Towards Canonical Quantum Gravity for G 1 Geometries in 2+1 Dimensions with a Λ –Term T. Christodoulakis, G. Doulis, Petros A. Terzis ∗ † ‡ Nuclear and Particle Physics Section, Physics Department, 9 University of Athens, GR 157–71 Athens 0 0 E. Melas § 2 Technological Educational Institution of Lamia n a Electrical Engineering Department, GR 35–100, Lamia J 7 Th. Grammenos¶ Department of Mechanical Engineering, University of Thessaly, ] c q GR 383–34 Volos - r G.O. Papadopoulos g k [ Department of Mathematics and Statistics, Dalhousie University 2 Halifax, Nova Scotia, Canada B3H 3J5 v 7 A. Spanou ∗∗ 3 1 School of Applied Mathematics and Physical Sciences, 0 National Technical University of Athens, GR 157–80, Athens . 6 0 8 0 : v i X r a ∗[email protected] †[email protected] ‡[email protected] §[email protected] ¶[email protected] [email protected] ∗∗[email protected] 1 Abstract The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. The proper impo- sition of the quantum analogues of the two linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components (and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi-superspace metric (inferred from the kinetic part of the quadratic (Hamiltonian) constraint) must define on the space of these states an induced metric whose components are given in terms of the same states, which is made possible through an appropriate re-normalization assumption, severely reduces the possible state vectors to three unique (up to gen- eralcoordinatetransformations)smoothscalarfunctionals. Thequantumanalogue of the Hamiltonian constraint produces a Wheeler-DeWitt equation based on this reduced manifold of states, which is completely integrated. PACS Numbers: 04.60.Ds, 04.60.Kz 1 Introduction Dirac’s seminal work on his formalism for a self-contained treatment of systems with constraints [1], [2], [3], [4] has paved the way for a systematic treatment of constrained dynamics. Some of the landmarks in the study of constrained systems have been the connection between constraints and invariances [5], the extension of the formalism to describe fields with half-integer spin through the algebra of Grassmann variables [6] and the introduction of the BRST formalism [7]. All the classical results obtained so far have made up an armory prerequisite for the quantization of gauge theories and there are several excellent reviews studying constraint systems with a finite number of degrees of 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 of quantum 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 to address the quantization of (classes of) simplified geometries. An elegant way to achieve a degree of simplification is to impose some symmetry. For example, the assumption of a G symmetry group acting simply transitively on the surfaces of simultaneity, i.e. 3 the existence of three independent space-like Killing vector fields, leads to classical and subsequently quantum homogeneous cosmology (see, e.g., [18], [19]). The imposition of lesser symmetry, e.g. fewer Killing vector fields, results in the various inhomogeneous cosmological models [20]. The canonical analysis under the assumption of spherical symmetry, which is a G group acting multiply transitively on two-dimensional space- 3 like subsurfaces of the three-slices, has been first considered in [21], [22]. Quantum black holes have also been treated, for instance, in [23], [24]while in[25]alattice regularization 2 has been employed to deal with the infinities arising due to the ill-defined nature of the quantum operator constraints. Another way toarrive atsimplified modelsistoconsider lower dimensions. Forexam- ple, there is a vast literature on (2+1)-dimensional gravity (see, e.g., [26], [27], [28] and references 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 canonical quantization 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 Hamiltonian formulation of the