Dynamically Implementing the $\overline{\mathit{\mu }}$-Scheme in Cosmological and Spherically Symmetric Models in an Extended Phase Space Model

Abstract

We consider an extended phase space formulation for cosmological and spherically symmetric models in which the choice of a given $\overline{\mu }$-scheme can be implemented dynamically. These models are constructed in the context of the relational formalism by using a canonical transformation on the extended phase space, which provides a Kuchař decomposition of the extended phase space. The resulting model can be understood as a gauge-unfixed model of a given $\overline{\mu }$-scheme. We use this formalism to investigate the restrictions to the allowed $\overline{\mu }$-scheme from this perspective and discuss the differences in the cosmological and spherically symmetric case. This method can be useful, for example, to obtain a $\overline{\mu }$-scheme in a top-down derivation from full LQG to symmetry-reduced effective models, where, for some models, only the ${\mu }_0$-scheme has been obtained thus far.

1. Introduction

The investigation of symmetry-reduced models for loop quantum gravity (LQG) in the context of cosmological or spherically symmetric spacetimes is a topic of growing interest in recent years; for spherically symmetric models, see, for instance, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] or [33] for a recent review and [34] for a review on loop quantum cosmology (LQC). In this context, effective models for LQC or quantum black holes are a useful framework to obtain insights on the underlying symmetry-reduced quantum model and its characteristic properties.

A common feature that such effective models share is that holonomy operators involved in full LQG are replaced by so-called polymerization functions that mimic the loop-inspired quantization of these symmetry-reduced models. Such a polymerization function is always accompanied by a so-called polymerization parameter, usually denoted by $\mu$, which is the relevant scale for, e.g., holonomy corrections in these effective models. The first formulation of such models used a fixed polymerization parameter that is independent of the phase space variables—the so-called ${\mu }_0$-scheme.

The usage of a fixed parameter is inspired by the regularization of holonomies of connections used in full loop quantum gravity [35]. One problem of the ${\mu }_0$-scheme in effective LQC dynamics is that the critical density at the bounce depends on the initial values and is possibly non-Planckian. In the context of the so-called improved dynamics, a $\overline{\mu }$-scheme for LQC was then proposed to overcome this problem [36], in which the polymerization parameter is related to the minimal area gap and, thus, depends on the triad operators.

Going beyond cosmological models, improved dynamics were then proposed for spherically symmetric models [4,9,10,14,28,29,30,31,37]. We note that there are different implementations of the improved dynamics in these models, unlike in LQC where the choice is unique if we require scale invariance. To establish a link between full LQG and the LQC effective dynamics, a model for a coherent state-path integral for LQG was introduced in [38].

However, obtaining the improved $\overline{\mu }$-scheme from such top-down derivations of full LQG is still an open problem [38,39]. If one generalizes the models in [38,39] to the extended phase space introduced here, we expect that these models will yield corresponding effective models with a $\overline{\mu }$-scheme in cosmology or spherically symmetric models. Thus, in these top-down derivations, the difference is whether one works with or without the extended phase space to obtain a model in either a ${\mu }_0$- or $\overline{\mu }$-scheme.

There have been some attempts to generalize models in different directions [40,41,42] to obtain effective models with an $\overline{\mu }$-scheme, which involves either a different regularization of the dynamics or an ensemble of microstates labeled by different graphs. The approach to the implementation of the $\overline{\mu }$-scheme in this work is more along the lines introduced in the effective cosmological models—namely, as a canonical transformation on the phase space of gravitational degrees of freedom.

In this paper, we aim at constructing a method based on an extended phase space that allows the implementation of a general $\overline{\mu }$-scheme dynamically, and we investigate the restrictions for the allowed $\overline{\mu }$-scheme. We discuss the model in the framework of the relational formalism [43,44,45,46,47] where we can formulate the model in terms of Dirac observables with respect to the constraints related to the choice of a given $\overline{\mu }$-scheme. The construction of such Dirac observables allows us to define a canonical transformation on the extended phase space, which maps the original kinematical variables into a new set of variables where the constraints related to the $\overline{\mu }$-scheme become new elementary variables in phase space.

By this, we achieve a so-called Kuchař-decomposition [48,49] of the phase space in which the constraints as well as the Dirac observables with respect to these constraints become the new elementary-phase space variables. Working in extended phase space allows the choice of a particular $\overline{\mu }$-scheme to be implemented as a canonical transformation. Although this is also possible directly in the reduced phase space of cosmological models, it is no longer possible in reduced phase space for general $\overline{\mu }$-schemes in spherically symmetric models.

With the detailed construction of the cosmological model in Section 2 and spherically symmetric model in Section 3, we show that the model in terms of these Dirac observables can be understood as the gauge-unfixed version of a given $\overline{\mu }$ model. The model includes constraints that fix $\overline{\mu }$ to some chosen phase space functions that is implementing a specific $\overline{\mu }$-scheme, and we call such functions gauge-fixing functions in the following.

By requiring that the elementary canonical variables encoding the constraints vanish, we obtain the Hamiltonian constraint in the $\overline{\mu }$-scheme characterized by the choice of the gauge-fixing functions, which can be taken as the starting point to further study the quantum theory and effective dynamics. In addition, we also investigate the corresponding gauge-fixed model for both the cosmological and spherically symmetric case. The method that we propose here can, for instance, be applied to the path integral formulation of full LQG [38], thereby, allowing us to obtain $\overline{\mu }$-like effective dynamics from the top-down derivations of full LQG.

As our analysis shows, in the cosmological model, the gauge-fixing function is encoded in the model, and thus the choice for $\overline{\mu }$ has no restrictions to the form of the $\overline{\mu }$ function. Any gauge-fixing function that one chooses, which depends on the scale factor and extrinsic curvature, can lead to a well-defined Hamiltonian in the partially reduced phase space equipped with the standard canonical commutation relations (CCR). The form of the $\overline{\mu }$-scheme is fixed only when we further require scale invariance of the polymerization function, which then reduces to the standard $\overline{\mu }$-scheme used in LQC.

In the spherically symmetric model, which is a $1+1$-dimensional field theory, this is no longer the case. Due to the requirement for a consistent density weight of the effective constraints, the dynamical implementation of a chosen $\overline{\mu }$-scheme is only possible if the polymerization depends on a combination of the extrinsic curvature and the triad such that they have density weight zero. This gives a restriction to the possible gauge-fixing functions for spherical symmetric models.

We note that, if one only works with the interior of a black hole, which can be described as a homogeneous Kantowski–Sachs spacetime, the gauge-fixing functions are again free from any restriction similar to the case of cosmological models.

The paper is structured as follows: In Section 2, we present the dynamical implementation of the $\overline{\mu }$-scheme for cosmological models. In Section 2.1, we use the framework of the relational formalism and discuss, in detail, how a canonical transformation can be constructed. Using the transformed variables, we show on the one hand that we obtain a gauge-unfixed model for each given choice of a $\overline{\mu }$-scheme, including also the special choice of ${\mu }_0$ and, on the other hand, how the reduced phase space can be obtained.

In Section 2.2, we present the corresponding gauge-fixed model for the cosmological case. In Section 3, we generalize our strategy to spherically symmetric models where in general more than one polymerization parameter is present and we consider field theory models. The sections on the cosmological and the spherically symmetric models should be self-contained, so that the reader interested in only one of the two cases can go directly to the appropriate section.

As shown in Section 3.1 a similar canonical transformation on the extended phase space can be constructed using the relational formalism. To show that the transformation is indeed canonical requires a slightly more effort compared to the cosmological case but works analogously. Again, the resulting model can be understood as a gauge-unfixed model of a given $\overline{\mu }$-scheme. In addition, we present, in Section 3.2, the corresponding gauge-fixed model. We summarize and conclude in Section 3.2 and provide further details for some calculations in the appendix.

2. Implementing the $\overline{\mathit{\mu }}$-Scheme in Cosmology

We aimed to construct a model on an extended phase space that allows us to implement the $\overline{\mu }$-scheme in cosmology dynamically. Note that we will not consider a further gauge fixing or the construction of Dirac observables with respect to the Hamiltonian constraint but see the model presented here rather as a starting point for a cosmological model with a given Hamiltonian constraints for which on the usual kinematical phase space a $\overline{\mu }$-scheme needs to be chosen. Afterwards, one can still consider a further gauge fixing of the Hamiltonian constraint or construct the fully reduced phase space that also involves the Hamiltonian constraint because such a step is usually performed within an already given $\overline{\mu }$-scheme.

