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Signature change in 2-dimensional black-hole models of loop quantum gravity

Signature change has been identified as a generic consequence of holonomy modifications in spherically symmetric models of loop quantum gravity with real connections, which includes modified Schwarzschild solutions. Here, this result is extended to 2-dimensional dilaton models and to different choices of canonical variables, including in particular the Callan-Giddings-Harvey-Strominger (CGHS) solution. New obstructions are found to coupling matter and to including operator-ordering effects in an anomaly-free manner.


Introduction
General covariance is deformed in most of the existing midisuperspace models of loop quantum gravity (LQG), in the sense that structure functions in the hypersurface deformation algebroid do not have the classical form [1]. In particular, non-singular signature change is possible in the presence of holonomy corrections in models based on real connection variables; for complex connections see [2,3]. These results have been derived by effective [4,5,6,7] and operator methods [8], and they appear in a related form in cosmological perturbation theory [9,10].
Since the effective signature of space-time determines the form of well-posed initial or boundary value problems, modified space-time structures that allow for signature change have an important influence on the causal behavior of black-hole models in loop quantum gravity. Signature change results from the presence of holonomy modifications in the theory, which are the same effects often used to argue that singularities are removed in such models. However, even if curvature invariants remain bounded, signature change in the high-curvature region would imply that there is no deterministic evolution through high curvature. A black-hole model rather different from simple bounce models is then obtained, with problematic aspects owing to the presence of new Cauchy horizons [11]. While diverging curvature may be avoided, a serious problem (much like a naked singularity) persists in the sense of a space-time incompletely determined by initial data.
In order to extend space-time across the Euclidean region, one requires additional data on the part of its boundary that borders on the future Lorentzian space-time. This additional requirement is reminiscent of other proposals of black-hole models, for instance 'stretched horizons', introduced in the context of black hole complementarity (see, for instance [12]), except that, with signature change, unexpected degrees of freedom are not located at a horizon. In any case, the detailed analysis of anomaly-free black hole models in loop quantum gravity points towards a much more subtle non-singular description of quantum space-time than usually postulated in simplified bounce models. Just as an outside observer finds the stretched horizon as a membrane storing and later releasing information in the form of microphysical degrees of freedom, additional information is encountered once an observer moves into the future of a Euclidean region embedded in space-time. However, in the case of black-hole models of loop quantum gravity, there is as yet no microscopic theory that would restrict or determine possible data around Euclidean regions. Geometrically, signature change does indeed demonstrate the existence of novel non-classical geometries in loop quantum gravity, from which classical space-time emerges in an effective picture [13].
It is important to note that the same modifications of the classical dynamics that lead to singularity resolution in terms of bounded curvature are also responsible for signature change. Bounce models based on real connections without signature change have been described, following [14]. However, this has been possible only by using classical gauge fixing or other assumptions about the structure of space-time, and therefore comes at the expense of not being able to discuss the all-important anomaly problem. Without a handle on the anomaly problem, it is not clear whether there is any consistent space-time structure at all. (In some cases, these models have been shown to also manifest signature change once the anomaly problem has been suitably addressed [6].) In this paper, we extend the analysis of signature change in modified Schwarzschild space-times in two ways. First, we use the formalism of [15] in order to study signature change for all 2-dimensional dilaton models with holonomy modifications. Secondly, we incorporate a canonical transformation that may be used to modify the kinetic term of the Hamiltonian constraint (quadratic in extrinsic curvature). Although some of the formal aspects relevant for signature change seem to be absent for some choices of such canonical transformations, we will show that signature change is still realized. Finally, we obtain new partial obstructions to anomaly-free formulations with holonomy modifications when operator effects or matter terms are included.

2-Dimensional dilaton gravity
In two space-time dimensions, the general form of dilaton gravity models has the first-order action [16] for a dyad e a with volume form ǫ, the dilaton field φ and a connection 1-form ω which appears in the covariant derivative D. The fields X a are Lagrange multipliers that ensure torsion-freedom. The potential V (φ) is an arbitrary function and characterizes different models.

First-order variables and Poisson Sigma model
After integrating by parts (ignoring boundary terms), we obtain As used in Poisson Sigma models [17,18,16], it is convenient to collect the fields in two triplets, X i = (X − , X + , φ) and A i = (e + x , e − x , ω x ) of canonical fields, and one triplet Λ i = (e + t , e − t , ω t ) of multipliers. The action then takes the compact form with a Poisson tensor The canonical formulation leads to Poisson brackets and three first-class constraintsC The componentC 3 generates SO(1,1) rotations of the Lorentzian dyad and connection, while the linear combination D = A iC i = A i (X i ) ′ generates spatial diffeomorphisms. The remaining independent combination ofC + andC − serves as a Hamiltonian constraint.

