Loop quantization of the Gowdy model with local rotational symmetry Daniel Mart´ın de Blas,1, ∗ Javier Olmedo,2, 3, † and Tomasz Pawlowski1, 4, ‡
arXiv:1509.09197v1 [gr-qc] 30 Sep 2015
1
Departamento de Ciencias F´ısicas, FCE, Universidad Andres Bello, Rep´ ublica 220, Santiago 8370134, Chile 2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA 3 Instituto de F´ısica, Facultad de Ciencias, Igu´ a 4225, esq. Mataojo, 11400 Montevideo, Uruguay 4 Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, Pasteura 5, 02-093 Warszawa, Poland, EU. We provide a full quantization of the vacuum Gowdy model with local rotational symmetry. We consider a redefinition of the constraints where the Hamiltonian Poisson-commutes with itself. We then apply of the canonical quantization program of loop quantum gravity within an improved dynamics scheme. We identify the exact solutions of the constraints and the physical observables, and we construct the physical Hilbert space. It is remarkable that quantum spacetimes are free of singularities. New quantum observables naturally arising in the treatment partially codify the discretization of the geometry. The preliminary analysis of the asymptotic future/past of the evolution indicates that the existing Abelianization technique needs further refinement.
Making realistic predictions on effects of quantum gravity in cosmological context –an element needed in particular to solve the singularity problem in cosmology– requires investigating models admitting inhomogeneous spacetimes, preferably at nonperturbative level. Among such settings, Gowdy spacetimes [1] in vacuum are particularly interesting since they are a natural extension of Bianchi I cosmologies [2] and admit nonperturbative inhomogeneities in the form of gravitational waves. As they capture essential properties of full general relativity (GR) and at the same time are relatively simple, these models have brought over the years a lot of attention of researchers delving upon various aspects of gravity quantization. For instance, a quantum (geometrodynamics) description was already considered in the 1970s by Misner [3] and Berger [4]. Further they were explored in the specific context of gravitational waves quantization by Pierri [5] which description was later shown to admit a unitary dynamics [6]. The canonical quantization of these models employing the (complex) Ashtekar variables was carried out by Mena-Marug´an [7]. These studies however, as treating the homogeneous background either classically or via geometrodynamics could not ’cure’ the singularity problem. The Gowdy model with linear polarization and T 3 spatial slices has been subsequently studied in terms of real Ashtekar–Barbero variables [8] via the midisuperspaces techniques [9]. There however, the difficulties in applying the conventional loop quantum gravity (LQG) techniques [10] did not allow to complete the quantization program and probe the dynamics. Fortunately, the problems hampering prior approaches were suitably addressed in a seminal work by Garay, Mart´ın-Benito and Mena-Marug´an [11] via the so-called hybrid quantization program. This approach, also suitable for perturbative cosmological scenarios, combines the standard Fock quantization for the gravitational waves with a polymeric quantization of the homogeneous degrees of freedom. All these models allow for a convenient partial gauge choice which reduces the set of local constraints to a global Hamiltonian and diffeomorphism constraints. The latter are sufficiently simple to allow finding their solutions (at least formally) and to construct the physical Hilbert space. This approach however, while successful, by the very nature of the hybrid quantization cannot be easily related with the standard LQG. In this work we follow a more orthodox approach, expanding upon the original midisuperspace program. In order to test new techniques we study a slight simplification of the full polarized, three-torus Gowdy model, namely its locally rotational symmetric (LRS) version. In this case, one identifies two of the anisotropy directions ending up in the model which in vacuo (while inhomogeneous) features just one free global degree of freedom. When coupled to, for example, a massless scalar field Gowdy LRS admits homogeneous and even isotropic solutions making the model viable for cosmology applications and interesting in