Spherically symmetric sector of self dual Ashtekar gravity coupled to matter: Anomaly-free algebra of constraints with holonomy corrections
aa r X i v : . [ g r- q c ] O c t Spherically symmetric sector of self dual Ashtekar gravity coupled to matter:Anomaly-free algebra of constraints with holonomy corrections
Jibril Ben Achour , ∗ Suddhasattwa Brahma , † and Antonino Marcian`o ‡ Center for Field Theory and Particle Physics,Fudan University, 200433 Shanghai, China
Using self dual Ashtekar variables, we investigate (at the effective level) the sphericallysymmetry reduced model of loop quantum gravity, both in vacuum and when coupled to ascalar field. Within the real Ashtekar-Barbero formulation, the system scalar field coupledto spherically symmetric gravity is known to possess a non closed (quantum) algebra ofconstraints once holonomy corrections are introduced, which forbids the loop quantizationof the model. Moreover, the vacuum case, while not anomalous, introduces modificationswhich are usually interpreted as an effective signature change of the metric in the deepquantum region. We show in this paper that both those complications disappear whenworking with self dual Ashtekar variables, both in the vacuum case and in the case of gravityminimally coupled to a scalar field. In this framework, the algebra of the holonomy correctedconstraints is anomaly free and reproduces the classical hypersurface deformation algebrawithout any deformations. A possible path towards quantization of this model is brieflydiscussed.
I. INTRODUCTION
In the last decade symmetry reduced models have played an increasing pivotal role in Loop Quan-tum Gravity (LQG). The full theory being not yet accessible, an important effort has been devotedto applying the polymer quantization procedure to some restricted sector of the phase space ofGeneral Relativity (GR). The most exciting sectors to apply the loop quantization are the oneswhich admit classical singularities, typically early universe cosmology and black hole geometries.Indeed, it is expected that taking into account the quantum nature of geometry, through loopquantization, will lead to regular quantum geometries where the singularities shall be naturallyresolved. While regular quantum cosmological geometries provide a very promising framework tostudy bouncing quantum cosmologies and extend the cosmological scenario to the Planck era [1–9],regular quantum dynamical black hole geometries would be the ideal platform to test ideas aboutthe gravitational collapse scenario, Hawking radiation and the tunneling from black holes to whiteholes recently proposed in [10–15].Let us briefly summarize the strategy used in such (effective) loop models. Once the symmetryreduced phase space is obtained, one follows the loop procedure and polymerize the (components ofthe) connection variable which survived the symmetry reduction. Physically, this is justified by thefact that, at the quantum level, the gravitational field is not well described by the connection butrather by its holonomy, which is an extended unidimensional object. Then one obtains a new phasespace where the symmetry reduced constraints are modified by the so called holonomy corrections.These quantum corrections encode the effect of the quantum nature of the geometry at the Planckscale. Using this strategy and loop techniques, one then quantizes this effective phase space whichis believed to be the correct physical one suitable to describe quantum gravity effects. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
However, the holonomy corrections introduced in the classical constraints have important con-sequences on the fate of the symmetries of the system described by the modified gauge generators.Indeed, the first class constraints of GR, i.e. the vectorial constraint H a and the scalar constraint H , generate the infinitesimal four dimensional diffeomorphisms and form therefore a closed alge-bra called the hypersurface deformation algebra. It is therefore natural to wonder what is thegeneralization of the underlying symmetry of the effective phase space with holonomy corrections.Putting it differently, does the algebra of the constraints remain closed after implementing theholonomy corrections? Such questions refer to the covariance of the effective phase space and aretherefore of primary importance. Indeed, if the algebra of the modified constraints does not close,it is then impossible to canonically quantize the system following the Dirac procedure since thenwe are violating the underlying gauge symmetry of the system, which in the case of gravity, isgeneral covariance. One then has lesser number of first class constraints left to generate gaugetransformations and we are left with spurious degrees of freedom in the modified system.Following [17], we will say that a system admits a covariant quantization if the following tworequirements are satisfied:1. The first class constraints remain first class after implementing the holonomy corrections.They correspond therefore to infinitesimal generators of some symmetries for the phase spacevariables and form a closed algebra (at the effective level).2. The modified first class constraints admit the right classical limit and lead to the usualhypersurface deformation algebra of GR in that limit.Up to now, the most important effort in the loop quantization of symmetry reduced modelshas been spared while focusing on the homogenous case, which have grown into the sub-field ofLoop Quantum Cosmology (LQC) [1–3]. From those models, one can exhibit some generic resultsinherited by the polymer quantization – the most important one being the resolution of the initialcosmological singularity. This singularity resolution allows one to extend the well known standardmodel of cosmology, starting at the beginning of inflation, to the so called Planck era and evenbeyond [4–6]. This extension to the pre-inflationary era led to a very large literature, and it isnow argued that the LQC framework is now mature enough to make contact with the observations[7–9].Yet, the symmetry reduction implemented within the context of LQC is a very drastic one,where only the Hamiltonian constraint survives in a very simplified form. In this over-simplifiedframework, the covariance of the holonomy corrected system becomes trivial due to vanishing ofthe spatial diffeomorphism constraint. However, the question of covariance can become highly nontrivial when applied to other symmetry reduced models, as pointed out in [17, 18]. In order toinvestigate this property within the holonomy corrected loop models, one needs to go beyond thehomogenous framework, and turn to less drastic symmetry reduced models where, at least, somecomponents of the Gauss constraint G i , the vectorial constraint H a and the scalar constraint H survive. Only in this context can one investigate the fate of the covariance of the effective systemby studying the closure of the algebra of modified constraints, and study possible deformations toit. This task was worked out for the spherical symmetric case in [17] and for the Gowdy model in[18], using real Ashtekar-Barbero variables.Let us summarize those results. In the spherically symmetric case, one needs to distinguishbetween the vacuum case, which does not have local degrees of freedom, and the system wherematter is coupled to gravity, which exhibits local physical degrees of freedom. Indeed, for thevacuum case, the algebra of the modified constraints remain closed although some modificationsshow up in the structure functions appearing in the Poisson bracket of the scalar constraint withitself {H [ N ] , H [ N ] } . This deformation in the algebra has been interpreted in the literature asan ‘effective’ change