*2.1. Algebroids*

Instead, the brackets have structure functions or, in a suitable mathematical formulation, form the higher algebraic structure of an *<sup>L</sup>*∞-algebroid rather than a Lie algebra [10–12]. An *<sup>L</sup>*∞-algebroid is defined as a vector bundle over a base manifold *M* with fiber *F* and bracket relations on bundle sections together with suitable anchor maps that map bundle sections to objects in the tangent bundle of *M*. A Lie algebroid [13], for instance, has a Lie bracket [·, ·] on its sections and an anchor *ρ* that maps (as a homomorphism) bundle sections to vector fields on the base manifold, such that the Lie bracket of vector fields is compatible with the algebroid bracket. The anchor map also appears in the Leibniz rule

$$[s\_1, fs\_2] = f[s\_1, s\_2] + s\_2 \mathcal{L}\_{\rho(s\_1)} f \tag{6}$$

where *s*1 and *s*2 are sections and *f* is a function on the base manifold. The anchor brings abstract algebraic relations on bundle sections in correspondence with geometrical transformations as vector fields on the base manifold. While an anchor that maps any section to the zero vector field is always consistent with the Lie-algebroid axioms (in which case the Lie algebroid is a bundle of Lie algebras given by the fibers), non-trivial transformations on the

base require a larger image of the anchor. A Lie algebroid with a non-trivial anchor generalizes bundles of Lie algebras. Yet more generally, and in particular in the case of structure functions, the brackets of bundle sections obey the axioms of an *<sup>L</sup>*∞-algebra, a generalized form of a Lie algebra in which the Jacobi identity is not required to hold strictly.

The introduction of the base manifold makes it possible to formalize brackets with structure functions in terms of an *<sup>L</sup>*∞-algebroid. In particular for gravity, the base manifold is (a suitable extension [6]) of the canonical phase space, given by the spatial metrics and momenta related to extrinsic curvature. The fibers are parameterized by the components *η* and of a gauge transformation. A section is then an assignment of spatial functions *η* and to any metric (or a pair of a metric and its momentum). In this way, the *q*-dependent structure function in (5) finds a natural home as a bracket of sections over the space of metrics (and momenta).

Constant sections, given by pairs of *η* and that are functions on space but do not depend on the phase-space degrees of freedom, have a bracket, implied by (4), that can be realized as a special case of sections of a Lie algebroid [5]. General, non-constant sections of this Lie algebroid have a bracket that may differ from what hypersurface deformations would suggest. Non-constant sections over phase space, discussed in more detail in [6], either violate some of the Lie-algebra relations on sections (in the controlled way of a specific *L*∞-structure, as it follows from a BV-BFV extension of general relativity [14,15]) or require a base manifold that extends the phase space of canonical gravity in a way that is not smooth. (The latter can be formulated by using the notion of a Lie-Rinehart algebra [16] in which functions on the base manifold are replaced with a suitable commutative algebra.

Phase-space dependent functions *η* and are also important for physics. They are often considered in specific gravitational applications, as in the simple case of cosmological evolution written in conformal time where the lapse function equals the scale factor, a metric component. More importantly for our purposes, the partial Abelianization of [7] relies on an application of phase-space dependent and *η*. Hypersurface deformations with such non-constant sections form a Lie algebroid only on-shell [6] when the constraints are solved. The partial Abelianization is therefore able to describe the solution space to all constraints and its covariance transformations, but it is not guaranteed that it correctly captures off-shell transformations which are relevant for general covariance.

Since the standard derivation of the brackets (5) assumes that *η* and are not phasespace dependent, the general brackets must be extended by additional terms that, heuristically, result from Poisson brackets of constraints with phase-space dependent *η* and . (A complete derivation is based on the BV-BFV analysis of [14,15]). The Poisson bracket of two diffeomorphism constraints, for instance, can still be written in the compact form

$$\{D[\mathfrak{e}\_1], D[\mathfrak{e}\_2]\} = D[\mathfrak{e}\_2 \mathfrak{e}\_1' - \mathfrak{e}\_1 \mathfrak{e}\_2'] \tag{7}$$

but with an application of the chain rule in the derivatives. Similarly, the mixed Poisson bracket of a Hamiltonian and a diffeomorphism constraint in general form reads