Variables invariant under SO(1,1)
A comparison with canonical variables commonly used in spherically symmetric models, based on a symmetry reduced ADM formulation of canonical gravity, is facilitated by first introducing combinations of the fields invariant under the SO(1,1) transformations generated byC 3 . Instead of X ± and e ± x , we follow [15] and work with and two boost parameters α and β such that X ± = X exp(±β) and e ± x = e exp(±α) .
One can turn these into canonical variables by introducing The original variables can be obtained from the canonical ones as We now have the constraints

Transformation to standard variables of spherical symmetry
The usual connection or extrinsic-curvature variables of spherically symmetric gravity [19] are finally obtained by a canonical transformation from (Q e , e; Q α , α; φ, ω x ) to (K ϕ , (See Eq. (42) in [15].) A suitable combination ofC + andC − , written in these variables, takes the usual form of the Hamiltonian constraint in spherically symmetric variables for the appropriate dilaton potential, V (E x ) = −2/ √ E x ; see Eq. (46) of [15]. For an arbitrary potential, the equations generalize the connection formulation of spherically symmetric canonical gravity to arbitrary dilaton models, including the CGHS model [20] for constant V .
At this point, we can compare the results with [21]. The expression for e in (14) is nothing but (51) in [21], Q e in (14) is (57), and ω x in (16) is (60), just in slightly different notations.
The formulation of the CGHS model presented [21] is different. In particular, the canonical transformation to connection variables is simpler than in the spherically symmetric model, and the Hamiltonian constraint has only a term of the form K x K ϕ in its kinetic part but no contribution of K 2 ϕ . It amounts to renaming the SO(1,1)-invariant variables by ω x ≡ K x , Q e ≡ K ϕ and e ≡ E ϕ . These are the variables introduced in Eq. (20) of [15], corresponding to a standard first-order formulation in variables as used in Poisson Sigma models. Since the Hamiltonian constraint then has only one term Q e ω x = K x K ϕ of variables identified with extrinsic-curvature components, there is no term of the form K 2 ϕ in the resulting Hamiltonian constraint, as is clear from (23) of [15] (our (12)) and rederived in [21].

Contribution to kinetic terms
As the examples discussed so far show, the form of the kinetic contribution to the Hamiltonian constraint is not invariant under canonical transformations. In general, if one starts with a generic combination some real number α, as it appears in the standard connection formulation of spherically symmetric canonical gravity, one can always transform to new canonical variables that amount to setting α = 0. To do so, we need a newK leave E x =Ẽ x unchanged, the tilde variables are canonical provided that β = α. The new kinetic term isK xKϕ without a contribution fromK 2 ϕ . The choice made in [22] for the CGHS model amounts to applying such a canonical transformation.
Previous derivations of signature change in spherically symmetric models with holonomy modifications used both contributions to the kinetic term of the Hamiltonian constraint, based on (17). If one of them can be removed by a canonical transformation, one may wonder whether the same conclusions can still be drawn. In the present paper, we answer this question in the affirmative, and also comment on possible operator effects as well as matter couplings.

Modified Schwarzschild Models
Spherically symmetric models have already been analyzed in detail [6]. At an effective level, one can work with different kinds of constraint algebras, given by the usual hypersurface deformation brackets on one hand and a partially Abelianized system without structure functions on the other [22]. The latter is easier to deal with when one attempts a full quantization, but as shown in [6], it obscures the important question of covariance because it is not obvious that a consistent set of hypersurface-deformation generators may be recovered in the modified or quantized system. If one analyzes possible realizations of hypersurfacedeformation brackets in the partially Abelianized system with quantum modifications, the structure functions turn out to be modified and signature change is obtained in the same way as in a direct treatment of the brackets. In [6], the analysis was done at an effective level; here we complement it with operator considerations, and further generalizations.

