Lattice refining loop quantum cosmology, anisotropic models and stability

TL;DR

This paper introduces a generalized loop quantum cosmology framework with lattice refinement for anisotropic models, analyzing stability and semiclassical behavior, and establishing constraints on refinement models through consistency tests.

Contribution

It presents a new class of lattice-refined loop quantizations for anisotropic models, addressing stability and semiclassical implications, and providing a broader framework for testing loop quantum gravity.

Findings

Lattice refinement affects the stability of difference equations.

Stability conditions constrain possible refinement models.

Quantum corrections are larger than previously estimated in isotropic models.

Abstract

A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.

Equations

$$A_a^i{\tau }_i\mathrm{d}{x}^a$$

$$E_i^a{\tau }^i\frac{\partial }{\partial x^a}$$

$$\mathrm{d}{s}^2=\frac{{\tilde{p}}_a^2+{\tilde{p}}_b^2}{|{\tilde{p}}_c|}\mathrm{d}{x}^2+|{\tilde{p}}_c|\mathrm{d}{\Omega }^2$$

$$e_a^i{\tau }_i\mathrm{d}{x}^a=e_c{\tau }_3\mathrm{d}x+(e_a{\tau }_1+e_b{\tau }_2)\mathrm{d}\vartheta +(-e_b{\tau }_1+e_a{\tau }_2)\sin \vartheta \mathrm{d}\phi$$

$$e_c=\frac{\mathrm{sgn}{\tilde{p}}_c\sqrt{{\tilde{p}}_a^2+{\tilde{p}}_b^2}}{\sqrt{|{\tilde{p}}_c|}},e_b=\frac{\sqrt{|{\tilde{p}}_c|}{\tilde{p}}_b}{\sqrt{{\tilde{p}}_a^2+{\tilde{p}}_b^2}}\mathrm{and}{e}_a=\frac{\sqrt{|{\tilde{p}}_c|}{\tilde{p}}_a}{\sqrt{{\tilde{p}}_a^2+{\tilde{p}}_b^2}}.$$

$$\{\tilde{a},{\tilde{p}}_a\}=\gamma G/L_0,\{\tilde{b},{\tilde{p}}_b\}=\gamma G/L_0,\{\tilde{c},{\tilde{p}}_c\}=2\gamma G/L_0$$

$$\frac{1}{8\pi \gamma G}\int {\mathrm{d}}^{3}x{\dot{A}}_a^i{E}_i^a=\frac{L_0}{2\gamma G}\dot{\tilde{c}}{\tilde{p}}_c+\frac{L_0}{\gamma G}\dot{\tilde{b}}{\tilde{p}}_b+\frac{L_0}{\gamma G}\dot{\tilde{a}}{\tilde{p}}_a$$

$$(b,c):=(\tilde{b},L_o\tilde{c}),(p_b,p_c):=(L_o{\tilde{p}}_b,{\tilde{p}}_c).$$

$$\{b,p_b\}=\gamma G,\{c,p_c\}=2\gamma G.$$

$$h_x^{(\tau )}(A)$$

$$h_{\vartheta }^{(\mu )}(A)$$

$$⟨b,c|\mu ,\tau ⟩=e^{\frac{i}{2}(\mu b+\tau c)}\mu ,\tau \in \mathbb{R}.$$

$${\hat{p}}_b=-i\gamma {\ell }_{\mathrm{P}}^2\frac{\partial }{\partial b},{\hat{p}}_c=-2i\gamma {\ell }_{\mathrm{P}}^2\frac{\partial }{\partial c}$$

$${\hat{p}}_b|\mu ,\tau ⟩=\frac{1}{2}\gamma {\ell }_{\mathrm{P}}^2\mu |\mu ,\tau ⟩,{\hat{p}}_c|\mu ,\tau ⟩=\gamma {\ell }_{\mathrm{P}}^2\tau |\mu ,\tau ⟩,$$

$$H=\frac{1}{{\gamma }^2}\int {\mathrm{d}}^{3}x{ϵ}_{ijk}(-{\underline{F}}_{ab}^k+{\gamma }^2{\Omega }_{ab}^k)\frac{E^{ai}{E}^{bj}}{\sqrt{|\det E|}}$$

$${ϵ}_{ijk}{\tau }^i\frac{E^{aj}{E}^{bk}}{\sqrt{|\det E|}}=-\frac{1}{4\pi \gamma G}\sum _{K\in \{x,\vartheta ,\phi \}}\frac{1}{{\ell }_0^K}{ϵ}^{abc}{\omega }_c^K{h}_K^{(\delta )}\{h_K^{(\delta )-1},V\}$$

$${\underline{F}}_{ab}^i(x){\tau }_i=\frac{{\omega }_a^I{\omega }_b^J}{{\mathcal{A}}_{(IJ)}}(h_{IJ}^{(\delta )}-1)+O((b^2+c^2{)}^{3/2}\sqrt{\mathcal{A}})$$

$$h_{IJ}^{(\delta )}=h_I^{(\delta )}{h}_J^{(\delta )}(h_I^{(\delta )}{)}^{-1}(h_J^{(\delta )}{)}^{-1}$$

$${\hat{H}}^{(\delta )}$$

$${\hat{H}}^{(\delta )}|\mu ,\tau ⟩$$

$${\hat{H}}_{\mathrm{symm}}^{(\delta )}|\mu ,\tau ⟩$$

$$V_{\mu ,\tau }=4\pi \sqrt{|({\hat{p}}_c{)}_{\mu ,\tau }|}({\hat{p}}_b{)}_{\mu ,\tau }=2\pi (\gamma {\ell }_{\mathrm{P}}^2{)}^{3/2}\sqrt{|\tau |}\mu ,$$

$$\frac{{\gamma }^{3/2}{\delta }^3}{\pi {\ell }_{\mathrm{P}}}({\hat{H}}_{\mathrm{symm}}^{(\delta )}|\psi ⟩{)}_{\mu ,\tau }=2\delta (\sqrt{|\tau +2\delta |}+\sqrt{|\tau |})\left({\psi }_{\mu +2\delta ,\tau +2\delta }-{\psi }_{\mu -2\delta ,\tau +2\delta }\right)$$

