Physics Letters B 747 (2015) 18-21

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A quantum reduction to spherical symmetry in loop quantum gravity

N. Bodendorfer, J. Lewandowski, J. Swiezewski *

Faculty of Physics, University of Warsaw, Pasteura 5,02-093, Warsaw, Poland

A R T I C L E I N F O A B S T R A C T

Based on a recent purely geometric construction of observables for the spatial diffeomorphism constraint, we propose two distinct quantum reductions to spherical symmetry within full 3 + 1-dimensional loop quantum gravity. The construction of observables corresponds to using the radial gauge for the spatial metric and allows to identify rotations around a central observer as unitary transformations in the quantum theory. Group averaging over these rotations yields our first proposal for spherical symmetry. Hamiltonians of the full theory with angle-independent lapse preserve this spherically symmetric subsector of the full Hilbert space. A second proposal consists in implementing the vanishing of a certain vector field in spherical symmetry as a constraint on the full Hilbert space, leading to a close analogue of diffeomorphisms invariant states. While this second set of spherically symmetric states does not allow for using the full Hamiltonian, it is naturally suited to implement the spherically symmetric midisuperspace Hamiltonian, as an operator in the full theory, on it. Due to the canonical structure of the reduced variables, the holonomy-flux algebra behaves effectively as a one parameter family of 2 + 1-dimensional algebras along the radial coordinate, leading to a diagonal non-vanishing volume operator on 3-valent vertices. The quantum dynamics thus becomes tractable, including scenarios like spherically symmetric dust collapse.

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Article history■ Received 12 May 2015 Accepted 14 May 2015 Available online 19 May 2015 Editor: A. Ringwald

Introduction

Loop quantum gravity [1,2] as a whole has matured into a serious candidate theory for quantum gravity in recent years. Progress has been especially strong in the areas of computing black hole entropy [3-5] and studying quantisations of mini- [6] or midisuperspace [7] models using techniques from the full theory. On the other hand, it has been notoriously hard to extract physics from computations directly in the full 3 + 1-dimensional theory. In order to avoid the "problem of time" associated with the underlying diffeomorphism invariance of general relativity, deparametri-sation [8,9] has been introduced within loop quantum gravity [10-12] in order to obtain a true Hamiltonian evolution. A certain form of deparametrisation, however, always puts restrictions on the physical situation that one can describe, due to a, in general, finite range of physical coordinates. It is therefore desirable to have different deparametrisation techniques at one's disposal, tailored to different interesting physical problems. Furthermore, deparametri-sation can in principle significantly alter the canonical structure,

* Corresponding author.

E-mail addresses: norbert.bodendorfer@fuw.edu.pl (N. Bodendorfer), jerzy.lewandowski@fuw.edu.pl (J. Lewandowski), swiezew@fuw.edu.pl (J. SwieZewski).

leading to different quantisation variables. While this does not happen for the standard example of dust [9], or only in a mild form of a possible rescaling for scalar fields [13] due to a Higgs-like "absorption" of matter degrees of freedom, we will encounter a more severe change of canonical structure due to a purely geometric deparametrisation in this article. As a direct consequence of this deparametrisation, we will obtain a family of holonomy-flux algebras labelled by the radial coordinate, each behaving effectively two-dimensional. Thus, we can use spin networks with three-valent vertices on which the volume operator is diagonal. The quantum dynamics thus becomes a lot more tractable than in the usual case. Also, the physical coordinate system introduced via this deparametrisation is ideally suited for introducing a quantum reduction to spherical symmetry. In order to simplify the presentation in this letter, we will gloss over some technical details which are addressed in our companion papers [14,15].

The radial gauge

Recently, a purely geometric construction of observables with respect to the spatial diffeomorphism constraint has been given [16], based on a physical coordinate system introduced by spatial geodesics outgoing from a central point a0. A point in the spatial slice £ is uniquely defined via the exponential map

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i e N, such that ri e (li-1, li), which correspond to lattice sites. We then approximate a radial integral as /0°drf (r) & ^¡=1 Al'f (ri). We furthermore restrict our cylindrical functions to have support only at the lattice sites ri.