of signature of the metric in the deep quantum regime, but the physicalnature of this modification is not yet fully understood at the fundamental level. Such phenomenonleads to drastic modifications such as the loss of metric structure and a change in the characterof the partial differential equations involved from hyperbolic to elliptic [19–21]. See e.g. [22–25]for some investigations of the consequences in LQC, and [26] for the resulting deformations in thePoincar´e algebra in the flat spacetime limit. Yet, it is also possible that such phenomenon is a puremathematical artifact due to the way non-perturbative quantum corrections are implemented. Wenote that this signature change shows up also in the so called anomaly free approach to computethe cosmological perturbations as developed in [27, 28]. See [29] for a detailed discussion. Despitethis point, ignoring the signature change phenomenon, the spherically symmetric vacuum can besafely quantized using loop techniques. However, the situation gets quite messy when one triesto couple some local matter degrees of freedom. Indeed, when coupling the spherically symmetricgravitational field to matter, the holonomy modification function prevents the Poisson brackets ofthe scalar constraint with itself from closing into the diffeomorphism constraint. Thus covarianceis violated in the case of the modified system gravity plus matter . Therefore, already for thespherically symmetric case, this partial no go result prevents from coupling any kind of matter togravity in the loop approach. The situation is much worse in the context of the Gowdy model wherethe same no go result show up already at the level of the vacuum case (which contains gravitationallocal degrees of freedom) [18]. Therefore, these powerful no go results derived in [17, 18] representa general obstruction to the development of a loop quantization of midisuperspace models thathave some local physical degrees of freedom.Having clarified the situation from the point of view of the underlying covariance of the model,let us now summarize what has been done in the loop quantization of the spherically symmetryreduced model. Since the vacuum spherical case remains covariant even after implementing theholonomy corrections, one can safely complete the quantization. The polymer quantization ofthe Schwarzschild interior spacetime was worked out first by Ashtekar and Bojowald almost tenyears ago [30]. Their quantization is based on the fact that the interior Schwarzschild spacetimecan be described as a homogenous contracting cosmology, i.e. the Kantowski-Sachs homogenousspace-time. This quantization was studied further by Modesto in [31, 32]. Following the improveddynamics introduced in LQC, Bohmer and Vandersloot introduced a new scheme in [33, 34], fixingsome drawbacks of the previous works, such as the dependency on some auxiliary unphysicalstructures as the fiducial cell. The interior problem was then revisited in [35] by Campiglia,Gambini and Pullin. The inhomogeneous exterior space-time was worked out in [37] by the sameauthors, some of whom extended the study to the whole space-time in [38]. In these works,the diffeomorphism constraint was removed by a suitable gauge fixing. In [39], Gambini andPullin introduced a new quantization procedure, based on an Abelianization of the constraints,avoiding therefore the previous gauge fixing of the diffeomorphism constraint. More recently, theinterior problem was revisited by Corichi and Singh in [36], improving the classical limit of theantecedent works aforementioned. As expected, a common conclusion of these works is that thecentral singularity is removed by the polymer quantization procedure, leading to a vacuum regularquantum geometry for the black hole. Unfortunately, as mentioned earlier, the loop quantization ofthe full dynamical system gravity plus matter remains elusive up to now, because of the difficultyof closing the modified algebra. An exception of this statement is the electro-vacuum case whichwas treated in [40]. Because of these difficulties, the first quantization of a scalar field coupledto spherical symmetric gravity were initiated in [41, 42] using another strategy called uniformdiscretization technique [43]. In [44, 45], a new strategy was proposed based on the new gaugefixing leading to a simplification of the modified constraints. Although interesting, such gaugefixings prevent us from inferring anything about the covariance of the system. Few years later,Gambini and Pullin introduced the first study of the Hawking radiation in this framework [46, 47],while the quantization of a test shell was presented in [48]. While very promising, it is importantto note that those conclusions rely on the study of a test scalar field over a quantum sphericallysymmetric vacuum geometry and thus not on the full quantization of a scalar field coupled tospherical symmetric gravity. These models typically have a non-matching version of covariance ofthe matter and the gravity sectors, as explained in [17]. Although those results on the vacuumblack hole geometry are very promising and lead to interesting insights, it seems mandatory to gobeyond the vacuum case and obtain a quantizable model for the full system scalar field coupled tospherically symmetric gravity . In order to so, one needs to by pass the partial no go results of [17]associated with the algebra of the modified constraints for this system.The first attempt to go beyond the test field approximation and study a full spherically selfgravitating collapsing shell was introduced recently in [49]. In this model, the authors succeeded tobypass the no go theorem of [17] and obtained an anomaly free algebra of the modified constraint.The strategy used in this work is to keep the Poisson bracket unchanged so that { K φ , E φ } = { f ( K φ ) , g ( K φ ) E φ } which is then a canonical transformation for a suitable choice of f and g ,namely f ( K φ ) = sin( ρK φ ) /ρ and g ( K φ ) = 1 / cos( ρK φ ). While this canonical transformationprovides indeed an anomaly free algebra of constraints, there are three difficulties arising withinthis approach. First, because we are dealing with a canonical transformation of the classicalcase, it is then difficult to interpret them as loop quantized models where, usually, the effectivePoisson bracket gets modified due to polymerization. Secondly, while the first modification function f ( K φ ) accounts for the usual polymerization of the connection component, the second modificationfunction g ( K φ ) remains quite unusual with respect to the loop quantization procedure. Indeed, itis not clear why one should implement a holonomy correction on the conjugate triad variable E φ in the process and this step remains to be justified from first principles. Finally, it is immediate tosee that at the singularity, where f ( K φ ) = sin( ρK φ ) /ρ becomes maximal, the correction g ( K φ ) =1 / cos( ρK φ ) is no more well defined. One, therefore, has to remove by hand some regions from thespectrum of the theory to obtain a well defined effective theory, which seems problematic if onewants to describe the bounce at the dynamical level. While interesting for evading the no go resultof [17], this proposal turns out to suffers from drawbacks that still need to be fixed or clarified,either through the quantization of the model, or through some new inputs. It is therefore naturalto look for another perspective in order to answer the question of the anomaly freeness of thissystem.In this paper, we introduced a different strategy to obtain an anomaly-free algebra of constraintswhich does not suffers from the same drawbacks. The initial observation is to wonder if the choiceof variables is responsible for