2.1. Extended Phase Space for Cosmology: Canonical Transformation and Dirac Observables

We consider the usual kinematical phase space of cosmology with elementary-phase space variables denoted by $\theta ,p,N,p_N,{\varphi }^A,{\pi }_A$, where N is the lapse function, $\theta ,p$ the gravitational variables and ${\varphi }^A,{\pi }_A$ some generic matter degrees of freedom where $A\in 1,\cdots ,L_{\#}$ is a generic label to allow more than one matter component in general. We extend this phase space by two additional canonical pairs $(\lambda ,p_{\lambda })$ and $(\mu ,p_{\mu })$. The non-vanishing CCR in the $(8+2L_{\#})$ dimensional phase space are given by

$$\begin{array}{c}\hfill \left\{\theta ,p\right\}=\frac{\gamma \kappa }{6}\left\{\lambda ,p_{\lambda }\right\}=\left\{\mu ,p_{\mu }\right\}=1,\{{\varphi }^A,{\pi }_B\}={\delta }_B^A,\{N,p_N\}=1,\end{array}$$

(1)

where $\gamma$ denotes the Barbero–Immirzi parameter and $\kappa =8\pi G$ with G being Newton’s constant. The model that consider has a primary Hamiltonian of the form

$$\begin{array}{c}\hfill H^P=N{H}_0(\theta ,p,\mu ,{\phi }^A,{\pi }_A)+\lambda (\mu -f(p,\theta ))+{\mathsf{\Lambda }}_{\lambda }{p}_{\lambda }+{\mathsf{\Lambda }}_{\mu }{p}_{\mu }+{\mathsf{\Lambda }}_N{p}_N,\end{array}$$

(2)

where ${\mathsf{\Lambda }}_{\lambda },{\mathsf{\Lambda }}_{\mu },{\mathsf{\Lambda }}_N$ are Lagrange multipliers, and $H_0(\theta ,p,\mu ,{\phi }^A,{\pi }_A)$ takes the following form

$$H_0(\theta ,p,\mu ,{\phi }^A,{\pi }_A)=-\frac{6\sqrt{p}h\left(\mu \theta \right)}{{\gamma }^2\kappa {\mu }^2}+H_0^{\mathrm{matter}}({\varphi }^A,{\pi }_A,p).$$

(3)

We consider a polymerization of the connection variable $\theta$ in the gravitational sector with a generic function $h\left(\mu \theta \right)$ satisfying

$$\begin{array}{c}\hfill \underset{\mu \to 0}{lim}\frac{h\left(\mu \theta \right)}{{\mu }^2}={\theta }^2.\end{array}$$

(4)

This provides the possibility to either consider this model as a classical starting point for a later loop quantization or to stay at the classical level and consider it as an effective model. Choosing this function to be $h\left(\mu \theta \right)=sin{\left(\mu \theta \right)}^2$ gives the usual LQC effective Hamiltonian. The function $f(\theta ,p)$ is an up to now arbitrary function and can be later chosen to have a specific form in order to reproduce the wanted $\overline{\mu }$-scheme. The system has three primary constraints

$$\begin{array}{c}\hfill C_{p_{\lambda }}=p_{\lambda }\approx 0C_{p_{\mu }}=p_{\mu }\approx 0C_{p_N}=p_N\approx 0.\end{array}$$

(5)

The stability of $C_{p_{\lambda }}$ leads to the following gauge fixing condition for $\mu$,

$$\begin{array}{ccc}\hfill C_{\mu }& =& {\dot{C}}_{p_{\lambda }}=\mu -f_{\mu }(\theta ,p)\approx 0\hfill \end{array}$$

(6)

The stability of $C_{p_{\mu }}$ gives

$$\begin{array}{c}\hfill C_{\lambda }={\dot{C}}_{p_{\mu }}=\lambda +\frac{12\sqrt{p}h\left(\mu \theta \right)}{{\gamma }^2\kappa {\mu }^3}-\frac{6\theta \sqrt{p}{h}^{\prime }\left(\mu \theta \right)}{{\gamma }^2\kappa {\mu }^2}=:\lambda +g(\theta ,p,\mu )\approx 0\end{array}$$

(7)

with $h^{\prime }\left(X\right)={\partial }_X{h\left(X\right)|}_{X=\mu \theta }$. Note that the matter contribution $H_0^{\mathrm{matter}}$ does not contribute to $C_{\lambda }$ because it does not depend on $\mu$ as long as we do not consider a polymerization of the matter part. Furthermore, the requirement of the stability for $C_{p_N}$ yields the Hamiltonian constraint

$$\begin{array}{c}\hfill C_0=H_0(\theta ,p,\mu ,{\phi }^A,{\pi }_A)\approx 0.\end{array}$$

(8)

The stability of $C_{\mu }$ and $C_{\lambda }$ fix Lagrangian multiplier ${\mathsf{\Lambda }}_{\mu }$ and ${\mathsf{\Lambda }}_{\lambda }$, respectively, whereas $C_0$ is already stable.

The canonical transformation that we want to construct is a so-called partial Kuchař decomposition of the original phase space. We call it partial here because we perform such a decomposition only with respect to part of the constraints, namely those related to the choice of the $\overline{\mu }$-scheme. Strictly speaking, we should also call the Dirac observable partial Dirac observables because they span the partially reduced phase space but not the fully reduced one, which needs to involve observables with respect to the Hamiltonian constraints.

In the following, we still call the partial Dirac observables simply Dirac observables but have in mind that the Hamiltonian constraint has not been considered yet but could easily be included following, for instance, the strategy used in [50] by using an additional observable map based on matter clocks and where details on the construction of such observable maps are presented.

For this purpose, we use the observable map as well as its dual version introduced in [51] where a Kuchař decomposition [48,49] of the phase space was constructed in the context of linearized gravity. This is necessary because originally we want to choose some of the involved constraints as new canonical coordinates, and this requires that each constraint commutes with all but one. As of this, we cannot use the original set of constraints for this purpose. Applying different observable maps successively will, however, yield new phase space coordinates that can be used for this purpose. We then show that the application of the different observables maps is indeed a canonical transformation on the extended phase space.

As a first step, we consider the two first class constraints $C_{p_{\mu }}$ and $C_{p_{\lambda }}$. As there are elementary momentum variables in phase space, they mutually commute with all remaining phase space variables except $\mu ,\lambda$ and as a consequence $C_{\mu },C_{\lambda }$. In particular $\theta$ and p are obviously Dirac observables with respect to $C_{p_{\mu }},C_{p_{\lambda }}$. This means that the corresponding observable maps are defined as

$${\mathcal{O}}_f^{p_{\mu }}:=\sum _{n=0}^{\infty }\frac{{(-1)}^n{\left(C_{\mu }\right)}^n}{n!}{\{f,C_{p_{\mu }}\}}_{\left(n\right)},{\mathcal{O}}_f^{p_{\lambda }}:=\sum _{n=0}^{\infty }\frac{{(-1)}^n{\left(C_{\lambda }\right)}^n}{n!}{\{f,C_{p_{\lambda }}\}}_{\left(n\right)},$$

(9)

where ${\{.,.\}}_{\left(n\right)}$ denotes the nested Poisson bracket with ${\{f,g\}}_{\left(0\right)}=f$ and ${\{f,g\}}_{(n+1)}=\{{\{f,g\}}_{\left(n\right)},g\}$ and f is a function on phase space that becomes the identity map for functions f that depend on $(\theta ,p)$ only. In the language of the relational formalism the constraints $C_{\mu },C_{\lambda }$ play the role of clocks. Now, given the observable map in (9), we can define its dual map along the lines of the work in [51] where the role of clocks and first class constraints is interchanged.

However, if we want to consider dual observable maps for both pairs $(C_{\mu },C_{p_{\mu }})$ and $(C_{\lambda },C_{p_{\lambda }})$ we need that the ‘clocks’ $C_{\mu },C_{\lambda }$ commute at least weakly. Considering the Dirac matrix in (19), we see that this is not the case. Since we aim at constructing a canonical transformation we even need the stronger requirement that the clocks strongly commute. This can be achieved by successively applying the dual observable map as follows: in the first step, we consider the following set of variables

$$\begin{array}{ccc}\hfill Q^1:=C_{\lambda },P_1:=C_{p_{\lambda }},Q^2:& =& {\mathcal{O}}_{C_{\mu }}^{C_{\lambda }},P_2:={\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }},\hfill \end{array}$$

(10)

with the dual observable map

$${\mathcal{O}}_f^{C_{\lambda }}:=\sum _{n=0}^{\infty }\frac{{\left(C_{p_{\lambda }}\right)}^n}{n!}{\{f,C_{\lambda }\}}_{\left(n\right)}.$$

(11)

By construction, we have $\{{\mathcal{O}}_f^{C_{\lambda }},C_{\lambda }\}=0$. Using the result in [47] on the Poisson algebra of the so-constructed Dirac observables and carrying it over to the dual observable map, we obtain

$$\{{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }},{\mathcal{O}}_{p_{\mu }}^{C_{\lambda }}\}={\mathcal{O}}_{{\{C_{\mu },p_{\mu }\}}_D^{C_{\lambda }}}={\mathcal{O}}_{\{C_{\mu },p_{\mu }\}}=1,$$

(12)

where ${\{.,.\}}_D^{C_{\lambda }}$ denotes the submatrix of the Dirac bracket in (19) that includes the Poisson brackets of the pair $(C_{\lambda },C_{p_{\lambda }})$ only. Given this, we already have $\{Q^j,P_k\}={\delta }_k^j$ with $j,k=1,2$. To extend that set by variables corresponding to the gravitational and matter degrees of freedom that have the property to mutually commute with $Q^j,P_j$ for $j=1,2$ and satisfy standard CCR, we define

$$Q^3:={\mathcal{O}}_{\theta }^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},P_3:={\mathcal{O}}_p^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},Q^4:=N,P_4:=p_N,Q^I:={\mathcal{O}}_{{\varphi }^A}^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},P_I:={\mathcal{O}}_{{\pi }_A}^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},$$

(13)

with $I\in 5,\cdots ,L_{\#}+4$ being an index set covering the matter degrees of freedom for all A at the level of the observables. The notation ${\mathcal{O}}_f^{C_{\lambda },{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}}$ means that, in the first step, we apply the dual observable map from (11) on f and afterwards the following dual observable map

$${\mathcal{O}}_f^{{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}}:=\sum _{n=0}^{\infty }\frac{{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n}{n!}{\{f,{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}\}}_{\left(n\right)}.$$

(14)

For the canonical pair $(N,p_N)$, we used that both variables trivially commute with $C_{\lambda },C_{\mu }$, and thus these maps act similar to the identity map on $N,p_N$.

