialized. In addition, note that thecanonical evolution maps provide the transformations I E t τ ( t f ,t ) I − E t : S → S in the covariantphase space (this notion of time evolution in the covariant description was employed in Ref.[11]).In order to present the evolution maps in explicit form, it is convenient to use the Fouriercomponents of the field ϕ and its momentum. We then define ϕ k := I d θ ϕe k , P kϕ := I d θ P ϕ e ∗ k . (13)It is clear from the form of the Hamiltonian (3) that modes with different values of | k | decouple. Furthermore, from now on we will concentrate ourselves on the infinite set ofinhomogeneous modes k = 0, since no relevant aspect of our discussion depends on thesingle zero mode (being single and decoupled, the quantum treatment of this mode can bemade independently by standard methods, and included in the final description by meansof a tensor product). 5mploying Eq. (10), one can check that the Fourier coefficients ϕ k and P kϕ are related tothose appearing in expression (7) by b k ( t ) = 1 p | k | (cid:0) | k | ϕ k + iP − kϕ (cid:1) , b ∗− k ( t ) = 1 p | k | (cid:0) | k | ϕ k − iP − kϕ (cid:1) . (14)We will adopt this convenient set of (complex) variables as alternative coordinates in Γ. Inthe following, to simplify the notation, we will let b k and b ∗− k denote the variables b k ( t )and b ∗− k ( t ), respectively, and collect them in the set of pairs { ( b k , b ∗− k ) } with k ∈ Z − { } . It is then straightforward to check that each of the considered pairs of variables decouplesin the evolution, so that the evolution transformations are 2 × τ ( t f ,t ) maps ( b k , b ∗− k ) to a new pair (cid:0) b k ( t f ) , b ∗− k ( t f ) (cid:1) [seenas new data at t = t , related to the new configuration and momentum of the field as in Eq.(14)], such that b k ( t f ) = α k ( t f , t ) b k + β k ( t f , t ) b ∗− k ,b ∗− k ( t f ) = β ∗ k ( t f , t ) b k + α ∗ k ( t f , t ) b ∗− k , (15)where α k ( t f , t ) = c ( | k | t f ) c ∗ ( | k | t ) − d ( | k | t f ) d ∗ ( | k | t ) ,β k ( t f , t ) = d ( | k | t f ) c ( | k | t ) − c ( | k | t f ) d ( | k | t ) . (16)Note that the functions c and d , given in Eqs. (9), satisfy | c | − | d | = 1, and we thus havethat | α k ( t f , t ) | − | β k ( t f , t ) | = 1 for all t f > t > α − k ( t f , t ) = α k ( t f , t ) , β − k ( t f , t ) = β k ( t f , t ) ,α k ( t , t f ) = α ∗ k ( t f , t ) , β k ( t , t f ) = − β k ( t f , t ) . (17) B. Fock quantization
Let us summarize now the Fock quantization of the model, i.e. the Fock quantizationof the sector of nonzero modes of the associated scalar field system, as performed in Refs.[7, 8]. We will call this sector of nonzero modes the inhomogeneous sector.By construction, the set of mode solutions { G ( t ) k ( t, θ ) , G ( t ) ∗ k ( t, θ ) } in Eq. (7) (with k ∈ Z − { } ) is complete in the inhomogeneous sector of the space of solutions S , and“orthonormal” in the product ( G ( t ) k , G ( t ) m ) = − i Ω( G ( t ) ∗ k , G ( t ) m ), in the sense that( G ( t ) k , G ( t ) m ) = δ km , ( G ( t ) ∗ k , G ( t ) ∗ m ) = − δ km , ( G ( t ) k , G ( t ) ∗ m ) = 0 . (18)Associated to the field decomposition (7), there is a natural Ω-compatible complex structure J : J h G ( t ) k ( t, θ ) i = iG ( t ) k ( t, θ ) , J h G ( t ) ∗ k ( t, θ ) i = − iG ( t ) ∗ k ( t, θ ) . (19) Note that the pairs with k > k < b k ( t ) = Ω( J G ( t ) ∗ k , ξ ) and b ∗ k ( t ) = Ω( J G ( t ) k , ξ ). We notice that J is invariant under thegroup of S translations T ω : θ θ + ω generated by the global constraint (2).Starting with ( S, J ), we can construct the so-called “one particle” Hilbert space H . Itis the Cauchy completion of the space of “positive” frequency solutions S + := (cid:26) ξ + = 12 ( ξ − iJ ξ ) (cid:27) (20)with respect to the norm || ξ + || = p h ξ + , ξ + i . Here, h· , ·i denotes the inner product h ξ +1 , ξ +2 i := − i Ω( ξ − , ξ +2 ) with ξ − = ( ξ + iJ ξ ) / ∈ ¯ H (the complex conjugate space of H ). The kinematical Hilbert space of the quantum theory is then the symmetric Fockspace F ( H ) = ∞ M n =0 (cid:16)O n ( s ) H (cid:17) , (21)where ⊗ n ( s ) H is the Hilbert space of all n -th rank symmetric tensors over H . Followingthis prescription, the formal field operator ˆ ξ yieldsˆ ξ ( t ; θ ) = X k ∈ Z −{ } h G ( t ) k ( t, θ )ˆ b k + G ( t ) ∗ k ( t, θ )ˆ b † k i . (22)Here, ˆ b k and ˆ b † k are, respectively, the annihilation and creation operators corresponding tothe “positive” and “negative” frequency decomposition defined by J , and represent theclassical variables b k and b ∗ k .A crucial aspect of this quantization is that the dynamics is unitarily implementable, i.e.for each symplectic transformation in the 1-parameter family τ ( t f ,t ) (15) defined by timeevolution ∀ t f >
