Towards Canonical Quantum Gravity for G1 Geometries in 2+1 Dimensions with a Lambda--Term
T. Christodoulakis, G. Doulis, Petros A Terzis, E. Melas, Th. Grammenos, G. O. Papadopoulos, A. Spanou
aa r X i v : . [ g r- q c ] J a n Towards Canonical Quantum Gravity for G Geometries in 2+1 Dimensions with a Λ –Term T. Christodoulakis ∗ , G. Doulis † , Petros A. Terzis ‡ Nuclear and Particle Physics Section, Physics Department,University of Athens, GR 157–71 Athens
E. Melas § Technological Educational Institution of LamiaElectrical Engineering Department, GR 35–100, Lamia
Th. Grammenos ¶ Department of Mechanical Engineering, University of Thessaly,GR 383–34 Volos
G.O. Papadopoulos k Department of Mathematics and Statistics, Dalhousie UniversityHalifax, Nova Scotia, Canada B3H 3J5
A. Spanou ∗∗ School of Applied Mathematics and Physical Sciences , National Technical University of Athens, GR 157–80, Athens ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] k [email protected] ∗∗ [email protected] bstract The canonical analysis and subsequent quantization of the (2+1)-dimensionalaction of pure gravity plus a cosmological constant term is considered, under theassumption of the existence of one spacelike Killing vector field. The proper impo-sition of the quantum analogues of the two linear (momentum) constraints reducesan initial collection of state vectors, consisting of all smooth functionals of thecomponents (and/or their derivatives) of the spatial metric, to particular scalarsmooth functionals. The demand that the midi-superspace metric (inferred fromthe kinetic part of the quadratic (Hamiltonian) constraint) must define on thespace of these states an induced metric whose components are given in terms ofthe same states, which is made possible through an appropriate re-normalizationassumption, severely reduces the possible state vectors to three unique (up to gen-eral coordinate transformations) smooth scalar functionals. The quantum analogueof the Hamiltonian constraint produces a Wheeler-DeWitt equation based on thisreduced manifold of states, which is completely integrated.
PACS Numbers : 04.60.Ds, 04.60.Kz
Dirac’s seminal work on his formalism for a self-contained treatment of systems withconstraints [1], [2], [3], [4] has paved the way for a systematic treatment of constraineddynamics. Some of the landmarks in the study of constrained systems have been theconnection between constraints and invariances [5], the extension of the formalism todescribe fields with half-integer spin through the algebra of Grassmann variables [6] andthe introduction of the BRST formalism [7]. All the classical results obtained so farhave made up an armory prerequisite for the quantization of gauge theories and thereare several excellent reviews studying constraint systems with a finite number of degreesof freedom [8] or constraint field theories [9], as well as more general presentations [10],[11], [12], [13], [14], [15]. In particular, the conventional canonical analysis approach ofquantum gravity has been initiated by B.S. DeWitt [16] based on earlier work of P.G.Bergmann [17].In the absence of a full theory of quantum gravity, it is reasonably important toaddress the quantization of (classes of) simplified geometries. An elegant way to achievea degree of simplification is to impose some symmetry. For example, the assumptionof a G symmetry group acting simply transitively on the surfaces of simultaneity, i.e.the existence of three independent space-like Killing vector fields, leads to classical andsubsequently quantum homogeneous cosmology (see, e.g., [18], [19]). The imposition oflesser symmetry, e.g. fewer Killing vector fields, results in the various inhomogeneouscosmological models [20]. The canonical analysis under the assumption of sphericalsymmetry, which is a G group acting multiply transitively on two-dimensional space-like subsurfaces of the three-slices, has been first considered in [21], [22]. Quantum blackholes have also been treated, for instance, in [23], [24] while in [25] a lattice regularization2as been employed to deal with the infinities arising due to the ill-defined nature of thequantum operator constraints.Another way to arrive at simplified models is to consider lower dimensions. For exam-ple, there is a vast literature on (2+1)-dimensional gravity (see, e.g., [26], [27], [28] andreferences there in). The role of non-commutative geometry in (2+1)-dimensional quan-tum gravity has been recently investigated in [29]. In this work we consider the canonicalquantization of all 2+1 geometries admitting one spacelike Killing vector field. In Sec-tion 2 we give the reduced metrics, the space of classical solutions and the Hamiltonianformulation of the reduced Einstein-Hilbert action principle, resulting in one (quadratic)Hamiltonian and two (linear) momentum first class constraints. In Section 3 we considerthe quantization of this constrained system following Dirac’s proposal of implementingthe quantum operator constraints as conditions annihilating the wave-function [4]. Ourguide-line is a conceptual generalization of the quantization scheme developed in [30],[31] for the case of constrained systems with finite degrees of freedom, to the present case.Even though after the symmetry reduction the system still represents a field theory (allremaining metric components depend on time and the radial coordinate), we manage toextract and subsequently completely solve a Wheeler-DeWitt equation in terms of threeunique smooth scalar functionals of the appropriate components of the reduced spatialmetric. This is achieved through an appropriate re-normalization assumption we adopt.Finally, some concluding remarks are included in the discussion. Our starting point is the action principle: I = Z d x √− g ( R − . (2.1)The equations of motion arising upon variation of this action are R IJ − g IJ R + Λ g IJ = 0 , (2.2)where I, J = 0 , ,