Deformed covariance
It is possible to modify (17) while keeping the system first-class. In spherically symmetric models, the possibility of anomaly-free holonomy corrections has been analyzed based on a Hamiltonian constraint with two free functions f 1 and f 2 of K ϕ . The holonomy modification function f (K ϕ ) is typically chosen to be sin (ρK ϕ ) /ρ, but may be kept more general. The system is anomaly- in which case the classical structure functions are modified by an additional factor of [4] β This function is negative around a local maximum of f 1 (K ϕ ), or in any regime where an upper bound on curvature is achieved by the modification.
Alternatively, as shown in [6], it is possible to construct a partially Abelianized system with holonomy modifications, using the methods of [22]. If one replaces the local Hamiltonian constraint H by a linear combination with the local diffeomorphism constraint D, the component K x cancels out and greatly simplifies the bracket of two constraints C: We have One can show that the Abelianization succeeds if C is a total derivative by x, which requires the same condition (20) that follows from anomaly freedom. If, on the other hand, one starts with the classical Abelianized system and modifies the dependence of C on K ϕ , as proposed in [22], hypersurface-deformation brackets can be recovered only if there is a modification function β in the structure functions, implying signature change [6]. However, obstructions to recovering hypersurface-deformation brackets, and therefore obstructions to covariance, are obtained if one tries to couple a matter field to the modified system. While this can be done within the Abelianized system [23], it is no longer possible to recover hypersurface-deformation brackets from the modified system [6]. Another question not yet addressed is whether additional quantum effects, such as ordering choices, may affect the outcome. This question is the topic of the following subsection.

Operator ordering in Abelianization of the constraints
It is possible to turn the modified Abelianized constraints into operators without anomalies. However, this quantization step could introduce additional properties that prevent one from recovering hypersurface-deformation brackets even in the vacuum case. This possibility turns out to be realized for the Schwarzschild model, and for general 2-dimensional dilaton models as well as we will show later.
We demonstrate the sensitivity to ordering questions by working with quantized constraints at a formal level. We only indicate the non-commuting nature of our variables by paying attention to their position in products, but do not go into details of regularizations such as point splitting.
We begin with the partially Abelianized system of constraints, as derived in [22] but written here as operators: The algebra of these constraints is such that any factor ordering choice is allowed forD SS whileĈ SS is free from operator ordering ambiguities. At the formal level we are working with here, it is easy to see that two smearedĈ SS commute because any non-zero commutator introduces a delta function which gives zero once the antisymmetric bracket is imposed. Let us start with a factor ordering choice for the diffeomorphism constraint as shown in Eq. (25). For simplicity, we will not write hats anymore, but all expressions in this subsection remain objects sensitive to ordering choices. We can recover the original Hamiltonian and diffeomorphism constraints as generators of the hypersurface deformation algebra, if we write them as linear combinations while D SS remains unmodified. (This procedure retraces the steps taken in [22].) We begin by rewriting C SS as Consequently, for our factor ordering choice for the diffeomorphism constraint, the Hamiltonian constraint (as defined in (26)) has the form This expression does not reduce to the usual one of the Hamiltonian constraint because the last term in the first line of Eq. (28) does not cancel out with the last term in the second line. Instead, a non-zero commutator [E ϕ , f (K ϕ )K ′ ϕ ] remains. Thus it is not possible to obtain the generators of the hypersurface deformation algebra from the newly defined system of constraints, once operator orderings are taken into account.
We could have started with the other factor ordering choice in the diffeomorphism constraint from the beginning. This is the only other choice left to be exploited because C SS does not have any factor order ambiguities in it. Had we started with the diffeomorphism constraint of the form we would have, for the Hamiltonian constraint, the expression Once again, the unwanted terms do not cancel out. We conclude that, with operator orderings, one cannot reproduce the hypersurface deformation algebra from the newly defined system of constraints with a (partially) Abelianized algebra.

Dilaton gravity models
For a general 2-dimensional dilaton gravity model with potential V (φ), the Hamiltonian constraint in the connection variables of [15] differs from the spherically symmetric one only in the term that does not depend on extrinsic curvature or spatial derivatives of the densitized triad; see Eq. (46) in [15], or (17) here. This term does not affect the constraint algebra, and therefore the same partial Abelianization found in [22] for spherically symmetric models can be applied to arbitrary 2-dimensional dilaton gravity models. The general Abelianized constraints are . This simple result can also be found in [24]. The same ordering obstructions as in the spherically symmetric model follow thanks to the closely related structure of the constraints, as do matter obstructions to be discussed in more detail in Sec. 5.