0, there exists a unitary quantum evolution operator ˆ U ( t f , t ) such thatˆ b k ( t f ) = α k ( t f , t )ˆ b k + β k ( t f , t )ˆ b †− k = ˆ U − ( t f , t )ˆ b k ˆ U ( t f , t ) , ˆ b †− k ( t f ) = β ∗ k ( t f , t )ˆ b k + α ∗ k ( t f , t )ˆ b †− k = ˆ U − ( t f , t )ˆ b †− k ˆ U ( t f , t ) . (23)As shown in Refs. [7, 8], this follows from the fact that the sequences { β k ( t f , t ) } are squaresummable. In addition, since J is invariant under the group of translations T ω , we have an invariantunitary implementation of the gauge group on the (kinematical) Fock space F ( H ).The physical Hilbert space F phys consists of all states in F ( H ) that belong to the kernelof the quantum constraint ˆ C = ∞ X k =1 k (cid:16) ˆ b † k ˆ b k − ˆ b †− k ˆ b − k (cid:17) . (24)Starting with the basis of “ n -particle” states determined by the annihilation and creationoperators { (ˆ b k , ˆ b † k ) } , one can then construct physical states by restricting the elements of Let us recall that a symplectic transformation is unitarily implementable with respect to a FR if, andonly if, its antilinear part is Hilbert-Schmidt on the “one particle” Hilbert space [12]. In the present casethis condition reduces to P k | β k ( t f , t ) | < ∞ . n -particle” states withzero field momentum P ∞ k =1 k ( N k − N − k ) = 0, where N k is the corresponding eigenvalue ofthe partial k -th number operator ˆ N k := ˆ b † k ˆ b k . Furthermore, it is straightforward to checkthat ˆ C is invariant under the time evolution (23). This invariance ensures that the dynamicsis unitarily implementable not just on F ( H ), but also on the space of physical states F phys .Let us conclude with a comment regarding an apparent ambiguity. Our fixed referencetime t certainly plays a role in the definition of J , and it is clear that, by changing t andkeeping the definition (19), one obtains new complex structures with the same properties of S -invariance and unitary dynamics, since the results of Refs. [7, 8] do not depend on thevalue of t . However, since these different complex structures are, by construction, relatedby evolution transformations, they give rise to unitarily equivalent quantizations, preciselybecause the evolution is unitary [8] (see also Subsec. IV B). Moreover, as we mentioned inthe introduction, much stronger results have indeed been proven regarding the uniquenessof the quantization [6, 9]. III. THE SCHR ¨ODINGER REPRESENTATION
We will now obtain the Schr¨odinger functional description of the quantum representationof the canonical commutation relations (CCRs) provided by the quantum fields of the systemat a given fixed time. Let us stress again that, just because of the unitary implementationof the field dynamics, the choice of this fixed time is irrelevant, in the sense that differentchoices lead to unitarily equivalent representations of the CCRs. So, for convenience, wewill take this fixed time to be our reference time t .The SR that we are going to construct is that defined by the specific complex structurethat is induced from J on the canonical phase space Γ by means of the isomorphism I E t .Taking into account that the complex structure J effectively declares that the classicalvariables { b k } and { b ∗ k } are to be quantized as the respective annihilation and creationoperators of the representation, and recalling Eq. (14), which gives the relation betweenthese variables and the field modes, it should not come as a surprise that the representationof the CCRs which we will obtain is essentially that associated with the free massless fieldin S . We will nevertheless present this construction in some detail, both for completenessand to clarify the relation that, for the quantization of the Gowdy model, exists betweenthe covariant approach adopted in Refs. [7, 8] and its canonical version. A. General framework
Let us start by considering the canonical phase space Γ (more precisely, its inhomogeneoussector). The set of elementary observables O is taken to be the vector space of linearfunctionals L λ ( Y ) := σ ( λ, Y ) = I d θ ( f ϕ + g P ϕ ) (25)and the unit functional , namely O = Span { , L λ } . Here, Y is a vector in Γ of the form( ϕ, P ϕ ) and λ denotes a pair of smooth test functions ( − g, f ) which have both a vanishingintegral on S . The set O is closed under Poisson brackets, { L λ ( Y ) , L ν ( Y ) } = L ν ( λ ), and iscomplete, in the sense that its elements separate points in (the inhomogeneous sector of) Γ.8he configuration and momentum observables are particular cases of functionals L λ .Whereas L λ | λ =(0 ,f ) defines the configuration observable ¯ ϕ ( f ) := I d θ f ϕ = X k ∈ Z −{ } f k ϕ k , (26)the momentum observable is defined by considering the label λ = ( − g, P ϕ ( g ) := I d θ g P ϕ = X k ∈ Z −{ } g k P kϕ . (27)From the Poisson brackets between the configuration and momentum observables (and set-ting ~ = 1), one obtains for their respective quantum operators ˆ¯ ϕ [ f ] and ˆ¯ P ϕ [ g ] the CCRs: h ˆ¯ ϕ [ f ] , ˆ¯ P ϕ [ g ] i = i ˆ X k ∈ Z −{ } f k g k . (28)At this point of the discussion and in order to make the analysis self-contained, it isconvenient to succinctly review how a Schr¨odinger functional representation