$$h_x=\exp ({\ell }_0\tilde{c}{\tau }_3)=\exp ({\ell }_{0}c{\tau }_3/L_0)=\exp (c{\tau }_3/{\mathcal{N}}_x)$$

$$h_x=\exp ({\ell }_0^x{L}_0^{-1}c{\tau }_3)=\exp (c{\tau }_3/{\mathcal{N}}_x),h_{\vartheta }=\exp ({\ell }_0^{\vartheta }b{\tau }_2)=\exp (b{\tau }_2/{\mathcal{N}}_{\vartheta }).$$

$$C_{+}(\mu ,\tau )\left({\psi }_{\mu +2\delta {\mathcal{N}}_{\vartheta }(\mu ,\tau {)}^{-1},\tau +2\delta {\mathcal{N}}_x(\mu ,\tau {)}^{-1}}-{\psi }_{\mu -2\delta {\mathcal{N}}_{\vartheta }(\mu ,\tau {)}^{-1},\tau +2\delta {\mathcal{N}}_x(\mu ,\tau {)}^{-1}}\right)$$

$$C_{\pm }(\mu ,\tau )$$

$$C_0(\mu ,\tau )$$

$${\psi }_{\mu +\delta /{\mathcal{N}}_1(\mu )}$$

$$=$$

$$\frac{{\mathcal{N}}_1^{\prime }}{{\mathcal{N}}_1^2}=\frac{q}{\mu {\mathcal{N}}_1(\mu )}=q{\left(\frac{4\pi \gamma {\ell }_{\mathrm{P}}^2}{3p}\right)}^{1+q}$$

$${\tilde{\psi }}^{\prime }=\frac{\mathrm{d}\tilde{\psi }}{\mathrm{d}\tilde{\mu }}=\frac{\mathrm{d}\mu }{\mathrm{d}\tilde{\mu }}\frac{\mathrm{d}\psi }{\mathrm{d}\mu }=\frac{1}{{\mathcal{N}}_1(\mu )}\frac{\mathrm{d}\psi }{\mathrm{d}\mu }=-\frac{i}{2}\frac{1}{{\mathcal{N}}_1(\mu )}\hat{c}\psi$$

Full text

1 Introduction

Loop quantum cosmology [1] was designed to test characteristic effects expected in the full framework of loop quantum gravity [2, 3, 4]. Implementing symmetries at the kinematical quantum level allows explicit treatments of the dynamical equations while preserving basic features such as the discreteness of spatial geometry [5]. (See also [6, 7, 8, 9, 10, 11] for recent work on symmetry reduction in quantum theories.) Indeed, several new, initially surprising results were derived in different applications in cosmology and black hole physics. By now many such models have been studied in detail.

As the relation of dynamics to that of a possible full framework without symmetries is not fully worked out, detailed studies can be used to suggest improvements of the equations for physically viable behavior. Comparing results with full candidates for quantum dynamics can then provide stringent self-consistency tests of the overall framework. It is to be seen if, and how, such alterations of quantization procedures naturally result from a full quantization. The first example of this type related to the stability behavior of solutions to the difference equations of isotropic loop quantum cosmology, which was studied in [12, 13] and was already restrictive for models with non-zero intrinsic curvature. Another limitation, realized early on [14], occurs in the presence of a positive cosmological constant $\Lambda$. In an exact isotropic model, the extrinsic curvature scale is given by $k=\dot{a}=\sqrt{8\pi Ga^{2}\Lambda/3}$ which, due to the factor of $a^{2}$, can be large in a late universe although the local curvature scale $\Lambda$ might be small. Extrinsic curvature plays an important role since in a flat isotropic model it appears in holonomies on which loop quantizations are based in such a way that only $e^{i\alpha k}$ with $\alpha\in{\mathbb{R}}$ can be represented as operators, but not $k$ itself [15]. Large values of $k$ would either require one to use extremely small $\alpha$ in the relevant operators, or imply unexpected deviations from classical behavior. In fact, holonomies as basic objects imply that the Hamiltonian constraint is quantized to a difference rather than differential equation [16] since $k$ in the Hamiltonian constraint (as in the Friedmann equation) is not directly quantized but only exponentials $e^{i\alpha k}$. These are shift operators instead of differential operators. For a large, semiclassical universe a Wheeler–DeWitt wave function should be a good approximation to the basic difference equation of loop quantum cosmology [17] which, in a representation as a function of the momentum $p=a^{2}$ conjugate to $k$, would be oscillating on scales of the order $(a\sqrt{\Lambda})^{-1}$. This scale becomes shorter and shorter in an expanding universe, eventually falling below the discreteness scale of the difference equation of loop quantum cosmology. At such a point, discreteness of spatial geometry would become noticeable in the behavior of the wave function (independently of how physical observables are computed from it) although the universe should be classical.

This does not pose a problem for the general formalism, because it only shows that the specific quantization of the exact isotropic model used reaches its limits. Physically, this can be understood as a consequence of a fixed spatial lattice being used throughout the whole universe evolution. Exponentials $e^{i\alpha k}$ in isotropic models derive from holonomies $h_{e}(A)={\cal P}\exp(\int_{e}A_{a}^{i}\tau_{i}\dot{e}^{a}{\mathrm{d}}t)$ of the Ashtekar connection along spatial curves $e$. All the freedom contained in choosing edges to capture independent degrees of freedom of the full theory reduces, in isotropic models, to the single parameter $\alpha$ which suffices to separate isotropic connections through all functions $e^{i\alpha k}$. The parameter $\alpha$, from the full perspective, is thus related to the edge length used in holonomies. Using a fixed and constant $\alpha$ is analogous to using only edges of a given coordinate length, as they occur, for instance, in a regular lattice. In the presence of a positive cosmological constant, for any $\alpha$ a value of $k$ will then be reached such that $e^{i\alpha k}$ differs strongly from $i\alpha k$. From the lattice perspective, this means that the local curvature radius becomes comparable to or smaller than the fixed lattice scale corresponding to $\alpha$. Such a fixed lattice ceases to be able to support all small-scale oscillations relevant for a semiclassical geometry.