studies of the relation of loop quantum cosmology (LQC) [12] with LQG. Our quantization program, unlike the previous approaches, will not involve gauge-fixing. We will be working instead with the constraint algebra, featuring (as in full GR) local structure functions, and will employ the Dirac program. To deal with the known difficulties of the latter we will follow a strategy already adopted in studies of spherically symmetric spacetimes [13, 14] (see Ref. [15] for discussion of the full polarized Gowdy model). That strategy is based on a specific redefinition of the constraints and consequently of their algebra structure, which makes the Hamiltonian constraint Abelian. Furthermore, in the construction of the quantum counterpart of this constraint we implement, for the first time in a loop quantized inhomogeneous model, an improved dynamics scheme. The solutions to the Hamiltonian constraint can be explicitly determined and can be equipped with a well defined Hilbert space structure, which in turn, together with applying the standard LQC treatment of the spatial diffeomorphisms, allows to unambiguously probe the dynamical sector of the model. It is remarkable that the resulting spacetimes are free of singularities. Furthermore, the area of the Killing orbits is quantized due to the discreteness of the spectrum of
2 a new observable emerging in this quantization. This treatment and its results open a new window for the quantization of cosmological scenarios in LQC featuring a natural connection with the full theory by means of more realistic models admitting non-perturbative inhomogeneities. Here we will follow the formalism of real Ashtekar-Barbero variables with notation introduced in [8], setting for simplicity the Barbero-Immirzi parameter to one. The phase space is coordinatized by two pairs of canonical variables E x , Kx and E, A, (densitized triads and connections) such that the spatial metric components are gθθ = (E x )2 E −1 , gxx = gyy = E, where the spatial coordinates take values on the circle, i.e. {θ, x, y} ∈ [0, 2π) and the function E corresponds to the area of the spatial Killing vectors. R In the specified notation the Hamiltonian takes the form HT = (κ)−1 dθ(N H + Nθ C), where κ = 2G/π, is a linear combination of the constraints E∂θ E 4 1 1 2 x 2 , C = 2 [(∂θ Kx )E x − (∂θ E)A] . (1) (∂θ E) + ∂θ √ H = − √ (Kx E ) − √ EAKx − √ E E 4 EE x EE x To the above system we apply the “Abelianization” procedure – the redefinition of the Hamiltonian constraint by √ subtracting the term 2Kx E(∂θ E)−1 C. By further rescaling with (E x )−1 we can rewrite the total Hamiltonian as " # √ Z ˜ √ 1 N E(∂θ E)2 2 ˜θ [(∂θ Kx )E x − (∂θ E)A] . + 2N (2) ∂θ −2 EKx + HT = dθ κ (∂θ E) 2(E x )2 ˜ = N E x and shift Nθ → N ˜ θ = Nθ + The above redefinition is equivalent to the change of lapse N → N √ −1 2N Kx E(∂θ E) . The crucial desired property of the new Hamiltonian constraint is that it Poisson commutes with itself while its Poisson bracket with the diffeomorphism constraint remains unchanged. The resulting 1 + 1 model is “on shell” equivalent to the original LRS Gowdy model. The above system is further quantized within the polymeric formalism. Here the basic objects of the description are: 1-dimensional closed graphs (along θ direction) as support of the wave functions, holonomies along disjoint edges of these graphs and the intertwiners on their vertices. A basis of states |~k, ~µi in the connection representation is given by ( ) Z n µ o Y kj j ~ exp i hA|k, ~ µi = dθA(θ) exp i Kx (θj ) . (3) 2 ej 2 j Here, ej is a collection of edges of the spin network with valences kj ∈ N ∪ {0} and vj its vertices with valences µj ∈ R, as we implement the corresponding components of the connection as point holonomies. Since we p will implement an improved dynamics scheme it is convenient to adopt a more appropriate state labeling µj → νj = kj µj /λ, where λ is a real (dimensionless) parameter related with the so-called area gap. The kinematical Hilbert space is constructed as the closure of the above space with respect to the inner product h~k, ~ν |~k ′ , ~ν ′ i = δ~k~k′ δ~ν ~ν ′ , further generalized by the rule that basis states belonging to different graphs are mutually orthogonal. On these states the action of basic (multiplicative) operators is ˆ ~k, ~ν i = ℓ2Pl kj(θ) |~k, ~ν i, E(θ)| X δ(θ − θ(vj ))νj |~k, ~ν i, Vˆ (θ)|~k, ~ν i = λℓ3Pl