Now, it remains to show that $Q^3,P_3,Q^4,P_4,Q^I,P_I$ satisfy standard CCR and have vanishing Poisson brackets with $Q^1,Q^2,P_1,P_2$. For the pair $(Q^4,P_4)$, this is the case because we have $\{Q^4,P_4\}=1$, and none of the constructed observables depends on $N,p_N$ so that they mutually commute with them. To show that it also for the remaining set $Q^3,P_3,Q^I,P_I$, somewhat more work is needed. By construction, $C_{\lambda }$ and ${\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}$ commute and hence the order of the application of the two (dual) observable maps is irrelevant and will yield a quantity that commutes with $Q^1$ as well as $Q^2$ by construction. To compute the Poisson bracket with the momenta, we use

$$\begin{array}{ccc}\hfill {\mathcal{O}}_f^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}}& =& \sum _{n=0}^{\infty }\frac{1}{n!}{\left({\mathcal{O}}_{p_{\mu }}^{C_{\lambda }}\right)}^n{\{{\mathcal{O}}_f^{C_{\lambda }},{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}\}}_{\left(n\right)}=\sum _{n=0}^{\infty }\frac{1}{n!}{\left({\mathcal{O}}_{p_{\mu }}^{C_{\lambda }}\right)}^n{\mathcal{O}}_{{\{f,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }}}^{C_{\lambda }},\hfill \end{array}$$

(15)

where ${\{f,g\}}_{D\left(n\right)}^{C_{\lambda }}$ denotes the iterated Dirac bracket with respect to the pair $(C_{\lambda },C_{p_{\lambda }})$ with ${\{f,g\}}_{D\left(0\right)}^{C_{\lambda }}=f$ and ${\{f,g\}}_{D(n+1)}^{C_{\lambda }}={\{{\{f,g\}}_{D\left(n\right)}^{C_{\lambda }},g\}}_D^{C_{\lambda }}$.

The last step in (15) can be easily shown using again the result on the observable algebra from [47] iteratively as well as a proof by induction, which we present in the Lemma 1 below. The Lemma 1 is not discussed in [47] but can be easily proven using the theorem presented in [47], and we present the proof in Appendix C.

Lemma 1.

For the iterated Poisson bracket of observables, we have

$${\{{\mathcal{O}}_f^{C_{\lambda }},{\mathcal{O}}_g^{C_{\lambda }}\}}_{\left(n\right)}={\mathcal{O}}_{{\{f,g\}}_{D\left(n\right)}^{C_{\lambda }}}^{C_{\lambda }}.$$

(16)

Using these results, the Poisson brackets with the momenta yield

$$\begin{array}{ccc}\hfill \{Q^3,P_2\}& =& \{{\mathcal{O}}_{\theta }^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\}=\sum _{n=0}^{\infty }\frac{1}{n!}\{{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n{\mathcal{O}}_{{\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }}}^{C_{\lambda }},{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\}\hfill \\ & =& \sum _{n=0}^{\infty }\frac{1}{n!}{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n\{{\mathcal{O}}_{{\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }}}^{C_{\lambda }},{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\}=\sum _{n=0}^{\infty }\frac{1}{n!}{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n{\mathcal{O}}_{{\{{\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }},C_{p_{\mu }}\}}_D^{C_{\lambda }}}^{C_{\lambda }},\hfill \end{array}$$

where we used the result of Lemma 1 in the second and one before the last step.

Now, we consider the Dirac bracket ${\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }}$. For $n=0$, we have ${\{\theta ,C_{\mu }\}}_{D\left(0\right)}^{C_{\lambda }}=\theta$. For $n\ge 1$, we find

$${\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }}={\{\theta ,C_{\mu }\}}_{\left(n\right)}={\{\theta ,f_{\mu }\}}_{\left(n\right)},$$

where, by assumption, the function $f_{\mu }$ depends on the gravitational degrees of freedom only, and thus ${\{\theta ,f_{\mu }\}}_{\left(n\right)}$ will do so as well. Here, we used, in the first step, that $\theta ,C_{\mu }$ both commute with $p_{\lambda }$. Since any partial derivative of f as well as $\theta$ commutes with $p_{\lambda }$ we obtain for $n\ge 1$

$${\{{\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }},C_{p_{\mu }}\}}_D^{C_{\lambda }}=\{{\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }},C_{p_{\mu }}\}=\frac{\partial }{\partial \mu }{\{\theta ,f_{\mu }\}}_{\left(n\right)}=0.$$

For $n=0$, the nested Dirac bracket also vanishes because $\{\theta ,C_{p_{\mu }}\}=0$. From these results, we immediately end up with

$$\{Q^3,P_2\}=\sum _{n=0}^{\infty }\frac{1}{n!}{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n{\mathcal{O}}_{{\{{\{\theta ,C_{\mu }\}}_{D\left(n\right)}^{C_{\lambda }},C_{p_{\mu }}\}}_D^{C_{\lambda }}}^{C_{\lambda }}=\sum _{n=0}^{\infty }\frac{1}{n!}{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n{\mathcal{O}}_0^{C_{\lambda }}=0.$$

(17)

For the momentum $P_1$, the calculation of the Poisson bracket leads to

$$\begin{array}{ccc}\hfill \{Q^3,P_1\}& =& \{{\mathcal{O}}_{\theta }^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},C_{p_{\lambda }}\}=\{{\mathcal{O}}_{\theta }^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},C_{p_{\lambda }}\}=\frac{\partial }{\partial \lambda }\sum _{n=0}^{\infty }\frac{{\left({\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\right)}^n}{n!}{\{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}\}}_{\left(n\right)}=0.\hfill \end{array}$$

The last step follows from the fact that, if we consider (15), we see that, for functions f that do not depend on $p_{\lambda }$, the term linear in $\lambda$ in $C_{\lambda }$ does not contribute to the nested Poisson bracket. Hence, neither ${\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}$ nor ${\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}$ depend on $\lambda$. Since also ${\partial }_{\lambda }{\mathcal{O}}_{\theta }^{C_{\lambda }}=0$, the nested Poisson bracket in the above equation is independent of $\lambda$ too.

The same steps can be performed if we replace $Q^3$ by $P_3$ as well as the matter degrees of freedom $Q^I,P_I$, and thus we have shown that, indeed, $Q^3,P_3,Q^I,P_I$ mutually commutes with $Q^1,Q^2,P_1,P_2$.

Finally, we still need to show that $Q^3,P_3$ as well as $Q^I,P_I$ satisfy standard CCR and mutually commute. For showing that they satisfy standard CCR, we restrict our discussion to the canonical pair $Q^3,P_3$ and discuss the matter variables afterwards. For this purpose it is more convenient to directly work with the second class constraints $(C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{{\mathrm{C}}_{\lambda }},C_{p_{\lambda }},{\mathcal{O}}_{p_{\mu }}^{C_{\lambda }})$ and the variables ${\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}$ and their corresponding algebra. Then, we obtain

$$\begin{array}{ccc}\hfill \{{\mathcal{O}}_{\theta }^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},{\mathcal{O}}_p^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}}\}& =& {\mathcal{O}}_{{\{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}}_D^{C_{\mu }}}={\mathcal{O}}_{\{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}}=\frac{\gamma \kappa }{6}\hfill \end{array}$$

(18)

with

$$\begin{array}{ccc}\hfill {\{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}}_D^{C_{\mu }}& :=& \{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}+\{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}\}\{{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}-\{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\}\{{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}\hfill \\ & =& \{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}=\frac{\gamma \kappa }{6}.\hfill \end{array}$$

Here, we used that

$$\begin{array}{ccc}\hfill \{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}& =& {\mathcal{O}}_{{\{\theta ,p\}}_D^{C_{\lambda }}}={\mathcal{O}}_{\{\theta ,p\}}=\frac{\gamma \kappa }{6},\hfill \\ \hfill \{{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }},{\mathcal{O}}_p^{C_{\lambda }}\}& =& {\mathcal{O}}_{{\{C_{p_{\mu }},p\}}_D^{C_{\lambda }}}={\mathcal{O}}_{\{C_{p_{\mu }},p\}}=0\hfill \\ \hfill \{{\mathcal{O}}_{\theta }^{C_{\lambda }},{\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }}\}& =& {\mathcal{O}}_{{\{\theta ,C_{p_{\mu }}\}}_D^{C_{\lambda }}}={\mathcal{O}}_{\{\theta ,C_{p_{\mu }}\}}=0,\hfill \end{array}$$

where, in all three cases, the Dirac bracket agrees with the Poisson bracket because $\theta ,p$ commute with $C_{p_{\lambda }}$, and we further use that $\theta ,p$ also Poisson commute with $C_{p_{\mu }}$. The result in (18) can be directly carried over to $Q^I,P_I$ the observables corresponding to the elementary matter phase space variables ${\varphi }^A,{\pi }_A$ because the crucial property needed for this result is that both $\theta ,p$ Poisson commute with $C_{p_{\lambda }}$ and $C_{p_{\mu }}$, which is also given for all ${\varphi }^A,{\pi }_A$. Moreover, this property is also sufficient to show that the sets $Q^3,P_3$$Q^I,P_I$ mutually commute because, also in this computation, the relevant Dirac brackets reduce to their corresponding Poisson brackets, and then we can use that the original phase space variables $p,\theta$ and ${\varphi }^A,{\pi }_A$ are sets of mutually commuting elementary variables in phase space.