of the CCRs isdetermined by a complex structure on the canonical phase space. We will start by describingthe most general form of a complex structure on Γ. This discussion can then be easily appliedto the general setting of a scalar field in a globally hyperbolic spacetime (see Refs. [13, 14])and, in particular, to the case of the Gowdy model.A ( σ -compatible) complex structure j on Γ has the generic form j ( ϕ, P ϕ ) = ( Aϕ + BP ϕ , CP ϕ + Dϕ ) , (29)where A , B , C and D are linear operators that satisfy A + BD = − , AB + BC = 0 ,C + DB = − , DA + CD = 0 (30)(so that j = − ), and ( f, Bf ′ ) = ( Bf, f ′ ) , ( g, Dg ′ ) = ( Dg, g ′ ) , ( f, Ag ) = − ( Cf, g ) , ( f, Bf ) < , ( g, Dg ) > g , g ′ , f and f ′ (so that j is σ -compatible). Here, we haveintroduced the notation ( f, g ) := H d θ f g . Notice that C and D can be obtained from A and B : indeed, from the two first relations in Eq. (30) one gets C = − B − A B and D = − B − ( + A ) (when B − exists). Thus, the set of all compatible complex structureson Γ can be parameterized by the operators A and B (assuming B is invertible); that is,this set can be identified with { j ( A,B ) } where (in matrix notation) j ( A,B ) = (cid:18) A B − B − ( + A ) − B − AB (cid:19) . (32) The Fourier components of f and g in λ are, respectively, f k = H d θ f e ∗ k and g k = H d θ ge k . j on the canonical phase space Γ, a Schr¨odinger, or “config-uration” wave functional representation – which we will call the j -SR – is determined asfollows. The j -SR consists of a representation of the basic operators of configuration andmomentum on a space of complex-valued functionals Ψ on the “quantum” configurationspace ¯ C (generally an extension of the classical configuration space). These functionals aresquare integrable with respect to a Gaussian measure µ with covariance − B/ On theHilbert space defined in this way, the basic operators of configuration and momentum are (cid:0) ˆ¯ ϕ [ f ]Ψ (cid:1) [ ¯ ϕ ] = ¯ ϕ ( f )Ψ[ ¯ ϕ ] , (33) (cid:16) ˆ¯ P ϕ [ g ]Ψ (cid:17) [ ¯ ϕ ] = − i δ Ψ δ ¯ ϕ [ g ] − i ¯ ϕ (cid:0) B − ( − iA ) g (cid:1) Ψ[ ¯ ϕ ] , (34)where ¯ ϕ ∈ ¯ C .It is worth noticing that, while the measure is determined just by B , there is an extrafreedom in the momentum operator, given by the operator A (see Ref. [15] for discussion).Finally, let us also recall that two complex structures j and j ′ on Γ lead to unitarily equivalentrepresentations of the CCRs if, and only if, j − j ′ defines a Hilbert-Schmidt operator on the“one particle” Hilbert space determined by j (or equivalently by j ′ ). B. The canonical complex structure
As we explained in Sec. II, given our RCS, which is determined by the chosen referencetime t , there is a preferred isomorphism between the canonical phase space and the spaceof solutions to the field equation (5). In order to simplify the notation, we will denote thisisomorphism by I E instead of I E t . Then, I E : Γ → S is such that S ∋ ξ I − E ( ξ ) = ( ϕ, P ϕ ) = ( ξ | t = t , ∂ t ξ | t = t ) . (35)Therefore, a complex structure J on the covariant phase space S determines (and is de-termined by) a corresponding complex structure j = I − E J I E on the canonical phasespace. In particular, the complex structure J of Sec. II has the canonical counterpart j = I − E J I E : Γ → Γ. The SR we are looking for is thus specified by j , following theprescription of the previous subsection. We will now obtain the explicit form of j .Recalling the field decomposition (7) (for the inhomogeneous sector) and employing Eq.(10), we get the explicit relation between ( ϕ , P ϕ ) and the set of pairs of variables { ( b k , b ∗− k ) } : ϕ = X k ∈ Z −{ } p | k | [ b k e ∗ k + b ∗ k e k ] , P ϕ = − i X k ∈ Z −{ } r | k | b k e ∗ k − b ∗ k e k ] . (36) We are only presenting the outcome, obtained under suitable regularity conditions. The full processinvolves the construction of an inner product from j and σ , which is used to determine a state of the Weylalgebra associated with the CCRs. The GNS representation defined by this state can be realized as anSR, since the restriction of the state to the Weyl configuration observables defines a measure. We define the covariance of a Gaussian measure as twice the positive bilinear form appearing in theexponential of the Fourier transform of the measure. We follow the standard practice of using the term“covariance” to refer not only to this bilinear form, but also to the operator which defines it with respectto a fiducial integration in the space of test functions, which in our case is given by d θ . ϕ, P ϕ ) ∈ Γ and the corresponding solution ξ = I E ( ϕ, P ϕ ) ∈ S , we obtainthe new canonical fields j ( ϕ, P ϕ ) = I − E J ( ξ ) ∈ Γ, which we will call ( ˜ ϕ, ˜ P ϕ ). Taking intoaccount that J ( ξ ) = iξ + − iξ − , with ξ + ( ξ − ) being the “positive” (“negative”) frequencypart spanned by { G ( t ) k } ( { G ( t ) ∗ k } ), with k ∈ Z − { } , we get˜ ϕ = J ( ξ ) | t = iξ + | t − iξ − | t , ˜ P ϕ = ∂ t J ( ξ ) | t = i∂ t