This is not problematic if it occurs in a quantum regime where dynamics is indeed expected to differ from the classical one, but it poses a problem in semiclassical regimes. A better treatment has to refer to changing lattices, which is not easy to implement in a straightforward quantization of purely homogeneous models. In a dynamical equation closer to what is expected from the full framework, lattice refinements would take place during the evolution since full Hamiltonian constraint operators generally create new vertices of a lattice state in addition to changing their edge labels [18, 19]. While $k$ increases with increasing volume, the corresponding $\alpha$ decreases since the lattice is being refined all the time. For a suitable lattice refinement, the increase in $k$ can be balanced by the decrease of $\alpha$ such that $\alpha k$ stays small and semiclassical behavior is realized for any macroscopic volume even with $\Lambda>0$. This provides an interesting relation between the fundamental Hamiltonian, which is responsible for the lattice refinement, and semiclassical properties of models. Testing whether an appropriate balance between increasing $k$ and lattice refinements can be reached generically can thus provide stringent tests on the fundamental dynamics even without using a precise full Hamiltonian constraint operator.

This feature of lattice refinements was not mimicked in the first formulations of loop quantum cosmology [20, 21, 14, 22, 15] since the main focus was to understand small-volume effects such as classical singularities [23, 24]. In this context, lattice refinements appear irrelevant because only a few action steps of the Hamiltonian, rather than long evolution, are sufficient to probe a singularity. By now, perturbative regimes around isotropic models have been formulated in loop quantum cosmology which are inhomogeneous and thus must take into account lattice states and, at least at an effective level, lattice refinements [9]. One special version, corresponding to lattices with a number of vertices growing linearly with volume in a specific way referring to the area operator, has been studied in detail in isotropic models with a free, massless scalar [25]. Although the complicated relation to a full, graph-changing Hamiltonian constraint is still not fully formulated, such models allow crucial tests of the local dynamics.

While isotropic models can easily be understood in terms of wave functions on a 1-dimensional discrete minisuperspace in terms of oscillation lengths [26], anisotropic models with higher-dimensional minisuperspaces can be more subtle. In such models, limitations similar to that of a cosmological constant have been observed as possible instabilities of solutions in classical regions or the lack of a sufficient number of semiclassical states [27, 28, 29]. For the partial difference equations of anisotropic models in loop quantum cosmology, stability issues can be much more severe than in isotropic models and thus lead to further consistency tests which might help to restrict possible quantization freedom (see, e.g., [30]). In this paper we therefore introduce the general setting of anisotropic models taking into account lattice refinements of Hamiltonian constraint operators, focusing mainly on the anisotropic model which corresponds to the Schwarzschild interior. As we will see, the type of difference equations in general changes since they can become non-equidistant. This leads to new mathematical problems which we address here briefly, leaving further analysis for future work. The examples presented here already show that one can distinguish different refinement models by their stability properties. The refinement model corresponding to [25] turns out to give unstable evolution of the Schwarzschild interior, while a new version, whose vertex number also grows linearly with volume, is stable. Compared to isotropic models which are sensitive only to how the vertex number of a state changes with volume, anisotropic models allow one to test much more detailed properties.

An appendix discusses subtleties in how homogeneous models faithfully represent inhomogeneous states, mainly regarding the magnitude of corrections arising from quantizations of inverse metric components which often plays a large role in cosmological applications.

2 Difference equation for the Schwarzschild interior with varying discreteness scale

Basic variables of a loop quantization are holonomies along lattice links and fluxes over transversal surfaces. For the Schwarzschild interior [31], the connection used for holonomies and the densitized triad used for fluxes take the form

[TABLE]

Coordinates $(x,\vartheta,\varphi)$ are adapted to the symmetry, with polar angles $\vartheta$ and $\varphi$ along orbits of the rotational symmetry subgroup, and $\tau_{j}=-\frac{i}{2}\sigma_{j}$ in terms of Pauli matrices. Spatial geometry is determined by the spatial line element, which in terms of the densitized triad components is

[TABLE]

obtained from $q^{ab}=E^{a}_{i}E^{b}_{i}/|\det E^{c}_{j}|$. We will also use the co-triad $e_{a}^{i}$, i.e. the inverse of $e^{a}_{i}=E^{a}_{i}/\sqrt{|\det E^{b}_{j}|}$,

[TABLE]

with components

[TABLE]

The phase space is spanned by the spatial constants $(\tilde{a},\tilde{b},\tilde{c},\tilde{p}_{a},\tilde{p}_{b},\tilde{p}_{c})\in{\mathbb{R}}^{6}$ with non-vanishing Poisson brackets

[TABLE]

where $G$ is the gravitational constant and $\gamma$ the Barbero–Immirzi parameter [32, 33]. Moreover, $L_{0}$ is the size of a coordinate box along $x$ used in integrating out the fields in

[TABLE]

to derive the symplectic structure. The SU(2)-gauge transformations rotating a general triad are partially fixed to U(1) by demanding the $x$-component of $E^{a}_{i}$ to point in the internal $\tau_{3}$-direction in (2). The U(1)-gauge freedom allows one to set $\tilde{a}=0=\tilde{p}_{a}$, still leaving a discrete residual gauge freedom $(\tilde{b},\tilde{p}_{b})\mapsto(-\tilde{b},-\tilde{p}_{b})$. The remaining variables can be rescaled by

[TABLE]

to make the canonical structure $L_{0}$-independent:

[TABLE]

This rescaling is suggested naturally by holonomies, as written below, and fluxes which are considered the basic objects in loop quantizations.