Summarizing the canonical transformation that allows to perform a (partial) Kuchař decomposition of the $(8+2L_{\#})$ dimensional phase space separating the physical degrees of freedom from the gauge degrees of freedom is given by

$$\begin{array}{ccc}\hfill Q^1& :=& C_{\lambda },P_1:=C_{p_{\lambda }},Q^2:={\mathcal{O}}_{C_{\mu }}^{C_{\lambda }},P_2:={\mathcal{O}}_{C_{p_{\mu }}}^{C_{\lambda }},Q^3:={\mathcal{O}}_{\theta }^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},P_3:={\mathcal{O}}_p^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},\hfill \\ \hfill Q^4& :=& N,P_4:=p_N,Q^I:={\mathcal{O}}_{{\varphi }^A}^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},P_I:={\mathcal{O}}_{{\pi }_A}^{C_{\lambda },{\mathcal{O}}_{C_{\mu }}^{C_{\lambda }}},\hfill \end{array}$$

with $I=5,\cdots ,L_{\#}+4$. In terms of the original variables $\theta ,p,\mu ,p_{\mu },\lambda ,p_{\lambda },{\varphi }^A,{\pi }_A$, these are rather complicated functions consisting of partly nested power series with phase space depended coefficients. However, as shown above, the new variables satisfy standard CCR

$$\begin{array}{ccc}\hfill \{Q^J,P_K\}& =& {\tilde{\delta }}_K^J,\{Q^J,Q^K\}=0,\{P_J,P_K\}=0,J,K=1,\dots ,L_{\#}+4\hfill \end{array}$$

with

$$\begin{array}{c}\hfill {\tilde{\delta }}_K^J={\delta }_K^J\mathrm{for}J,K=1,\dots ,L_{\#}+4\mathrm{if}J=K\ne 3\mathrm{and}{\tilde{\delta }}_3^3=\frac{\gamma \kappa }{6}.\end{array}$$

The primary Hamiltonian in the new variables expressed as a function in terms of the old canonical variables looks similarly complicated but has, in the new variables, the following form:

$$\begin{array}{ccc}\hfill H^P& =& H_0(Q^3,P_3,Q^1+f_{\mu }(Q^3,P^3),Q^I,P_I)+(Q^2-g(Q^3,P_3,Q^1+f_{\mu }(Q^3,P_3)))Q^1\hfill \\ & & +{\mathsf{\Lambda }}_{\lambda }{P}_2+{\mathsf{\Lambda }}_{\mu }{P}_1.\hfill \end{array}$$

If we consider the Hamiltonian constraint $C_0$ in (8) on the partially reduced phase space with respect to the constraints encoded in $Q^1,P_1,Q^2,P_2$ that we obtain by setting $Q^1=Q^2=P_1=P_2=0$, we end up with

$$\begin{array}{ccc}\hfill C_0& =& H_0(Q^3,P_3,f_{\mu }(Q^3,P^3),Q^I,P_I)\hfill \end{array}$$

showing that we obtain the Hamiltonian constraint in the $\overline{\mu }$-scheme characterized by the choice of the function $f_{\mu }$ in (6) and, in this sense, dynamically implemented the $\overline{\mu }$ by a canonical transformation on an extended phase space in terms of a Kuchař-decomposition and the corresponding partial reduction afterwards. This can now be taken as the starting point to either implement a Dirac quantization or reduced quantization with respect to the Hamiltonian constraint and the remaining primary constraint $p_N$, which are both first class or work with corresponding effective models.

2.2. Gauge-Fixed Model for Cosmology

For the corresponding gauge fixed version of the model introduced in Section 2.1, we use that the set of constraints $C=\left\{C_{\lambda },C_{\mu },C_{p_{\lambda }},C_{p_{\mu }}\right\}$ form a second class system. The Dirac matrix has the form

$$\begin{array}{c}\hfill M=\left(\begin{array}{cccc}0& A_{\lambda \mu }& 1& A_{\lambda p_{\mu }}\\ -A_{\lambda \mu }& 0& 0& 1\\ -1& 0& 0& 0\\ -A_{\lambda p_{\mu }}& -1& 0& 0\end{array}\right)M^{-1}=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& A_{\lambda p_{\mu }}& -1\\ 1& -A_{\lambda p_{\mu }}& 0& A_{\lambda \mu }\\ 0& 1& -A_{\lambda \mu }& 0\end{array}\right)\end{array}$$

(19)

with

$$\begin{array}{ccc}\hfill A_{\lambda p_{\mu }}& =& -\frac{6\sqrt{p}\left(6h\left(\theta \mu \right)+\theta \mu (-4h^{\prime }\left(\theta \mu \right)+\theta \mu h^{″}\left(\theta \mu \right))\right)}{{\gamma }^2\kappa {\mu }^4}\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill A_{\lambda \mu }& =& \frac{{\partial }_{\theta }{f}_{\mu }(2h\left(\theta \mu \right)-\theta \mu h^{\prime }\left(\theta \mu \right))+2\mu p{\partial }_p{f}_{\mu }(-h^{\prime }\left(\theta \mu \right)+\theta \mu h^{″}\left(\theta \mu \right))}{2\gamma {\mu }^3\sqrt{p}}.\hfill \end{array}$$

(21)

The Poisson brackets between $\theta ,p$ and constraints C are non-trivial for $C_{\lambda ,\mu }$ only. As a result, the Dirac brackets between $\theta$ and p recovers their Poisson brackets for generic gauge-fixing function f, namely

$$\begin{array}{c}\hfill {\left\{\theta ,p\right\}}_D=\frac{\gamma \kappa }{6}{\left\{\theta ,\theta \right\}}_D={\left\{p,p\right\}}_D=0\end{array}$$

(22)

Likewise, we obtain for the matter degrees of freedom

$$\begin{array}{c}\hfill {\left\{{\varphi }^A,{\pi }_B\right\}}_D={\delta }_B^A{\left\{{\varphi }_A,{\varphi }_B\right\}}_D={\left\{{\pi }_A,{\pi }_B\right\}}_D=0,A,B\in 1,\dots ,L_{\#}.\end{array}$$

(23)

The remaining Hamiltonian constraint has, in the partially reduced phase space, the following form

$$C=H_0(\theta ,p,f_{\mu }(\theta ,p),{\varphi }^A,{\pi }_A).$$

(24)

The canonical transformation in Section 2.1 and the gauge fixing here does not further restrict the function $f_{\mu }(\theta ,p)$ and particularly allows to choose the special case of a constant function, which corresponds exactly to the ${\mu }_0$-scheme in LQC.

Now, if we require in addition, invariance under the following transformation

$$\theta \to \alpha \theta ,p\to {\alpha }^{2}p$$

(25)

then this further restricts the form of $f_{\mu }(\theta ,p)$ to $f_{\mu }\left(p\right)=\frac{c_1}{\sqrt{\left|p\right|}}$ where $c_1$ is some constant. In the case of the cosmological models, this is the only possible scale invariant choice. Choosing this constant to be $c_1:=\sqrt{\Delta }$ with $\Delta :=2\sqrt{3}\pi \gamma {\ell }_p^2$, where $\gamma$ is the Barbero–Immirzi parameter and ${\ell }_p$ denotes the Planck length, we end up with the usual $\overline{\mu }$ scheme used in LQC [52].

In this section, we demonstrate that we can implement the choice of a generic phase-space dependent function in the argument of the polymerization function dynamically in the following sense: in Section 2.1, this was achieved in terms of a canonical transformation on the extended phase space that allows a Kuchař decomposition [48,49] of the extended phase space. We extended the usual kinematical phase space used in LQC by two additional canonical pairs, which come along with two first class constraints so that the number of physical degrees of freedom does not change.

A later reduction to the physical sector associated with these additional first class constraints then corresponds to a specific choice of a $\overline{\mu }$-scheme. Furthermore, we showed that, in this extended phase space, we can also understand the choice of a given $\overline{\mu }$-scheme as a gauge fixing. We extend our analysis to the spherically symmetric case in the next section.

3. Implementing the $\overline{\mathit{\mu }}$-Scheme in Spherically Symmetric Models

In this section, we want to generalize the formalism used for cosmology in Section 2.1 to spherically symmetric models. The main difference is that on the one hand, we deal with a $1+1$-dimensional field theory, and thus the diffeomorphism constraint is no longer trivial. Moreover, since the number of gravitational (field) variables is larger here, we need to extend the phase space by more variables in order to be able to allow an independent polymerization for the different connection variables involved. In case we describe the Schwarzschild black hole using isometry with Kantowski–Sachs vacuum cosmology, where the diffeomorphism is trivially satisfied, the system is similar to cosmological models with a larger phase space.