(4) (5)
vj ∈g
√ d ˆx where Vˆ = E E is the volume operator, and ℓ2Pl = G~. Here θ(vj ) is the position of the vertex vj and j(θ) is the ˆ integer corresponding to the edge ej going through θ. For the sake of simplicity, we extend the definition of E(θ) to the vertices of the graph by treating these vertices as points of the edges in their counterclockwise direction. In the next step we regularize (using Thiemann method) and quantized the Hamiltonian constraint, using the polymerization scheme of LQC, adopting, in particular, the improved dynamics prescription. It accounts for a minimum length ρj (kj ) of the point holonomies, singled out by the requirement ρ2j kj = λ2 . The final result is (in this case) mathematically equivalent to replacing Kx → sin (ρj Kx ) /ρj . On a given graph, the operator representing the Hamiltonian constraint becomes i h X 1 ˆ j + (kj−1 )3/2 ˆhj−1 Pˆ , ˆ )= (6) Nj Pˆ −(kj )3/2 h H(N ℓPl ∆kj j
3 Q 2 where Pˆ |~k, ~ν i = vj [sgn(kj )sgn(νj )] |~k, ~ν i has been introduced for convenience in order to decouple vertices with 2 kj = 0 and/or νj = 0. Besides, ∆kj = (kj − kj−1 ) amounts to the eigenvalue of ∂d θ E, which is given by ℓ ∆kj . Finally, Pl
ˆj = h
d1/2 1 V j
! 1/2 d d 1 1 1 2 2 2 2 ˆ − 2Ω ℓPl (ℓPl ∆kj ) . j 2 V j V j
(7)
Here we have adopted a regularization of the inverse triad operators ` a la Thiemann d1/2 1 1 1 |~k, ~ν i = 3/2 b(νj )|~k, ~ν i := 3/2 ||νj + 1|1/2 − |νj − 1|1/2 ||~k, ~ν i, 1/2 V j ℓPl ℓPl λ
(8)
and x x x x \) N \) |Vˆ |1/4 ˆ−2ρ ˆ2ρ ˆ−2ρ ˆ2ρ ˆ j = 1 |Vˆ |1/4 sgn(V − N − N N + sgn(V , Ω j j j j 4iλ θ=θ(vj )
(9)
x ˆ±2ρ ) |kj , νj i = |kj , νj ± 2i. A similar operator have has been constructed following ideas of [16], and where (N j θ=θ(vj ) been already studied in spherically symmetric spacetimes [14]. It commutes with itself and is free of anomalies. It is remarkable that the states with either kj = 0 or νj = 0, or both, are trivially annihilated by the constraints, thus they will be irrelevant for the dynamics. In this way the quantum analogues to the classical singularity can be decoupled from the theory. Besides, states with ∆kj = 0 yields an ill defined quantum constraint, so they do not belong to its P P domain of definition. The solutions to the Hamiltonian constraint, generally of the form (Ψ| = ~k ~ν h~k, ~ν |ψ(~k, ~ν ), ˆ ). It can be seen by direct inspection, admit a natural factorization on each vertex due to the particular form of H(N † ˆ that the solutions at each vertex must be annihilated by the operator [hj − h0 /(kj )3/2 ], where h0 is a global positive constant. As a consequence, in order to completely characterize the solutions it is enough to carry out a spectral ˆ † . It is further possible to see that the continuous part of the spectrum of the operator decomposition of the operator h j −2 ˆ† ˆ † is Sp ] and is nondegenerate. Thus, the (kinematical) Hilbert space can be spanned by a basis of h cont (hj ) = [0, 2λ j products of eigenstates of |ωj i (corresponding to generalized eigenvalues ωj ) of that operator. The states annihilated ˆ (denoted further as pre-physical ) can be obtained by group averaging techniques, which yield by H X ˜ ~k, ~ωh ; h0 ). ˜ h0 = (Ψ| (10) h~k, ~ωh0 |ψ( 0 ~ k
Here the vector ~ ωh0 is formed out of eigenvalues satisfying ωj =
h0 (kj )3/2
for each j. Due to boundedness of the
ˆ j the global variable h0 is further restricted, namely h0 < h∗ = min[~k] (continuous) spectrum of h , where min[~k] is λ2 ~ the minimum over components of k. ˜ g |h0 on h0 different choices of the latter yield distinct solutions, Due to a nontrivial dependence of the states (Ψ thus h0 should be considered an observable rather than a parameter. Indeed, classically (∂θ E) E 3/2 , (11) h(θ) = x 2AKx − ∂θ E 2(E x ) 3/2
is constant on-shell and coincides with h0 . As a consequence, it is natural to consider as (pre-)physical states the superpositions of (10) for different h0 . Therefore, ˜ = ˜ h0 → (Ψ| (Ψ|
Z
h∗
˜ h, dh(Ψ|
(12)
0
where we have replaced h0 with h for convenience. The inner product is then ˜ Φi ˜ = (Ψ|
Z
0
h∗
dh
X ~ k
˜ ~k, ~ωh , h) := ˜ ~k, ω (ψ( ~ h , h))∗ φ(
Z
h∗ 0
dh
X
˜ ~k, h). ˜ ~k, h))∗ φ( (ψ(
(13)
~ k
In the final step of the quantization we identify the diffeomorphism-invariant sector of the theory. This is done via the standard treatment of LQG [17] where one averages the graphs with respect to the group of finite diffeormophisms.