the anomaly of the modified algebra. Indeed, all the conclusionand partial no go results presented in [17, 18] were obtained using the real Ashtekar-Barberoformulation. How many of those conclusions might get modified if one uses instead the self dualvariables? (Interestingly, a similar comparison between the Schr¨odinger quantization of sphericallysymmetric models, using real variables and the self-dual variables, was done in [50]).Recently, some efforts have been developed to understand more precisely the impact of workingwith the self dual variables instead of the real ones. The result of those investigations, whichfocus mainly in the context of black hole thermodynamics suggest that the self dual variablescould be better behaved than the real ones with respect to the semi classical limit of the theory.It is well known that the computation of the entropy of a spherically symmetric isolated horizonbased on the real Ashtekar-Barbero variables leads to a semi-classical result which agrees with theBekenstein-Hawking area law only up to a fine tuning of the Immirzi parameter γ . On the otherhand, it was shown in [51, 52] that the dimension of the Hilbert space of a spherically isolatedhorizon, which is a function of γ , can be analytically continued to γ = ± i in a consistent way.Surprisingly, the result matches perfectly the Bekenstein-Hawking entropy without requiring anyfine tuning. Although this result has been obtained via an analytic continuation procedure andtherefore, there is little control on the underlying self dual quantum theory, it is quite striking.This conclusion was recently generalized to the case of rotating isolated horizon in [53]. Aside fromthis result, others investigations in the context of black holes thermodynamics [54–61], spinfoamsmodels [62] and in (2 + 1)-dimensional LQG in [63, 64], were worked out in the same period, all ofwhich suggest that the self dual variables could be more suited than the real ones with respect tothe semi classical limit of the theory.Moreover, from the point of view of the symmetries, it is well known that the real Ashtekar-Barbero connection does not transform as a true space-time connection under the action of theHamiltonian constraint, contrary to the self dual connection [65]. This fact could be responsible foranomalies when going to the quantum theory. In this paper, we will show that working with the selfdual variables instead of the real ones indeed allows to preserve the covariance of the system scalarfield coupled to spherically symmetric gravity , once the holonomy corrections have been introduced.More precisely, it turns out that using the self dual variables allows one to naturally bypass thepartial no go result obtained in [17] with the real variables, and obtain a closed algebra for themodified constraints. Thus, it is possible to obtain an anomaly free algebra for the system scalarfield coupled to spherically symmetric gravity , which moreover does not exhibit any deformations,reproducing exactly the classical hypersurface deformation algebra of GR . It represents thereforea first step towards the construction of anomaly free LQG model with local physical degrees offreedom, which was up to now out of reach.From the point of view of the flat spacetime limit, it is well known that the classical hypersur-face deformation algebra of GR reduces uniquely to the Poincar´e algebra [16]. Since the holonomycorrected algebra of constraint studied in this paper reproduces without deformations the hyper-surface deformation algebra of GR, it is natural to conclude that its flat spacetime limit will alsoreduces to the Poincar´e algebra. Whether some deformations occurs at the level of the co-algebrain the flat spacetime limit is still an open question [68]. However, since our interest is to build aquantizable model for studying the gravitational collapse in the context of LQG, the flat spacetimelimit of our modified algebra is of no interest for our purposes.The paper is presented as follows. In section II, we recall the partial no go results obtained in[17, 18]. In section III, we first introduce the formulation of the spherically symmetric sector ofGR written in terms of the self dual Ashtekar variables. This first part relies heavily on the workof Thiemann and Kastrup [69]. Then we implement holonomy corrections and obtain our modifiedeffective phase space. In section IV, we study the algebra of the modified constraints and presentour main result. Finally, section V is devoted to a discussion of the plausible future direction toquantize this model. II. SUMMARY OF THE PARTIAL NO GO RESULTS OBTAINED USING THE REALASHTEKAR-BARBERO FORMALISM
In this section, we recall some previously established facts and summarize the partial no go resultsobtained in [17]. It is not necessary that quantum corrections must lead to a deformed notion of general covariance. Indeed, it isknown that perturbative higher curvature corrections do not modify the hypersurface deformation algebra [66, 67].
1. The classical framework
Let us consider the phase space of GR written in terms of the real SU(2) Ashtekar-Barbero variables { A ia , E bj } = γδ ij δ ba , A ia = Γ ia + γK ia , E ai = √ q e ai , (2.1)where q is the determinant of the induced metric over the spatial slices q ab = e ia e jb δ ij and K ia = K ab e bi is the extrinsic curvature of these spatial slices. The canonical variables are constrained tosatisfy the Gauss constraint G i , the vectorial constraint H a and the scalar (Hamiltonian) constraint H , G i = D a E ai , H a = E bi F iab , H = 1 √ E E ai E bj ( ǫ ijk F kab − γ ) K [ ia K j ] b ) , (2.2)where F iab = ∂ a A ib − ∂ b A ia + ( A a ∧ A b ) i is the curvature of the Ashtekar-Barbero connection.While the Gauss constraint generates the SU(2) gauge transformations, the linear combination D [ N a ] = N a ( H a − γ − A a . G ) generates the infinitesimal spatial diffeomorphisms and the scalarconstraint H generates the infinitesimal diffeomorphisms in the time direction as selected by theinitial slicing of the four dimensional manifold M . Once smeared, those constraints form thefollowing algebra { D [ M a ] , D [ M a ] } = D [ L M M a ] , (2.3) { H [ N ] , D [ M a ] } = − H [ L M N ] , (2.4) { H [ N ] , H [ N ] } = D [ q ab ( N ∂ b N − N ∂ b N )] . (2.5)We do not rewrite the Poisson bracket involving only the Gauss constraint, since they will not playa role in what follow.The spherically symmetric sector of the phase space of GR in terms of the real Ashtekar Barberovariables is given by three pairs of canonically conjugated variables which are constrained to satisfythe three first class constraints G i , H x and H (there is only one nontrivial component of the Gaussand the diffeomorphism constraint each). Therefore, the number of local physical degrees of freedomis d = 6 − × ds = ( E φ ) | E x | dx + | E x | ( dθ + sin θdφ ) , (2.6)where x is the non compact direction which matches the radial direction at infinity. Once theGauss constraint G i is solved, we obtain a phase space given by two pairs of canonically conjugatedvariables denoted (once we set G = 1) as { K x ( x ) , E x ( y ) } = δ ( x, y ) , { K φ ( x ) , E φ ( y ) } = δ ( x, y ) . (2.7)The remaining constraints are given by D [ M ] = Z dx M ( x ) ( 12 ( E x ) ′ K x + K ′ φ E φ ) , (2.8) H [ N ] = 12 Z dx N ( x ) ( E φ | E x | / K φ + 2 | E x | / K φ K x + E φ | E x | / (1 − Γ φ ) + 2Γ ′ φ | E x | / ) . (2.9)Their algebra is simply the hypersurface deformation algebra adapted to the midisuperspacemodel, given by { D [ M ] , D [ M ] } = D [ L M M ] (2.10) { H [ N ] , D [ M ] } = − H [ L M N ] (2.11) { H [ N ] , H [ N ] } = D [ q xx ( N ∂ x N − N ∂ x N )] . (2.12)Let us now study the holonomy corrected constraints.