reduced Einstein-Hilbert action principle, resulting in one (quadratic) Hamiltonian and two (linear) momentum first class constraints. In Section 3 we consider the quantization of this constrained system following Dirac’s proposal of implementing the quantum operator constraints as conditions annihilating the wave-function [4]. Our guide-line is a conceptual generalization of the quantization scheme developed in [30], [31]forthecaseofconstrained systems withfinitedegrees offreedom, tothepresent case. Even though after the symmetry reduction the system still represents a field theory (all remaining metric components depend on time and the radial coordinate), we manage to extract and subsequently completely solve a Wheeler-DeWitt equation in terms of three unique smooth scalar functionals of the appropriate components of the reduced spatial metric. This is achieved through an appropriate re-normalization assumption we adopt. Finally, some concluding remarks are included in the discussion. 2 Possible Metrics and Hamiltonian Formulation Our starting point is the action principle: I = d3x√ g(R 2Λ). (2.1) − − Z The equations of motion arising upon variation of this action are 1 R g R+Λg = 0, (2.2) IJ IJ IJ − 2 where I,J = 0,1,2. Of course, since in three dimensions the Riemann curvature tensor is 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 topological considerations are taken into account, the above space might be “enriched” containing, for example, the stationary BTZ “black” hole [32], [33] J2 −1 ds2 = (M Λr2)dt2 Jdtdφ+ M Λr2 + dr2 +r2dφ2 (2.3) − − − − 4r2 (cid:18) (cid:19) 3 or the “cosmological” solutions [34], [35] 1 1 ds2 = dt2 + (dr2 +dφ2), (2.4) −4t2Λ 2t√Λ 4 2 4 4e 4r ds2 = dt2 + dr2 + − dφ2. (2.5) − 16t2 Λ 16t2 Λ 16t2 Λ (cid:18) − (cid:19) − − Note that all these three line elements are locally AdS3 and therefore admit six local Killingfields. Theirdifferencesconsistinthetopologicalidentifications. Atthispoint,we deem it pertinent to explain our view concerning the issue of the bearing of topology on a local theory: The Hamiltonian formulation is by itself implying a space-time topology R Σ2. Consequently, what we are concerned with is the topology of the 2-slices. Since × the theory is local, it is implicitly assumed that the entire analysis holds in a coordinate patch. Different topologies can only affect the number of patches needed to cover the space and, therefore, can only impose restrictions on the range of the coordinates and/or the range of validity of local fields, such as the symmetry generators admitted by these metrics; The paradigm of the cylinder may help clarify our point: The integral curves of rotations in the plane are circles, but if one tries to draw a circle of radius R 2πL ≥ 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 situation one can, as many do, drop rotations altogether; this is the case in [32], [33], where four of the six Killing fields are considered as non-valid symmetries. On the other hand one can accept integral curves (circles) of radius R < 2πL (by suitably restricting the range of validity of the Killing field), which would simply result in the need of two patches to cover the cylinder with these lines. We adopt this latter point of view, as it seems to us much 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 our results. In this spirit we can say that the above metrics admit a G symmetry group. In what 6 follows, we consider a generalization consisting in the imposition of a G symmetry only, 1 i.e we impose one Killing vector field, say ξ = ∂ . Subsequently, all components of the ∂φ metric become functions of both the time and the radial coordinate only. The canonical decomposition of such a metric is given in terms of the spatial metric g (t,r), the lapse ij function No(t,r) and the shift “vector” N (t,r) [10]: i ds2 = (No)2 +gijN N dt2 +2N dtdxi +g dxidxj, (2.6) i j i ij − where (cid:0) (cid:1) σ σχ ρ2 +σ2χ2 σχ ρ2 −ρ2 g = σ , gij =   (2.7) ij   σχ ρ2 +σ2χ2 σχ σ     −ρ2 σρ2      with i,j = 1,2, and xi = (r,φ). The particular parametrization of g above has been ij chosen in such a way as to simplify the second linear constraint (see below), and conse- quently the resulting algebra. 