3.1. Extended Phase Space for Spherically Symmetric Models: Canonical Transformation and Dirac Observables

In spherically symmetric symmetry-reduced case after implementing the Gauß constraint, the Ashtekar–Barbero variables $(A_a^j,E_j^a)$ have the following form

$$\begin{array}{ccc}\hfill A_a^j{\tau }_j\mathrm{d}{X}^a& =& 2\gamma K_x\left(x\right){\tau }_1\mathrm{d}x+\left(\gamma K_{\phi }\left(x\right){\tau }_2+\frac{{\partial }_x{E}^x\left(x\right)}{2E^{\phi }\left(x\right)}{\tau }_3\right)\mathrm{d}\theta \hfill \\ & & +\left(\gamma K_{\phi }\left(x\right){\tau }_3-\frac{{\partial }_x{E}^x\left(x\right)}{2E^{\phi }\left(x\right)}{\tau }_2\right)sin\left(\theta \right)\mathrm{d}\phi +cos\left(\theta \right){\tau }_1\mathrm{d}\phi \hfill \\ \hfill E_j^a{\tau }^j\frac{\partial }{\partial X^a}& =& E^x\left(x\right)sin\left(\theta \right){\tau }_1{\partial }_x+\left(E^{\phi }\left(x\right){\tau }_2\right)sin\left(\theta \right){\partial }_{\theta }+\left(E^{\phi }\left(x\right){\tau }_3\right){\partial }_{\phi },\hfill \end{array}$$

where $X^a=(x,\theta ,\phi )$, $a=1,2,3$ denote spherical coordinates, as before, $\gamma$ the Barbero–Immirzi parameter and ${\tau }_j=-\frac{1}{2}{\sigma }_j$ with ${\sigma }_j$ being the Pauli matrices. The with respect to the Gauß constraint partially gauge-fixed phase space includes two canonical pairs denoted by $\left(K_x\left(x\right),E^x\left(x\right)\right)$ and $\left(K_{\phi }\left(x\right),E^{\phi }\left(x\right)\right)$. In addition, as in the cosmological case, we consider a set of matter degrees of freedom that we do not further specify whose phase space variables are $({\varphi }^A\left(x\right),{\pi }_A\left(x\right))$ with $A\in 1,\dots ,L_{\#}$. Their non-vanishing Poisson brackets read

$$\{K_x\left(x\right),E^x\left(y\right)\}=G\delta (x,y)\{K_{\phi }\left(x\right),E^{\phi }\left(y\right)\}=G\delta (x,y)\{{\varphi }^A\left(x\right),{\pi }_B\left(y\right)\}={\delta }_B^A\delta (x,y),$$

with G being Newton’s constant. We consider this phase space as the starting point for our extension. We introduce the following additional set of canonical pairs

$$\begin{array}{c}\hfill (e_1\left(x\right),p_{e_1}\left(x\right)),(e_2\left(x\right),p_{e_2}\left(x\right)),({\mu }_x\left(x\right),p_{{\mu }_x}\left(x\right)),({\mu }_{\phi }\left(x\right),p_{{\mu }_{\phi }}\left(x\right)).\end{array}$$

with standard CCR

$$\begin{array}{c}\hfill \{e_I\left(x\right),p_{e_J}\left(y\right)\}={\delta }_J^I\delta (x,y),I,J=1,2\{{\mu }_K\left(x\right),p_{{\mu }_M}\left(y\right)\}={\delta }_L^K\delta (x,y),K,L=x,\phi .\end{array}$$

Note that, here, we considered already the partially reduced phase space with respect to the constraint $p_N\left(x\right)$ associated with the momentum of the lapse function $N\left(x\right)$ such that the latter becomes a Lagrange multiplier here and is not part of the elementary-phase space variables. We start with the following primary Hamiltonian

$$\begin{array}{ccc}\hfill H& =& \int \mathrm{d}x[{\mathcal{C}}_{\mathrm{tot}}^{\Delta }+N^x{\mathcal{C}}_x^{\mathrm{tot}}+e_1({\mu }_x-f_{{\mu }_x}(E^x,E^{\phi },K_x,K_{\phi }))+e_2({\mu }_{\phi }-f_{{\mu }_{\phi }}(E^x,E^{\phi },K_x,K_{\phi }))\hfill \\ & & +{\mathsf{\Lambda }}_{N^x}{p}_{N^x}+{\mathsf{\Lambda }}_{e_1}{p}_{e_1}+{\mathsf{\Lambda }}_{e_2}{p}_{e_2}+{\mathsf{\Lambda }}_{{\mu }_x}{p}_{{\mu }_x}+{\mathsf{\Lambda }}_{{\mu }_{\phi }}{p}_{{\mu }_{\phi }}]\left(x\right),\hfill \end{array}$$

(26)

where we considered a partial gauge fixing that yields for the lapse function $N=1$. This can, for instance, be achieved by choosing a suitable external matter field that is chosen as a temporal reference field such that $N=1$ is implemented. As a consequence, in contrast to the cosmological model in Section 2.1 here the polymerized contribution ${\mathcal{C}}_{\mathrm{tot}}^{\Delta }$ is not a secondary constraint but plays the role of a physical Hamiltonian instead, once the reduction with respect to the spatial diffeomorphism constraint has also been performed. We consider here a partial gauge fixing because, in general, we also expect further restrictions on the possible polymerizations from the closure of the algebra of constraints. Such a general analysis is beyond the scope of this article and is planned for future work, although the analysis performed here will be useful as a first step.

Here, ${\mathcal{C}}_{\mathrm{tot}}^{\Delta }$ and ${\mathcal{C}}_x^{\mathrm{tot}}$ are given by

$${\mathcal{C}}_{\mathrm{tot}}^{\Delta }={\mathcal{C}}^{\Delta }+{\mathcal{C}}^{\mathrm{matter}}(E^x,E^{\phi },{\varphi }^A,{\pi }_A){\mathcal{C}}_x^{\mathrm{tot}}={\mathcal{C}}_x+{\mathcal{C}}_x^{\mathrm{matter}}({\varphi }^A,{\pi }_A)$$

(27)

with

$${\mathcal{C}}^{\Delta }\left(x\right)=\frac{1}{2G}\left[\frac{E^{\phi }}{\sqrt{E^x}}\left(\frac{f_1\left({\mu }_x{K}_x{\mu }_{\phi }{K}_{\phi }\right)}{{\mu }_{\phi }^2}+{\mathsf{\Gamma }}^2-1-4E^x\frac{f_2\left({\mu }_x{K}_x{\mu }_{\phi }{K}_{\phi }\right)}{{\mu }_x{\mu }_{\phi }{E}^{\phi }}\right)+2\sqrt{E^x}{\mathsf{\Gamma }}^{\prime }\right]\left(x\right)$$

(28)

$${\mathcal{C}}_x=\frac{1}{G}\left(E^{\phi }{K_{\phi }}^{\prime }-K_x{E^x}^{\prime }\right)\left(x\right)$$

(29)

with $\mathsf{\Gamma }\left(x\right)=\left(\frac{{E^x}^{\prime }}{2E^{\phi }}\right)\left(x\right)$ the spin connection, and $f_1,f_2$ represent general polymerization functions satisfying

$$\begin{array}{c}\hfill \underset{{\mu }_{\phi }\to 0}{lim}\frac{f_1({\mu }_x{K}_x,{\mu }_{\phi }{K}_{\phi })}{{\mu }_{\phi }^2}=K_{\phi }^2\underset{{\mu }_x,{\mu }_{\phi }\to 0}{lim}\frac{f_2({\mu }_x{K}_x,{\mu }_{\phi }{K}_{\phi })}{{\mu }_x{\mu }_{\phi }}=K_x{K}_{\phi }\end{array}$$

(30)

In order that ${\mathcal{C}}^{\Delta }$ is a scalar density of weight 1, we are forced to choose ${\mu }_x$ being as scalar with density weight $-1$ since $K_x$ has density weight 1. Furthermore, ${\mu }_{\phi }$ needs to have density weight 0. As a result, the only allowed functions $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$ will be of the form

$$\begin{array}{cc}\hfill & f_{{\mu }_x}(E^x,E^{\phi },K_x,K_{\phi })={\left(E^{\phi }\right)}^{-1}{\tilde{f}}_{{\mu }_x}(E^x,K_{\phi },{\left(E^{\phi }\right)}^{-1}{K}_x),\end{array}$$

(31)

$$\begin{array}{cc}\hfill & f_{{\mu }_{\phi }}(E^x,E^{\phi },K_x,K_{\phi })={\tilde{f}}_{{\mu }_{\phi }}(E^x,K_{\phi },{\left(E^{\phi }\right)}^{-1}{K}_x).\end{array}$$

(32)

Here, the appearance of $E^{\phi }$, and the combination of ${\left(E^{\phi }\right)}^{-1}{K}_x$ are essential in order to have density weight of one and zero for $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$, respectively. This completely removes the possibility of the usual ${\mu }_0$ scheme in the effective dynamics if we have polymerization of $K_x$. If we require, in addition, the scaling invariance of $f_1$ and $f_2$ and remove the K dependence in the gauge-fixing function, the only possible ansatz for $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$ is the $\overline{\mu }$ scheme used in [4,29,31,41], which is given by

$$\begin{array}{c}\hfill f_{{\mu }_x}(E^x,E^{\phi })={\alpha }_1\sqrt{E^x}{\left(E^{\phi }\right)}^{-1},f_{{\mu }_{\phi }}(E^x,E^{\phi })={\alpha }_2\sqrt{E^x}\end{array}$$

(33)

for some constants ${\alpha }_1,{\alpha }_2$ proportional to the minimal area gaps introduced in LQG.