2. The holonomy corrected constraint algebra for the vacuum case
At the effective level, the polymerization of the connection variables (in this case the extrinsiccurvature components), can be implemented as the following transformations K x → f ( K x ) and K φ → f ( K φ ) . (2.13)However, the holonomy corresponding to K x is an extended one represented along the edgesof spin-network states and is actually difficult to implement explicitly, requiring suitable non-localfunctions (see e.g. [70] for a negative result in this regard). Therefore, this holonomy correction isusually disregarded in the first attempt. While this might seem to be an oversimplification at first,it can be shown that it is possible to rewrite the constraints in such a manner that the Hamiltonianconstraint does not depend on K x . Therefore, the correction K x → f ( K x ) can be safely ignoredfor our purposes .In [30–39], the function f was chosen such that f ( K φ ) = sin ( λK φ ) /λ , in analogy with theresult of the polymerization procedure that is implemented within LQC. For a formal derivation ofthis correction, we should first find a regularization formula for the curvature or for the connectionin term of the holonomies, such as the Baker Campbell Haussdorf formula used in LQC, and thenderive the precise form of the polymerization function f from it. In order to avoid this difficulty,we will work with a general function f without fixing its expression and allowing for quantizationambiguities. This is the strategy used in [17], which ensures that conclusions will be general.Under this polymerization, the modified algebra of the basic variables is as follow { K x ( x ) , E x ( y ) } = δ ( x, y ) , { f ( K φ )( x ) , E φ ( y ) } = d f d K φ ( x ) δ ( x, y ) . (2.14)As in [30–39], we only polymerize the scalar constraint and therefore obtain D [ M ] = Z dx M ( x ) ( 12 ( E x ) ′ K x − ( K φ ) ′ E φ ) , (2.15) H [ N ] = 12 Z dx N ( x ) ( E φ | E x | / f ( K φ ) + 2 | E x | / g ( K φ ) K x + E φ | E x | / (1 − Γ φ ) + 2Γ ′ φ | E x | / ) , (2.16)where we have introduced a new function g in order to be general. One can show [17] that thefunctions f and g are not independent, but are actually related through (which, of course, is truefor the classical case as well) g = 12 ∂f ∂K φ . (2.17) Although the diffeomorphism constraint does still depend on K x , it is usually left unchanged. Even the K φ component appearing in D [ N x ] is not modified. See [17] for details on why this is justified. This property also allows one to Abelianize the constraints, as first proposed in [37].Following [17], we compute the algebra of the modified constraints and finally obtains { D [ M ] , D [ M ] } = D [ L M M ] , (2.18) { H [ N ] , D [ M ] } = − H [ L M N ] , (2.19) { H [ N ] , H [ N ] } = D [ β ( K φ ) q xx ( N ∂ x N − N ∂ x N )] , where β ( K φ ) = 12 ∂ f ∂K φ . (2.20)We can see immediately that the algebra of the modified constraints is still closed, allowing fora complete quantization of the spherically symmetric vacuum case. However, the last bracket ismodified by the function β , which depends only on the extrinsic curvature K φ . If the singularityis resolved when f ( K φ ) is maximal, as in LQC, the function β , given by the second derivative of f , flips its sign at the ‘bounce’. This can then be interpreted as an ‘effective’ signature changesince then it has the same signature as in the Euclidean case.Let us now describe the case where a scalar field is coupled to the spherically symmetric vacuum,thus including some local degrees of freedom within the analysis.
3. Adding a minimally coupled scalar field
The situation is radically different when we couple a scalar field Φ to the vacuum model. In thiscase, the system inherits some local degrees of freedom and the canonical pairs are given by { K x ( x ) , E x ( y ) } = δ ( x, y ) , { K φ ( x ) , E φ ( y ) } = δ ( x, y ) , { Φ( x ) , P Φ ( y ) } = δ ( x, y ) . (2.21)The total constraints are written as D T [ M ] = D g [ M ] + D m [ M ] = Z dx M ( x ) ( 12 ( E x ) ′ K x + K ′ φ E φ ) + 4 π Z dx M ( x ) P Φ Φ ′ ,H T [ N ] = H g [ M ] + H m [ M ]= 12 Z dx N ( x ) ( E φ | E x | / K φ + 2 | E x | / K φ K x + E φ | E x | / (1 − Γ φ ) + 2Γ ′ φ | E x | / )+ 4 π Z dx N ( x ) ( P | E x | / E φ + | E x | / E φ Φ ′ + 12 | E x | / E φ V (Φ) ) . Proceeding to the same polymerization mentioned above, we obtain the following holonomy cor-rected constraints D T [ M ] = D g [ M ] + D m [ M ] = Z dx M ( x ) ( 12 ( E x ) ′ K x + K ′ φ E φ ) + 4 π Z dx M ( x ) P Φ Φ ′ H T [ N ] = H g [ M ] + H m [ M ]= 12 Z dx N ( x ) ( E φ | E x | / f ( K φ ) + 2 | E x | / g ( K φ ) K x + E φ | E x | / (1 − Γ φ ) + 2Γ ′ φ | E x | / )+ 4 π Z dx N ( x ) ( P | E x | / E φ + | E x | / E φ Φ ′ + 12 | E x | / E φ V (Φ) ) , where the functions f and g satisfy (2.17). After a lengthy but straightforward computation, weobtain the algebra of those modified constraints supplemented with holonomy corrections { D T [ M ] , D T [ M ] } = D T [ L M M ] , (2.22) { H T [ N ] , D T [ M ] } = − H T [ L M N ] , (2.23) { H T [ N ] , H T [ N ] } = D g [ β ( K φ ) q xx ( N ∂ x N − N ∂ x N )] + D m [ q xx ( N ∂ x N − N ∂ x N )] (2.24) = D T [ β ( K φ ) q xx ( N ∂ x N − N ∂ x .N )] . (2.25)While the former two brackets, (2.22) and (2.23), reproduce the corresponding subalgebra of thehypersurface deformation algebra, the latter one, (2.25), does not close anymore. Indeed, modifica-tions on the gravitational part occurring through the function β ( K φ ) do not show up in the matterpart. Thus we cannot factorize these modifications, and finally obtain the full diffeomorphism con-straint D T . One can still try to introduce some modifications through some functions dependingon K φ in the matter part, in such a way that the function β ( K φ ) factorize, but now the bracketbetween the gravitational and the matter parts is not zero anymore which will prevent the algebrafrom closing. Note that these results do not depend on the precise form of the correction function f . Therefore, this is a very general obstruction preventing the matter part of the constraint fromclosing the algebra. See [17] for a detailed proof of these statements.As a consequence, one is forced to conclude that within this context, the holonomy correctedsystem scalar field coupled to spherically symmetric gravity is not covariant. This no go resultprevents from quantizing this loop model following the procedure adopted for the vacuum case in[30–39]. It is therefore mandatory, as a first step, to find a way to by pass those partial no goresults and find an effective covariant model for this system. In a second step, one should find away to get the physical Hilbert space this system. If one succeeds to do so, it would provide a veryexciting simplified platform to investigate the fate of a gravitational collapsing scalar quantumshell, its evaporation through Hawking radiation and problems related to those phenomena, suchas the information loss paradox.In this paper, we present a way out for the first step, i.e. the possibility to obtain a covariantholonomy corrected phase space for this system. The second step, i.e. the full quantization of thismodel, will be discussed at the end but it will require some novel non trivial steps which are stillunder development and are beyond the scope of this paper. III. THE SPHERICAL SYMMETRIC SECTOR OF SELF DUAL ASHTEKAR GRAVITY
In this section, we present the spherically symmetric phase space of GR in terms of the self dualAshtekar variables. The formulation of the spherically symmetric sector was first presented in[69] where the authors proceeded to the quantization of the spherically symmetric vacuum inthe Schr¨odinger representation. We will keep their notations in our presentation. The detailsconcerning the symmetry reduction procedure and the construction of the adapted variables canbe found in [71–76].