To express the elementary variables through holonomies, which unlike connection components will be promoted to operators, it suffices to choose curves along the $x$-direction of coordinate length $\tau L_{0}$ and along $\vartheta$ of coordinate length $\mu$ since this captures all information in the two connection components,

[TABLE]

The quantum Hilbert space is then based on cylindrical states depending on the connection through countably many holonomies, which can always be written as almost periodic functions $f(b,c)=\sum_{\mu,\tau}\,f_{\mu,\tau}\exp{\frac{i}{2}\,(\mu b+\tau c)}$ of two variables. These form the set of functions on the double product of the Bohr compactification of the real line, which is a compact Abelian group. Its Haar measure defines the inner product of the (non-separable) Hilbert space, in which states

[TABLE]

form an orthonormal basis. Holonomies simply act by multiplication on these states, while densitized triad components become derivative operators

[TABLE]

using the Planck length $\ell_{\mathrm{P}}=\sqrt{G\hbar}$. They act as

[TABLE]

immediately showing their eigenvalues.

To formulate the dynamical equation, one has to quantize the Hamiltonian constraint

[TABLE]

where $\Omega_{ab}^{k}\tau_{k}{\mathrm{d}}x^{a}\wedge{\mathrm{d}}x^{b}=-\sin\vartheta\tau_{3}{\mathrm{d}}\vartheta\wedge{\mathrm{d}}\varphi$ is the intrinsic curvature of 2-spheres, while $\underline{F}_{ab}^{k}$ is the curvature computed from $A_{a}^{i}$ ignoring the spin connection term $\sin\vartheta\tau_{3}{\mathrm{d}}\varphi$. Following standard procedures a Hamiltonian constraint operator can be expressed in the basic operators. First, one replaces the inverse determinant of $E^{a}_{i}$ by a Poisson bracket, following [19],

[TABLE]

with edge lengths $\ell_{0}^{x}=\delta L_{0}$ and $\ell_{0}^{\vartheta/\varphi}=\delta$, and left-invariant 1-forms $\omega_{c}^{K}$ on the symmetry group manifold. For curvature components $\underline{F}_{ab}^{k}$ one uses a holonomy around a closed loop

[TABLE]

with

[TABLE]

and ${\cal A}_{IJ}$ being the coordinate area of the loop, using the corresponding combinations of $\ell_{0}^{I}$. In these expressions, a parameter $\delta$ has been chosen which specifies the length of edges with respect to the background geometry provided by the symmetry group. Putting all factors together and replacing Poisson brackets by commutators, one has

[TABLE]

which acts as

[TABLE]

on basis states. This operator can be ordered symmetrically, defining $\hat{H}_{\rm symm}^{(\delta)}:=\frac{1}{2}(\hat{H}^{(\delta)}+\hat{H}^{(\delta)\dagger})$, whose action is111Note that the first factor of 2 in the next-to-last line was missing in [31] and analogous places in subsequent formulas. This turns out to be crucial for the stability analysis below. In particular, with the corrected coefficient the quantization of the Schwarzschild interior in [31] is unstable for all values of $\gamma$. Possible restrictions on $\gamma$, as suggested in [30] based on a difference equation with the wrong coefficient, then do not follow easily but could be obtained from a more detailed analysis.

[TABLE]

Transforming this operator to the triad representation obtained as coefficients of a wave function $|\psi\rangle=\sum_{\mu,\tau}\,\psi_{\mu,\tau}|\mu,\tau\rangle$ in the triad eigenbasis and using the volume eigenvalues

[TABLE]

a difference equation

[TABLE]

results for physical states. (For small $\mu$ the equation has to be specialized further due to the remaining gauge freedom; see [31]. This is not relevant for our purposes.)

2.1 Relation to fixed lattices

Although there are no spatial lattices appearing in the exactly homogeneous context followed here, the construction of the Hamiltonian constraint mimics that of the full theory. States are then associated with spatial lattices, and holonomies refer to embedded edges and loops. The parameter $\delta$ is the remnant of the loop size (in coordinates) used to act with holonomies on a spatial lattice. As one can see, this parameter is important for the resulting difference equation, determining its step-size. The above construction, using a constant $\delta$, can be seen as corresponding to a lattice chosen once and for all such that the loop size is not being adjusted even while the total volume increases. As described in the introduction, this ignores the possible creation of new lattice vertices and links, and can be too rigid in certain semiclassical regimes.

To express this clearly, we now construct holonomies which are not simply along a single edge of a certain length $\delta$, but which are understood as holonomies along lattice links. We keep our coordinate box of size $L_{0}$ in the $x$-direction as well as the edge length $\ell_{0}$. If this is a link in a uniform lattice, there are ${\cal N}_{x}=L_{0}/\ell_{0}$ lattice links in this direction, and a link holonomy appears in the form

[TABLE]

when computed along whole lattice edges. Thus, a constant coefficient $1/{\cal N}_{x}$ in holonomies corresponds to a fixed lattice whose number of vertices does not change when the volume increases. Lattice refinements of an inhomogeneous lattice state, on the other hand, can be mimicked by a parameter ${\cal N}_{x}$ which depends on the phase space variables, most importantly the triad components. If this is carried through, as we will see explicitly below, the step-size of the resulting difference equation is not constant in the triad variables anymore.

2.2 Lattice refinements

Let us now assume that we have a lattice with ${\cal N}$ vertices in a form adapted to the symmetry, i.e. there are ${\cal N}_{x}$ vertices along the $x$-direction (whose triad component $p_{c}$ gives rise to the label $\tau$) and ${\cal N}_{\vartheta}^{2}$ vertices in spherical orbits of the symmetry group (whose triad component $p_{b}$ gives rise to the label $\mu$). Thus, ${\cal N}={\cal N}_{x}{\cal N}_{\vartheta}^{2}$.