ξ + | t − i∂ t ξ − | t . (37)Hence, it is easy to check that˜ ϕ = X k ∈ Z −{ } i p | k | [ b k e ∗ k − b ∗ k e k ] , ˜ P ϕ = X k ∈ Z −{ } r | k | b k e ∗ k + b ∗ k e k ] . (38)From Eqs. (36) and (38), one obtains that ˜ ϕ = − ( − ∆) − / P ϕ and ˜ P ϕ = ( − ∆) / ϕ , where∆ is the second order differential operator d / d θ . The explicit expression for the canonicalcounterpart of J is then j = (cid:18) − ( − ∆) − / ( − ∆) / (cid:19) . (39)A comparison with Eq. (32) shows that, in this case, A = 0 and B = − ( − ∆) − / . Therefore,the momentum operators are completely determined by the covariance ( − ∆) − / / { ( ϕ k , P − kϕ ) } with k ∈ Z − { } , the complex structure(39) yields ( j ) k = (cid:18) − | k | | k | (cid:19) . (40)So, in this alternative description of Γ provided by the Fourier components of ϕ and P ϕ ,the counterparts of A and B are given by A k = 0 and B k = − | k | , respectively (recall that k = 0). C. The functional representation of the Gowdy cosmologies
Let us now complete the construction of the j -SR. We will call T our space of testfunctions, i.e. the space of smooth real functions on S with vanishing integral. By standardarguments in the theory of measures in infinite dimensional spaces (see e.g. Ref. [16]), thespace T can be equipped with a so-called nuclear topology, and the covariance ( − ∆) − / / µ on the topological dual of T , namely the real vector space T ⋆ of continuous linear functionals on T . This will be the quantum configuration space ¯ C .Designating a generic element of T ⋆ as ¯ ϕ and its action on elements of T as f ¯ ϕ ( f ),the measure µ is defined by its Fourier transform Z T ⋆ e i ¯ ϕ ( f ) d µ = exp (cid:20) −
14 ( f, ( − ∆) − / f ) (cid:21) . (41)The “configuration” wave functional representation of ˆ¯ ϕ and ˆ¯ P ϕ on H s := L ( T ⋆ , d µ ) isthen (cid:0) ˆ¯ ϕ [ f ]Ψ (cid:1) [ ¯ ϕ ] = ¯ ϕ ( f )Ψ[ ¯ ϕ ] , (42) (cid:16) ˆ¯ P ϕ [ g ]Ψ (cid:17) [ ¯ ϕ ] = − i δ Ψ δ ¯ ϕ [ g ] + i ¯ ϕ (( − ∆) / g )Ψ[ ¯ ϕ ] . (43)11n alternative description is obtained in Fourier space as follows. By means of theFourier correspondence f
7→ { f k } = { H d θf e ∗ k } , one can identify T with the space of rapidlydecreasing complex sequences { f k } with k ∈ Z − { } , i.e. sequences such that k r f k goesto zero as | k | → ∞ , for all r > f k ∗ = f − k , so that thecorresponding functions f are real). Likewise, the dual space T ⋆ can be identified with asubspace (of sequences of appropriate behavior) of the space of all complex sequences { ϕ k } with k ∈ Z −{ } and ϕ − k = ϕ ∗ k . This correspondence is given by T ⋆ ∋ ¯ ϕ ↔ { ϕ k } := { ¯ ϕ ( e k ) } ,so that ¯ ϕ ( f ) = X k =0 f k ϕ k = X k> f k ϕ k + X k> (cid:0) f k ϕ k (cid:1) ∗ . (44)In order to present the measure without unnecessary complications, we note that, since thesequences { f k } ∈ T and { ϕ k } ∈ T ⋆ are both determined by their values for k >
0, one cansimply work with sequences whose index is defined in N , rather than in Z − { } . Actually,one can view µ as a measure on the space of all complex sequences { ϕ k } with k ∈ N thathappens to be supported on the subspace T ⋆ . In this description, µ is a product measure on (a subset of) the product space C N ofcomplex sequences { ϕ k } with k ∈ N :d µ = Y k ∈ N | k | π exp (cid:0) − | k | | ϕ k | (cid:1) d µ k , (45)where d µ k is the Lebesgue measure on the plane coordinatized by ( ϕ k , ϕ ∗ k ). It is easily seenthat this measure corresponds to that appearing in Eq. (41).Note that we are using here complex canonical variables. This accounts for the factors2 in Eq. (45), which no longer appear when the quantization is recasted in terms of realcanonical variables, namely the coefficients in the Fourier decompositions of ϕ and P ϕ interms of normalized sine and cosine functions.It is worth pointing out that one can reinterpret the measure µ described above as ameasure on the original space of sequences { ϕ k } with integer index ( k ∈ Z − { } ) and suchthat ϕ − k = ϕ ∗ k . Using the one-to-one correspondence between these sequences and theirrestrictions to k ∈ N , one can define both the measurable sets and the measure.The operators which present the simplest expressions correspond to the Fourier com-ponents of the field operators, ˆ¯ ϕ [ e k ] and ˆ¯ P ϕ [ e ∗ k ], i.e. to the quantization of the classicalvariables ϕ k and P kϕ : ˆ ϕ k Ψ = ϕ k Ψ , (46)ˆ P kϕ Ψ = − i ∂ Ψ ∂ϕ k + i | k | ϕ − k Ψ , (47)where Ψ is a functional of the Fourier components ϕ k .The CCRs (28) are clearly satisfied. Moreover, the same happens with the reality condi-tions ˆ ϕ † k = ˆ ϕ − k and ˆ P k † ϕ = ˆ P − kϕ with respect to the L ( T ⋆ , d µ )-inner product. Equivalently,the operators ˆ¯ ϕ [ f ] and ˆ¯ P ϕ [ g ] are symmetric, leading to self-adjoint operators on an appro-priate domain of definition. On the other hand, one can certainly find proper subsets of T ⋆ which support the measure. See e.g. Ref.[17] for reviews of results and for techniques concerning support properties of field measures.