A. The classical framework
Let us first focus on the pure gravity case. Using the self dual Ashtekar formulation of GR, theHolst action for pure gravity reads (for details, we refer the reader to [77]). S = 1 κ Z { ǫ IJKL e I ∧ e J ∧ F KL ( A ) + 1 γ e I ∧ e J ∧ F ( A ) IJ } , (3.1)0where γ = ± i and the gauge group is SL(2 , C ).The interest of working with self dual variables in the action is that the canonical analysis turnsout to be much simpler, and while we end up again with first class constraints, the form of thescalar constraint H simplifies drastically. In terms of these constraints, the precedent action canbe written as S = 1 κ Z dt Z dx { Θ L − ( iλ i G i − iN a D a + 12 ˜ N H ) } , (3.2)where the different terms are respectively the canonical variables part, i.e. the Liouville form Θ L ,the Gauss constraint G i enforcing the SL(2 , C ) symmetry, the spatial diffeomorphism constraint D a and finally the Hamiltonian constraint H . The self dual canonical variables are given by A ia = Γ ia + iK ia , E ai = ǫ abc ǫ ijk e jb e kc , { A ia , E bj } = iδ ij δ ba , (3.3)and satisfy the first class constraints G i = D a E ai , H a = ǫ abc E bi B ci , H = 12 ǫ abc ǫ ijk E bi E cj B ak , (3.4)where the ˜ N = N/ (det E ) − is the rescaled lapse function. The ‘magnetic’ field variable B hasbeen defined for simplicity and is related to the curvature of the self dual Ashtekar connection as B ai = 12 δ ij ǫ abc F jbc = 12 δ ij ǫ abc ( ∂ b A jc − ∂ c A jb + ( A b × A c ) j ) . (3.5)Finally, we have also to impose some reality conditions to the canonical variables in order torecover GR: E ia E b i − ¯ E ia ¯ E b i = 0 and A ia − ¯ A ib = 2Γ ia . (3.6)However, as discussed later, there are alternative ways to implement the reality conditions, one ofwhich shall be employed by us. B. Reduction to spherical symmetry
Let us now select the spherically symmetric sector of this phase space. The procedure for performingthis symmetry reduction was presented and discussed in [71–76]. We refer the reader to thosereferences for more details. The spatial metric is given by ds = E E dx + E ( dθ + sin θdφ ) , where E = ( E ) + ( E ) . (3.7)Using this procedure, we end up with the symmetry reduced connection adapted to the spheri-cally symmetric case, as well as its conjugated momentum. They read( E x , E θ , E φ ) = ( E sin θn x , √ E n θ + E n φ ) sin θ, √ E n φ − E n θ ) ) , (3.8)( A x , A θ , A φ ) = ( A n x , √ A n θ + ( A − √ n φ ) , √ A n φ − ( A − √ n θ ) sin θ ) . (3.9)We can then compute the ‘magnetic’ field B ( B x , B θ , B φ ) = ( B sin θn x , √ B n θ + B n φ ) sin θ, √ B n φ − B n θ ) ) , (3.10)1where we have B = 12 ( ( A ) + ( A ) − ,B = A ′ + A A ,B = − A ′ + A A . Having obtained our variables, we can now give the expression of the constraints in the reducedsymmetry model. The Liouville form reduces toΘ L = − i sin θ { E ˙ A + E ˙ A + E ˙ A } . We deduce that we have three canonically conjugate pair of variables, which define our uncon-strained phase space. Those variables are algebraically constrained by the three first class con-straints, which therefore lead to D = 6 − × G x = sin θ { ( E ) ′ − E A + E A } , H x = sin θ { B E − B E } , H = sin θ { E (2 E B + E B ) + E (2 E B + E B ) } . We can rewrite the diffeomorphism constraint D as the following linear combination of the vectorialconstrant H x and the Gauss constraint G x D = H x − A G x = sin θ { A ′ E + A ′ E − A ( E ) ′ } (3.11)This latter will generate the residual diffeomorphisms along the radial direction x . Note that theoverall factor sin θ will disappear once integrating over the angular part of the action. Finally,the spherically symmetric version of the reality conditions will not be useful in what follow, thuswe refrain from writing them explicitly (see the last section for more details about the realityconditions). Having described our classical symmetry reduced phase space, we can now implementthe holonomy corrections. C. Implementing the holonomy corrections
In this section, we implement the holonomy corrections, which encode the quantum corrections atan effective level inherited from the polymer or loop quantization. However, instead of followingexactly what is done in the real formulation [17], K → f ( K ), we follow an equivalent procedurewhere we introduce corrections of the type B → f ( B ). This implies that we end up modifyingthe curvature functions on using the holonomies (instead of the connection coefficients) in them.Thus, instead of introducing modification functions for the Ashtekar connection components, A i ,we therefore choose to modify the dual of the curvature components, B i . Note that this is anequivalent prescription since the dualized curvature components are functions of A i alone anddoes not depend on the triad components and, thus, there is a one-to-one correspondence betweenthem. The holonomy corrections, in a very general way, can be implemented by the followingtransformations B → f ( B ) , B → f ( B ) , B → f ( B ) . (3.12)2However, all of these transformations simultaneously turn out to be too general, and cannot beimplemented consistently . Instead, we will mimic what is usually done in the real sphericallysymmetric model, and introduce only point-wise local holonomy modifications, namely B → f ( B ) while f ( B ) = B f ( B ) = B . (3.13)As in the real case, we are only modifying the angular part of the curvature, B ∼ F . Indeed,since one cannot explicitly compute the holonomy of the x -component of the connection A , oneshould not modify the curvature component which involve it, i.e. F and F . One can alsounderstand that point by noticing that implementing local (point-wise) holonomy corrections isequivalent to modifying the angular part of the Ashtekar connection, as in the real variables case.This means introducing general functions of the form f ( A + A ) in the self dual case, which isequivalent to using a modified version of B that represents regularized version of the angular partof the curvature. The dependence of the other curvature components on A and A are not ofthe same form and thus modifying those would not be akin to modifying the angular part of theconnection alone. This means that if we modify the A and A components in the other curvaturecomponents ( B and B ), this would imply modifying also the radial components as well, whichcan only be corrected using non-local functions.In addition, one can easily check to find that replacing A → h ( A ) and A → h ( A ) in the B and B immediately requires that such modification functions have to be the same as the in theclassical case from requirements of anomaly freedom of the constraint algebra (See the Appendixfor clarifications of this assertion).With this effective modification, the deformed constraints are therefore given by D = [ A ′ E + A ′ E − A ( E ) ′ ] , (3.14) H = 12 [ E (2 E B + E f ( B )) + E (2 E B + E g ( B )) ] , (3.15)where we have introduced a new function g to remain general. We will see that, in order to closethe algebra, we must require f = g . Note also that, once again, we have modified only the scalarconstraint, following what was done in [17] . We are now ready to compute the algebra of themodified constraint and investigate the covariance of this new holonomy-corrected system. IV. INVESTIGATING THE COVARIANCE OF THE MODEL IN THE SELF DUALCONTEXT
In this section we present the computation of the algebra of the modified constraints. We firstdemonstrate our results for the vacuum case, and then extend it to the system composed by thescalar field minimally coupled to spherically symmetric gravity.