2. Of course, since in three dimensions the Riemann curvature tensoris expressible in terms of both the Ricci tensor and scalar, the space of solutions to(2.2) consists simply of all maximally symmetric 3D metrics (AdS3). If topologicalconsiderations are taken into account, the above space might be “enriched” containing,for example, the stationary BTZ “black” hole [32], [33] ds = − ( M − Λ r ) dt − J dtdφ + (cid:18) M − Λ r + J r (cid:19) − dr + r dφ (2.3)3r the “cosmological” solutions [34], [35] ds = − t Λ dt + 12 t √ Λ ( dr + dφ ) , (2.4) ds = − (cid:18) t − Λ (cid:19) dt + 416 t − Λ dr + 4 e − r t − Λ dφ . (2.5)Note that all these three line elements are locally AdS3 and therefore admit six localKilling fields. Their differences consist in the topological identifications. At this point, wedeem it pertinent to explain our view concerning the issue of the bearing of topology ona local theory: The Hamiltonian formulation is by itself implying a space-time topology R × Σ . Consequently, what we are concerned with is the topology of the 2-slices. Sincethe theory is local, it is implicitly assumed that the entire analysis holds in a coordinatepatch. Different topologies can only affect the number of patches needed to cover thespace and, therefore, can only impose restrictions on the range of the coordinates and/orthe range of validity of local fields, such as the symmetry generators admitted by thesemetrics; The paradigm of the cylinder may help clarify our point: The integral curvesof rotations in the plane are circles, but if one tries to draw a circle of radius R ≥ πL on the cylinder (L being the cylinder’s radius), crossings (or a pinch in case of equality)will occur, indicating that the corresponding generator is ill-defined. In such a situationone can, as many do, drop rotations altogether; this is the case in [32], [33], where fourof the six Killing fields are considered as non-valid symmetries. On the other hand onecan accept integral curves (circles) of radius R < πL (by suitably restricting the rangeof validity of the Killing field), which would simply result in the need of two patches tocover the cylinder with these lines. We adopt this latter point of view, as it seems to usmuch more reasonable. We shall thus not specify any ranges for our coordinates ( t, r, φ )precisely to allow for different topological options, which are not otherwise affecting ourresults.In this spirit we can say that the above metrics admit a G symmetry group. In whatfollows, we consider a generalization consisting in the imposition of a G symmetry only,i.e we impose one Killing vector field, say ξ = ∂∂φ . Subsequently, all components of themetric become functions of both the time and the radial coordinate only. The canonicaldecomposition of such a metric is given in terms of the spatial metric g ij ( t, r ), the lapsefunction N o ( t, r ) and the shift “vector” N i ( t, r ) [10]: ds = (cid:0) − ( N o ) + g ij N i N j (cid:1) dt + 2 N i dtdx i + g ij dx i dx j , (2.6)where g ij = ρ + σ χ σ σχσχ σ , g ij = σρ − σχρ − σχρ ρ + σ χ σρ (2.7)with i, j = 1 ,
2, and x i = ( r, φ ). The particular parametrization of g ij above has beenchosen in such a way as to simplify the second linear constraint (see below), and conse-quently the resulting algebra. 4or the Hamiltonian formulation of the system (2.6) (see, e.g., chapter 9 of [10]), wefirst define the vectors η I = 1 N o (cid:0) , − N i (cid:1) , N i ≡ g ik N k F I = η J ; J η I − η I ; J η J where I , J are space-time indices and “ ; ” stands for covariant differentiation with respectto (2.6). Then, utilizing the Gauss-Codazzi equation (see, e.g., [36]), we eliminate allsecond time-derivatives from the Einstein-Hilbert action and arrive at an action quadraticin the velocities, I = R d x √− g ( R − − F I ; I ). The application of the Dirac algorithmresults firstly in the three primary constraints P o ≡ δLδ ˙ N o ≈ , P i ≡ δLδ ˙ N i ≈ H = Z (cid:0) N o H o + N i H i (cid:1) dr, (2.8)where H o , H i are given by H o = 12 G αβ π α π β + V (2.9a) H = σ ′ π σ − ρ π ′ ρ − χ π ′ χ (2.9b) H = − π ′ χ , (2.9c)the indices ( α, β ) take the values ( ρ, σ, χ ) and ′ ≡ ∂∂r . The Wheeler-DeWitt midi-superspace metric G αβ reads G αβ = − ρ − σ − σ ρσ , (2.10)while the potential V is V = 2Λ ρ + (cid:18) σ ′ ρ (cid:19) ′ . (2.11)The requirement for preservation, in time, of the primary constraints leads to the sec-ondary constraints H o ≈ , H ≈ , H ≈ . (2.12)At this stage, a tedious but straightforward calculation produces the following “open”5oisson bracket algebra of these constraints: {H o ( r ) , H o (˜ r ) } = [ g j ( r ) H j ( r ) + g j (˜ r ) H j (˜ r )] δ ′ ( r, ˜ r ) {H ( r ) , H o (˜ r ) } = H o ( r ) δ ′ ( r, ˜ r ) {H ( r ) , H o (˜ r ) } = 0 (2.13) {H ( r ) , H (˜ r ) } = H ( r ) δ ′ ( r, ˜ r ) − H (˜ r ) δ ( r, ˜ r ) ′ {H ( r ) , H (˜ r ) } = H ( r ) δ ′ ( r, ˜ r ) {H ( r ) , H (˜ r ) } = 0indicating that they are first class and also signaling the termination of the algorithm.Thus, our system is described by (2.12); the “dynamical” Hamilton-Jacobi equations d π ρ d t = { π ρ , H } , d π σ d t = { π σ , H } , d π χ d t = { π χ , H } are satisfied by virtue of the timederivatives of (2.12). One can readily check (as one must always do with reduced actionprinciples) that these three equations, when expressed in the velocity phase-space withthe help of the definitions d ρd t = { ρ, H } , d σd t = { σ, H } , d χd t = { χ, H } , are completelyequivalent to the three independent Einstein’s field equations satisfied by (2.6).We end up this section by noting a few facts concerning the transformation propertiesof ρ ( t, r ) , σ ( t, r ) , χ ( t, r ) and their spatial derivatives under changes of the radial variable r of the form r → ˜ r = h ( r ). As it can easily be inferred from (2.6) and (2.7):˜ ρ (˜ r ) = ρ ( r ) d rd ˜ r , ˜ σ (˜ r ) = σ ( r ) , ˜ χ (˜ r ) = χ ( r ) d rd ˜ r ,d ˜ σ (˜ r ) d ˜ r = d σ ( r ) d r d rd ˜ r , dd ˜ r (cid:18) ˜ χ (˜ r )˜ ρ (˜ r ) (cid:19) = dd r (cid:18) χ ( r ) ρ ( r ) (cid:19) d rd ˜ r , (2.14)where the t -dependence has been omitted for the sake of brevity. Thus, under the abovecoordinate transformations, σ, χρ are scalars, while ρ, χ and the derivatives of σ, χρ arecovariant rank 1 tensors (one-forms), or, equivalently in one dimension, scalar densities ofweight −