12n addition, from Eq. (14) the variables b k and b ∗ k are quantized asˆ b k = 1 p | k | ∂∂ϕ − k , ˆ b † k = − p | k | ∂∂ϕ k + p | k | ϕ − k . (48)These are precisely the annihilation and creation operators of the j -SR. By construction,the “zero particle” state of the j -SR, which we will call the vacuum, is the unit constantfunctional Ψ [ ¯ ϕ ] = 1 (up to a constant phase).As we have already mentioned, the invariance of J – and therefore of j – under thegroup of S -translations T ω : θ → θ + ω , ω ∈ S , provides us with corresponding unitaryoperators ˆ T ω which leave the vacuum invariant, and whose explicit action, in the Fourierdescription, is given by (cid:16) ˆ T ω Ψ (cid:17) [ ϕ k ] = Ψ[ e − i ω k ϕ k ] , Ψ ∈ H s . (49)The generator of the unitary group ˆ T ω ,ˆ C = ∞ X k =1 | k | (cid:18) ϕ − k ∂∂ϕ − k − ϕ k ∂∂ϕ k (cid:19) , (50)is the quantum constraint operator in the functional approach.The space of physical states consists of all states in H s which are invariant under theaction of ˆ T ω for every ω ∈ S . That is, physical states are invariant under the group ofphase transformations ϕ k → e − i ω k ϕ k ∀ ω ∈ S . This property allows a characterizationof physical states alternative to that presented at the end of Sec. II. One can obtain theHilbert space of physical states H phys as the quotient of the kinematical Hilbert space H s by the action of the considered gauge group. Since this group is compact, the projection ofany kinematical state Ψ onto the space of physical states can then be easily determined bya group averaging procedure (see e.g. Ref. [18]):Ψ phys [ ϕ k ] = I d ω π (cid:16) ˆ T ω Ψ (cid:17) [ ϕ k ] . (51)It is important to emphasize that, because the gauge group is unitary and compact, thephysical state Ψ phys has a finite norm for any Ψ ∈ H s . Therefore, the space of physicalstates is just a Hilbert subspace of the kinematical Hilbert space.In summary, the j -SR consists of a (kinematical) Hilbert space H s defined by a Gaussianmeasure of covariance ( − ∆) − / /
2, on which the CCRs are implemented by the operatorsˆ¯ ϕ and ˆ¯ P ϕ (42)-(43) [or equivalently, by ˆ ϕ k and ˆ P kϕ (46)-(47)]. The physical Hilbert spaceconsists of the invariant subspace under S -translations. It follows from the results of Refs.[6–9] that the j -SR is the (essentially) unique S -invariant “configuration” wave functionalrepresentation with a unitary dynamics. Finally, it is worth emphasizing that the SR herepresented is not equivalent to the Schr¨odinger representations (SRs) constructed in Ref. [19],where the considered basic field was ¯ ξ = ξ/ √ t instead of ξ [6–8].13 V. TIME EVOLUTION
In this section we will address the issue of how time evolution is implemented in ourmodel in the context of the functional representation.
A. Creation and annihilation operators and the vacuum
In the J -Fock quantization, classical dynamics is implemented in the Heisenberg pictureby a unitary operator ˆ U ( t f , t ) relating annihilation and creation operators at different timesas in Eq. (23). Recalling that ˆ U − ( t f , t ) = ˆ U ( t , t f ) and the last two relations in Eq. (17),we can now introduce the annihilation and creation operators corresponding to evolution“backwards in time”,ˆ¯ b k ( t f ) = ˆ U ( t f , t )ˆ b k ˆ U − ( t f , t ) = α ∗ k ( t f , t )ˆ b k − β k ( t f , t )ˆ b †− k , ˆ¯ b † k ( t f ) = ˆ U ( t f , t )ˆ¯ b † k ˆ U − ( t f , t ) = α k ( t f , t )ˆ b † k − β ∗ k ( t f , t )ˆ b − k . (52)Obviously, ˆ¯ b k ( t ) and ˆ¯ b † k ( t ) coincide with ˆ b k and ˆ b † k , respectively.Because of the mixing of annihilation and creation operators, the Heisenberg vacuumstate | i H which is annihilated by all the operators ˆ b k fails to be in the kernel of all thetime-evolved operators ˆ¯ b k ( t f ) for any t f = t . Instead, these operators annihilate the state | , t f i := ˆ U ( t f , t ) | i H , (53)which is just the time-evolved vacuum, i.e. the counterpart of the state | i H in theSchr¨odinger picture. Of course, | , t i = | i H .We will refer to | , t f i and to states of the form | n, t f i = ˆ¯ b † k ( t f )ˆ¯ b † k ( t f ) . . . ˆ¯ b † k n ( t f ) | , t f i (54)as the t f -vacuum and the t f “ n -particle” states, respectively. From Eqs. (52) and (54) oneconcludes that the t f “ n -particle” states are related with the Heisenberg “ n -particle” states | n i H := | n, t i as follows | n, t f i = ˆ U ˆ b † k ˆ b † k . . . ˆ b † k n ˆ U − | , t f i = ˆ U ˆ b † k ˆ b † k . . . ˆ b † k n | i H = ˆ U | n i H , (55)where we have used ˆ U as an abbreviation for ˆ U ( t f , t ). The t f “ n -particle” states are thus theresult of evolving the states | n i H from t to t f . Therefore, in order to specify the evolutionto time t f of all Heisenberg “ n -particle” states –and hence determine the time evolutionoperator–, we only need to supply the operators (52). In this respect, we note that anequivalent condition for unitarity of the evolution to time t f is the existence of a vectorwhich is annihilated by all the operators ˆ¯ b k ( t f ). If this vector exists, then it is unique (upto a constant phase), so that the considered annihilation operators contain indeed all thenecessary information to fix the evolved vacuum (53).Turning back to the functional description, let