Since holonomies in such a lattice setting are computed along single links, rather than through all of space (or the whole cell of size $L_{0}$), basic ones are $h_{x}=\exp(\ell_{0}^{x}\tilde{c}\tau_{3})$ and $h_{\vartheta}=\exp(\ell_{0}^{\vartheta}\tilde{b}\tau_{2})$, denoting the edge lengths by $\ell_{0}^{I}$ and keeping them independent of each other in this anisotropic setting. Edge lengths are related to the number of vertices in each direction by $\ell_{0}^{x}=L_{0}/{\cal N}_{x}$ and $\ell_{0}^{\vartheta}=1/{\cal N}_{\vartheta}$. With the rescaled connection components $c=L_{0}\tilde{c}$ and $b=\tilde{b}$ we have basic holonomies

[TABLE]

Using this in the Hamiltonian constraint operator then gives a difference equation whose step-sizes are $1/{\cal N}_{I}$.

So far, we only reinterpreted $\delta$ in terms of vertex numbers. We now turn our attention to solutions to the Hamiltonian constraint which, in the full theory, usually changes the lattice by adding new edges and vertices while triad eigenvalues increase. For larger $\mu$ and $\tau$, the Hamiltonian constraint thus acts on a finer lattice than for small values, and the parameter ${\cal N}$ for holonomies appearing in the constraint operator is not constant on phase space but triad dependent. Due to the irregular nature of lattices with newly created vertices such a refinement procedure is difficult to construct explicitly. But it is already insightful to use an effective implementation, using the derivation of the Hamiltonian constraint for a fixed lattice, but assuming the vertex number ${\cal N}(\mu,\tau)$ to be phase space dependent. Moreover, we include a parameter $\delta$ as before, which now takes a value $0<\delta<1$ and arises because a graph changing Hamiltonian does not use whole lattice edges but only a fraction, given by $\delta$.222A precise value can be determined only if a precise implementation of the symmetry for a fixed full constraint operator is developed. Currently, both the symmetry reduction for composite operators and a unique full constraint operator are lacking to complete this program and we have to work with $\delta$ as a free parameter. This parameter is sometimes related to the lowest non-zero eigenvalue of the full area operator [15, 25]. From the inhomogeneous perspective of lattice states used here, however, there is no indication for such a relation. Effectively assuming in this way that the lattice size is growing through the basic action of the Hamiltonian constraint, we will obtain a difference equation whose step-size $\delta/{\cal N}$ is not constant in the original triad variables.

For the Schwarzschild interior, we have step sizes $\delta/{\cal N}_{\vartheta}$ for $\mu$ and $\delta/{\cal N}_{x}$ for $\tau$. Going through the same procedure as before, we end up with an operator containing flux-dependent holonomies instead of basic ones, e.g., ${\cal N}_{x}(\mu,\tau)h_{x}={\cal N}_{x}(\mu,\tau)\exp(c\tau_{3}/{\cal N}_{x}(\mu,\tau))$ which reduces to an ${\cal N}_{x}$-independent connection component $c$ in regimes where curvature is small. Keeping track of all prefactors and holonomies in the commutator as well as the closed loop, one obtains the difference equation

[TABLE]

with

[TABLE]

(A total factor ${\cal N}_{x}{\cal N}_{\vartheta}^{2}$ for the number of vertices drops out because the right hand side is zero in vacuum, but would multiply the left hand side in the presence of a matter term.)

3 Specific refinement models

For further analysis one has to make additional assumptions on how exactly the lattice spacing is changing with changing scales $\mu$ and $\tau$. To fix this in general, one would have to use a full Hamiltonian constraint and determine how its action balances the creation of new vertices with increasing volume. Instead of doing this, we will focus here on two geometrically motivated cases. Technically simplest is a quantization where the number of vertices in a given direction is proportional to the geometrical area of a transversal surface. Moreover, the appearance of transversal surface areas is suggested by the action of the full Hamiltonian constraint which, when acting with an edge holonomy, creates a new vertex along this edge (changing ${\cal N}_{I}$ for this direction) and changes the spin of the edge (changing the area of a transversal surface). It also agrees with [25, 34], although the motivation in those papers, proposing to use geometrical areas rather than coordinate areas ${\cal A}_{IJ}$ in (16), is different.

Geometrically more intuitive is the case where the number of vertices in a given direction is proportional to the geometrical extension of this direction.333This behavior is introduced independently in [35] where “effective” equations, obtained by replacing connection components in the classical constraint by sines and cosines of such components according to how they occur in the quantized constraint, are analyzed for the Schwarzschild interior. The results are complementary to and compatible with our stability analysis of the corresponding difference equations below. We thank Kevin Vandersloot for discussions on this issue. The resulting difference equation will be more difficult to deal with due to its non-constant step-size, but naturally gives rise to Misner-type variables. This case will also be seen to have improved stability properties compared to the first one using areas. In both cases, ${\cal N}\propto V$ is assumed, i.e. the lattice size increases proportionally to volume. This is not necessary in general, and we choose these two cases mainly for illustrative purposes. In fact, constant ${\cal N}$ as in [15] and ${\cal N}\propto V$ first used in [25] are two limiting cases from the full point of view, the first one without creating new vertices and the second one without changing spin labels along edges since local lattice volumes $V/{\cal N}$ remain constant. In general, both spin changes and the creation of new vertices happen when acting with a Hamiltonian constraint operator. Thus, one expects ${\cal N}\propto V^{\alpha}$ with some $0<\alpha<1$ to be determined by a detailed analysis of the full constraint and its reduction to a homogeneous model. Even assuming a certain behavior of ${\cal N}(V)$ without analyzing the relation to a full constraint leaves a large field to be explored, which can give valuable consistency checks. We will not do this systematically in this paper but rather discuss a mathematical issue that arises in any such case: initially, one has to deal with difference equations of non-constant step-size which can be treated either directly or by tranforming a non-equidistant difference equation to an equidistant one. We first illustrate this for ordinary difference equations since partial ones, as they arise in anisotropic models, can often be reduced to this case.