$$\{H[\eta], D[\epsilon]\} = H[-\epsilon \eta'] + D[\eta \mathcal{L}\_n \epsilon] \tag{8}$$

where the normal derivative L*n* of a spatial function is defined by the Poisson bracket with the Hamiltonian constraint, *η*1L*nη*<sup>2</sup> = {*H*[*η*1], *η*2}. For two Hamiltonian constraints, we have the Poisson bracket

$$\{H[\eta\_1], H[\eta\_2]\} = D[q^{-1}(\eta\_1 \eta\_2' - \eta\_2 \eta\_1')] + H[\eta\_1 \mathcal{L}\_u \eta\_2 - \eta\_2 \mathcal{L}\_u \eta\_1].\tag{9}$$

In general, the extra terms implied by phase-space dependent *η* and , such as those in = *∂x* + (*∂xqi*)(*∂qi* )+(*∂xki*)(*∂ki* ) summing over the two independent components *qi*, *i* = 1, 2, of a spherically symmetric spatial metric as well as two components *ki* of extrinsic curvature, introduce further structure functions, such as *∂xqi* and *∂xki*, that depend on the metric as well as its momenta.

While these Poisson brackets illustrate the additional complications encountered with phase-space dependent and *η*, they do not immediately show the algebraic nature of general non-constant sections of hypersurface deformations. In particular, Poisson brackets do not directly mirror relevant *L*∞-structures. In our following discussion, we will not need the full algebraic structure and instead perform a comparison of different versions of constant and non-constant sections in gravitational applications.

#### *2.2. Partial Abelianization*

As noticed in [7], certain linear combinations of *<sup>H</sup>*[*η*] and *<sup>D</sup>*[] have vanishing Poisson brackets in spherically symmetric models. In order to specify these combinations, we have to refer to explicit variables that determine the spatial metric and its momenta. Following Refs. [17–19], this is conveniently done in triad variables (*Ex*, *Eϕ*) such that the spatial metric is given by the line element

$$\mathrm{d}s^2 = \frac{(E^\circ)^2}{E^\times} \mathrm{d}x^2 + E^\times (\mathrm{d}\vartheta^2 + \sin^2\vartheta \mathrm{d}\varphi^2) \tag{10}$$

in standard spherical coordinates. (For our purposes, it is sufficient to assume *Ex* > 0, fixing the orientation of the triad.) The triad components are canonically conjugate (up to constant factors) to components of extrinsic curvature, (*Kx*, *<sup>K</sup>ϕ*), such that

$$\{K\_x(\mathbf{x}), E^x(y)\} = 2G\delta(\mathbf{x}, y) \quad , \quad \{K\_\varphi(\mathbf{x}), E^\varphi(y)\} = G\delta(\mathbf{x}, y) \tag{11}$$

with Newton's constant *G*. (We keep a factor of two in the first relation. As implicitly done in [7,8], this factor can easily be eliminated by a rescaling of *Kx*. Since this procedure would not affect the main equations and conclusions shown below, we do not make use of this rescaling and instead keep the original components of extrinsic curvature).

The delta functions disappear in Poisson brackets of integrated (smeared) expressions, resulting in well-defined brackets. In particular, the diffeomorphism constraint

$$D[M] = \frac{1}{G} \int \mathrm{d}x M(x) \left( -\frac{1}{2} (E^x)' K\_x + K\_\varphi' E^\varphi \right),\tag{12}$$

and Hamiltonian constraint

$$H[\mathbf{N}] = \frac{-1}{2G} \int d\mathbf{x} N(\mathbf{x}) \left( |\mathbf{E}^{\mathbf{x}}|^{-1/2} E^{\mathbf{q}} K\_{\mathbf{q}}^2 + 2|\mathbf{E}^{\mathbf{x}}|^{1/2} K\_{\mathbf{q}} K\_{\mathbf{x}} + |\mathbf{E}^{\mathbf{x}}|^{-1/2} (1 - \Gamma\_{\mathbf{q}}^2) \mathbf{E}^{\mathbf{q}} + 2 \Gamma\_{\mathbf{q}}' |\mathbf{E}^{\mathbf{x}}|^{1/2} \right) \tag{13}$$