4 Here, this procedure corresponds to removing the embedding data – the information about the position of each vertex while keeping their order on the circle (note that the quantum numbers h and ~k are preserved under both the Hamiltonian constraint and the spatial diffeomorphisms). ˆ once it has been suitably represented. For instance, One of the observables of the model then corresponds to h ˆ ˆ = i~h ˆ with sˆ = −i~(h ˆ ∂ˆh + ∂ˆh h)/2. since the classical theory seems to require h to be positive, we adopt [ˆ s, h] ˆ ˆ This guaranties that both h and sˆ operators are well defined assuming that h is positive definite (with a continuous spectrum). Remarkably the sequence of integers ~k allows one to construct another set of observables (as it was ~k, hiphys = ℓ2 ~ ˆ done in case of spherically symmetric vacuum spacetimes [13, 14]) O(z)| Planck kInt(nz) |k, hiphys where |~k, hiphys is a basis of eigenstates of the observables, z ∈ [0, 1], Int(nz) is the integer part of nz and n is the number of vertices. These observables contain information about areas of the Killing orbits. Indeed, given a particular embedding z(t, θ) one can write the metric component gxx (t, θ) = E(t, θ) (by its nature breaking diffeomorphism invariance) as ˆ θ)|~k, hiphys = O(z(t, ˆ E(t, θ))|~k, hiphys . The time coordinate t (representing the time reparametrization freedom) can be chosen, for instance, to be t = s – the conjugate momentum of h. Another physically relevant component of the metric gθθ (t, θ) can be specified as an embedding dependent function of the above observables and the polymer analog of Kx . Since the gauge freedom of the system can be realized through associating with the latter a (gauge) function v(t, θ) the component gθθ (t, θ) can be written as gˆθθ (t, θ) :=
b θ))′ ]2 [(E(t, 1 Eˆ 4v 2 (t, θ) − √ 2h
.