Note that the conditions (31) and (32) come from the fact that $K_x$ has a non-trivial density weight, in contrast to $K_{\varphi }$, which has density weight 0. If we exclude the polymerization of $K_x$ and consider the polymerization of $K_{\varphi }$ only:

$$\begin{array}{c}\hfill f_1({\mu }_x{K}_x,{\mu }_{\phi }{K}_{\phi })={\tilde{f}}_1\left({\mu }_{\phi }{K}_{\phi }\right),f_2({\mu }_x{K}_x,{\mu }_{\phi }{K}_{\phi })={\mu }_x{K}_x{\tilde{f}}_2\left({\mu }_{\phi }{K}_{\phi }\right)\end{array}$$

(34)

e.g., as in the models presented in [6,9,10,16,17]. In this case, we can remove the condition on $f_{{\mu }_x}$, and a gauge fixing leading to a ${\mu }_0$-scheme is still allowed.

The model has five primary constraints

$$\begin{array}{c}\hfill C_{\mathrm{Pri}}=[p_{N^x},p_{e_1},p_{e_2},p_{{\mu }_x},p_{{\mu }_{\phi }}]\approx 0.\end{array}$$

(35)

Starting from (26), we obtain the following five secondary constraints:

$$\begin{array}{c}\hfill C_{\mathrm{Sec}}=\{H,C_{\mathrm{Pri}}\}=[{\mathcal{C}}_x^{\mathrm{tot}},C_{{\mu }_x},C_{{\mu }_{\phi }},C_{e_1},C_{e_2}]\approx 0.\end{array}$$

(36)

Here, $C_{{\mu }_x,{\mu }_{\phi }}$ are the constraints related to implementing the $\overline{\mu }$-scheme, they read

$$\begin{array}{c}\hfill C_{{\mu }_x}={\mu }_x-f_{{\mu }_x},C_{{\mu }_{\phi }}={\mu }_{\phi }-f_{{\mu }_{\phi }}\end{array}$$

(37)

and $C_{e_1,e_2}$ fix $e_1,e_2$ to be:

$$\begin{array}{ccc}\hfill C_{e_1}& =& e_1+\frac{4E^x{f}_2{\mu }_{\phi }-K_x{\mu }_x\left(E^{\phi }{\mu }_x{\partial }_X{f}_1+4E^x{\mu }_{\phi }{\partial }_X{f}_2\right)}{2\sqrt{E^x}{{\mu }_x}^2{{\mu }_{\phi }}^2},\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill C_{e_2}& =& e_2+\frac{E^{\phi }{\mu }_x\left(2f_1-K_{\phi }{\mu }_{\phi }{\partial }_Y{f}_1\right)+4E^x{\mu }_{\phi }\left(f_2-K_{\phi }{\mu }_{\phi }{\partial }_Y{f}_2\right)}{2\sqrt{E^x}{\mu }_x{{\mu }_{\phi }}^3}\hfill \end{array}$$

(39)

where we defined $X:={\mu }_x{K}_x,Y:={\mu }_{\phi }{K}_{\phi }$. The stability of $C_{e_1},C_{e_2},C_{{\mu }_x},C_{{\mu }_{\phi }}$ can be ensured by choosing the Lagrangian multipliers ${\mathsf{\Lambda }}_{e_1},{\mathsf{\Lambda }}_{e_2},{\mathsf{\Lambda }}_{{\mu }_x},{\mathsf{\Lambda }}_{{\mu }_{\phi }}$ appropriately, while the stability of ${\mathcal{C}}_x^{\mathrm{tot}}$ is automatically satisfied,

$$\begin{array}{c}\hfill \{H,{\mathcal{C}}_x^{\mathrm{tot}}\}\approx 0.\end{array}$$

(40)

The sets $C_{\mathrm{Pri}}$ and $C_{\mathrm{Sec}}$ from (35) and (36) include all constraints of the model, so we end up with ten constraints

$$\begin{array}{c}\hfill C_{\mathrm{all}}=[p_{N^x},p_{e_1},p_{e_2},p_{{\mu }_x},p_{{\mu }_{\phi }},{\mathcal{C}}_x^{\mathrm{tot}},C_{{\mu }_x},C_{{\mu }_{\phi }},C_{e_1},C_{e_2}]\approx 0.\end{array}$$

It remains to classify the constraints (or suitable combinations thereof) as first or second class. We immediately see that $p_{N^x}$ is first class. Since ${\mathcal{C}}_x^{\mathrm{tot}}$ may not be first class as it will have non-trivial brackets with the constraints $C_{{\mu }_x},C_{{\mu }_{\phi }}$ as well as the polymerization functions in $C_{e_1},C_{e_2},C_{{\mu }_x},C_{{\mu }_{\phi }}$. However, to circumvent this, we introduce an extended diffeomorphism ${\tilde{C}}_x^{\mathrm{tot}}$ on the entire phase space by taking into account the density weight of the individual phase space variables:

$$\begin{array}{c}{\tilde{\mathcal{C}}}_x^{\mathrm{tot}}\left(x\right)={\mathcal{C}}_x^{\mathrm{tot}}\left(x\right)-2p_{{\mu }_x}\left(x\right){\mu }_x^{\prime }\left(x\right)-p_{{\mu }_x}^{\prime }\left(x\right){\mu }_x\left(x\right)-p_{{\mu }_{\phi }}\left(x\right){\mu }_{\phi }^{\prime }\left(x\right)\\ +p_{e_1}\left(x\right)e_1^{\prime }\left(x\right)+2p_{e_1}^{\prime }\left(x\right)e_1\left(x\right)+p_{e_2}^{\prime }\left(x\right)e_2\left(x\right)\end{array}$$

(41)

One can verify that ${\tilde{C}}_x^{\mathrm{tot}}$ is first class, the explicit derivation of this result is presented in Appendix A. Note that this extension is always possible and does not change the constraint structure because the linear momenta involved in the extra terms are all constraints themselves. The remaining constraints are second class. Hence, from now on, we work with the following set of constraints

$$\begin{array}{c}\hfill {\tilde{C}}_{\mathrm{all}}=[p_{N^x},p_{e_1},p_{e_2},p_{{\mu }_x},p_{{\mu }_{\phi }},{\tilde{\mathcal{C}}}_x^{\mathrm{tot}},C_{{\mu }_x},C_{{\mu }_{\phi }},C_{e_1},C_{e_2}]\approx 0,\end{array}$$

including two first class and eight second class constraints.

In an analogous manner to the cosmological case in Section 2.1, we construct a canonical transformation on the extended phase space that allows to perform a Kuchař decomposition of the extended phase space introduced above. For this purpose, we consider the subset of second class constraints $C=\left[p_{e_1},p_{e_2},p_{{\mu }_x},p_{{\mu }_{\phi }},C_{e_1},C_{e_2},C_{{\mu }_x},C_{{\mu }_{\phi }}\right]$. Similar to the cosmological case, $p_{{\mu }_x},p_{{\mu }_{\phi }},C_{{\mu }_x},C_{{\mu }_{\phi }}$ do not commute with $C_{e_1},C_{e_2}$, in addition, $C_{e_1},C_{e_2}$ do not mutually commute. Therefore, as a first step, we introduce the following set of variables

$$\begin{array}{ccc}\hfill Q^1& :=& C_{e_2},P_1:=p_{e_2},Q^2:={\mathcal{O}}_{C_{e_1}}^{C_{e_2}}={\mathcal{O}}_{C_{e_1}}^{Q^1},P_2:={\mathcal{O}}_{C_{p_{e_1}}}^{C_{e_2}}={\mathcal{O}}_{C_{p_{e_1}}}^{Q^1},\hfill \\ \hfill Q^3& :=& {\mathcal{O}}_{C_{{\mu }_x}}^{C_{e_2},{\mathcal{O}}_{C_{e_1}}^{C_{e_2}}}={\mathcal{O}}_{C_{{\mu }_x}}^{Q^1,Q^2},P_3:={\mathcal{O}}_{p_{{\mu }_x}}^{C_{e_2},{\mathcal{O}}_{C_{e_1}}^{C_{e_2}}}={\mathcal{O}}_{p_{{\mu }_x}}^{Q^1,Q^2},\hfill \\ \hfill Q^4& :=& {\mathcal{O}}_{C_{{\mu }_{\phi }}}^{C_{e_2},{\mathcal{O}}_{C_{e_1}}^{C_{e_2}},{\mathcal{O}}_{C_{{\mu }_x}}^{C_{e_2},{\mathcal{O}}_{C_{e_1}}^{C_{e_2}}}}={\mathcal{O}}_{C_{{\mu }_{\phi }}}^{Q^1,Q^2,Q^3},P_4:={\mathcal{O}}_{p_{{\mu }_{\phi }}}^{C_{e_2},{\mathcal{O}}_{C_{e_1}}^{C_{e_2}},{\mathcal{O}}_{C_{{\mu }_x}}^{C_{e_2},{\mathcal{O}}_{C_{e_1}}^{C_{e_2}}}}={\mathcal{O}}_{p_{{\mu }_{\phi }}}^{Q^1,Q^2,Q^3}.\hfill \end{array}$$