1. Therefore, the scalar derivative is not dd r but rather dρ d r or dχ d r ≡ ρχ dρ d r .Finally, if we consider an infinitesimal transformation r → ˜ r = r − η ( r ), it is easily seenthat the corresponding changes induced on the basic fields are: δ ρ ( r ) = ( ρ ( r ) η ( r )) ′ , δ σ ( r ) = σ ′ ( r ) η ( r ) , δ χ ( r ) = ( χ ( r ) η ( r )) ′ (2.15)i.e., nothing but the one-dimensional analogue of the appropriate Lie derivatives.With the use of (2.15), we can reveal the nature of the action of H on the basicconfiguration space variables as that of the generator of spatial diffeomorphisms: (cid:26) ρ ( r ) , Z d ˜ r η (˜ r ) H (˜ r ) (cid:27) = ( ρ ( r ) η ( r )) ′ , (cid:26) σ ( r ) , Z d ˜ r η (˜ r ) H (˜ r ) (cid:27) = σ ′ ( r ) η ( r ) , (cid:26) χ ( r ) , Z d ˜ r η (˜ r ) H (˜ r ) (cid:27) = ( χ ( r ) η ( r )) ′ . (2.16)6hus, we are justified to consider H as the representative, in phase-space, of an ar-bitrary infinitesimal reparametrization of the radial coordinate. As far as H is con-cerned, the situation is a little more complicated: the imposition of the symmetry gen-erated by the Killing vector field ξ = ∂/∂φ has left all configuration variables withoutany φ dependence; subsequently we can not expect H to generate arbitrary infinites-imal reparametrization of φ . Nevertheless, we can identify a property of H whichlinks its existence to the existence of ξ . This property is described by the relation: {H ( r ) , {H ( r ) , {H ( r ) , g ij (˜ r ) }}} = 0 correspo ⇐⇒ ndence L ξ g ij = 0. We are now interested in attempting to quantize this Hamiltonian system followingDirac’s general spirit of realizing all the classical first class constraints (2.12) as quantumoperator constraint conditions annihilating the wave functional. The main motivationbehind such an approach is the justified desire to construct a quantum theory manifestlyinvariant under the “gauge” generated by the constraints. To begin with, let us first notethat, despite the simplification brought by the imposition of the symmetry ξ = ∂/∂φ ⇔L ξ g IJ = 0, the system is still a field theory in the sense that all configuration variablesand canonical conjugate momenta depend not only on time (as is the case in homogeneouscosmology), but also on the radial coordinate r . Thus, to canonically quantize the systemin the Schr¨odinger representation, we first realize the classical momenta as functionalderivatives with respect to their corresponding conjugate fieldsˆ π ρ ( r ) = − i δδ ρ ( r ) , ˆ π σ ( r ) = − i δδ σ ( r ) , ˆ π χ ( r ) = − i δδ χ ( r ) . We next have to decide on the initial space of state vectors. To elucidate our choice,let us consider the action of a momentum operator on some function of the configurationfield variables, say ˆ π ρ ( r ) ρ (˜ r ) = − iρ (˜ r ) δ (˜ r, r ) . The Dirac delta-function renders the outcome of this action a distribution rather than afunction. Also, if the momentum operator were to act at the point at which the functionis evaluated, i.e. if ˜ r = r , then its action would produce a δ (0) and would therefore beill-defined. Both of these unwanted features are rectified, as far as expressions linear inmomentum operators are concerned, if we choose as our initial collection of states all smooth functionals (i.e., integrals over r ) of the configuration variables ρ ( r ) , σ ( r ) , χ ( r )and their derivatives of any order. Indeed, as we infer from the previous example,ˆ π ρ ( r ) Z d ˜ rρ (˜ r ) = − i Z d ˜ rρ (˜ r ) δ (˜ r, r ) = − iρ ( r );thus the action of the momentum operators on all such states will be well-defined (no δ (0)’s) and will also produce only local functions and not distributions. However, even7o, δ (0)’s will appear as soon as local expressions quadratic in momenta are considered,e.g.,ˆ π ρ ( r ) ˆ π ρ ( r ) Z d ˜ rρ (˜ r ) = ˆ π ρ ( r )( − i Z d ˜ rρ (˜ r ) δ (˜ r, r )) = ˆ π ρ ( r )( − iρ ( r )) = − iδ ( r, r ) . Another problem of equal, if not greater, importance has to do with the number ofderivatives (with respect to r ) considered: A momentum operator acting on a smoothfunctional of degree n in derivatives of ρ ( r ) , σ ( r ) , χ ( r ) will, in general, produce a functionof degree 2 n , e.g.,ˆ π ρ ( r ) Z d ˜ rρ ′′ (˜ r ) = − i Z d ˜ rρ ′′ (˜ r ) δ ′′ (˜ r, r ) = − iρ (4) ( r ) . Thus, clearly, more and more derivatives must be included if we desire the action ofmomentum operators to keep us inside the space of integrands corresponding to theinitial collection of smooth functionals; eventually, we have to consider n → ∞ . This,in a sense, can be considered as the reflection to the canonical approach, of the non-re-normalizability results existing in the so-called covariant approach. The way to dealwith these problems is, loosely speaking, to regularize (i.e., render finite) the infinitedistribution limits, and re-normalize the theory by, somehow, enforcing n to terminateat some finite value.In the following, we are going to present a quantization scheme of our system which:(a) avoids the occurrence of δ (0)’s, (b) reveals the value n = 1, as the only possibilityto obtain a closed space of state vectors, and (c) extracts a finite-dimensional Wheeler-DeWitt equation governing the quantum dynamics. The scheme closely parallels, con-ceptually, the quantization developed in [30],[31] for finite systems with one quadraticand a number of linear first class constraints. Therefore, we deem it appropriate andinstructive to present a brief account of the essentials of this construction.To this end, let us consider a system described by a Hamiltonian of the form H ≡ µX + µ i χ i = µ (cid:18) G AB ( Q Γ ) P A P B + U A ( Q Γ ) P A + V ( Q Γ ) (cid:19) + µ i φ Ai ( Q Γ ) P A , (3.1)where A, B, Γ . . . = 1 , . . . , M count the configuration space variables and i = 1 , , . . . , N < ( M −
1) numbers the super-momenta constraints χ i ≈
0, whichalong with the super-Hamiltonian constraint X ≈ { X, X } = 0 , { X, χ i } = XC i + C ji χ j , { χ i , χ j } = C kij χ k , (3.2)where the first (trivial) Poisson bracket has been included only to emphasize the differ-ence from the first of (2.13).The physical state of the system is unaffected by the “gauge” transformations gen-erated by ( X, χ i ), but also under the following three changes:8I) Mixing of the super-momenta with a non-singular matrix¯ χ i = λ ji ( Q Γ ) χ j (II) Gauging of the super-Hamiltonian with the super-momenta¯ X = X + κ ( Ai ( Q Γ ) φ B ) i ( Q Γ ) P A P B + σ i ( Q Γ ) φ Ai ( Q Γ ) P A (III) Scaling of the super-Hamiltonian ¯ X = τ ( Q Γ ) X Therefore, the geometrical structures on the configuration space that can be inferred fromthe super-Hamiltonian are really equivalence classes under actions (I), (II) and (III); forexample (II), (III) imply that the super-metric G AB is known only up to conformalscalings and additions of the super-momenta coefficients ¯ G AB = τ ( G AB + κ ( Ai φ B ) i ). Itis thus mandatory that, when we Dirac-quantize the system, we realize the quantumoperator constraint conditions on the wave-function in such a way as to secure that thewhole scheme is independent of actions (I), (II), (III). This is achieved by the followingsteps:(1) Realize the linear operator constraint conditions with the momentum operators tothe right ˆ χ i Ψ = 0 ↔ φ Ai ( Q Γ ) ∂ Ψ( Q Γ ) ∂ Q A = 0 , which maintains the geometrical meaning of the linear constraints and produces the M − N independent solutions to the above equations q α ( Q Γ ) , α = 1 , , . . . , M − N called physical variables, since they are invariant under the transformations gener-ated by ˆ χ i .