3.1 Ordinary difference equations of varying step-size

Let us assume that we have an ordinary difference equation for a function $\psi_{\mu}$, which appears in the equation with $\mu$-dependent increments $\psi_{\mu+\delta{\cal N}_{1}(\mu)^{-1}}$. To transform this to a fixed step-size, we introduce a new variable $\tilde{\mu}(\mu)$ such that $\tilde{\mu}(\mu+\delta/{\cal N}_{1}(\mu))=\tilde{\mu}(\mu)+\delta\tilde{\mu}^{\prime}/{\cal N}_{1}(\mu)+O(\delta^{2})$ has a constant linear term in $\delta$. (For the isotropic equation, ${\cal N}_{1}$ is the vertex number only in one direction. The total number of vertices in a 3-dimensional lattice is given by ${\cal N}={\cal N}_{1}^{3}$.) This is obviously satisfied if we choose $\tilde{\mu}(\mu):=\int^{\mu}{\cal N}_{1}(\nu){\mathrm{d}}\nu$. We then have

[TABLE]

where ${\cal N}_{1}^{(i)}$ denotes the $i$-th derivative of ${\cal N}_{1}$. Thus, up to terms of order at least $\delta^{2}$ the new equation will be of constant step-size for the function $\tilde{\psi}_{\tilde{\mu}}:=\psi_{\mu(\tilde{\mu})}$. (The derivative $\tilde{\psi}^{\prime}$ by $\tilde{\mu}$ may not be defined for any solution to the difference equation. We write it in this form since such terms will be discussed below in the context of a continuum or semiclassical limit where derivatives would exist.)

It is easy to see that, for refining lattices, the additional terms containing derivatives of the wave function are of higher order in $\hbar$ and thus correspond to quantum corrections. For ${\cal N}_{1}(\mu)\propto\mu^{q}$ as a positive power of $\mu$, which is the expected case from lattice refinements related to the increase in volume, we have

[TABLE]

relating $\mu$ to an isotropic triad component $p=4\pi\gamma\ell_{\mathrm{P}}^{2}\mu/3$ as it occurs in isotropic loop quantum gravity [14]. Moreover,

[TABLE]

in terms of a curvature operator $\hat{c}=8\pi i\gamma G\hbar/3{\mathrm{d}}/{\mathrm{d}}p=2i{\mathrm{d}}/{\mathrm{d}}\mu$ which exists in a continuum limit [17]. Thus,

[TABLE]

With $q$ positive (or just larger than $-1/2$) for a refining lattice, there is a positive power of $\hbar$, showing that additional terms arising in the transformation are quantum corrections.

This has two important implications. First, it shows that the correct classical limit is obtained if lattices are indeed refined, rather than coarsened, since $q$ is restricted for corrections to appear in positive powers of $\hbar$. In anisotropic models, as we will see, the behavior is more complicated due to the presence of several independent variables. An analysis of the semiclassical limit can then put strong restrictions on the behavior of lattices. Secondly, we can implicitly define a factor ordering of the original constraint giving rise to the non-equidistant difference equation by declaring that all quantum correction terms arising in the transformation above should cancel out with factor ordering terms. We then obtain a strictly equidistant equation in the new variable $\tilde{\mu}$. For example, a function ${\cal N}_{1}(\mu)=\sqrt{|\mu|}$ gives $\tilde{\mu}\propto|\mu|^{3/2}$ such that the transformed difference equation will be equidistant in volume rather than the densitized triad component. For this special case, factor orderings giving rise to a precisely equidistant difference equation have been constructed explicitly in [25, 34].

3.2 Number of vertices proportional to transversal area

A simple difference equation results if the number of vertices is proportional to the transversal area in any direction.444Since this refers to the area, it is the case which agrees with the motivation of [25, 34]. In the $x$-direction we have transversal surfaces given by symmetry orbits of area $p_{c}$, using the line element (3), and thus ${\cal N}_{x}\propto\tau$. Transversal surfaces for an angular direction are spanned by the $x$- and one angular direction whose area is $p_{b}$, giving ${\cal N}_{\vartheta}\propto\mu$. Each minisuperspace direction has a step-size which is not constant but independent of the other dimension. Moreover, due to the simple form one can transform the equation to constant step-size by using independent variables $\tau^{2}$ and $\mu^{2}$ instead of $\tau$ and $\mu$. Illustrating the general procedure given before, a function $\tilde{\psi}_{\tau^{2},\mu^{2}}$ acquires constant shifts under the basic steps,

[TABLE]

up to terms which can be ignored for large $\tau$ and $\mu$. This is sufficient for a straightforward analysis in asymptotic regimes. Moreover, higher order terms in the above equation come with higher derivatives of the wave function in the form

[TABLE]

since $q=1$ compared to the discussion in Sec. 3.1. Due to the extra factors of $\hbar$ (or even higher powers in further terms in the Taylor expansion) any additional term adding to the constant shift of $\tilde{\psi}_{\tau^{2},\mu^{2}}$ can be attributed to quantum corrections in a semiclassical limit. Accordingly, such terms can be avoided altogether by a judicious choice of the initial factor ordering of operators.

3.3 Number of vertices proportional to extension

Geometrically more intuitive, and as we will see below dynamically more stable, is the case in which the number of vertices in each direction is proportional to the extension of that direction measured with the triad itself. This gives ${\cal N}_{\vartheta}\propto\sqrt{|\tau|}$ and ${\cal N}_{x}\propto\mu/\sqrt{|\tau|}$, using the classical co-triad (4). (One need not worry about the inverse $\tau$ since the effective treatment of lattice refinements pursued here is not valid close to a classical singularity where an already small lattice with a few vertices changes. Singularities in general can only be discussed by a direct analysis of the resulting difference operators. Since only a few recurrence steps are necessary to probe the scheme around a classical singularity, equidistant difference operators are not essential in this regime. They are more useful in semiclassical regimes where one aims to probe long evolution times as in the examples below. Similar remarks apply to the horizon at $\mu=0$ which, although a classical region for large mass parameters, presents a boundary to the homogeneous model used for the Schwarzschild interior.) The behavior is thus more complicated than in the first case since the step size of any of the two independent variables depends on the other variable, too. First, it is easy to see, as before with quadratic variables, that the volume label $\omega=\mu\sqrt{|\tau|}$ changes (approximately) equidistantly with each iteration step which is not equidistant for the basic variables $\mu$ and $\tau$. But it is impossible to find a second, independent quantity which does so, too. In fact, such a quantity $f(\mu,\tau)$ would have to solve two partial differential equations in order to ensure that

[TABLE]

changes only by a constant independent of $\tau$ and $\mu$. This implies $\partial_{\mu}f(\mu,\tau)\propto\sqrt{|\tau|}$ and $\partial_{\tau}f(\mu,\tau)\propto\mu/\sqrt{|\tau|}$ whose only solution is $f({\mu,\tau})\propto\mu\sqrt{|\tau|}$ which is the volume $\omega$.