Following [23,24], we then arrive at the volume operator

V (R )Vy = J2

V v Vy

veV (y,R)

V v Vy:= fei-

J2 sgn(e, e')eiikRejRek\ Vy

\e,e'eE(y,v)

where Vy denotes a cylindrical function expressed on a graph y chosen such that at non-trivial (= not only parallel tangents) vertices all edges are ingoing, V(y, R) is the set of vertices of y contained in R, E (y, v) denotes the set of edges incident at the vertex v, sgn(e, e') denotes the sign of eABeAe'B at v, where eA, e'B are the tangents of the edges e, e', and Re are the standard right invariant vector fields. As expected, this operator is identical to the one of [23] up to the factor of Ali. The special properties of the volume operator in 2 + 1 dimensions [23] directly transfer to (8). It generically does not vanish on two- or three-valent vertices, as long as there are at least two edges with non-parallel tangents.

Strategy for spherical symmetry

1. Strategy: Due to the coordinate system defined by the map x1 ^ expff0(x'e'/), it is a natural approach to group-average the quantum states over rotations, defined in £ as the image of rotations in Tao £ via the exponential map. Due to our choice of spherical coordinates, a rotation in £ just corresponds to changing the angular coordinates (A, B) = 6 in the same way at all radial coordinates. Due to the spherically symmetric setting, the coordinates xl have maximal range and span all of £ (up to non-trivial topology). This strategy, which is detailed in our companion paper [15], retains most degrees of freedom, as we will discuss later. Spherically symmetric cylindrical functions can now be defined by demanding invariance under such rotations, which have a straightforward action as moving holonomies in the quantum theory. They can be obtained by taking arbitrary cylindrical functions and group averaging them. The same applies to operators.

2. Strategy: We recall [25] that the ADM formulation can be reduced to spherical symmetry via the ansatz ds2 = A2(r, t)dr2 + R2(r, t)dQ2 for the spatial line element, leading to the Poisson brackets {R(r), PR(r')| = S(r, r') and {A(r), PA(r')} = S(r, r'). A necessary consequence of this reduction is that prA = 0, which already follows from the vanishing of spherically symmetric vector fields on S2. Comparing with (4), we find that the generator of a certain class of spatial diffeomorphisms has to vanish. Furthermore restricting a = A in (4), it follows that the involved vector fields span the set of vector fields tangential to spheres Sr2, the image of the exponential map for radial coordinate r [14]. Thus, prA = 0 can be imposed in the quantum theory by demanding invariance with respect to spatial diffeomorphisms which preserve the S2. Technically, this is equivalent to the usual problem of implementing the spatial diffeomorphism constraint in loop quantum gravity [26], however this time the conceptual difference is that we are averaging with respect to physical coordinates, which results in a reduction of physical degrees of freedom. The resulting quantum states live in the dual of the Hilbert space and consist essentially of diffeomorphism equivalence classes of spin networks lying on individual Sr2.

Comparison of the two proposals

Degrees of freedom: In the first strategy, the group averaging is performed with respect to the action of the compact group SO(3). Specifying the edges of the spin networks thus retains an uncountable amount of information. The situation is different in the second strategy, which reduces spin networks to their corresponding diffeomorphism equivalence classes with respect to the physical coordinate system (6), resulting in far less degrees of freedom [27]. The difference is essentially given by spherically symmetric correlations such as [tp(r,61)^(r',62) = [<p(r, g61)$(r', g62), where <p can represent e.g. a matter field or curvature scalar, and g is a rotation. While the Hilbert space from the first strategy contains full information about such correlations, it is drastically reduced in the second proposal to diffeomorphism invariant correlations at r = r'.

Dynamics: Quantum Hamiltonians (deparametrised, as constraints, or master constraints) in the radial gauge can be constructed using slight extensions of the quantisation techniques of [22,23,28], as shown in [15,29]. The regulators in the angular directions can always be removed, while integrals in the radial direction are approximated as before. For derivatives in the radial direction, we use standard finite difference approximations. Details are provided in [15]. For angle-independent lapse, the Hamiltoni-ans preserve the spherically symmetric states from the first strategy, which ultimately results from using quantisation techniques for diffeomorphism invariant theories for the angular directions. However, the diffeomorphisms preserving the Sj from the second strategy do not commute with such Hamiltonians, since they do not map radial geodesics into radial geodesics with respect to the original metric, interfering with the reduced phase space structure of the Hamiltonian, which contains non-local contributions from integrals along the radial direction [14].