(2) In order to make the final states physical with respect to the “gauge” generated bythe quadratic constraint ˆ X as well:Define the induced structure g αβ ≡ G AB ∂ q α ∂ Q A ∂ q β ∂ Q B and realize the quadratic inmomenta part of X as the conformal Laplace-Beltrami operator based on g αβ . Notethat in order for this construction to be self consistent, all components of g αβ mustbe functions of the physical coordinates q γ . This can be proven to be so by virtue ofthe classical algebra the constraints satisfy (for specific quantum cosmology examplessee [19]).We are now ready to proceed with the quantization of our system, in close analogyto the scheme above outlined. In order to realize the equivalent to step 1, we first definethe quantum analogue of H ( r ) ≈ H ( r )Φ = 0 ↔ − ρ ( r ) ( δ Φ δ ρ ( r ) ) ′ + σ ′ ( r ) δ Φ δ σ ( r ) − χ ( r ) ( δ Φ δ χ ( r ) ) ′ = 0 . (3.3)9s explained in the beginning of the section, the action of ˆ H ( r ) on all smooth functionalsis well defined, i.e., produces no δ (0)’s. It can be proven that, in order for such afunctional to be annihilated by this linear quantum operator, it must be scalar, i.e. havethe formΦ = Z ρ (˜ r ) f (cid:0) Σ (0) , Σ (1) , . . . , Σ ( n ) , X (0) , X (1) , . . . , X ( n ) (cid:1) d ˜ r (3.4a)Σ (0) ≡ σ (˜ r ) , Σ (1) ≡ σ ′ (˜ r ) ρ (˜ r ) , . . . , Σ ( n ) ≡ ρ (˜ r ) dd ˜ r . . . |{z} n − σ (˜ r ) ! (3.4b) X (0) ≡ χ (˜ r ) ρ (˜ r ) , X (1) ≡ ρ (˜ r ) (cid:18) χ (˜ r ) ρ (˜ r ) (cid:19) ′ , . . . , X ( n ) ≡ ρ (˜ r ) dd ˜ r . . . |{z} n − χ (˜ r ) ρ (˜ r ) ! (3.4c)where f is any function of its arguments. We note that, as it is discussed at the end ofthe previous section, σ ′ ρ is the only scalar first derivative of σ , and likewise for the higherderivatives. The proof of this statement is analogous to the proof of the correspondingresult concerning full gravity [37]: consider an infinitesimal r -reparametrization ˜ r = r − η ( r ). Under such a change, the left-hand side of (3.4), being a number, must remainunaltered. If we calculate the change induced on the right-hand side we arrive at0 = Z (cid:20) f δρ + ρ δfδσ δσ + ρ δfδ ( χ/ρ ) δ (cid:18) χρ (cid:19)(cid:21) dr = Z [ ρ ˆ H ( f )] η ( r ) dr, (3.5)where use of (2.15) and a partial integration has been made. Since this must hold forany η ( r ), the result sought for is obtained.We now turn to the second linear constraint and try to see what are the restrictionsit brings into our space of state vectors. We defineˆ H ( r )Φ = 0 ↔ ( δ Φ δ χ ( r ) ) ′ = 0 ↔ δ Φ δ χ ( r ) = k, (3.6)where k is any constant (with respect to r) independent of the basic fields and theirderivatives, and Φ is given by (3.4a) − (3.4c). As we argued before, the functional deriva-tive δδχ ( r ) acting on X ( n ) will produce, upon partial integration of the n th derivative of theDirac delta function, a term proportional to X (2 n ) . Since the arguments of f in (3.4a)reach only up to X ( n ) , it is evident that f must be such that the coefficient of X (2 n ) vanishes; more precisely δ Φ δ χ ( r ) = k ↔ . . . + Z ρ (˜ r ) ∂f∂X ( n ) (˜ r ) δX ( n ) (˜ r ) δχ ( r ) d ˜ r = k ↔ . . . + Z ρ (˜ r ) ∂f∂X ( n ) (˜ r ) 1 ρ (˜ r ) dd ˜ r . . . |{z} n − δ ( r, ˜ r ) ρ (˜ r ) ! d ˜ r = k ↔ . . . +( − n Z ∂ f∂ ( X ( n ) (˜ r )) X (2 n ) (˜ r ) δ ( r, ˜ r ) d ˜ r = k ↔ . . . +( − n ∂ f∂ ( X ( n ) ) X (2 n ) = k. . . . do not involve X (2 n ) and (3.6) must be satisfiedidentically for all X k ’s k = 0 , , ... n , we conclude that ∂ f∂ ( X ( n ) ) = 0 in order for thisequation to have a possibility to be satisfied. Subsequently: f = f (cid:0) Σ (0) , . . . , Σ ( n ) , X (0) , . . . , X ( n − (cid:1) X ( n ) + f (cid:0) Σ (0) , . . . , Σ ( n ) , X (0) , . . . , X ( n − (cid:1) . Now, the term in Φ corresponding to f is, up to a surface term, equivalent to a generalterm depending on X (0) , . . . , X ( n − only: indeed,Φ = Z ρ (˜ r ) f ρ (˜ r ) dd ˜ r X ( n − d ˜ r, which upon subtraction of the surface term A = Z d ˜ r dd ˜ r (cid:18)Z dX ( n − f (cid:19) produces a smooth functional with arguments up to X ( n − only. Since a surface term inΦ does not affect the outcome of the variational derivative δ Φ δ χ ( r ) , we conclude that only f is important for the local part of Φ. The entire argument can be repeated successivelyfor n − , n − , . . . ,
1; Therefore all X ( n ) ’s are suppressed from f except when n = 0.Thus, finally, upon inserting into (3.6) the resulting functional:Φ = Z ρ (˜ r ) h (cid:0) Σ (0) , . . . , Σ ( n ) , X (0) (cid:1) d ˜ r we obtain δ Φ δ χ ( r ) = k ↔ Z ρ (˜ r ) ∂h∂X (0) δ ( r, ˜ r ) ρ (˜ r ) d ˜ r = k ↔ ∂h∂X (0) = k ↔ h = k χ ( r ) ρ ( r ) + L (cid:0) Σ (0) , . . . , Σ ( n ) (cid:1) . We have thus reached the conclusion that the imposition of both linear quantum opera-tors ˆ H and ˆ H dictates the form of the smooth functional to be:Φ = k Z d ˜ r χ (˜ r ) + Z d ˜ r ρ (˜ r ) L (cid:0) Σ (0) , . . . , Σ ( n ) (cid:1) . (3.7)We now try to realize step 2 of the programm previously outlined. We have to definethe equivalent of Kuchaˇr’s induced metric on the so far space of “physical” states Φdescribed by (3.7) which are the analogues, in our case, of Kuchaˇr’s physical variables q α . Let us start our investigation by considering one initial candidate of the above form.Then, generalizing the partial to functional derivatives, the induced metric will be givenby g ΦΦ = G αβ δ Φ δ x α δ Φ δ x β , (3.8)11here ( x α , x β ) = ( ρ, σ, χ ) and G αβ is given by (2.10). Note that this metric is well de-fined since it contains only first functional derivatives of the state vectors, as opposedto any second order functional derivative operator that might have been considered as aquantum analogue of the kinetic part of H o . Nevertheless, g ΦΦ is a local function andnot a smooth functional. It is thus clear that, if we want the induced metric g ΦΦ tobe composed out of the “physical” states annihilated by ˆ H , ˆ H , we must establish acorrespondence between local functions and smooth functionals. A way to achieve thisis to adopt the following ansatz: Assumption:
We assume that, as part of the re-normalization procedure, we are per-mitted to map local functions to their corresponding smeared expressions e.g., χ ( r ) ↔ R d ˜ rχ (˜ r ) . Let us be more specific, concerning the meaning of the above Assumption. Let F be thespace which contains all local functions, and define the equivalence relations ∼ : { f ( r ) ∼ f (˜ r ) , ˜ r = g ( r ) } , ≈ : { h ( r ) ≈ h (˜ r ) d ˜ rd r , ˜ r = g ( r ) } (3.9)for scalars and densities respectively.Now let F o = { f ∈ F , mod ( ∼ , ≈ ) } and F I the space of the smeared functionals.We define the one to one maps G , G − G : F o
7→ F I : χ ( r ) Z χ ( r ) dr, G − : F I
7→ F o : Z χ ( r ) dr χ ( r ) (3.10)The necessity to define the maps G , G − on the equivalence classes and not on the indi-vidual functions, stems out of the fact that we are trying to develop a quantum theoryof the geometries (2.6), (2.7) and not of their coordinate representations. If we hadtried to define the map G from the original space F to F I we would end up with stateswhich would not be invariant under spatial coordinate transformations ( r - reparameter-izations). Indeed, one can make a correspondence between local functions and smearedexpressions, but smeared expressions must contain another arbitrary smearing function,say s ( r ). Then the map between functions and smeared expressions is one to one (as isalso the above map) and is given by multiplying by s ( r ) and integrating over r ; whilethe inverse map is given by varying w.r.t. s ( r ). However, this would be in the oppositedirection from that which led us to the states (3.7) by imposition of the linear operatorconstraints. As an example consider the action of these operators on two particular casesof the states (3.7), containing the structure s ( r ) :ˆ H ( r ) Z s (˜ r ) ρ (˜ r ) σ (˜ r ) d ˜ r = − s ′ ( r ) ρ ( r ) σ ( r ) = 0 for arbitrary s ( r ) (3.11)ˆ H ( r ) Z s (˜ r ) χ (˜ r ) d ˜ r = s ′ ( r ) = 0 for arbitrary s ( r ) (3.12)12hus, every foreign to the geometry structure s ( r ) is not allowed to enter the physicalstates.Now, after the correspondence has been established, we can come to the basic prop-erty the induced metric must have. In the case of finite degrees of freedom the inducedmetric depends, up to a conformal scaling, on the physical coordinates q α by virtueof (3.2). In our case, due to the dependence of the configuration variables on the ra-dial coordinate r , the above property is not automatically satisfied; e.g. the functionalderivative δδσ ( r ) acting on Σ ( n ) will produce, upon partial integration of the n th derivativeof the Dirac delta function, a term proportional to Σ (2 n ) . Therefore, since L in (3.7)contains derivatives of σ ( r ) up to Σ ( n ) , the above mentioned property must be enforced .The need for this can also be traced to the substantially different first Poisson bracketin (2.13), which signals a non trivial mixing between the dynamical evolution generator H o and the linear generators H i .Thus, according to the above reasoning, in order to proceed with the generalization ofKuchaˇr’s method, we have to demand that: Requirement: L (cid:0) Σ (0) , . . . , Σ ( n ) (cid:1) must be such that g ΦΦ becomes a general function, say F (cid:0) k χ ( r ) + ρ ( r ) L (Σ (0) , . . . , Σ ( n ) ) (cid:1) of the integrand of Φ , so that it can be considered afunction of this state: g ΦΦ Assumption ⇐⇒ F (cid:0) k R χ (˜ r ) d ˜ r + R ρ (˜ r ) L (Σ (0) , . . . , Σ ( n ) ) d ˜ r (cid:1) = F (Φ).At this point, we must emphasize that the application of the Requirement in the sub-sequent development of our quantum theory will result in very severe restrictions on theform of (3.7). Essentially, χ ( r ) as well as all higher derivatives of σ ( r ) (i.e Σ (2) . . . Σ ( n ) ))are eliminated from Φ (see (3.14), (3.23)). This might, at first sight, strike as odd; in-deed, the common belief is that all the derivatives of the configuration variables shouldenter the physical states. However, before the imposition of both the linear and thequadratic constrains there are no truly physical states. Thus, no physical states are lostby the imposition of the Requirement ; ultimately the only true physical states are thesolutions to (3.24).Having clarified the way in which we view the
Assumption and
Requirement above, we now proceed to the restrictions implied by their use.A first consequence of the requirement that g ΦΦ = F (cid:0) k χ ( r ) + ρ ( r ) L (Σ (0) , . . . , Σ ( n ) ) (cid:1) is the vanishing of k . This follows from (a) the property that g ΦΦ is homogenous in thefunctional derivative δδχ ( r ) , (b) that G αβ in (2.10) does not contain any χ ( r ); namely g ΦΦ = . . . + G δ Φ δχ ( r ) δ Φ δχ ( r ) ↔ g ΦΦ = . . . |{z} no χ + ρσ k ≡ F ( k χ + ρ L ) . Since . . . are terms not involving χ ( r ), the final identification is possible iff k = 0. Thus,Φ is reduced to: Φ = Z d ˜ r ρ (˜ r ) L (cid:0) Σ (0) , . . . , Σ ( n ) (cid:1) . (3.13)We now turn to the degree of derivatives ( n ) of σ ( r ). The situation is similar to13he corresponding case with X ( n ) considered before; again the functional derivative δδσ ( r ) acting on Φ will bring a maximum term Σ (2 n ) while δδρ ( r ) a corresponding term Σ (2 n − .More precisely g ΦΦ = . . . + 2 G δ Φ δρ ( r ) δ Φ δσ ( r ) . Where the functional derivatives are: δ Φ δσ = . . . + Z ρ ∂L∂ Σ ( n ) δ Σ ( n ) δσ d ˜ r = . . . + Z ρ ∂L∂ Σ ( n ) ρ dd ˜ r . . . |{z} n − δ ( r, ˜ r ) ! d ˜ r == . . . − Z dd ˜ r (cid:18) ∂L∂ Σ ( n ) (cid:19) ρ dd ˜ r . . . |{z} n − δ ( r, ˜ r ) ! d ˜ r == . . . − Z ρ ∂ L∂ (Σ ( n ) ) Σ ( n +1) ρ dd ˜ r . . . |{z} n − δ ( r, ˜ r ) ! d ˜ r == . . . + ( − n Z ρ (˜ r ) ∂ L∂ (Σ ( n ) ) Σ (2 n ) δ ( r, ˜ r ) d ˜ r == . . . + ( − n ρ ∂ L∂ (Σ ( n ) ) Σ (2 n ) and δ Φ δρ = . . . + Z ρ ∂L∂ Σ ( n ) δ Σ ( n ) δρ d ˜ r = . . . + Z ρ ∂L∂ Σ ( n ) ρ dd ˜ r . . . |{z} n − − δ ( r, ˜ r ) ρ (˜ r ) σ ′ (˜ r ) ! d ˜ r == . . . + Z ρ ∂L∂ Σ ( n ) ρ dd ˜ r . . . |{z} n − − δ ( r, ˜ r ) ρ (˜ r ) Σ (1) ! d ˜ r = . . . − Z dd ˜ r (cid:18) ∂L∂ Σ ( n ) (cid:19) ρ dd ˜ r . . . |{z} n − − δ ( r, ˜ r ) ρ (˜ r ) Σ (1) ! d ˜ r == . . . + ( − n − Z ∂ L∂ (Σ ( n ) ) Σ (2 n − Σ (1) δ ( r, ˜ r ) d ˜ r == . . . + ( − n − ∂ L∂ (Σ ( n ) ) Σ (2 n − Σ (1) . Therefore g ΦΦ = . . . − ρ σ ( − n − ∂ L∂ (Σ ( n ) ) ! Σ (1) Σ (2 n − Σ (2 n ) , where the . . . stand for all other terms, not involving Σ (2 n ) . Now, according to theaforementioned Requirement we need this to be a general function, say F ( ρL ), and for14his to happen the coefficient of Σ (2 n ) must vanish, i.e. ∂ L∂ (Σ ( n ) ) = 0 ⇔ L = L (cid:0) Σ (0) , . . . , Σ ( n − (cid:1) Σ ( n ) + L (cid:0) Σ (0) , . . . , Σ ( n − (cid:1) . Again the term of Φ corresponding to L is, up to a total derivative, equivalent toa local smooth functional containing Σ (0) , . . . , Σ ( n − . The argument can be repeatedfor ( n − , ( n − , . . . ,