We thus have to deal with non-equidistant partial difference equations in this case which in general can be complicated. A possible procedure to avoid this is to split the iteration in two steps since an ordinary difference equation can always be made equidistant as above (cancelling quantum corrections by re-ordering). We first transform $\tau$ to the volume variable $\omega$ which gives, up to quantum corrections, constant iteration steps for this variable. With the second variable still present, a higher order difference equation

[TABLE]

results with

[TABLE]

derived from the original coefficients (24). The structure of this difference equation is quite different from the original one: not only is it of higher order, but now only one value of the wave function appears at each level of $\omega$, rather than combinations of values at different values of $\mu$. Note also that only the coefficient of the unshifted $\psi_{\mu,\omega}$ depends on $\mu$. This form of the difference equation is, however, a consequence of the additional rotational symmetry and is not realized in this form for fully anisotropic Bianchi models as we will see below.

Proceeding with this specific case, we have to look at wave functions evaluated at shifted positions $\mu(1+m\delta/\omega)$ with integer $m$. At fixed $\omega=\omega_{0}$, we are thus evaluating the wave function at values of $\mu$ multiplied with a constant, instead of being shifted by a constant as in an equidistant difference equation. This suggests to use the logarithm of $\mu$ instead of $\mu$ itself as an independent variable, which is indeed the result of the general procedure. After having transformed from $\tau$ to $\omega$ already, we have to use $\tau$ as a function of $\mu$ and $\omega$ in the vertex number ${\cal N}_{\vartheta}$, which is $\tau(\mu,\omega)=(\omega/\mu)^{2}$ after using $\omega=\mu\sqrt{\tau}$. Thus, ${\cal N}_{\vartheta}(\mu,\tau(\mu,\omega))=\sqrt{\tau(\mu,\omega)}=\omega/\mu$ now is not a positive power of the independent variable $\mu$ and we will have to be more careful in the interpretation of correction terms after performing the transformation. (The lattice is coarsened with increasing anisotropy at constant volume.) Naively applying the results of Sec. 3.1 to $q=-1$ would suggest that corrections come with inverse powers of $\hbar$ which would certainly be damaging for the correct classical limit. However, the factors change due to the presence of the additional variable $\omega_{0}$ even though it is treated as a constant. We have ${\cal N}_{\vartheta}^{\prime}/{\cal N}_{\vartheta}^{2}=-1/\omega_{0}=-(\gamma\ell_{\mathrm{P}}^{2}/2)^{3/2}/V_{0}$ in terms of the dimensionful volume $V$, while it would just be a constant $-1$ without the presence of $\omega$. The additional factor of $\hbar^{3/2}$ ensures that corrections come with positive powers of $\hbar$ for the correct classical limit to be realized.

For any $\omega_{0}$, we thus transform $\tilde{\psi}_{\mu(1+m\delta/\omega_{0})}$ to equidistant form by using $\tilde{\tilde{\psi}}_{\tilde{\mu}}=\tilde{\psi}_{\mu(\tilde{\mu})}$ with $\tilde{\mu}(\mu)=\log\mu$. This transformation is possible since the second label $\omega_{0}$ is now treated as a constant, rather than an independent variable of a partial difference equation. (Recall that for the type of difference equation discussed here there is only one variable, the volume, which is equidistant under all of the original discrete steps.) Despite of negative powers of some variables in the vertex numbers, we have the correct classical limit in the presence of $\omega$. As before, the transformation is exact up to higher order terms which are quantum and higher order curvature corrections. Defining the original constraint operator ordering implicitly by the requirement that all those terms are cancelled allows us to work with an equidistant difference equation.

3.4 Bianchi models

As mentioned before, the transformed difference equation does not become higher order for fully anisotropic Bianchi models. In this case, we have three independent flux labels $\mu_{I}$, $I=1,2,3$, and vertex numbers ${\cal N}_{I}$. Using vertex numbers proportional to the spatial extensions for each direction gives ${\cal N}_{1}=\sqrt{\mu_{2}\mu_{3}/\mu_{1}}$, ${\cal N}_{2}=\sqrt{\mu_{1}\mu_{3}/\mu_{2}}$ and ${\cal N}_{3}=\sqrt{\mu_{1}\mu_{2}/\mu_{3}}$. As in the difference equation for the Schwarzschild interior, the difference equation for Bianchi models [22] uses values of the wave function of the form $\psi_{\mu_{1}+2\delta/{\cal N}_{1},\mu_{2}+2\delta/{\cal N}_{2},\mu_{3}}$. One can again see easily that the volume $\omega=\sqrt{|\mu_{1}\mu_{2}\mu_{3}|}$ behaves equidistantly under the increments,

[TABLE]

The leading order term of the difference equation in $\omega$ results from a combination

[TABLE]

where we used $1/{\cal N}_{1}=\sqrt{\mu_{1}/\mu_{2}\mu_{3}}=\mu_{1}/\omega$ and defined the operator $\hat{C}_{+}$ acting on the dependence of $\psi$ on $\mu_{1}$ and $\mu_{2}$. Thus, unlike for the Schwarzschild interior the difference equation does not become higher order in $\omega$, and the highest order term does have a difference operator coefficient in the remaining independent variables.