Simplified kinetic term
Using a canonical transformation, as shown in Sec. 2.4, the Hamiltonian and diffeomorphism constraints of 2-dimensional dilaton models can be brought to the form (The two dimensional Newton's constant G 2 has been set to one for simplicity.) Classically, the hypersurface deformation algebra does not change by a canonical transformation and is still given by The only non-vanishing component of the (inverse of the) 1-dimensional spatial metric, q xx = 1/ (E ϕ ) 2 , appears as a structure function. (After the canonical transformation of Sec. 2.4, the variables used here should be identified withẼ ϕ = E ϕ / √ E x with α = 1/2. Therefore, 1/(E ϕ ) 2 , dropping the tilde after the transformation, is the same as the usual structure function E x /(E ϕ ) 2 in spherically symmetric models.) If one follows the common steps to use loop quantum gravity as a motivation of a specific quantization of this model [19,25], one has well-defined holonomy operators which are exponentiated versions of the extrinsic-curvature components but are not weakly continuous in those variables any longer. However, the holonomies corresponding to the K x variables are extended along the edge of a (one-dimensional) spin network. As a consequence, the holonomy corrections arising from these are non-local in nature and difficult to implement [26]. (However, using partial Abelianization techniques as in [22], a suitable redefinition of the constraints is possible to eliminate the K x variable.) On the other hand, the point-wise holonomy operators, corresponding to the K ϕ component, act at the nodes of the spin networks. These are local modifications and can be included in our Hamiltonian constraint (37) The diffeomorphism constraint remains unmodified as suggested by the simple geometrical action on states of finite diffeomorphisms.
The hypersurface deformation brackets that do not contain structure functions remain unaltered while the only modified bracket is As expected, the dilaton potential does not contribute to the bracket. The structure function q xx is modified by the presence ofḟ := df /dK ϕ , which is equal to one for the classical case but not if holonomy modifications are present. The underlying space-time symmetry is deformed for this system as in the Schwarzschild case. However, it is not as straightforward to conclude that this definition can give rise to signature change: The first derivative of the modification function appears here in the structure function, which unlike the second derivative in (21) does not necessarily change sign near a local maximum. Nevertheless, the same relationship can be established as we will do now.

Partial Abelianization of the constraint algebra
The unsmeared modified constraints are We redefine the system of constraints by keeping the diffeomorphism constraint the same while defining a new constraint as a linear combination of the (old) Hamiltonian constraint and the diffeomorphism constraint as where, as before, dW (E x )/dE x = V (E x ). We have introduced a new function g(K ϕ ) via f (K ϕ ) = dg(K ϕ )/dK ϕ . The smeared version of the new constraint is obtained after integrating by parts and using N ′ as the new smearing function: The new constraint, so defined, is such that it commutes with itself while its Poisson bracket with the diffeomorphism constraint is The Poisson bracket between two diffeomorphism constraints remains unaltered. The (partial) Abelianization of the constraint algebra helps us in demonstrating signature change in all dilaton models, including the CGHS black hole model. If one were to follow [22], one would first Abelianize the algebra classically and then insert the holonomy modification function in the new constraint. This procedure, sometimes called 'polymerization', replaces the extrinsic curvature component K ϕ with a bounded function of K ϕ , usually sin(ρK ϕ )/ρ, in the new constraint. Claims about singularity resolution, as in [22,27], are based on this boundedness property. However, as we have seen in Eq. (38), what appears in the deformation of the structure function is df /dK ϕ = d 2 g/dK 2 ϕ , the second derivative of the holonomy modification function. At the point of the classical singularity, the bounded function reaches a maximum and therefore its second derivative must be negative. We obtain hypersurface-deformation brackets with the same sign for normal deformations as one would have in the case of Euclidean gravity. Thus, singularity resolution in the vacuum CGHS black hole in the framework of loop quantum gravity, just like for the vacuum Schwarzschild black hole, cannot be divorced from signature-change.

Operator ordering in the Abelianization
For the spherically symmetric case, starting from the partially Abelianized constraints, we have shown that none of the possible factor ordering choices lead us to the required form of the hypersurface-deformation brackets. One can repeat the analysis for all 2-dimensional dilaton models, including the CGHS black hole model, following the same procedure and show that there is a similar obstruction. Instead, we shall arrive at the same conclusion following a different approach. We shall find a suitable operator ordering for the generators of the hypersurface-deformation brackets, the original Hamiltonian and diffeomorphism constraints, and then try to define the new, partially Abelianized constraint while keeping in mind that we are now dealing with objects sensitive to ordering choices. (Again, our treatment of operators is formal.) Our first task is to find a factor ordering for the original system of constraints such that the hypersurface-deformation brackets are realized. Following [8], one factor ordering choice with closed brackets iŝ The operator ordering of the Hamiltonian constraint is subtle as can be seen from the first term in the Hamiltonian constraint, where there is no triad component E ϕ classically, but it is important to introduce this into the quantum operator. The resulting operator version of the hypersurface-deformation brackets takes the form The details of the operator ordered constraints closing into an algebra are shown in the Appendix. The next step is to start from these constraint operators and try to define a factor ordered new set of constraints for a (partially) Abelianized system. In fact what we shall find below is that there is no consistent way to carry out the Abelianization procedure anymore, once we start from these factor ordered constraint operators. We try to define the new constraint in such a way that we cancel out the E x K x term coming from the Hamiltonian constraint operator with a similar term from the diffeomorphism constraint operator.Ĉ As is evident from Eq. (50), operator ambiguities play a major role in choosing the prefactor of the second term. This nontrivial operator ordering is chosen so as to make the cancellation mentioned above possible. However, the term which was a total derivative of the holonomy correction function earlier is more complicated due to the structure of the factor ordering chosen The last term in this expression is the reason why we do not have an Abelianized system of constraints any longer.