us now write the operators ˆ¯ b k ( t f ) anddetermine the explicit form of the state | , t f i in the j -SR. From Eqs. (48) and (52) one14btains ˆ¯ b k ( t f ) = α ∗ k + β k p | k | ∂∂ϕ − k − p | k | β k ϕ k . (56)Here, α k and β k denote α k ( t f , t ) and β k ( t f , t ), respectively, a simplified notation that wewill use in the following. It is straightforward to see that, formally, the solution of the setof conditions ˆ¯ b k ( t f )Ψ = 0 ( ∀ k ∈ Z − { } ) is given byΨ ( t f )0 := Y k ∈ N | α ∗ k + β k | exp (cid:18) | k | β k α ∗ k + β k | ϕ k | (cid:19) , (57)where we have already normalized each of the factors in the infinite product. Actually, owingto the summability of the sequences {| β k | } (i.e. thanks to unitarity), one can check thatthe normalized sequence formed by the finite number of factors 1 ≤ k ≤ K with K ∈ N is a Cauchy sequence in the L ( T ⋆ , d µ )-norm. Hence, the t f -vacuum | , t f i in the j -SR is(up to a constant phase) the state Ψ ( t f )0 , rigorously defined as the L -limit of the sequenceof products with a finite number of factors. B. Complex structures induced by time evolution
Regardless of its unitary implementability in the quantum theory, the classical evolution,being defined by a family of symplectic transformations, generates a family of representationsof the CCRs starting from a given one. In the present case, this family of representations isassociated with the family of complex structures j t f := τ ( t f ,t ) j τ − t f ,t ) , j t f : Γ → Γ , (58)obtained by evolving the complex structure j . Here, τ ( t f ,t ) is the classical evolution operatorfor an arbitrary time t f >
0. Clearly, the condition of unitary implementability of time evo-lution in the j -representation translates into the condition of unitary equivalence betweenthat representation and the representations defined by the complex structures j t f , ∀ t f > j t f . The relationship betweenthe members of this family of representations provides us with an alternative, equivalentdescription of the time evolution. In the present case, given the unitary implementabilityof the evolution, established in Refs. [7, 8], we obtain a family of unitarily equivalent rep-resentations. In particular, the family of SRs defined by the complex structures j t f , whichwe will refer to as the family of j t f -SRs, is associated with a family of mutually absolutelycontinuous Gaussian measures.Before determining explicitly the complex structures j t f and the corresponding j t f -SRs,we will give an equivalent characterization of them which is related to the discussion in theprevious subsection. Let us consider the set of (pairs of) coefficients { (¯ b k ( t f ) , ¯ b ∗ k ( t f )) } whichis obtained from { ( b k , b ∗ k ) } by applying τ − t f ,t ) [i.e. the relation between the two sets is thedirect classical counterpart of Eq. (52)]. It is clear that, when expressed in terms of the One may also obtain the operators ˆ¯ b † k ( t f ) in the same way. { (¯ b k ( t f ) , ¯ b ∗ k ( t f )) } , the complex structure j t f adopts the same form as j in terms ofthe pairs { ( b k , b ∗ k ) } [namely, it is given by a block-diagonal matrix with the 2 × j t f ) k = diag( i, − i )]. Therefore, the j t f representation is such that the classical variableswhich are quantized as the creation and annihilation operators are { ¯ b ∗ k ( t f ) } and { ¯ b k ( t f ) } ,respectively, rather than { b ∗ k } and { b k } . Returning to the covariant description for a moment, the family { j t f } determines a familyof complex structures on the covariant phase space via the isomorphism I E (35). Theseare given by J t f = ¯ τ ( t f ,t ) J ¯ τ − t f ,t ) = I E j t f I − E , where ¯ τ ( t f ,t ) = I E τ ( t f ,t ) I − E is the classicalevolution map in covariant phase space. Just as J is associated with the field decomposition(7), J t f can be understood as being associated with the decomposition ξ ( t, θ ) = X k =0 h ¯ b k ( t f ) G ( t f ) k ( t, θ ) + ¯ b ∗ k ( t f ) G ( t f ) ∗ k ( t, θ ) i , (59)where G ( t f ) k = ¯ τ ( t f ,t ) G ( t ) k are the time-evolved modes. One can thus see that, as commentedabove, changing the time used to define our fiducial complex structure on the covariantphase space corresponds in fact to evolution. C. The family of unitarily equivalent functional representations
Explicit expressions for the complex structures j t f (58) are obtained quite straightfor-wardly. Taking into account expression (40) for j , relations (14) and the evolution (15),one concludes that j t f , given in terms of the Fourier coefficients { ( ϕ k , P − kϕ ) } , is defined bythe following 2 × j t f ) k = α k β k ) − | α ∗ k + β k | | k | | k || α ∗ k − β k | − α k β k ) ! . (60)One can now easily determine the corresponding family of j t f -SRs. Comparing with thecase (40) for j , and referring to the general form (32), we find a change in the terms B k ,which now become B k = −| α ∗ k + β k | / | k | and correspond to a new Gaussian measure. Inaddition, we note the appearance of the term A k = 2Im( α k β k ) (owing to the mixing between“positive” and “negative” frequency parts during evolution). The respective contribution − iA k in the general expression for the momentum operators (34) can be