A. The vacuum model
The Hamiltonian constraint, for this case, reads H [ N ] = (cid:18) (cid:19) Z d xN ( x ) (cid:8) E ( x ) (cid:0) B ( x ) E ( x ) + B ( x ) E ( x ) (cid:1) + E ( x ) (cid:0) B ( x ) E ( x ) + B ( x ) E ( x ) (cid:1)(cid:9) . (4.1) In our case, this seems natural as we are only modifying the curvature components and not the connectioncomponents directly. Indeed, the magnetic field component B doesn’t enter in the expression of the diffeomorphismconstraint. H [ N ] = (cid:18) (cid:19) Z d xN (cid:8) E (cid:0) B E + f (cid:0) B (cid:1) E (cid:1) + E (cid:0) B E + g (cid:0) B (cid:1) E (cid:1)(cid:9) , (4.2)where we have introduced the two different modification functions f and g and have suppressedthe argument of each of the phase space variables on the radial coordinate. The other constraints,the diffeomorphism one and the Gauss one, remain unmodified from the classical ones. Our firstgoal is to calculate the [ H [ N ] , H [ M ]] bracket with such correction functions. We look at thebrackets of the Hamiltonian constraint with itself and with the diffeomorphism constraint. Theother bracket involving the Hamiltonian constraint with the Gauss constraint remains obviouslyunmodified. [ H, H ] bracket We start with the calculation of the [
H, H ] bracket, since this is the only one which gets deformedin the real Ashtekar-Barbero case by a modification of the structure functions appearing on theright hand side of the expression. In such calculations, it is useful to remember that we have anon zero bracket between two conjugate variables only when one of them have a spatial derivativeon it. (For instance, note that B is independent of spatial derivatives whereas B , B are not.)Thus, the contribution to this bracket should come from the commutator between the first andthird term, the second and third term and the first and fourth term in the Hamiltonian constraint.The bracket between the first and third term is Z d x d yM ( x ) N ( y ) E ( x ) E ( y ) (cid:2) E ( x ) B ( x ) , E ( y ) B ( y ) (cid:3) − ( x ↔ y )= i Z d x d yM ( x ) N ( y ) E ( x ) E ( y ) (cid:26) E ( x ) B ( y ) dd x [ δ ( x, y )] + B ( x ) E ( y ) dd y [ δ ( x, y )] (cid:27) − ( x ↔ y )= i Z d x (cid:0) M ′ ( x ) N ( x ) − N ′ ( x ) M ( x ) (cid:1) n(cid:0) E ( x ) (cid:1) E ( x ) B ( x ) − (cid:0) E ( x ) (cid:1) E ( x ) B ( x ) o . (4.3)The above bracket does not involve any of the modification functions and is exactly what it wouldhave been for the classical case. Turning our attention to the bracket between the first and fourthterm, we find the appearance of the holonomy corrections Z d x d yM ( x ) N ( y ) E ( x ) E ( x ) f (cid:0) B ( y ) (cid:1) h B ( x ) , (cid:0) E ( y ) (cid:1) i − ( x ↔ y )= Z d x d yM ( x ) N ( y ) E ( x ) E ( x ) f (cid:0) B ( y ) (cid:1) E ( y ) dd x [ δ ( x, y )] − ( x ↔ y )= i Z d x (cid:0) M ′ ( x ) N ( x ) − N ′ ( x ) M ( x ) (cid:1) E ( x ) E ( x ) E ( x ) f (cid:0) B ( y ) (cid:1) . (4.4)Proceeding similarly, we find that the bracket between the second and third terms gives − i Z d x (cid:0) M ′ ( x ) N ( x ) − N ′ ( x ) M ( x ) (cid:1) E ( x ) E ( x ) E ( x ) g (cid:0) B ( y ) (cid:1) . (4.5)We know that [ H [ N ] , H [ M ]] must close into one of the other first class constraints. In theclassical case we have[ H [ N ] , H [ M ]] = i Z d x (cid:0) M ( x ) N ′ ( x ) − N ( x ) M ′ ( x ) (cid:1) (cid:0) E ( x ) (cid:1) (cid:8) B ( x ) E ( x ) − B ( x ) E ( x ) (cid:9) , (4.6)4where the vector constraint is given by H x [ N x ] = R d xN x ( x ) (cid:8) B ( x ) E ( x ) − B ( x ) E ( x ) (cid:9) . Inorder to have a similar closure of the [ H [ N ] , H [ M ]] commutator, it is immediately obvious thatwe require f (cid:0) B (cid:1) = g (cid:0) B (cid:1) for (4.4) to cancel (4.5), just as in the classical case. Thus we findthat the the commutator of Hamiltonian constraints give us the exact same result as we have inthe classical case, even in the presence of holonomy modifications!There are a few observations to make from this rather astonishing result. Firstly, from amathematical point of view, the reason for this can be explained as follows. The modification tothe structure functions in the real variables case is primarily due the presence of second (spatial)derivatives of the triad components appearing in the Hamiltonian constraint [78]. Those termsappear from the presence of the spin connection term, which does not appear in the self-dual case.The coefficient of the spin connection term is (1 + γ ) and that goes to zero when γ = i . (Thisalso tells us that this result is a rather special case and would not be valid for a general imaginaryImmirzi parameter.) The other thing to point out is that we should really look at the hypersurfacedeformation algebra which really has the [ H, H ] bracket closing into a diffeomorphism constraint.We can easily rewrite the vector constraint as a combination of the diffeomorphism constraint andthe Gauss constraint. We then need to make sure that the bracket of the Hamiltonian constraintcloses with both these other first class constraints. The one with the Gauss constraint remainsobviously unmodified whereas the one with the diffeomorphism constraint is shown in the nextsection. [ D, H ] bracket Rewriting the diffeomorphism constraint D [ N x ] = Z d xN x [ − A ( E ) ′ + A ′ E + A ′ E ] , (4.7)we want to evaluate the bracket [ D [ N x ] , H [ N ]]. Instead of explicitly showing the full calculationinvolving all the terms, let us only focus on the bracket of the second term from the Hamiltonianconstraint with the diffeomorphism constraint. It is enough to do so in this case since a particularterm of the Hamiltonian constraint must reproduce that specific term from the bracket with the5diffeomorphism constraint: (cid:20) D [ N x ] , − Z d yN f ( B )( E ) (cid:21) = 12 Z d x d yN x ( x ) N ( y ) (cid:2) − A ( x )( E ) ′ ( x ) + A ( x ) ′ E ( x ) + A ′ ( x ) E ( x ) , f ( B ( y ))( E ) ( y ) (cid:3) = − i Z d xN x ′ N f ( B )( E ) − i d xN x N f ( B )( E ) ′ E − i Z d xN x N A ′ ∂f ∂A ( E ) − i Z d xN x N A ′ ∂f ∂A ( E ) = − i Z d x (cid:0) N x ′ N − N x N ′ (cid:1) ( E ) f ( B ) − i Z d xN x ′ N ( E ) f ( B ) − i Z d xN x N ′ ( E ) f ( B ) − i Z d xN x N ( E ) [ f ( B )] ′ − i Z d xN x N f ( B )( E ) ′ E = − i Z d x (cid:0) ( N x ) ′ N − N x N ′ (cid:1) ( E ) f ( B ) − i Z d x ( N x N ) ′ ( E ) f ( B ) − i Z d xN x N ( E ) [ f ( B )] ′ − i Z d xN x N f ( B ) (cid:0) ( E ) (cid:1) ′ = − i Z d x (cid:0) ( N x ) ′ N − N x N ′ (cid:1) ( E ) f ( B ) + total derivative . (4.8)Similarly, the bracket of all the other terms of the Hamiltonian constraint with the diffeomorphismconstraint reproduces the whole Hamiltonian constraint.Thus, we have shown that starting from the holonomy corrected constraints D = [ A ′ E + A ′ E − A ( E ) ′ ] , (4.9) H = 12 [ E (2 E B + E f ( B )) + E (2 E B + E g ( B )) ] , (4.10)their algebra is given by { D [ M x ] , D [ M x ] } = D [ L M x M x ] , (4.11) { H [ N ] , D [ M x ] } = − H [ L M x N ] , (4.12) { H [ N ] , H [ N ] } = H x [( E ( x )) ( N ∂ x N − N ∂ x N )] , (4.13)reproducing without any modifications the classical algebra of constraints. However, to reproducethe hypersurface deformation algebra of GR, we need to rewrite the vector constraint in terms ofthe diffeomorphism constraint and the Gauss constraint as well as restore the rescaling of the lapsefunctions in terms of the determinant of the spatial metric. Looking at the RHS of (4.13), Z d x (cid:0) E ( x ) (cid:1) ( N ∂ x N − N ∂ x N ) H x = Z d x (cid:0) E ( x ) (cid:1) ˜ N (det E ) ∂ x ˜ N (det E ) ! − ˜ N (det E ) ∂ x ˜ N (det E ) !! ( D + A G x )= Z d x q xx (cid:16) ˜ N ∂ x ˜ N − ˜ N ∂ x ˜ N (cid:17) ( D + A G x ) , (4.14)it is clear that we can recover the usual form of the classical hypersurface deformation algebra,once we solve for the Gauss constraint G x ≈
0. (We have used the expression for q xx in the lastline above.)6More precisely, the modifications function β ( K φ ) that shows up in the Poisson bracket betweentwo Hamiltonian constraints (2.20) in the real formulation has disappeared in the self dual formu-lation. Therefore, the question of the interpretation of this modification simply drops out in thecontext of the self dual formulation. B. Adding a minimally coupled scalar field
Let us now couple a scalar field to the spherically symmetric vacuum. The preceding holonomycorrected constraints for the vacuum case are now supplemented by terms arising from the matterpart. The matter part of the constraint comes in turn from the corresponding ones for a minimallycoupled scalar to a spherically symmetric space-time. The Hamiltonian constraint for a scalar fieldis given by H scalar = Z d xN (cid:26) P √ q − √ qq xx Φ ′ + √ qV (Φ) (cid:27) , (4.15)whereas the diffeomorphism constraint in this case is D scalar = Z d xN x Φ ′ p Φ . (4.16)We have one more canonical pair in the new phase space given by { P Φ ( x ) , Φ( y ) } = δ ( x, y ) / π .Thus the full constraint is as follow D T [ M x ] = D grav [ M x ] + D scalar [ M x ] = Z dx M ( x ) ( A ′ E + A ′ E − A ( E ) ′ ) + 4 π Z dx M ( x ) P Φ Φ ′ ,H T [ N ] = H grav [ N ] + H scalar [ N ]= 12 Z dx N ( x ) ( E (2 E B + E f ( B )) + E (2 E B + E f ( B )) )+ 2 π Z dx N ( x ) (cid:0) P + ( E ) Φ ′ + E (( E ) + ( E ) ) V (Φ) (cid:1) . It is easy to observe that there shall be no holonomy modification functions appearing in thematter sector since they do not have any dependence on the connection components but ratheron only triad variables. Thus the commutator of the scalar Hamiltonian would close into thescalar diffeomorphism, as in the classical case. However, deviations from the classical form of thehypersurface deformation algebra can still take place only if there are cross terms between thegravity and matter sectors, i.e. if [ H grav [ N ] , H scalar [ M ]] = 0. In the classical case, such crossterms were zero and thus the bracket closed into the vector constraint. The holonomy correctionsintroduced by us have an argument of B , and do not depend on the other curvature components.However, B does not have any spatial derivatives on the connection components, which has nonon-zero contribution to the above bracket. Since this is the only modification introduced by us,and it does not contribute to the bracket, we can easily conclude [ H grav [ N ] , H scalar [ M ]] = 0. Thusthe full Hamiltonian constraint commutator still closes into constraints without any deformationof the structure function as in the classical case.All the other brackets between the full Hamiltonian constraint and the full diffeomorphismconstraint, and between the full Hamiltonian constraint and the full Gauss constraint remains thesame as in the classical case. We can see this without explicit calculation by just noticing thatall the cross terms between the gravitational and the matter parts of the constraints vanish, i.e.7[ H grav [ N ] , D scalar [ N x ]] = 0 and [ H grav [ N ] , G scalar [ λ ]] = 0. Thus we now have a model with localdegrees of freedom which does have holonomy corrections but the underlying covariance of thesystem is the same as the classical case and is given by { D T [ M ] , D T [ M ] } = D T [ L M M ] , (4.17) { H T [ N ] , D T [ M ] } = − H T [ L M N ] , (4.18) { H T [ N ] , H T [ N ] } = D T [ q xx ( N ∂ x N − N ∂ x N )] , (4.19)where we have written H T = H grav + H scalar and D T = D grav + D scalar . V. DISCUSSION
Let us now discuss the results obtained in this paper. We have shown that the partial no go resultsobtained in [17] within the context of spherically symmetry reduced loop model can be overcomeby working with the self dual variables instead of the real ones. We comment briefly on the statusof the self dual variables and then discuss the possibility to quantize this model.