Obstructions to add matter to the holonomy-modified CGHS model
In this section, we refer specifically to the CGHS model because of its importance in discussions of Hawking radiation, which require a scalar field. However, the same conclusions can easily be achieved for general 2-dimensional dilaton models. If we add the simplest type of matter, a minimally coupled scalar, to the CGHS model, the total constraints take the form with the gravitational and matter parts of the constraints given by We have extended the gravitational phase space by a scalar field, satisfying the Poisson bracket {ϕ(x), P ϕ (y)} = δ(x, y). The classical total constraints satisfy the usual hypersurface-deformation brackets, as expected. However, once we incorporate holonomy effects in the constraints, H grav contains a modification function (K ϕ → f (K ϕ )) whereas H matter does not since there are no K ϕ components in the matter part of the Hamiltonian constraint. Thus, we have with the gravitational and matter parts of the Hamiltonian constraint now satisfying different versions of covariance. The Poisson bracket of the total Hamiltonian constraint with itself does not close into the full diffeomorphism constraint and one does not have a first-class system any longer. One natural recourse might be to introduce a holonomy correction function in the matter part of the Hamiltonian constraint, by hand, but then we have a non-vanishing cross term between the gravitational and matter parts, which still leads to anomalies. For details, see [6]. Another way to address this problem would be to try and (partially) Abelianize the full constrained system (with both gravity and matter), as we did before. In this case, for the classical constraints, the Abelianization goes through due to some subtle cancellations as shown below. That the classical constraint algebra remains Abelianized has already been shown in [24]. We show crucial parts of the calculation here which help us to emphasize why the quantum algebra is not anomaly free. We write where we have defined The . This cancellation can take place precisely due to the linear factor of K ϕ in the last term of the matter part of the new constraint. If this factor is replaced by the holonomy modification function as we should have to in case of holonomy modifications, these two terms do not cancel anymore and we, once again, end up with an anomaly. This is just another way of expressing that we cannot add a massless scalar to the holonomy-modified CGHS model in a covariant quantization, at least in the standard regularization procedure of loop quantum gravity. Our result extends a set of no-go theorems which had previously been proved for the Schwarzschild black hole [6] and Gowdy models [7], to the CGHS black hole and all other 2-dimensional dilaton models. An open question was left in [24,27] whether one could add matter to the vacuum CGHS model and still have an anomaly free quantum algebra. We answer this question in the negative in this article and, as a consequence, provide an obstruction to studying Hawking radiation in this context. It is possible to show that this obstruction can be generalized to other possible matter models, but this calculation is essentially the same as what has already been shown for the spherically symmetry case in [6].

Conclusions
We have obtained new obstructions to anomaly-free midisuperspace models with holonomy modifications, given by ordering effects and matter terms. The second set of obstructions is particularly important because it spoils attempts to discuss Hawking radiation. Such a discussion is possible within a partially Abelianized system, only when one considers a background treatment, but such models cannot be covariant 1 . In cases in which anomaly freedom can be achieved, holonomy modifications are accompanied by signature change. According to [13], new non-classical space-time structures are then obtained. dx dy N(x)M(y) 4(E ϕ (x)) −1 f (K ϕ (x))E ϕ (x)K x (x) , 1 4 ((E x′ (y))(E ϕ (y)) −2 (E ϕ′ (y)) . (69) Performing an intergration by parts on the last term, we have The required cancellation between the second term of (70) and the second term of (68), as it happens in the classical case, is obtained here with our choice of the factor ordering for the Hamiltonian constraint: The term on the right-hand side is the required diffeomorphism constraint, with the proper operator ordering, as proposed in (45).