where Γ*ϕ* = −(*Ex*)/(2*E<sup>ϕ</sup>*) have Poisson brackets

$$\{D[M\_1], D[M\_2]\} \quad = \quad D[M\_1M\_2'] \tag{14}$$

$$\{H[N], D[M]\} \quad = \quad -H[MN'] \tag{15}$$

$$\left\{ H[\mathbf{N}\_1], H[\mathbf{N}\_2] \right\} \quad = \quad D[E^{\mathbf{x}} (E^{\mathbf{g}})^{-2} (\mathbf{N}\_1 \mathbf{N}\_2' - \mathbf{N}\_2 \mathbf{N}\_1')] \tag{16}$$

(for spatial functions *Mi* and *Ni*, *i* = 1, 2, that do not depend on the phase-space variables) of the correct form for hypersurface deformations in spherically symmetric space-times. Simple algebra and integration by parts shows that the linear combinations

$$\mathbb{C}[L] = H[(E^{\mathbf{x}})'(E^{\boldsymbol{\varrho}})^{-1} \int E^{\boldsymbol{\varrho}} L \mathbf{dx}] - 2D[K\_{\boldsymbol{\varrho}} \sqrt{E^{\boldsymbol{\varrho}}} (E^{\boldsymbol{\varrho}})^{-1} \int E^{\boldsymbol{\varrho}} L \mathbf{dx}],\tag{17}$$

where *<sup>E</sup>ϕL*d*x* is understood as a function of *x* obtained by integrating *EϕL* from a fixed starting point up to *x*, have zero Poisson brackets with one another for different *L*:

$$\{\mathcal{C}[L\_1], \mathcal{C}[L\_2]\} = 0 \tag{18}$$

for all functions *L*1 and *L*2 on a spatial slice. To see this, it is sufficient to notice that the combination eliminates any dependence on *Kx* and on spatial derivatives of *Eϕ*. The antisymmetric nature of the Poisson bracket then implies that it must vanish. Explicitly, the new combination of constraints takes the form

$$\mathbb{C}[L] = -\frac{1}{G} \int \mathrm{d}x L(x) E^{\varrho} \left( \sqrt{|E^x|} \left( 1 + K\_{\varrho}^2 - \Gamma\_{\varrho}^2 \right) + \text{const.} \right). \tag{19}$$

A free constant appears because a constant *<sup>E</sup>ϕL*d*x* implies a non-vanishing lapse function in (17), and therefore a non-trivial constraint, but corresponds to a vanishing *EϕL* in (19). The new constraint *C*[*L*] therefore constrains one degree of freedom less than the original *<sup>H</sup>*[*N*]. The free constant in (19) can be determined through boundary conditions, which would also restrict the lapse functions allowed in gauge transformations.

At first sight, it seems that the partial Abelianization eliminates structure functions from the brackets and may simplify quantization and the preservation of symmetries and therefore covariance. However, the importance of metric-dependent structure functions in the standard brackets, which make sure that deformations are defined with respect to a unit normal that is in fact normalized, raises the question of whether an elimination of these structure functions and their metric dependence by redefined generators can still capture the full picture of general covariance. To answer this question, it is instructive to place the partial Abelianization of the brackets in the context of the hypersurface-deformation structure. Several features of the full mathematical construction are then relevant.

First, the integration of *EϕL* required to define *C*[*L*] as a combination of *H*[*N*] and *D*[*M*] may seem unusual, but while this means that the relevant *N* and *M* are non-local in space, they are local within both the fiber (spatial functions *N* and *M*) and the base (the gravitational phase space with independent functions *Ex*, *Eϕ*, *Kx* and *<sup>K</sup>ϕ* or a suitable extension) that may be used to construct a corresponding *<sup>L</sup>*∞-algebroid. The combination (17) therefore defines an admissible set of sections.