In the spherically symmetric models, the individual dual observable maps can be defined as

$${\mathcal{O}}_f^{Q^I}\left(x\right):=\sum _{n=0}^{\infty }\int d{y}_1\cdots \int d{y}_n\int \frac{(P_I\left(y_1\right)\cdots P_I\left(y_n\right))}{n!}\{\dots \{\{f\left(x\right),Q_I\left(y_1\right)\},Q_I\left(y_2\right)\}\dots \}\},Q_I\left(y_n\right)\}.$$

(42)

For these subsets of canonical variables, we have

$$\begin{array}{c}\hfill \{Q^I\left(x\right),P_J\left(y\right)\}={\delta }_J^I\delta (x,y),\{Q^I\left(x\right),Q^J\left(y\right)\}=0,\{P_I\left(x\right),P_J\left(y\right)\}=0,I=1\cdots 4.\end{array}$$

This is explicitly shown in Appendix D for the canonical pair $(Q^4,P_4)$. If we consider any of the other pairs $(Q^I,P_I)$ the computations work similarly with fewer steps since for the construction of these observables fewer individual observable maps have been applied. As the computation in Appendix D show in these cases the number of observable maps fit again well such that the involved Poisson brackets can be mapped to their corresponding Dirac brackets successively. By construction the observables $Q^I$ mutually commute. It only remains to show that all observables related to momenta mutually commute and that these commute with all but one $Q^I$. In the two latter cases, the number of applied observable maps for a given $Q^I$ and $P_I$ will not be the same. Therefore, to show that these Poisson brackets vanish works slightly differently in these cases. As an example, we consider

$$\begin{array}{ccc}\hfill \{P_4\left(x\right),P_2\left(y\right)\}& =& \{{\mathcal{O}}_{p_{{\mu }_{\phi }}\left(x\right)}^{Q^1,Q^2,Q^3},{\mathcal{O}}_{p_{e_1}}^{Q^1}\left(y\right)\}=\{{\mathcal{O}}_{p_{{\mu }_{\phi }}}^{Q^2,Q^3,Q^1}\left(x\right),{\mathcal{O}}_{p_{e_1}}^{Q^1}\left(y\right)\}={\mathcal{O}}_{{\{{\mathcal{O}}_{p_{{\mu }_{\phi }}}^{Q^2,Q^3}\left(x\right),p_{e_1}\left(y\right)\}}_D^{Q^1}}^{Q^1}\hfill \\ & =& {\mathcal{O}}_{\{{\mathcal{O}}_{p_{{\mu }_{\phi }}}^{Q^2,Q^3}\left(x\right),p_{e_1}\left(y\right)\}}^{Q^1}\hfill \end{array}$$

In the second step, we used that the $Q^I$s mutually commute and in the last step that $p_{e_1}$ commutes with $Q^1=C_{e_2},P_1=p_{e_2}$. Finally, from

$$\{{\mathcal{O}}_{p_{{\mu }_{\phi }}}^{Q^2,Q^3}\left(x\right),p_{e_1}\left(y\right)\}=\frac{\delta }{\delta e_1\left(y\right)}{\mathcal{O}}_{p_{{\mu }_{\phi }}}^{Q^2,Q^3}\left(x\right)=0\mathrm{we}\mathrm{find}\{P_4\left(x\right),P_2\left(y\right)\}=0.$$

That the Poisson bracket above vanishes follows from the fact that $e_1\left(y\right)$ is involved in $C_{e_1}$ and when it is used in the observable map for other phase space variables than $p_{e_1}$ the part linearly in $e_1\left(y\right)$ does not contribute to the nested Poisson bracket. Now, we extend the set of canonical pairs in (42) by the corresponding observables with respect to $Q^1,\cdots ,Q^4$ of the geometrical and matter degrees of freedom. For this purpose let us introduce the notation ${\mathcal{O}}_f^{Q^{1\cdots 4}}$ for the observable of the function f with respect to the constraints $Q^{1\cdots 4}$. Then, we consider the following canonical transformed variables

$$\begin{array}{ccc}\hfill Q^5& :=& {\mathcal{O}}_{K_x}^{Q^{1\cdots 4}},P_5:={\mathcal{O}}_{E^x}^{Q^{1\cdots 4}},Q^6:={\mathcal{O}}_{K_{\phi }}^{Q^{1\cdots 4}},P_6:={\mathcal{O}}_{E^{\phi }}^{Q^{1\cdots 4}},\hfill \\ \hfill Q^J& :=& {\mathcal{O}}_{{\varphi }^A}^{Q^{1\cdots 4}},P_J:={\mathcal{O}}_{{\pi }_A}^{Q^{1\cdots 4}},J=7,\cdots ,L_{\#}+6\hfill \end{array}$$

As shown in the Appendix D with these additional variables, we obtain a set of canonical variables satisfying standard CCR given by

$$\begin{array}{c}\hfill \{Q^I\left(x\right),P_J\left(y\right)\}={\tilde{\delta }}_J^I\delta (x,y),\{Q^I\left(x\right),Q^J\left(y\right)\}=0,\{P_I\left(x\right),P_J\left(y\right)\}=0,I,J=1,\cdots ,L_{\#}+6.\end{array}$$

with

$$\begin{array}{c}\hfill {\tilde{\delta }}_J^I={\delta }_J^I\mathrm{for}I,J=1,\dots ,L_{\#}+6\mathrm{if}I=J\ne 5,6\mathrm{and}{\tilde{\delta }}_5^5={\tilde{\delta }}_6^6=G.\end{array}$$

In the new coordinates $(Q^I,P_I)$, a Kuchař decomposition [48,49] of the kinematical phase space can be achieved. In the explicit computations of the individual Poisson brackets one sees that the reason why they satisfy CCR is very close to the case of the cosmological model but generalized to the field theory case and a larger number of phase space variables. In both cases it turns out to be crucial that the constraints involving the gauge-fixing functions f in the cosmological case and $f_{{\mu }_x},f_{{\mu }_{\phi }}$ are given in a form in which they split into a part depending on the variables $\mu$ and ${\mu }_x,{\mu }_{\phi }$, respectively, and a further part that depends on the gravitational degrees of freedom only, where the latter, in principle, can be generalized to a dependence on the matter degrees of freedom as well. Furthermore, also the fact that $\lambda$ and $e_1,e_2$, respectively, only enter linearly in the respective constraints is important.

3.2. Gauge-Fixed Model

Similar to the cosmological model, we also discuss the corresponding gauge-fixed model in the spherically symmetric case. As discussed above the set of constraints $C=\left\{p_{e_1},p_{e_2},p_{{\mu }_x},p_{{\mu }_{\phi }},C_{e_1},C_{e_2},C_{{\mu }_x},C_{{\mu }_{\phi }}\right\}$ forms a second class system. The Dirac matrix has a similar form as (19), which reads

$$\begin{array}{c}\hfill D_{BH}=\left(\begin{array}{cccccccc}0& 0& 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& D_{35}& D_{36}& -1& 0\\ 0& 0& 0& 0& D_{45}& D_{46}& 0& -1\\ 1& 0& -D_{35}& -D_{45}& 0& D_{56}& D_{57}& D_{58}\\ 0& 1& -D_{36}& -D_{46}& -D_{56}& 0& D_{67}& D_{68}\\ 0& 0& 1& 0& -D_{57}& -D_{67}& 0& D_{78}\\ 0& 0& 0& 1& -D_{58}& -D_{68}& -D_{78}& 0\end{array}\right),\end{array}$$

(43)

where the non-trivial components $D_{ij}$ are given explicitly in Appendix B. Note that $D_{78}=0$ if the functions $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$ in (31) do not contain $K_x,K_{\phi }$. The inverse of the $D_{BH}$ reads

$$\begin{array}{c}\hfill D_{BH}^{-1}=\left(\begin{array}{cccccccc}0& \tilde{D}& D_{57}-D_{45}{D}_{78}& D_{58}+D_{35}{D}_{78}& 1& 0& D_{35}& D_{45}\\ -\tilde{D}& 0& D_{67}-D_{46}{D}_{78}& D_{68}+D_{36}{D}_{78}& 0& 1& D_{36}& D_{46}\\ D_{45}{D}_{78}-D_{57}& D_{46}{D}_{78}-D_{67}& 0& D_{78}& 0& 0& 1& 0\\ -D_{58}-D_{35}{D}_{78}& -D_{68}-D_{36}{D}_{78}& -D_{78}& 0& 0& 0& 0& 1\\ -1& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0\\ -D_{35}& -D_{36}& -1& 0& 0& 0& 0& 0\\ -D_{45}& -D_{46}& 0& -1& 0& 0& 0& 0\end{array}\right)\end{array}$$

(44)

with

$$\begin{array}{c}\hfill \tilde{D}=D_{56}+D_{36}{D}_{57}+D_{46}{D}_{58}-D_{35}{D}_{67}-D_{45}{D}_{68}-D_{36}{D}_{45}{D}_{78}+D_{35}{D}_{46}{D}_{78}.\end{array}$$

(45)