4 For the Hamiltonian formulation of the system (2.6) (see, e.g., chapter 9 of [10]), we first define the vectors 1 ηI = 1, Ni , Ni gikN No − ≡ k (cid:0) (cid:1) FI = ηJ ηI ηI ηJ ;J − ;J whereI,J arespace-timeindicesand“;”standsforcovariantdifferentiationwithrespect to (2.6). Then, utilizing the Gauss-Codazzi equation (see, e.g., [36]), we eliminate all secondtime-derivativesfromtheEinstein-Hilbertactionandarriveatanactionquadratic in the velocities, I = d3x√ g(R 2Λ 2FI). The application of the Dirac algorithm − − − ;I results firstly in the three primary constraints P δL 0, Pi δL 0 and the R o ≡ δN˙o ≈ ≡ δN˙i ≈ Hamiltonian H = No +Ni dr, (2.8) o i H H Z (cid:0) (cid:1) where , are given by o i H H 1 = Gαβπ π +V (2.9a) o α β H 2 = σ π ρπ χπ (2.9b) H1 ′ σ − ρ′ − χ′ = π , (2.9c) H2 − χ′ ∂ the indices (α,β) take the values (ρ,σ,χ) and . The Wheeler-DeWitt midi- ′ ≡ ∂r superspace metric Gαβ reads ρ σ 0 − −   Gαβ = σ 0 0 , (2.10) −    ρ   0 0    σ2     while the potential V is σ′ ′ V = 2Λρ+ . (2.11) ρ (cid:18) (cid:19) The requirement for preservation, in time, of the primary constraints leads to the sec- ondary constraints 0, 0, 0. (2.12) o 1 2 H ≈ H ≈ H ≈ At this stage, a tedious but straightforward calculation produces the following “open” 5 Poisson bracket algebra of these constraints: (r), (r˜) = [g1j(r) (r)+g1j(r˜) (r˜)]δ (r,r˜) o o j j ′ {H H } H H (r), (r˜) = (r)δ (r,r˜) 1 o o ′ {H H } H (r), (r˜) = 0 (2.13) 2 o {H H } (r), (r˜) = (r)δ (r,r˜) (r˜)δ(r,r˜) 1 1 1 ′ 1 ′ {H H } H −H (r), (r˜) = (r)δ (r,r˜) 1 2 2 ′ {H H } H (r), (r˜) = 0 2 2 {H H } indicating 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 dπ dπ dπ ρ σ χ = π ,H , = π ,H , = π ,H are satisfied by virtue of the time ρ σ χ dt { } dt { } dt { } derivatives of (2.12). One can readily check (as one must always do with reduced action principles) that these three equations, when expressed in the velocity phase-space with dρ dσ dχ the help of the definitions = ρ,H , = σ,H , = χ,H , are completely dt { } dt { } dt { } equivalent to the three independent Einstein’s field equations satisfied by (2.6). We end upthis section by noting a few factsconcerning thetransformationproperties of 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): → dr dr ρ˜(r˜) = ρ(r) , σ˜(r˜) = σ(r), χ˜(r˜) = χ(r) , dr˜ dr˜ (2.14) dσ˜(r˜) dσ(r) dr d χ˜(r˜) d χ(r) dr = , = , dr˜ dr dr˜ dr˜ ρ˜(r˜) dr ρ(r) dr˜ (cid:18) (cid:19) (cid:18) (cid:19) where the t-dependence has been omitted for the sake of brevity. Thus, under the above coordinate transformations, σ, χ are scalars, while ρ, χ and the derivatives of σ, χ are ρ ρ covariantrank1tensors(one-forms),or, equivalently inonedimension, scalar densities of d d d ρ d weight 1. Therefore, the scalar derivative is not but rather or . − dr ρdr χdr ≡ χρdr Finally, if we consider an infinitesimal transformation r r˜= r η(r), it is easily seen → − that the corresponding changes induced on the basic fields are: δρ(r) = (ρ(r)η(r)), δσ(r) = σ (r)η(r), δχ(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 on the basic 1 H configuration space variables as that of the generator of spatial diffeomorphisms: ρ(r), dr˜η(r˜) (r˜) = (ρ(r)η(r)), 1 ′ H (cid:26) Z (cid:27) σ(r), dr˜η(r˜) (r˜) = σ (r)η(r), (2.16) 1 ′ H (cid:26) Z (cid:27) χ(r), dr˜η(r˜) (r˜) = (χ(r)η(r)). 1 ′ H (cid:26) Z (cid:27) 6 Thus, we are justified to consider as the representative, in phase-space, of an ar- 1 H bitrary infinitesimal reparametrization of the radial coordinate. As far as is con- 2 H 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 without any φ dependence; subsequently we can not expect to generate arbitrary infinites- 2 H imal reparametrization of φ. Nevertheless, we can identify a property of which 2 H links its existence to the existence of ξ. This property is described by the relation: correspo (r), (r), (r),g (r˜) = 0 g = 0. 