Secondly, while the section defined by (17) makes use of phase-space dependent *N* and *M* in the Hamiltonian and diffeomorphism constraints, which are therefore not constant over the base manifold, an Abelian bracket (18) is obtained only for functions *L*1 and *L*2 that do not have the full phase-space dependence allowed for general sections. In particular, if *L*1 or *L*2 are allowed to depend on (*Eϕ*) or *Kx*, the bracket {*C*[*<sup>L</sup>*1], *<sup>C</sup>*[*<sup>L</sup>*2]} no longer vanishes, and it can then have structure functions. Partial Abelianization is therefore obtained for a restricted class of sections, defined such that *L* does not depend on (*Eϕ*) and *Kx* (while it may still have an unrestricted spatial dependence). If *L* does not depend on (*Eϕ*) and *Kx* but on the other independent phase-space variables, *<sup>K</sup>ϕ* as well as *Ex* or on *Eϕ* but not its derivatives, the bracket {*C*[*<sup>L</sup>*1], *<sup>C</sup>*[*<sup>L</sup>*2]} remains zero, but there are then structure functions in the bracket of *C*[*L*] with the diffeomorphism constraint, analogously to (8). Therefore, structure functions are eliminated from the brackets only for a restricted class of sections. This observation raises the question whether full covariance can still be realized.

A restriction to constant sections over the base manifold is not unusual, for certain purposes. A similar assumption is made in the standard form (14)–(16) of hypersurfacedeformation brackets, in which case the original *N* and *M* are often assumed to be constant over the base (while their spatial dependence remains unrestricted). There is, however, a crucial difference between assuming constant *N* and *M* over the base and assuming constant *L* over the base: In the former case, allowing for non-constant sections produces additional terms in the brackets, shown in (7)–(9) , that follow directly from an application of the product rule of Poisson brackets. The partial Abelianization, however, relies on cancellations between different structure functions in the original brackets that are no longer realized once non-constant sections with phase-space dependent *L* are allowed.

In particular, allowing for phase-space dependent *L* and *M* in the (*D*[*M*], *C*[*L*]) system makes the transformation from (*<sup>N</sup>*, *M*) to (*<sup>M</sup>*, *L*) invertible. It is then possible to write the original *H*[*N*] as a combination of *D*[*M*] and *C*[*L*] in the partial Abelianization, regaining the full non-Abelian brackets with metric-dependent structure functions. Restricting the system to phase-space independent *L*, by contrast, implies that the transformation from the original hypersurface-deformation structure to the brackets of *D*[*M*] and *C*[*L*] is not

invertible. It is then unclear whether hypersurface deformations and general covariance can be recovered from a partial Abelianization, in particular if the latter has been modified by quantum corrections.

#### *2.3. Modified Deformations*

It has been known for some time [20–22] that spherically symmetric hypersurface deformations can be modified consistently, maintaining closed brackets while modifying the structure functions. The dependence on *<sup>K</sup>ϕ* in (13) can be generalized to

$$H[\mathbf{N}] = \frac{-1}{2\mathbf{G}} \int d\mathbf{x} \mathcal{N}(\mathbf{x}) \left( |E^x|^{-1/2} E^q f\_1(\mathbf{K}\_q) + 2|E^x|^{1/2} f\_2(\mathbf{K}\_q) \mathbf{K}\_x + |E^x|^{-1/2} (1 - \Gamma\_q^2) \mathcal{E}^q + 2\Gamma\_q' |E^x|^{1/2} \right) \tag{20}$$

where *f*1 and *f*2 are functions of *<sup>K</sup>ϕ* related by

$$f\_{\mathbb{Z}}(\mathbb{K}\_{\mathbb{Q}}) = \frac{1}{2} \frac{\mathrm{d}f\_1(\mathbb{K}\_{\mathbb{Q}})}{\mathrm{d}\mathbb{K}\_{\mathbb{Q}}}.\tag{21}$$

If this equation is satisfied, the bracket of two Hamiltonian constraints is still closed,

$$\{H[\mathcal{N}\_1], H[\mathcal{N}\_2]\} = D[\beta(\mathcal{K}\_\Psi) E^x (E^\Psi)^{-2} (\mathcal{N}\_1 \mathcal{N}\_2' - \mathcal{N}\_2 \mathcal{N}\_1')] \tag{22}$$

for phase-space independent *N*1 and *N*2. In this bracket, *D*[*M*] is the unmodified diffeomorphism constraint, but the structure function is multiplied by a new factor of

$$\beta(K\_{\varphi}) = \frac{\mathrm{d}f\_2(K\_{\varphi})}{\mathrm{d}K\_{\varphi}} = \frac{1}{2} \frac{\mathrm{d}^2 f\_1(K\_{\varphi})}{\mathrm{d}K\_{\varphi}^2}. \tag{23}$$

Additional terms in the bracket for non-constant sections follow immediately from the product rule for Poisson brackets.