A different strategy is thus necessary for the second approach, based on operators which are invariant with respect to S;2-preserving diffeomorphisms. Our proposal for such Hamiltoni-ans consists in defining the spherically symmetric midisuperspace Hamiltonians, e.g. the one of [25], as operators on the full theory Hilbert space. The radial gauge corresponds to setting A = 1 and solving for PA in terms of R and PR via the spatial diffeomorphism constraint, see [14] for details. We then have to represent only R and P R as operators, which can be easily done again using standard techniques [22,23]. Since both R(r) and PR(r) can be obtained from averages of functions of the unreduced variables qAB and pAB over S2, they are naturally invariant with respect to S;2-preserving diffeomorphisms. Hamiltonians constructed along the above lines thus preserve the spherically symmetric states of the second strategy. This strategy is suited best for a true Hamil-tonian resulting from deparametrisation, since no issues related to a proper reduction of degrees of freedom at the quantum level occur.

In both strategies, the issue of anomalies in the Hamiltonian constraint is so far unresolved. This problem can however be improved upon by using non-rotating dust to deparametrise the Hamiltonian constraint [9,12,30], leading to a single true Hamilto-nian, and thus the absence of the anomaly issue. For pure general relativity, a suitable deparametrisation of the Hamiltonian constraint, leading to a manageable true Hamiltonian, is not known so far. In this case, we would resort to the Master constraint approach [28].

Choice of spin networks and comparison to other approaches

A natural question to ask now is the relation of the presented reduction strategies to previous work based on quantising classically reduced models, such as [7,25,31]. Here, the choice of spin

networks in our proposal is crucial: in the radial gauge, a spherically symmetric midisuperspace model (coupled to additional matter) has R (r) and PR (r) as gravitational degrees of freedom. Our spin networks at the radial lattice points ri however gener-ically encode more degrees of freedom than just the total area AS2 = 4n R2. Interestingly, due to the peculiar properties of the 2 + 1-dimensional volume operator [23], there exists a choice of spin network encoding exactly one degree of freedom, being a Wilson loop with exactly one kink, labelled only by the SU(2) spin j. If we restrict ourselves only to such Wilson loops, one for each lattice site ri , we obtain a maximally simple model, for which closed and manageable formulas for the action of the Hamiltonians from both strategies can be obtained, since the volume operator is diagonal and non-trivial on the kink and we choose a graph-preserving regularisation for the field strength of the connection. The corresponding object in a midisuperspace quantisation is then the quantum number associated to R2 (e.g. ki associated to Ex in [7]) at a given lattice site. A comparison of the explicit dynamics of our models to a midisuperspace quantisation using the radial gauge has not been performed so far and will be left for further research.

Another interesting aspect of the dynamics, which has received much attention in recent years, is the issue of coarse graining and choice of appropriate spin networks to describe physical situations, see [32] and references therein. In our model, this issue reappears in the choice of spin networks for the spheres S2. While the total volume can be encoded in a Wilson loop as above, it is unclear how the dynamics would be affected by choosing a more refined spin network encoding the same total volume, but more local information. Due to the simplified dynamics (the volume operator is still diagonal on three-valent vertices), this issue can be investigated using spin networks based on graphs which are triangulations of the Sr2.

Conclusion

We have outlined the construction of a computable framework to study spherically symmetric quantum gravity dynamics. The key classical ingredient has been a geometric deparametrisation of the spatial diffeomorphism constraint, leading to a family of effectively 2 + 1-dimensional holonomy-flux algebras. The quantum dynamics becomes tractable because the spin network vertices for the quantum states can be chosen to be two- or three-valent due to the special properties of the 2 + 1-dimensional volume operator. This work opens the possibility to study spherically symmetric quantum dynamics within full loop quantum gravity, such as dust collapse in the Lemaitre-Tolman-Bondi model.

Acknowledgements

This work was partially supported by the grant of Polish Nar-odowe Centrum Nauki No. 2011/02/A/ST2/00300 and by the grant of Polish Narodowe Centrum Nauki No. 2013/09/N/ST2/04299. N.B. was supported by a Feodor Lynen Research Fellowship of the Alexander von Humboldt-Foundation and gratefully acknowledges discussions with Antonia Zipfel.

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