2. The case n = 1 needs separate consideration since, uponelimination of the linear in Σ (2) term we are left with a local function of Σ (1) , and thusthe possibility arises to meet the Requirement by solving a differential equation for L .In more detail, if Φ ≡ Z ρ (˜ r ) L (cid:0) σ, Σ (1) (cid:1) d ˜ r, (3.14) g ΦΦ reads g ΦΦ = − ρ (cid:18) L − Σ (1) ∂L∂ Σ (1) (cid:19) (cid:20) L − Σ (1) ∂L∂ Σ (1) + 2 σ (cid:18) ∂L∂σ − Σ (1) ∂ L∂σ ∂ Σ (1) (cid:19)(cid:21) ++2 ρ σ (cid:18) L − Σ (1) ∂L∂ Σ (1) (cid:19) ∂ L∂ (Σ (1) ) Σ (2) . (3.15)Through the definition H ≡ L − Σ (1) ∂L∂ Σ (1) (3.16)we obtain ∂H∂σ = ∂L∂σ − Σ (1) ∂ L∂σ ∂ Σ (1) ,∂H∂ Σ (1) = − Σ (1) ∂ L∂ (Σ (1) ) . Thus (3.15) assumes the form g ΦΦ = − ρ (cid:18) H + 2 σ H ∂H∂σ + 2 σ Σ (1) H ∂H∂ Σ (1) Σ (2) (cid:19) , which upon addition, by virtue of the Assumption , of the surface term A = ddr (cid:18)Z σ Σ (1) H ∂H∂ Σ (1) d Σ (1) (cid:19) gives g ΦΦ = − ρ (cid:18) H + 2 σ H ∂H∂σ − Σ (1) ∂∂σ Z σ Σ (1) H ∂H∂ Σ (1) d Σ (1) (cid:19) . (3.17)Since in the last expression we have only a multiplicative ρ ( r ), it is obvious that the Requirement g ΦΦ = F ( ρ L )15an be satisfied only by g ΦΦ = − κ ρ L, (3.18)with g ΦΦ given by (3.17). Upon differentiation of this equation with respect to Σ (1) weget − ∂∂σ Z σ Σ (1) H ∂H∂ Σ (1) = κ ∂L∂ Σ (1) . Multiplying the last expression by Σ (1) and subtracting it from (3.18) we end up withthe autonomous necessary condition for H ( σ, Σ (1) ): H (cid:18) H + 2 σ ∂H∂σ − κ (cid:19) = 0 , where (3.16) was also used. The above equation can be readily integrated giving H = 0 ,H = κ + a (Σ (1) ) √ σ , where a (Σ (1) ) is an arbitrary function of its argument. The first possibility gives accord-ing to (3.16) L = λ Σ (1) which, however, contributes to Φ a surface term, and can thusbe ignored. Inserting the second solution into (3.16) we construct a partial differentialequation for L , namely L − Σ (1) ∂L∂ Σ (1) = κ + a (Σ (1) ) √ σ , which upon integration gives L = κ − Σ (1) √ σ Z a (Σ (1) )Σ (1)2 d Σ (1) + c ( σ ) Σ (1) . Since this form of L emerged as a necessary condition, it must be inserted (along with H ) in (3.18). The result is that c ( σ ) = 0. Thus L reads L = κ − Σ (1) √ σ Z a (Σ (1) )Σ (1)2 d Σ (1) . (3.19)By assuming that the Σ (1) –dependent part of L equals b (Σ (1) ), i.e. − Σ (1) Z a (Σ (1) )Σ (1)2 d Σ (1) = b (Σ (1) ) , we get, upon a double differentiation with respect to Σ (1) , the ordinary differential equa-tion − a ′ (Σ (1) )Σ (1) = b ′′ (Σ (1) )16ith solution a (Σ (1) ) = b (Σ (1) ) + κ − Σ (1) b ′ (Σ (1) ) , where κ is a constant. Substituting this equation into (3.19) and performing a partialintegration we end up with L = κ + κ √ σ + b (Σ (1) ) √ σ . (3.20) κ , κ and b (Σ (1) ) being completely arbitrary and to our disposal; the two simpler choices κ = 0 , b (Σ (1) ) = 0 and κ = 0 , b (Σ (1) ) = 0 lead respectively to the following two basiclocal smooth functionals: q = Z d ˜ rρ (˜ r ) , q = Z d ˜ r ρ (˜ r ) p σ (˜ r ) . (3.21)The next simplest choice κ = 0 , κ = 0 and b (Σ (1) ) arbitrary leads to a generic q = R d ˜ rρ (˜ r ) b (Σ (1) ) √ σ (˜ r ) . However, it can be proven that, for any choice of b (Σ (1) ), the correspond-ing renormalized induced metric g AB = G αβ δq A δx α δq B δx β where A, B = 1 , , g AB gives: g = G αβ δq δx α δq δx β = − ρ Assumption ⇐⇒ g ren = − q ,g = G αβ δq δx α δq δx β = − ρ √ σ Assumption ⇐⇒ g ren = − q ,g = G αβ δq δx α δq δx β = 0 Assumption ⇐⇒ g ren = 0 ,g = G αβ δq δx α δq δx β = ρ (cid:18) − b √ σ + Σ (1) √ σ b ′ + √ σ Σ (2) b ′′ (cid:19) Assumption ⇐⇒ g ren = Z drρ (cid:18) − b √ σ + Σ (1) √ σ b ′ + √ σ Σ (2) b ′′ (cid:19) − Z dr ddr (cid:18)Z d Σ (1) √ σ b ′′ (cid:19) == − Z drρ b √ σ = − q ,g = G αβ δq δx α δq δx β = ρ Σ (2) b ′′ = ddr b ′ Assumption ⇐⇒ g ren = 0 ,g = G αβ δq δx α δq δx β = 2 ρ (cid:0) b − Σ (1) b ′ (cid:1) Σ (2) b ′′ Assumption ⇐⇒ g ren = 2 Z drρ (cid:0) b − Σ (1) b ′ (cid:1) Σ (2) b ′′ − Z dr ddr (cid:20)Z d Σ (1) (cid:0) b − Σ (1) b ′ (cid:1) b ′′ (cid:21) = 0 , ′ we denote differentiation with respect to Σ (1) . Thus the renormalized inducedmetric reads g ABren = − q q q q q . Effecting the transformation (˜ q , ˜ q , ˜ q ) = (cid:16) q , q , ln q q (cid:17) we bring g ABren into a manifestlydegenerate form: g ABren = − q q q . So, it seems as though the relevant part of the renormalized metric is described by theupper 2 × V = 2 Λ q which indeed does not contain any Σ (1) term.However, this is not the end of our investigation for a suitable space of state vectors:the argument leading to q , q depends upon the original choice of one initial candidatesmooth scalar functional (3.14); to complete the search we must close the circle bystarting with the two already secured smooth functionals ( q , q ), and a third of thegeneral form q = Z dr ρ L (Σ (1) ) , since the σ dependence has already been fixed to either 1 or √ σ . The calculation of the,related to q , components of the induced metric g AB gives: g = ρ (cid:0) − L + Σ (1) L ′ + σ Σ (2) L ′′ (cid:1) Assumption ⇐⇒ g ren = Z drρ (cid:0) − L + Σ (1) L ′ + σ Σ (2) L ′′ (cid:1) − Z dr ddr (cid:18)Z d Σ (1) σ L ′′ (cid:19) = − Z drρ L == − q ,g = ρ (cid:18) − L √ σ + Σ (1) √ σ L ′ + √ σ Σ (2) L ′′ (cid:19) Assumption ⇐⇒ g ren = Z drρ (cid:18) − L √ σ + Σ (1) √ σ L ′ + √ σ Σ (2) L ′′ (cid:19) − Z dr ddr (cid:18)Z d Σ (1) √ σ L ′′ (cid:19) == − Z drρ L √ σ Assumption = − q q q ,g = − ρ (cid:0) L − Σ (1) L ′ (cid:1) + 2 ρ σ (cid:0) L − Σ (1) L ′ (cid:1) Σ (2) L ′′ .