Since $K_x,K_{\phi },E^x,E^{\phi }$ clearly commute with $p_{e_1},p_{e_2},p_{{\mu }_x},p_{{\mu }_{\phi }}$, and we have a vanishing $D_{BH}^{-1}$ for its sub-matrix corresponding to $C_{e_1},C_{e_2},C_{{\mu }_x},C_{{\mu }_{\phi }}$, we recover the standard CCR between the gravitational degrees of freedom $K_x,K_{\phi },E^x,E^{\phi }$

$$\begin{array}{ccc}& & {\{K_x\left(x\right),E^x\left(y\right)\}}_D={\{K_{\phi }\left(x\right),E^{\phi }\left(y\right)\}}_D=G\delta (x-y)\hfill \end{array}$$

(46)

$$\begin{array}{ccc}& & {\{K_x\left(x\right),K_{\phi }\left(y\right)\}}_D={\{E^x\left(x\right),E^{\phi }\left(y\right)\}}_D=0\hfill \end{array}$$

(47)

and, for matter contributions, we have

$$\begin{array}{ccc}& & {\{{\varphi }^A\left(x\right),{\pi }_B\left(y\right)\}}_D={\delta }_B^A\delta (x,y),{\{{\varphi }^A\left(x\right),{\varphi }^B\left(y\right)\}}_D={\{{\pi }_A\left(x\right),{\pi }_B\left(y\right)\}}_D=0,\hfill \end{array}$$

(48)

where $A,B\in 1,\dots ,L_{\#}$. With the allowed choice of the functions $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$ in (31), that can be understood as a gauge fixing for ${\mu }_x$ and ${\mu }_{\phi }$, respectively, the gravitational contribution to the Hamiltonian in the gauge-fixed model reads

$$\begin{array}{ccc}\hfill {\mathcal{C}}^{\Delta }\left(x\right)& =& \frac{1}{2G}\left[\frac{E^{\phi }}{\sqrt{E^x}}\left(\frac{f_1\left({\tilde{f}}_{{\mu }_x}\frac{K_x}{E^{\phi }},{\tilde{f}}_{{\mu }_{\phi }}{K}_{\phi }\right)}{{\tilde{f}}_{{\mu }_{\phi }}^2}-4E^x\frac{f_2\left({\tilde{f}}_{{\mu }_x}\frac{K_x}{E^{\phi }},{\tilde{f}}_{{\mu }_{\phi }}{K}_{\phi }\right)}{{\tilde{f}}_{{\mu }_x}{\tilde{f}}_{{\mu }_{\phi }}}+{\mathsf{\Gamma }}^2-1\right)+2\sqrt{E^x}{\mathsf{\Gamma }}^{\prime }\right]\left(x\right)\hfill \end{array}$$

We can rewrite it as

$$\begin{array}{ccc}\hfill {\mathcal{C}}^{\Delta }\left(x\right)& =& \frac{1}{2G}\left[\frac{E^{\phi }}{\sqrt{E^x}}\left(f(E^x,K_{\phi },{\left(E^{\phi }\right)}^{-1}{K}_x)+{\mathsf{\Gamma }}^2-1\right)+2\sqrt{E^x}{\mathsf{\Gamma }}^{\prime }\right]\left(x\right),\hfill \end{array}$$

(49)

where we combine $f_1,f_2$ and the gauge fixing conditions into a single polymerization function $f(E^x,K_{\phi },{\left(E^{\phi }\right)}^{-1}{K}_x):=f_1/{\tilde{f}}_{{\mu }_{\phi }}^2-4E^x{f}_2/\left({\tilde{f}}_{{\mu }_{\phi }}{\tilde{f}}_{{\mu }_x}\right)$. The polymerization function depends on the combination of ${\left(E^{\phi }\right)}^{-1}{K}_x$ rather than depending on either of them only. We remark that, in the gauge-fixed model with (49), we have

$$\begin{array}{c}\hfill {\{{\mathcal{C}}^{\Delta }\left(x\right),{\mathcal{C}}_x\left(y\right)\}}_D=-{\partial }_x\delta (x-y){\mathcal{C}}^{\Delta }\left(y\right)\end{array}$$

(50)

which confirms that ${\mathcal{C}}^{\Delta }$ has density weight one. Further requiring scale invariance under the transformation

$$\begin{array}{c}\hfill E^x\to {\mu }^2{E}^x,E^{\phi }\to {\mu }^2{E}^{\phi },K_x\to \mu K_x,K_{\phi }\to \mu K_{\phi },\end{array}$$

(51)

leads to the following $C^{\Delta }$,

$$\begin{array}{ccc}\hfill {\mathcal{C}}^{\Delta }\left(x\right)& =& \frac{1}{2G}\left[\frac{E^{\phi }}{\sqrt{E^x}}\left(f\left(\frac{K_{\phi }}{\sqrt{E^x}},\frac{\sqrt{E^x}{K}_x}{E^{\phi }}\right)+{\mathsf{\Gamma }}^2-1\right)+2\sqrt{E^x}{\mathsf{\Gamma }}^{\prime }\right]\left(x\right),\hfill \end{array}$$

(52)

which is compatible with the choice of $\overline{\mu }$-scheme used in [4,28,29,31,37]. We note that there are special cases where one or both $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$ are Dirac observables and constants of motion under the effective dynamics, where $f_{{\mu }_x}$ and $f_{{\mu }_{\phi }}$ can still appear as constants, e.g., similar to the cases in [15].

4. Conclusions

In this article, we demonstrated that we can implement the choice of a given $\overline{\mu }$-scheme dynamically. Such an implementation corresponds to choosing a phase-space dependent function in the argument of a generic polymerization function, which replaces holonomies in effective models. We perform the analysis in homogeneous cosmological models as well as spherical symmetric models, which are $1+1$-dimensional field theories. To construct the models, we extended the usual kinematical phase space by additional canonical pairs, which come along with a set of primary constraints so that the number of physical degrees of freedom does not change.

These additional constraints build a set of second class constraints for which the corresponding gauge-unfixed model can be constructed in the relational formalism. The dynamical implementation of a $\overline{\mu }$-scheme can be achieved in terms of a canonical transformation on the extended phase space that allows a Kuchař decomposition [48,49] of the extended phase space where all constraints related to the dynamical implementation as well as the Dirac observables with respect to these constraints are the elementary-phase space variables.

Furthermore, we showed that, in this extended phase space, we can also understand the choice of a given $\overline{\mu }$-scheme as a gauge fixing, which gives the polymerized Hamiltonian with the given $\overline{\mu }$-scheme in reduced phase space. This gives a possible way from a loop holonomy on a fixed lattice to a polymerization with a $\overline{\mu }$-scheme. Such extension of the phase space can be included in the path integral formulation of full LQG [38] to obtain $\overline{\mu }$-effective dynamics.

In cosmological models, the canonical transformation on the extended phase space and the gauge fixing does not further restrict the choice of gauge-fixing function. In particular, the special case of a constant function that corresponds exactly to the ${\mu }_0$-scheme in LQC is allowed. The $\overline{\mu }$-scheme used in LQC is determined only when we require in addition, scale invariance. In contrast, in black hole models, the requirement of keeping the density weight of the scalar Hamiltonian restricts the choice of the gauge-fixing functions and, hence, the possible $\overline{\mu }$-schemes.

In such case, a constant ${\mu }_0$-scheme is generally not allowed, unless one does not polymerize the densitized connection $K_x$ carrying a density weight of one, e.g., as in the models presented in [6,9,10,16,17]. A special case is if the gauge-fixing functions themselves corresponding to some constant of motion under the effective dynamics, e.g., similar to the model presented in [14].

In this work, we only consider the polymerization of the gravitational variables. The analysis can be generalized to the models with polymerized matter degrees of freedom by introducing additional functions $f_{{\mu }_{\mathrm{matter}}}$. With this, we can go beyond the standard ${\mu }_0$ and $\overline{\mu }$ scheme and allow more general gauge-fixing functions, which depend on all variables containing both matter and geometry. Our results here, in particular, the construction of the canonical transformation on the extended phase space, will work similarly in this more general situation.

This becomes relevant for models, such as those in [21,53] in which the matter sector is also polymerized. In these cases, because the momentum of the matter variables is a quantity of density weight, one could also use it to construct quantities that yield a Hamiltonian with density weight one. Another interesting situation is to extend the analysis to cases beyond spherically symmetric spacetimes. We expect the canonical transformation on the extended phase space and the gauge-fixing method will still work, similar to the spherically symmetric case if the involved gauge-fixing functions depend on the matter and gravitational degrees of freedom only.

In the general case, the requirement to have a Hamiltonian with density weight one may give us more restrictions on the gauge-fixing functions, thus, constraining the allowed $\overline{\mu }$-schemes. In the models studied in this work, the Gauß constraints were already gauge fixed. The gauge fixing of the Gauß constraint was also required in some early attempts to obtain a $\overline{\mu }$-scheme effective dynamics from full LQG [41,42]. It is interesting to generalize the current approach to models that emerge from the LQG theory where the Gauß constraints have not been gauge fixed.

The presence of Gauß constraints may give strong constraints to the possible form of such gauge fixing conditions. Moreover, in this work, we did not consider a polymerization of the diffeomorphism constraints. Such an assumption must be dropped if, for example, we work with models based on a lattice theory on a fixed graph. We expect that work with polymerized diffeomorphism constraints will provide further constraints on admissible $\overline{\mu }$-schemes.