written in thiscase as ( α k + β ∗ k )( α ∗ k − β k ). Thus, B − k ( − iA k ) = −| k | α ∗ k − β k α ∗ k + β k . (61)Adopting the same Fourier space description as in Subsec. III C, the j t f -SR is then realizedin the Hilbert space L ( T ⋆ , d µ t f ) defined by the Gaussian product measured µ t f = Y k ∈ N | k | π | α ∗ k + β k | exp (cid:18) − | k || α ∗ k + β k | | ϕ k | (cid:19) d µ k , (62) Let us point out that a different but equivalent way to recast time evolution is with the family of repre-sentations arising from the set of complex structures { ˜ j t f = τ − j τ } . In that case, the annihilation andcreation-like variables defined by ˜ j t f are { b k ( t f ) } and { b ∗ k ( t f ) } , introduced in Eq. (15). µ k is again the Lebesgue measure in C .The (Fourier components of the) basic field operators are now represented byˆ ϕ k Ψ = ϕ k Ψ , (63)ˆ P kϕ Ψ = − i ∂ Ψ ∂ϕ k + i | k | α ∗ k − β k α ∗ k + β k ϕ − k Ψ . (64)Notice that, in order to avoid an excessively complicated notation, we have used the samesymbols as in Eqs. (46) and (47) to denote quantum operators and states in the j t f -SR. Forcompleteness, let us also present the form of the annihilation and creation operators of the j t f -SR, which are given byˆ¯ b k ( t f ) = α ∗ k + β k p | k | ∂∂ϕ − k , ˆ¯ b † k ( t f ) = − α k + β ∗ k p | k | ∂∂ϕ k + p | k | α ∗ k + β k ϕ − k . (65)As we have discussed above, they represent the classical variables { (¯ b k ( t f ) , ¯ b ∗ k ( t f )) } . Thequantization of the variables { ( b k , b ∗ k ) } in this representation can be obtained from (theinverse of) relations (52), or from Eqs. (63) and (64), using relation (14).Let us now analyze the issue of unitarity in this context, namely, the unitary equivalencebetween the j -SR and the j t f -SRs. We first remark that, since unitarity is granted forany finite number of degrees of freedom, unitary equivalence (for a case of compact spatialtopology such as the present one) rests just on the behavior of the high frequency modes. Inour case, the asymptotic limit for large k of the sequences β k ( t, t ) and α k ( t, t ) is zero andone, respectively. Therefore, the factors in the measure (62) and the momentum operators(64) approach the corresponding expressions for the j -SR. Actually, this is a necessarycondition for unitarity, but not sufficient. Unitary equivalence between the j t f and the j representations amounts to requiring that j t f − j be a Hilbert-Schmidt operator. In turn,this is equivalent to the summability of the sequences {| β k | } , a condition which is indeedsatisfied, as shown in Refs. [7, 8]. So, all the representations in the 1-parameter family of j t f -SRs are equivalent to the j -SR, and hence any two members of the family are equivalentto each other.Consider now in more detail the momentum operators (64), and in particular the extramultiplicative term (that cannot be obtained from the measure) − B − k A k = 2 | k | Im( α k β k ) | α ∗ k + β k | , (66)coming from the diagonal component 2Im( α k β k ) in ( j t f ) k . The presence of this term meansthat the unitary group generated by the momentum operators is not simply the naturalunitary implementation in L ( T ⋆ , d µ t f ) of translations (by elements of T ) in T ⋆ . In additionto the contribution coming from the transformation under translations of the quasi-invariantmeasure µ t f [which corresponds to the term − iB − k in Eq. (64)], the elements of thatunitary group carry additional (nonconstant) phases. Such phases, responsible for the extraterm in Eq. (64), can be viewed in our case as generated by the unitary transformation17 : L ( T ⋆ , d µ t f ) → L ( T ⋆ , d µ t f ), with( T Ψ)[ ϕ k ] = exp i ∞ X k =1 B − k A k | ϕ k | ! Ψ[ ϕ k ]= exp − i ∞ X k =1 | k | Im( α k β k ) | α ∗ k + β k | | ϕ k | ! Ψ[ ϕ k ] . (67)In fact, one can check that T − maps the j t f -SR to the representation defined by the complexstructure j t f , with ( j t f ) k = − | α ∗ k + β k | | k || k || α ∗ k + β k | ! . (68)It is also worth noting that the summability of {| β k | } guarantees that the unitary transfor-mation T is well defined. Adopting the above perspective, the unitary transformation mapping the j t f -SR to the j -SR can be obtained as the composition of T − with the natural unitary transformationbetween the j t f -SR and the j -SR, namely Ψ (cid:0) d µ t f / d µ (cid:1) / Ψ. We also notice that theexistence of both derivatives d µ t f / d µ and d µ/ d µ t f , i.e. the mutual absolute continuity ofthe Gaussian measures, depends on whether the operator C t f C − − is Hilbert-Schmidt,where C and C t f denote the covariances of µ and µ t f , respectively. In the present case thisleads to the condition that {| α ∗ k + β k | − } be a square summable sequence. Again, this isensured by the summability of {| β k | } .Summarizing, the unitary transformation mapping the j t f -SR to the j -SR is the multi-plicative transformation L ( T ⋆ , d µ t f ) ∋ Ψ (cid:18) d µ t f d µ (cid:19) / exp i ∞ X k =1 | k | Im( α k β k ) | α ∗ k + β k | | ϕ k | ! Ψ ∈ L ( T ⋆ , d µ ) . (69)Of course, the multiplicative factor in this expression is simply the image of the unit func-tional Ψ [ ϕ k ] = 1 of the j t f -SR, and therefore supplies the state | , t f i (53) in the j -SR,namely, it coincides with Ψ ( t f )0 given in Eq. (57) [one can check this by introducing theexplicit form of d µ t f / d µ obtained from Eqs. (45) and (62)].Finally, we want to comment that any unitary transformation between two SRs admitsa form like that displayed in Eq. (69). In fact, given two normalized measures µ and µ (not necessarily Gaussian), if a unitary transformation U exists such that it maps one SR tothe other, then it is necessarily of the multiplicative form Ψ Ψ Ψ, where Ψ is the image In general, the presence of phases in the unitary representation of the group of (appropriate) translationsin the quantum configuration space is a source of unitary inequivalence in quantum field theory, in additionto the existence of nonequivalent quasi-invariant measures in infinite dimensions (see, e.g., Refs. [20, 21]).From the viewpoint of the momentum operators, rather than from that of the corresponding unitarygroup, this issue was addressed more recently in Ref. [15], where the possible lack of unitary equivalencebetween representations with and without an extra linear term in the momentum operators was discussed,and related to the possibility or impossibility of defining unitarity transformations of the type (67). U of the unit functional. Moreover, the identity R | Ψ | d µ = R | Ψ | | Ψ | d µ , valid ∀ Ψ, implies that µ is continuous with respect to µ , with d µ / d µ = | Ψ | . By interchangingthe roles of µ and µ , one concludes that the measures are mutually continuous. Thus, theequivalence of the measures is a necessary condition for the unitary equivalence between twoSRs, and any possible unitary equivalence is of the form Ψ (d µ / d µ ) / e iF Ψ, where F is a real functional. As one can easily realize from the discussion of Ref. [15], in the case oftwo representations defined by equivalent complex structures, the functional F is a bilinearform of the type appearing in Eq. (67) and its introduction results in a modification of theaction of the momentum operators by linear terms. V. CONCLUSION
In full canonical quantum gravity formulated on a compact spatial section Σ, there is nofundamental notion of time. There is no Hamiltonian, and therefore no time with respect towhich one might define evolution (this is one of the manifestations of the notorious problemof time). The Gowdy model that we have considered here is somewhat special in this respectsince, through a partial gauge fixing, a particular notion of internal time is introduced inorder to “de-parametrize” the theory. Even when this parameter has no physical meaning inthe final description, it is used as an intermediate step in order to construct the correspondingphysical operators that define the true quantum geometry. This is the strategy that hasalso been followed in the quantization of homogeneous cosmologies [1]. Therefore, withinthe model, it is important to implement this notion of time evolution in a unitary way.Furthermore, the strategy that we have followed of implementing the internal notion of timeat the quantum level, together with the remaining gauge group, receives support from thefact that a quantization with such properties exists [7, 8] and is essentially unique [6, 9].This consistent quantization has to be contrasted to a previous proposal [4] that does notadmit an unitary time evolution [5].The purpose of this paper was to bridge the gap between the formalism of Refs. [7, 8] andthe standard formulation of canonical quantum gravity, and thus to recast the quantizationof the Gowdy model into the Schr¨odinger functional representation, where the states of thetheory are functionals Ψ( ϕ ) on the quantum configuration space. Let us now summarizethe results found here. First, we have constructed the Schr¨odinger functional version of thisquantum Gowdy model, and analyzed the (unitary) time evolution in this context. Second,we have solved the remaining constraint that is present in the model. In this way, we havebeen able to define the space of physical states in the Sch¨odinger picture, where unitaryevolution is again well defined.As a general strategy, we have approached the problem from a functional perspective.In this fashion, we have constructed explicitly the 1-parameter family of representationsthat gives rise to the quantum description at any time. These different representationsare unitarily equivalent precisely because time evolution is unitarily implementable. We This can be seen using the fact that, by construction, configuration operators such as the unitary groupsgenerated by the basic field operators generate a dense set when applied to the unit functional. On theother hand, the action of the configuration operators is the same in both representations. Thus, forΨ = e i ¯ ϕ ( f ) , U Ψ =
U e i ˆ¯ ϕ [ f ] U e i ˆ¯ ϕ [ f ] U − Ψ = e i ¯ ϕ ( f ) Ψ . The general expression follows from linearityand continuity. Acknowledgements
This work was supported by the Spanish MEC Projects FIS2005-05736-C03-02 andFIS2006-26387-E/, the CONACyT U47857-F grant, the Joint CSIC/CONACyT Project2005MX0022, the Portuguese FCT Project POCTI/FIS/57547/2004, the NSF PHY04-56913grant and the Eberly Research Funds of Penn State. [1] C. W. Misner, in
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