Similarly, the Abelianized constraint *C*[*L*] can be generalized in its dependence on *<sup>K</sup>ϕ*, using the same function *f*1 as before:

$$\mathbb{C}[L] = -\frac{1}{G} \int \mathrm{d}x L(\mathbf{x}) E^{\varrho} \left( \sqrt{|E^{x}|} \left( 1 + f\_{1}(K\_{\varrho}) - \Gamma\_{\varrho}^{2} \right) + \mathrm{const.} \right). \tag{24}$$

Its brackets remain Abelian for phase-space independent *L*. There is no obvious term in *C*[*L*] where the second function *f*2 might appear or the important consistency condition (21). It therefore seems easier to modify (or quantize) the constraint *C*[*L*] compared with *<sup>H</sup>*[*N*]. However, for full hypersurface deformations and covariance to be realized in the modified setting, we still have to make sure that the transformation from (*<sup>N</sup>*, *M*) to (*<sup>L</sup>*, *M*) can be inverted. As shown in [23], this is possible only if we also modify the transformation (17) to

$$\mathbb{C}[L] = H[(E^{\mathbf{x}})'(E^{\boldsymbol{\uprho}})^{-1} \int E^{\boldsymbol{\uprho}} L \mathbf{dx}] - 2D[f\_{\boldsymbol{\uprho}}(\mathcal{K}\_{\boldsymbol{\uprho}}) \sqrt{E^{\mathbf{x}}} (E^{\boldsymbol{\uprho}})^{-1} \int E^{\boldsymbol{\uprho}} L \mathbf{dx}] \tag{25}$$

where *f*2 obeys the same consistency condition with *f*1, (21), as derived from the modified Hamiltonian constraint. The partial Abelianization and the original form of hypersurface deformations therefore imply equivalent results, provided one makes sure that the transformation of sections can be inverted. Only then can access to full hypersurface deformations and covariance be realized.

#### **3. Non-Covariant Modifications of Abelianized Brackets**

A recent paper [8] by Gambini, Olmedo and Pullin (GOP) argues that general covariance can be realized in modified versions of spherically symmetric models, for which a partial Abelianization of the brackets plays a crucial role: As the abstract claims, "We show explicitly that the resulting space-times, obtained from Dirac observables of the quantum theory, are covariant in the usual sense of the way—they preserve the quantum line element—for any gauge that is stationary (in the exterior, if there is a horizon). The con-

struction depends crucially on the details of the Abelianized quantization considered, the satisfaction of the quantum constraints and the recovery of standard general relativity in the classical limit and suggests that more informal polymerization constructions of possible semi-classical approximations to the theory can indeed have covariance problems."

These claims raise several questions. For instance, how can the construction depend "crucially on the details of the Abelianized quantization considered" if a partial Abelianization is either completely equivalent to the non-Abelian orignal version of hypersurface deformations (if the transformation is made sure to be invertible) or gives access to only a subset of hypersurface deformations (if the transformation is not invertible owing to a restriction to a subset of sections)?

A closer inspection of technical calculations performed by GOP shows that spherically symmetric hypersurface deformations are, in fact, violated in the construction. GOP use two different kinds of modifications, a generalized dependence of *C*[*L*] on *<sup>K</sup>ϕ* of the form (24), and a spatial discretization of phase-space functions and their derivatives. Because the authors use a certain combination of solutions to the constraints and gauge-fixing conditions, it turns out that only the latter modification survives in the final expressions for line elements that are supposed to be invariant.