2 2 2 ij ξ ij {H {H {H }}} n⇐de⇒nce L 3 Quantization We are now interested in attempting to quantize this Hamiltonian system following Dirac’s general spirit of realizing all the classical first class constraints (2.12) as quantum operator constraint conditions annihilating the wave functional. The main motivation behind such an approach is the justified desire to construct a quantum theory manifestly invariant under the “gauge” generated by the constraints. To begin with, let us first note that, despite the simplification brought by the imposition of the symmetry ξ = ∂/∂φ ⇔ g = 0, the system is still a field theory in the sense that all configuration variables ξ IJ L andcanonicalconjugatemomentadependnotonlyontime(asisthecaseinhomogeneous cosmology), but alsoontheradialcoordinater. Thus, tocanonicallyquantize thesystem in the Schr¨odinger representation, we first realize the classical momenta as functional derivatives with respect to their corresponding conjugate fields δ δ δ πˆ (r) = i , πˆ (r) = i , πˆ (r) = i . ρ σ χ − δρ(r) − δσ(r) − δχ(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 configuration field variables, say πˆ (r)ρ(r˜)2 = 2iρ(r˜)δ(r˜,r). ρ − The Dirac delta-function renders the outcome of this action a distribution rather than a function. Also, if the momentum operator were to act at the point at which the function is evaluated, i.e. if r˜ = r, then its action would produce a δ(0) and would therefore be ill-defined. Both of these unwanted features are rectified, as far as expressions linear in momentum 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, πˆ (r) dr˜ρ(r˜)2 = 2i dr˜ρ(r˜)δ(r˜,r) = 2iρ(r); ρ − − Z Z 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, even 7 so, δ(0)’s will appear as soon as local expressions quadratic in momenta are considered, e.g., πˆ (r)πˆ (r) dr˜ρ(r˜)2 = πˆ (r)( 2i dr˜ρ(r˜)δ(r˜,r)) = πˆ (r)( 2iρ(r)) = 2iδ(r,r). ρ ρ ρ ρ − − − Z Z Another problem of equal, if not greater, importance has to do with the number of derivatives (with respect to r) considered: A momentum operator acting on a smooth functional of degree n in derivatives of ρ(r),σ(r),χ(r)will, in general, produce a function of degree 2n, e.g., πˆ (r) dr˜ρ (r˜)2 = 2i dr˜ρ (r˜)δ (r˜,r) = 2iρ(4)(r). ρ ′′ ′′ ′′ − − Z Z Thus, clearly, more and more derivatives must be included if we desire the action of momentum operators to keep us inside the space of integrands corresponding to the initial 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 deal with these problems is, loosely speaking, to regularize (i.e., render finite) the infinite distribution limits, and re-normalize the theory by, somehow, enforcing n to terminate at 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 possibility to 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 quadratic and a number of linear first class constraints. Therefore, we deem it appropriate and instructive 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 +µiχ i ≡ 1 = µ GAB(QΓ)P P +UA(QΓ)P +V(QΓ) +µiφA(QΓ)P , (3.1) 2 A B A i A (cid:18) (cid:19) where A,B,Γ... = 1,2...,M count the configuration space variables and i = 1,2,...,N < (M 1) numbers the super-momenta constraints χ 0, which i − ≈ along with the super-Hamiltonian constraint X 0 are assumed to be first class: ≈ X,X = 0, X,χ = XC +Cjχ , χ ,χ = Ckχ , (3.2) { } { i} i i j { i j} ij k 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 ), but also under the following three changes: i 8 (I) Mixing of the super-momenta with a non-singular matrix χ¯ = λj(QΓ)χ i i j (II) Gauging of the super-Hamiltonian with the super-momenta X¯ = X +κ(Ai(QΓ)φB)(QΓ)P P +σi(QΓ)φA(QΓ)P i A B i A (III) Scaling of the super-Hamiltonian X¯ = τ2(QΓ)X