(14)
b E(t,θ)
At this point we have at our disposal the completed quantization program with well defined physical Hilbert space and the set of physically relevant observables. It is thus natural to ask about the dynamical sector of the theory, essentially its semiclassical regime. Due to inherent discreteness of the polymer quantization (here reflected in the spectra of the operators Eˆ and Vˆ ), it is not obvious whether the present model can reproduce GR in its low curvature ˆ and O(z)) ˆ limit. For that one needs to select/construct states which (for defined observables, h are sharply peaked about classical trajectories in some epoch of the evolution. Another necessary condition for these states is to produce smooth manifolds. This could involve that the number of vertices in graphs supporting them is large and, for instance, that the differences of the expectation values of the metric components between two consecutive vertices are small. Among the semiclassical geometries, the most interesting ones are those that approximate homogeneous cosmologies. One possible way is to consider suitable superpositions in the quantum numbers h and ~k, together with conditions that the gauge parameters are homogeneous. These two additional requirements appear to be sufficient to pick out the states providing homogeneous settings to a good approximation. These states are particularly relevant in the understanding of the relation of the presented approach with nonsingular (bouncing) scenarios studied in LQC [12]. Such comparison however is out of scope of this manuscript and it is left for future research. However certain observations can be made in the context of the low curvature limit of the theory. ˆ j admit a well defined large νj limit –an analog Indeed, since (similarly to the isotropic LQC) the eigenfunctions of h of standing waves– one can use the scattering description of [18]. One then arrives to the asymptotic future/past ˜ ω , h) as state characterized by the spectral profile Ψ(~ ~ = einπ/4 Ψǫ (s, ~ν ; A)
Z
n k (h,ω ) R λσ(ω ) Pn Y ˜ ω , h) iπ Ψ(~ iǫ √8 i νj −is ln h+i j 2 j e dθA(θ) j h1/2 e 2 [ j=1 βj (~ω,h)] , e dhdn ~ ω √ 2π j=1
(15)
σ λ where 16βj2 := (kj − kj−1 )2 − 1, ωj = λ22 sin2 √j 2 and ǫ = ∓1 for the future/past respectively. A critical consequence of the above form of the state regards the number of degrees of freedom. Classically, the LRS Gowdy spacetime is completely characterized by a global degree of freedom, whereas an inspection of (15) shows that it features local degreesR of freedom. Indeed, one can choose a convenient family of partial observables (evolving ˜ ω , h), construct the semiclassical state peaked constants parametrized by ej Adθ) and, by appropriate choice of Ψ(~ about any arbitrary sequence of νj . In consequence the space of solutions (and consequently of quantum trajectories) is much larger than that of GR. Thus, in the present form of the model, GR does not emerge solely as the low energy limit of the (loop) quantum description. This excessive freedom can be traced back to the conjunction of the procedure of Abelianization (making the Hamiltonian constraint ultralocal) and the qualitative differences in the treatment of Hamiltonian and diffeomorphism constraints. Since the original Hamiltonian constraint relates the quantum data at distinct vertices of the graph, in order to recover the correct count of the degrees of freedom one may be forced to implement the diffeomorphism constraint in the same footing as the Hamiltonian one, that is by building the quantum
5 counterpart of the regularized infinitesimal diffeomorphism generator [19]. This would provide the additional operator constraint, now mixing the data on distinct vertices of the graph. To summarize, we have carried out a full quantization (within the LQG framework) of the polarized LRS Gowdy model in vacuum with T 3 topology. In the process no gauge fixing was implemented – the treatment remains diffeomorphism invariant. Our strategy is based on a suitable redefinition of the Hamiltonian constraint in such a way that it commutes with itself on both classical (under Poisson brackets) and quantum level. The resulting ultralocality of the Hamiltonian constraint allows then to find the solutions to it that are invariant under spacetime diffeomorphisms and to construct the physical Hilbert space. The observables of the model correspond to a global degree of freedom as in the classical theory and a new observable without classical analogue codifying the areas of the consecutive Killing orbits. An analogous observable has been already identified in spherically symmetric loop gravity [13]. The treatment allows to probe the dynamics in an unambiguous way and the system admits a large semiclassical sector. A remarkable property of the dynamics is the singularity resolution with a mechanism similar to the one observed in LQC [11, 20]. The preliminary analysis of the asymptotic future/past epoch of the states suggests the necessity of implementing the infinitesimal diffeomorphism constraint. These results will on the one hand allow to verify the existing LQC frameworks against the genuine quantum nonperturbative dynamics of the inhomogeneous model and on the other hand provide a crucial information for the programs of probing the dynamical sector in full LQG, indicating new possible avenues for improving/completing the existing treatments. We wish to thank R. Gambini, G. A. Mena Marug´an and J. Pullin for comments. J. O. acknowledges the partial support by Pedeciba and by grant NSF-PHY-1305000. T. P. thanks the Polish Narodowe Centrum Nauki (NCN) grant 2012/05/E/ST2/03308 and the Chilean grant CONICYT/FONDECYT/REGULAR/1140335. D. M.-dB. is supported by the project CONICYT/FONDECYT/POSTDOCTORADO/3140409 from Chile. The authors thank the grants MICINN/MINECO FIS2011-30145-C03-02 and FIS2014-54800-C2-2-P from Spain.
∗ † ‡
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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