1. Undeformed covariance with self-dual variables
From the point of view of the self dual variables, our result reinforces the idea that the self dualvariables are more natural in the context of the loop quantum theory of black holes. Indeed, aspointed in the introduction, it has been shown recently that the self dual variables reproduce in amore satisfying way (without any fine tuning) the expected semi-classical results in the context ofblack holes thermodynamics, such as the Bekenstein-Hawking area law for the entropy of sphericallyand rotating isolated horizons [51–53], or the thermal character of the partition function for thehorizon [54–56]. In this paper, we have shown that, additionally, the use of the self dual variablesallows to define a quantizable model for the gravitational collapse, which does not suffer from thedrawbacks of [49] and by pass naturally the no go result found in [17].While the model we studied in this work is an effective quantum one, where the holonomycorrections are partially implemented (recall that we only modified the angular part of the curva-ture), it represents the first holonomy corrected model of spherically symmetric gravity coupled toa scalar field which is fully covariant. It seems that one is forced to use the self dual variables inorder to define a quantizable holonomy corrected model for this system due to the no-go theoremof [17].Even more remarkably, we have shown that the holonomy corrected algebra has the exact sameform as the classical hypersurface deformation algebra, which is an unexpected outcome. Indeed,it is commonly believed that classical diffeomorphism symmetry of general relativistic spacetimeswill be deformed at the quantum level. While it is the case when one uses the real Ashtekar-Barbero variables resulting in the so called signature change phenomenon [17], it turns out thatno symmetry deformation occurs in the self dual case, at least in the spherically symmetric sector.Whether this conclusion extends to other sectors where the signature change phenomenon alsooccurs, needs to be worked out in the future. If this conclusion holds for all such models, then oneshould interpret signature change as an artificial characteristic of using the real Ashtekar-Barberovariables.Moreover, the fact that we have obtained an algebra of the modified constraints which is ex-actly the same than the hypersurface deformation algebra, as arising for ordinary GR, could seemproblematic at first. Indeed, the well known theorem obtained by Hojman, Kuchar and Teitelboim[79, 80] states that starting from the hypersurface deformation algebra, one can uniquely derive the8Einstein-Hilbert action, up to a cosmological constant, when the constraints contain no higher thansecond derivatives of the metric. However, that result has been derived for full (3 + 1)-gravity andwe have considerable more freedom in our case since we are working in a symmetry reduced model.(Once again, this is related to the fact that we have only one non-zero component of the spatialdiffeomorphism constraint in our case.) Therefore, the Hojman-Kuchar-Teitelboim theorem, in itsoriginal form, doesn’t apply to our symmetry reduced system.Finally, from the point of view of the anomaly free algebra, this result opens up a very promisingpath towards constructing quantum theories of inhomogeneous midisuperspace models with localphysical degrees of freedom, which have been out of reach up till now. While our results pointtowards the need of using self dual variables for such non trivial models , a definitive conclusionis not yet possible and one has to investigate some other reduced loop model in order to obtain amore robust generic result along these lines. The next interesting reduced models to investigatewould be polarized Gowdy models for which partial no go theorem have already been proven in [18]using the real Ashtekar-Barbero formulation. We plan to address this question in a future work.While very encouraging, the anomaly free algebra presented in this work will be useful only ifone is either able to extract a concrete effective theory from it, by picking up an explicit form ofthe holonomy correction f , or, even better if one is able to quantize this model based on the selfdual variables. Let us now comment on this point.
2. On the quantization of this self-dual model
There are two main outstanding technical complications that are encountered within the quanti-zation procedure of this model:1. one has to derive the explicit expression of the holonomy correction function f ( B );2. one has to find a way to implement the reality conditions inherent to the self dual formulation.While the first point represents the most important difficulty, the second one can be overcome moreeasily in this spherically symmetry reduced case. Let us comment on this latter point first. Whilethe imposition of the reality conditions at the quantum level for the full theory is a highly non trivialproblem, which remains open since the very advent of the self dual variables, the situation is quitedifferent in symmetry reduced models. Indeed, if one knows what are the quantum observablesof the system studied, which is a non trivial question for a diffeomorphism invariant system,one can simply require those observables to be self adjoint with respect to the scalar producton the physical Hilbert space of the quantum theory. This was precisely the strategy used in[69] where the spherically symmetric self-dual Ashtekar gravity was quantized in the Schr¨odingerrepresentation. Therefore, it seems reasonable to infer that the problem generally associated tothe reality conditions will not be difficult to be solved within our symmetry-reduced model either.For the first obstruction, the situation is more subtle. Since we are now working in the self-dualformulation of gravity, the gauge group is non compact and given by SL(2 , C ). At first sight,it could seem hopeless to expect a resolution of the singularity because the holonomy correctedfunctions can be unbounded contrary to the SU(2) case. However, it has been shown in [82] thatsuch preconceptions can be misleading. In this work, a proposal for defining a self dual modelof LQC through an analytic continuation procedure was introduced. It was possible to exhibit abounded holonomy correction function involving hyperbolic functions, which still preserved the Indeed, we have demonstrated a similar result for cosmological scalar perturbations in the self dual context wherewe end up with an undeformed algebra, as opposed to the system with real-valued variables [81]. , C ). It is then straightforward to seethat we cannot work in the fundamental representation if we want to obtain singularity resolutionsince the curvature will be replaced by an unbounded function. Instead, we need to investigatethe higher dimensional representations of the group in order to obtain holonomy corrections whichremains bounded (albeit this comes at the expense of working with more complicated lookingfunctions). Implementing this program is currently under investigation with promising initialresults. Acknowledgements
We are indebted to Martin Bojowald for a careful reading of an earlier version of the draftand for many helpful suggestions on it. We would also like to thank Miguel Campiglia, RodolfoGambini, Jorge Pullin and Javier Olmedo for useful comments, as well as Julien Grain, GiovanniAmelino-Camelia and Michele Ronco for critical remarks. This work has been supported by theShanghai Municipality, through the grant No. KBH1512299, and by Fudan University, throughthe grant No. JJH1512105.
Appendix
Let us suppose that we do not consider local holonomy corrections alone and introduce modificationfucntions in A and A wherever we see them in B , B and B . The correction functions wouldthen take the form B = f ( A + A ) (5.1) B = d g ( A )d A A ′ + A h ( A ) (5.2) B = d g ( A )d A A ′ + A h ( A ) . (5.3)The correction functions g and g an be immediately ruled out by looking at Eqns (4.4) and (4.5).If we have g and g different from the classical case, then these two terms cannot cancel out anymore (which we require for the closure of the brackets).However, the modification functions h and h do not cause any problems as far as the closingof the brackets are concerned. But if we look at Eqn (4.3), we shall find that the new term leftover is not going to be proportional to the diffeomorphism (or vector) constraint any longer. Thenew version of Eqn (4.3) shall be given by i Z d x (cid:0) M ′ ( x ) N ( x ) − N ′ ( x ) M ( x ) (cid:1) n(cid:0) E ( x ) (cid:1) E ( x ) B ( x ) − (cid:0) E ( x ) (cid:1) E ( x ) B ( x ) o , (5.4)0with the modification coming from the fact that B and B are now modified by the presenceof h and h . One might think that this is okay if we redefine our vector constraint to havethese modifications in built in it. However, in that case, the diffeomorphism constraint shall alsoget modified by the presence of these functions h and h (remember that the diffeomorphismconstraint is just a linear combination of the vector and Gauss constraints). It is then easy toconvince one self that the { D, D } bracket does close with any modification functions in it (andthis is partly the reason why we do not modify the diffeomorphism constraint). 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