18y following the procedure presented between (3.15) and (3.17) we end up with theexpression g = − ρ (cid:20)(cid:0) L − Σ (1) L ′ (cid:1) − Σ (1) Z d Σ (1) Σ (1) ∂∂ Σ (1) (cid:0) L − Σ (1) L ′ (cid:1) (cid:21) , the expression inside the square brackets being a generic function of Σ (1) and therefore,also of L : let this function be parameterized as L (cid:0) Σ (1) (cid:1) − F [ L ( Σ (1) ) ] F ′ [ F [ L ( Σ (1) ) ]] ; this “peculiar”parametrization of the arbitrariness in L (cid:0) Σ (1) (cid:1) has been chosen in order to facilitate thesubsequent proof that this freedom is a pure general coordinate transformation (gct) ofthe induced re-normalized metric. Indeed, let us first take the simplest non trivial choice L (cid:0) Σ (1) (cid:1) ≡ Σ (1)2 which results in the re-normalized metric g ABren = − q q q q q q q q q q q − q ) q , g ABren = 12 q − q − q − q q ( q ) − q q q ) . (3.22)Considering a generic L (cid:0) Σ (1) (cid:1) , i.e. x = R dr ρ L (Σ (1) ) (along with (3.21)) we are led to g = − x , g = − q x q and g = − ρ (cid:20) L − F [ L ] F ′ [ F [ L ]] (cid:21) = − ( ρ L ) ρ + 4 ρ F [ ρ Lρ ] F ′ [ F [ ρ Lρ ]] Assumption ⇐⇒ g ren = − ( x ) q + 4 q F [ x q ] F ′ [ F [ x q ]] Remarkably enough, the new induced re-normalized metric can be put in gct equivalencewith the metric (3.22) through the transformation( q , q , x ) = ( q , q , q F − ( q q )) , with F − denoting the function inverse to F, i.e. F − ( F ( x )) ≡ x .We can therefore consider, without loss of generality, the reduced re-normalized man-ifold to be parameterized by the following three smooth scalar functionals: q = Z d ˜ rρ (˜ r ) , q = Z d ˜ r ρ (˜ r ) p σ (˜ r ) , q = Z d ˜ r σ ′ (˜ r ) ρ (˜ r ) . (3.23)19ny other functional, say q = R d ˜ r ρ (˜ r ) L ( σ (˜ r ) , Σ (1) (˜ r )), can be considered as a func-tion of q , q , q ; indeed, since the scalar functions appearing in the integrands of q , q form a base in the space of σ, Σ (1) , we can express the generic L in q as F ( ρρ √ σ , ρ Σ (1)2 ρ )),which (through the Assumption ) gives q = q F ( q q , q q ).The geometry of this space is described by the induced re-normalized metric (3.22).Any function Ψ( q , q , q ) on this manifold is of course annihilated by the quantum linearconstraints, i.e.ˆ H Ψ( q , q , q ) = ∂ Ψ( q , q , q ) ∂q ˆ H q + ∂ Ψ( q , q , q ) ∂q ˆ H q + ∂ Ψ( q , q , q ) ∂q ˆ H q = 0ˆ H Ψ( q , q , q ) = ∂ Ψ( q , q , q ) ∂q ˆ H q + ∂ Ψ( q , q , q ) ∂q ˆ H q + ∂ Ψ( q , q , q ) ∂q ˆ H q = 0since the derivatives with respect to r are transparent to the partial derivatives of Ψ(which are, just like the q A ’s, r-numbers).The final restriction on the form of Ψ will be obtained by the imposition of thequantum analog of the quadratic constraint H o . According to the above expositionwe postulate that the quantum gravity of the geometries given by (2.6), (2.7) will bedescribed by the following partial differential equation (in terms of the q A ’s)ˆ H o Ψ ≡ [ − ✷ c + V ren ] Ψ( q , q , q ) = 0 (3.24)with ✷ c = ✷ + d −
24 ( d − R (3.25)being the conformal Laplacian based on g AB ren , R the Ricci scalar, and d the dimensionsof g AB ren . Metric (3.22) is conformally flat with Ricci scalar R = q , and its dimensionis d = 3. The re-normalized form of the potential (2.11) offers us the possibility tointroduce, in a dynamical way, topological effects into our wave functional: Indeed,under our Assumption , the first term becomes 2 Λ q while the second, being a totalderivative, becomes A T ≡ σ ′ ρ | βα ( if α < r < β ). In the spirit previously explained weshould drop this term, however one could keep it, thus arriving at V ren = 2 Λ q + A T and the Wheeler-deWitt equation is finally given as2 q Λ Ψ( q , q , q ) + A T Ψ( q , q , q ) − q Ψ( q , q , q ) + q q ∂ Ψ( q , q , q ) ∂q + q q ∂ Ψ( q , q , q ) ∂q + 34 ∂ Ψ( q , q , q ) ∂q + q q q ∂ Ψ( q , q , q ) ∂q ∂q + q ∂ Ψ( q , q , q ) ∂q ∂q + q ∂ Ψ( q , q , q ) ∂q ∂q + q ∂ Ψ( q , q , q ) ∂ ( q ) − ( q ) q ∂ Ψ( q , q , q ) ∂ ( q ) = 0 . (3.26)20he change to new coordinates ( x , x , x ) described by( q , q , q ) = (cid:16) e x , e ( x + x ) , e x + √ x (cid:17) transforms the metric into the manifestly conformally flat form diag { e x , − e x , − e x } and brings (3.26) into the form2 e x Λ Ψ( x , x , x ) + A T e x Ψ( x , x , x ) −
132 Ψ( x , x , x ) + 14 ∂ Ψ( x , x , x ) ∂x − ∂ Ψ( x , x , x ) ∂ ( x ) + 12 ∂ Ψ( x , x , x ) ∂ ( x ) + 12 ∂ Ψ( x , x , x ) ∂ ( x ) = 0 . (3.27)This equation is readily solved by the method of separation of variables: assumingΨ( x , x , x ) = X ( x ) X ( x ) X ( x ) and dividing (3.27) by Ψ we get the three ordinarydifferential equations:12 X ( x ) d X ( x ) d ( x ) + 132 = m + n, (3.28a)12 X ( x ) d X ( x ) d ( x ) + 14 X ( x ) dX ( x ) dx + 2 e x Λ + A T e x = n, (3.28b)12 X ( x ) d X ( x ) d ( x ) = m, (3.28c)where m and n are separation constants. Their solutions for A T = 0 are: X ( x ) = c e √ m +32 n − x + c e − √ m +32 n − x , (3.29a) X ( x ) = c e − x / J − √ n +1 (cid:16) e x √ Λ (cid:17) + c e − x / J √ n +1 (cid:16) e x √ Λ (cid:17) (3.29b) X ( x ) = c e √ m x + c e −√ m x , (3.29c)where J ± √ n +1 (cid:16) e x √ Λ (cid:17) Bessel functions of the first kind and of non-integral order.