Therefore, thegeometricalstructuresontheconfigurationspacethatcanbeinferredfrom the super-Hamiltonian are really equivalence classes under actions (I), (II) and (III); for example (II), (III) imply that the super-metric GAB is known only up to conformal scalings and additions of the super-momenta coefficients G¯AB = τ2(GAB +κ(AiφB)). It i is thus mandatory that, when we Dirac-quantize the system, we realize the quantum operator constraint conditions on the wave-function in such a way as to secure that the whole scheme is independent of actions (I), (II), (III). This is achieved by the following steps: (1) Realize the linear operator constraint conditions with the momentum operators to the right ∂Ψ(QΓ) χˆ Ψ = 0 φA(QΓ) = 0, i ↔ i ∂QA which maintains the geometrical meaning of the linear constraints and produces the M N independent solutions to the above equations qα(QΓ), α = 1,2,...,M N − − called physical variables, since they are invariant under the transformations gener- ated by χˆ . i (2) In order to make the final states physical with respect to the “gauge” generated by the quadratic constraint Xˆ as well: ∂qα ∂qβ Define the induced structure gαβ GAB and realize the quadratic in ≡ ∂QA ∂QB momenta part of X as the conformal Laplace-Beltrami operator based on g . Note αβ that in order for this construction to be self consistent, all components of g must αβ be functions of the physical coordinates qγ. This can be proven to be so by virtue of theclassicalalgebratheconstraintssatisfy(forspecificquantumcosmologyexamples see [19]). We are now ready to proceed with the quantization of our system, in close analogy to the scheme above outlined. In order to realize the equivalent to step 1, we first define the quantum analogue of (r) 0 as 1 H ≈ δΦ δΦ δΦ ˆ (r)Φ = 0 ρ(r)( ) +σ (r) χ(r)( ) = 0. (3.3) 1 ′ ′ ′ H ↔ − δρ(r) δσ(r) − δχ(r) 9 Asexplainedinthebeginningofthesection, theactionof ˆ (r)onallsmoothfunctionals 1 H is well defined, i.e., produces no δ(0)’s. It can be proven that, in order for such a functional to be annihilated by this linear quantum operator, it must be scalar, i.e. have the form Φ = ρ(r˜)f Σ(0),Σ(1),...,Σ(n),X(0),X(1),...,X(n) dr˜ (3.4a) Z (cid:0) (cid:1) σ (r˜) 1 d Σ(0) σ(r˜), Σ(1) ′ ,..., Σ(n) ... σ(r˜) (3.4b) ≡ ≡ ρ(r˜) ≡ ρ(r˜)dr˜ ! n 1 − X(0) χ(r˜), X(1) 1 χ(r˜) ′,..., X(n) |{z1} d ... χ(r˜) (3.4c) ≡ ρ(r˜) ≡ ρ(r˜) ρ(r˜) ≡ ρ(r˜)dr˜ ρ(r˜) ! (cid:18) (cid:19) n 1 − where f is any function of its arguments. We note that, as it is discussed at the end of |{z} the previous section, σ′ is the only scalar first derivative of σ, and likewise for the higher ρ derivatives. The proof of this statement is analogous to the proof of the corresponding result 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 remain − unaltered. If we calculate the change induced on the right-hand side we arrive at δf δf χ 0 = fδρ+ρ δσ +ρ δ dr = [ρ ˆ (f)]η(r)dr, (3.5) 1 δσ δ(χ/ρ) ρ H Z (cid:20) (cid:18) (cid:19)(cid:21) Z where use of (2.15) and a partial integration has been made. Since this must hold for any η(r), the result sought for is obtained. We now turn to the second linear constraint and try to see what are the restrictions it brings into our space of state vectors. We define δΦ δΦ ˆ (r)Φ = 0 ( ) = 0 = k, (3.6) 2 ′ H ↔ δχ(r) ↔ δχ(r) where k is any constant (with respect to r) independent of the basic fields and their derivatives, and Φ is given by (3.4a) (3.4c). As we argued before, the functional deriva- − tive δ acting onX(n) will produce, upon partialintegration of the nth derivative of the δχ(r) Dirac delta function, a term proportional to X(2n). 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(2n) vanishes; more precisely δΦ ∂f δX(n)(r˜) = k ... + ρ(r˜) dr˜= k δχ(r) ↔ ∂X(n)(r˜) δχ(r) ↔ Z ∂f 1 d δ(r,r˜) ... + ρ(r˜) ... dr˜= k ∂X(n)(r˜)ρ(r˜)dr˜ ρ(r˜) ↔ ! Z n 1 − ∂2f ... +( 1)n X(2n)|({r˜z)}δ(r,r˜)dr˜= k − ∂(X(n)(r˜))2 ↔ Z ∂2f ... +( 1)n X(2n) = k. − ∂(X(n))2 10