However, also the former (a generalized dependence on *<sup>K</sup>ϕ*) is relevant because, as we have seen, the correct form of a modification must appear in two different places, in the constraint *C*[*L*] and in the transformation back to unrestricted hypersurface deformations. These two appearances are clear but somewhat implicit in [8]: The modified *C*[*L*] is implied by the modified solutions in Equation (14) in [8] (or, equivalently, (21) there, referring to the preprint version) where *f*1(*<sup>K</sup>ϕ*) = sin<sup>2</sup>(*ρKϕ*)/*ρ*<sup>2</sup> with a spatial function *ρ*. The modified transformation back to unrestricted hypersurface deformations is implied by Equation (20) in [8] which in our notation amounts to replacing *<sup>K</sup>ϕ* in (17) with / *f*1(*<sup>K</sup>ϕ*). Using the same function *f*1(*<sup>K</sup>ϕ*) is crucial for the constructions in [8] because the partial gauge fixing employed there replaces / *f*1(*<sup>K</sup>ϕ*) with a fixed function on space (rather than phase space). The same gauge-fixing function is then used in both places, in the constraint *C*[*L*] or its solutions and in the transformation back to unrestricted hypersurface deformations from which a line element can be constructed. However, this construction, which is equivalent to assuming *f*2(*<sup>K</sup>ϕ*) = / *f*1(*<sup>K</sup>ϕ*) in (25), violates the condition (21) required for unrestricted hypersurface deformations to follow for the modified constraint. (For the specific *f*1(*<sup>K</sup>ϕ*) considered by GOP, *f*2 should have an additional cosine factor, or equivalently have a doubled argumen<sup>t</sup> of the sine function.) The constructions of [8] therefore violate hypersurface deformations.

How can GOP then claim to have performed crucial steps toward demonstrating general covariance in this setting? Unfortunately, much of the constructions are obscured by an application of incompletely defined mixtures of gauge fixings and idiosyncratic notions of observables. Here, it suffices to highlight only a few of the shortcomings found in the GOP analysis. (For more details, see [24].) Continuing with the replacement of / *f*1(*<sup>K</sup>ϕ*) by a gauge-fixing function that depends only on space, GOP replace any appearance of / *f*1(*<sup>K</sup>ϕ*) with gauge-fixing functions (on space) derived from the classical solutions for *<sup>K</sup>ϕ* in two specific slicings. Implicitly, the authors simply remove the modification in this way because they indirectly equate / *f*1(*<sup>K</sup>ϕ*) with *<sup>K</sup>ϕ*, mediated by the gauge-fixing function. As a result, they do not test how non-classical *f*1(*<sup>K</sup>ϕ*) can be consistent with covariance. It is also problematic that this step in a rather careless gauge-fixing procedure replaces a phase-space function *<sup>K</sup>ϕ* that does not Poisson commute with the constraints with a spatial function that does obey this commutation property. The procedure turns a *<sup>K</sup>ϕ*-dependent expression for *Eϕ*, obtained by solving *C*[*L*] = 0, into a function that Poisson commutes with *<sup>C</sup>*[*L*]. GOP then call the result a Dirac observable, even though *Eϕ* is not gauge invariant.

After replacing *<sup>K</sup>ϕ* with a spatial function, the resulting expression for *Eϕ* still does not Poisson commute with the diffeomorphism constraint and is therefore not a Dirac observable, even if *<sup>K</sup>ϕ* could meaningfully be replaced. The same expression for *Eϕ* also depends on *Ex*, which is not a spatial invariant. Indeed, unlike *<sup>C</sup>*[*L*], the diffeomorphism constraint (12) depends on *Kx* and therefore does not Poisson commute with *Ex*. GOP arrive at their conclusion about *Eϕ* being a Dirac observable by misidentifying *Ex* as a Dirac observable because the (loop) quantization procedure they use establishes a correspondence between an operator *E*ˆ *x* and labels of a spherically symmetric spin network state [17,25] that are unchanged by the spatial shifts of a finite diffeomorphism. However, having a correspondence between a classical object, *Ex*, that is not a Dirac observable and a quantum operator, *E*ˆ *x*, that is a Dirac observable may indicate that the theory fails to have the correct classical limit. Since this way of imposing the diffeomorphism constraint is directly inherited from more general constructions in the full theory of loop quantum gravity [26,27], the issues revealed by our analysis of [8] might hint at deeper problems within the kinematics of loop quantum gravity.