We have considered the canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term, under the as-sumption of the existence of one Killing vector field. The implementation of the Diracalgorithm for this action results, at the classical level, in two linear (momentum) andone quadratic (Hamiltonian) first class constraints. The first linear constraint (2.9b) isshown to correspond to arbitrary changes of the radial coordinate. The second linearconstraint (2.9c) owes its existence to the G symmetry imposed, a fact that is by itselfworth mentioning. The quadratic constraint (2.9a) is, as usual, the generator of thetime evolution (using the classical equation of motion, see pp. 21 of [26]). To avoid21n ill-defined action of the quantum analogues of the linear constraints, we adopt asour initial collection of state vectors all smooth (integrals over the radial coordinate r )functionals. The first quantum linear constraint entails a reduction of this collection toall smooth scalar functionals (3.4). The subsequent imposition of the second quantumlinear constraint further reduces these states to (3.7). At this stage the need emergesto somehow obtain, through the midi-superspace metric (2.10), an induced metric (3.8)whose components are given in terms of the same states. This leads us to firstly adopt aparticular (formal) re-normalization prescription (see Assumption pp. 11) and secondlyimpose the
Requirement . As a result, the final collection of state vectors is reducedto the three unique smooth scalar functionals (3.23). The quantum analogue of the ki-netic part of (2.9a) is then realized as the conformal Laplace-Beltrami operator basedon the induced re-normalized metric (3.22), resulting in the Wheeler-DeWitt equation(3.26). Effecting an appropriate change of variables the equation is made separable and,subsequently, completely integrated.We now come to two issues we deem worth-wile discussing:The first has to do with the apparent absence of the quantum analogue of the classicalPoisson algebra (2.13). It seems to us that the primary purpose of searching for a (self-adjoint) representation of this algebra on a Hilbert space is to secure, through Frobenius’Theorem, the consistency of the quantum theory emanating from the chosen operatorconstraints (3.3), (3.6) and (3.24). But this aim is superseded by the finding of thecommon kernel, i.e. the solutions (3.29a). Furthermore, if, after the issue of the measureis resolved, the Hilbert space is to be composed out of these states, the algebra of theoperator constraints will be reduced to an Abelian one.The second concerns our choice of following Dirac’s Proposal to implement the firstclass constraints (2.9b),(2.9c) and (2.9a) as operator conditions annihilating the wavefunctional, rather than ”imposing” them at the classical level, as is the case for the vastmajority of relevant works in 2 + 1 gravity. Within Dirac’s Theory for constraint systemsthe only correct way we are aware of to impose the first class constraints at the classicallevel is to choose a “gauge”, i.e. to select a phase-space function for each first classconstraint so that constraints plus “gauge” fixing conditions become second class: thenand only then one is allowed to solve them all, at the very important expense of beingobliged there-afterwards to use Dirac rather than Poisson brackets. Since the construc-tion of these brackets makes use of the matrix formed by the second class constraints, itis obvious that one will, in general, be carrying to the subsequent quantization procedureproperties of the choice made. In such a situation one is never certain of how and/or towhat extent the “gauge” fixing chosen will infiltrate and affect the emanating quantumtheory, especially if the “gauge” involved is so immense and complicated as the groupof space-time coordinate transformations. This constitutes our primary motivation forfollowing Dirac’s proposal which we interpret as an elimination of the “gauge” freedomat the quantum level. The fact that in 2+1 dimensions it seems more easy to classicallyseparate the “gauge” from the “true” degrees of freedom does not at all diminish thestrength of this motivation, much more in view of the fact that our method is meant tobe applicable to spherically symmetric 3+1 geometries as well.22enerally (and somewhat loosely) speaking, the point of the exercise as we see it is,at a first stage, to assign a unique number between 0 and 1 to each and every geometry(2.6)-(2.7), in a way that is independent of the coordinate system used to represent themetric. Of course, at the present status of things we cannot do this, since the followingtwo problems remain to be solved: i) render finite the three smooth functionals (3.23)and ii) select an appropriate inner product.The first will need a final regularization of q , q , q , but most probably, the detailedway to do this will depend upon the particular geometry under consideration. Forexample, it is obvious that for the metric (2.3) three segments of the range (0 , ∞ ) of theradial coordinate have to be separately considered, while for the metrics (2.4), (2.5) onesegment (the entire range) is enough.For the second, a natural choice would be the determinant of the induced re-normalizedmetric, although the problem with the positive definiteness may dictate another choice.An analogous treatment of the (3+1)-dimensional spherically symmetric configura-tions can be carried through, a task that we have already under active consideration.23 cknowledgments One of the authors (T. C.) acknowledges pleasant and fruitful discussions with Ass. Prof.C. Chiou and Prof. C. N. Ktorides. Another author (G. O. Papadopoulos) is a KillamPostdoctoral Fellow and acknowledges the relevant support from the Killam Foundation.
References [1] P.A.M. Dirac, Can. J. Math. , (1950) 129.[2] P.A.M. Dirac, Can. J. Math. , (1951) 1.[3] P.A.M. Dirac, Proc. R. Soc. (London) A , (1958) 326.[4] P.A.M. Dirac, Lectures on Quantum Mechanics , Yeshiva University, AcademicPress, New York (1964).[5] P.G. Bergmann, Rev. Mod. Phys. , (1961) 510.[6] F.A. Berezin and M.S. Marinov, Ann. Phys. (N.Y.) , (1977) 336.[7] C. Becchi, C. Rouet, and R. Stora, Ann. Phys. (N.Y.) , (1976) 287.[8] E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective ,Wiley, New York (1974).[9] A.J. Hanson, T. Regge and C. Teitelboim,
Constrained Hamiltonian Systems , Ac-cademia Nazionale dei Lincei, Vatican (1976).[10] K. Sundermeyer,
Constrained Dynamics , Lecture Notes in Physics , Springer-Verlag, Heidelberg (1982).[11] D.M. Gitman and I.V. Tyutin,
Quantization of Fields with Constraints , SpringerSeries in Nuclear and Particle Physics, (1991).[12] J. Govaers,
Hamiltonian Quantization and Constrained Dynamics , Leuven Univer-sity Press, Leuven (1991).[13] M. Henneaux and C. Teitelboim,
Quantization of Gauge Systems , Princeton Uni-versity Press, Princeton (1992).[14] A. Wipf,
Hamilton’s Formalism for Systems with Constraints , hep-th/9312078.[15] Th. Thiemann,
Modern Canonical Quantum General Relativity , Cambridge Univer-sity Press, Cambridge (2007).[16] B.S. DeWitt, Phys. Rev. (5), (1967) 1113.2417] P.G. Bergmann, Phys. Rev. , (1966) 1078.[18] M.P. Ryan Jr. and L.C. Shepley,
Homogeneous Relativistic Cosmologies , PrincetonUniversity Press, Princeton (1975).[19] T. Christodoulakis,
Quantum Cosmology , Lect.Notes Phys. : 318-350, SpringerVerlag (2002); gr-qc/0109059.[20] A. Krasinski,
Inhomogeneous Cosmological Models , Cambridge University Press,Cambridge (1997).[21] P. Thomi, B. Isaak and P. Hajicek, Phys. Rev. D , (1984) 1168.[22] P. Hajicek, Phys. Rev. D , (1984) 1178.[23] L.M.C.S. Rodrigues, I.D. Soares and J. Zanelli, Phys. Rev. Lett. , (1989) 989.[24] C. Kiefer, Quantum Gravity , 2nd ed., Oxford University Press, Oxford (2007), ch.7.[25] C. Kiefer, J. M¨uller-Hill and C. Vaz, Phys. Rev. D , (2006) 044025.[26] S. Carlip, Quantum Gravity in 2+1 dimensions , Cambridge University Press, Cam-bridge (2003).[27] S. Carlip, Class. Quantum Grav. , (2005) R85.[28] S. Carlip, Living Rev. Rel. , (2005) 1.[29] B.J. Schroers, Lessons from (2+1)-dimensional quantum gravity, arXiv:gr-qc/0710.5844.[30] P. Hajicek and K.V. Kuchaˇr, Phys. Rev. D , (1990) 1091.[31] P. Hajicek and K.V. Kuchaˇr, J. Math.Phys. , (1990) 1723.[32] M. Ba˜nados, C.Teitelboim, and J. Zanelli, Phys. Rev. Lett. , (1992) 1849.[33] M. Ba˜nados, M. Henneaux, C.Teitelboim, and J. Zanelli, Phys. Rev. D , (1993)1506.[34] T. Christodoulakis and G.O. Papadopoulos, arXiv:gr-qc / , (2006) 5291.[36] L.P. Eisenhart, Riemannian Geometry , Princeton University Press, Princeton, NewJersey, 5th printing (1964), pp.146.[37] T. Y. Thomas,