The recurrence proceeds as follows: We have a partial difference equation of the form

[TABLE]

with difference operators $\hat{C}_{\pm}$ and $\hat{C}_{0}$ acting on the dependence on $\mu_{1}$ and $\mu_{2}$. In terms of initial data at two slices of $\omega$ we can solve recursively for $\hat{C}_{0}\tilde{\psi}_{\omega}(\mu_{1},\mu_{2})+\hat{C}_{-}\tilde{\psi}_{\omega-2\delta}(\mu_{1},\mu_{2})=:\phi(\mu_{1},\mu_{2})$ and then, in each $\omega$-step, use boundary conditions to solve the ordinary difference equation

[TABLE]

Although the operator $\hat{C}_{+}$ itself is not equidistant, this remaining ordinary difference equation can be transformed to an equidistant one by transforming $\mu_{1}$ and $\mu_{2}$ as in Sec. 3.1 (using that $\omega$ is constant and fixed for this equation at any recursion step). With $\mu_{3}(\mu_{1},\mu_{2},\omega)=\omega^{2}/\mu_{1}\mu_{2}$, we have lattice spacings ${\cal N}_{1}(\mu_{1},\mu_{2},\omega)=\omega/\mu_{1}$ and ${\cal N}_{2}(\mu_{1},\mu_{2},\omega)=\omega/\mu_{2}$ in terms of $\omega$ which are already independent of each other. The two remaining variables $\mu_{1}$ and $\mu_{2}$ are thus transformed to equidistant ones by taking their logarithms as encountered before.

Note the resemblance of the new variables, volume and two logarithms a metric components at constant volume, to Misner variables [36]. This observation may be of interest in comparisons with Wheeler–DeWitt quantizations where Misner variables have often been used, making the Wheeler–DeWitt equation hyperbolic.

4 Application: Stability of the Schwarzschild interior

Now that we have several possibilities for the lattice spacings, we consider their effect on the solutions of the Hamiltonian constraint. In particular, these solutions may have undesirable properties reminiscent of numerical instabilities, as it was indeed noticed for the original quantization of the Schwarzschild interior in [28]. Also problems in the presence of a positive cosmological constant, described in the introduction, are of this type. Recall that when one wishes to solve an ordinary differential equation, for example, there are various discrete schemes that ensure errors do not propagate as the number of time steps increases. Here we are in the opposite situation – instead of having the freedom to pick the discrete version of a continuous equation, the discrete equation itself is what is fundamental. Thus, like a badly chosen numerical recipe, some choices of the functions $N_{\tau}$ and $N_{\vartheta}$ in the constraint equation may quickly lead to solutions that are out of control, and increase without bound. To test for this, we will use a von Neumann stability analysis [28] on the possible recursion relations. The essential idea is to treat one of the relation parameters as an evolution parameter, and decompose the rest in terms of orthogonal functions, representing “spatial” modes of the solution. This will give rise to a matrix that defines the evolution of the solution; if the matrix eigenvalues are greater than unity for a particular mode, that mode is unstable. In particular, a relation $\sum_{k=-M}^{M}a_{n+k}\psi_{n+k}=0$ is equivalent to a vector equation of the form ${\vec{v}}_{n}=Q(n){\vec{v}}_{n-1}$, where the column vector ${\vec{v}}_{n}=(\psi_{n+M},\psi_{n+M-1},\cdots,\psi_{n-M+1})^{T}$. The evolution of an eigenvector ${\vec{w}}$ of the matrix $Q(n)$ is given by ${\vec{w}}_{n}=\lambda_{w}{\vec{w}}_{n-1}$. Thus, when the size of the corresponding eigenvalue $|\lambda_{w}|>1$, the values in the sequence associated to ${\vec{w}}$ will grow as well.

With this in mind, we consider the choices of $N_{x}$ and $N_{\vartheta}$ discussed previously, starting with the case $N_{x}=\tau$ and $N_{\vartheta}=\mu$. In the large $\mu,\tau$ limit for this choice, the coefficients of the Hamiltonian constraint become

[TABLE]

In the asymptotic limit, the coefficients of the $\psi_{\mu\pm 2\delta/\mu,\tau}$ and $\psi_{\mu,\tau}$ terms go to $C_{0}(\mu,\tau)\mu$. As we saw in Section 3.2, we can choose a different set of variables in which the step sizes are constant (up to ordering of the operators). Plugging these asymptotic values into the Hamiltonian constraint, and changing variables to ${\tilde{\mu}}=\mu^{2}/2$ and ${\tilde{\tau}}=\tau^{2}/2$ gives

[TABLE]

Because all the step sizes now are constants depending on $\delta$, we define new parameters $m,n$ such that ${\tilde{\mu}}=2m\delta$ and ${\tilde{\tau}}=2n\delta$. Using $m$ as our evolution parameter and $n$ as the “spatial” direction, we decompose the sequence as $\psi_{2m\delta,2n\delta}=u_{m}\exp(in\omega)$. With this new function, the recursion relation is written as

[TABLE]

This is equivalent to the vector equation

[TABLE]

The eigenvalues of the matrix $Q$ are

[TABLE]

When the discriminant $16n^{2}-m^{2}\sin^{2}\theta\geq 0$, then $|\lambda|=1$, and the solution is stable; however, there are unstable modes when $16n^{2}-m^{2}\sin^{2}\theta<0$. The most unstable mode corresponds to the choice $\sin\theta=1$, giving instabilities in terms of the original variables when $\mu>2\tau$. In this regime, all solutions behave exponentially rather than oscillating. This region includes parts of the classical solutions for the Schwarzschild interior even for values of $\mu$ and $\tau$ for which one expects classical behavior to be valid. The presence of instabilities implies, irrespective of the physical inner product, that quantum solutions in those regions cannot be wave packets following the classical trajectory, and the correct classical limit is not guaranteed for this quantization, which is analogous to that introduced in [25, 34].

The situation is different when we consider the choices $N_{x}=\sqrt{|\tau|}$ and $N_{\vartheta}=\mu/\sqrt{|\tau|}$, where we will find a lack of instability. There is no choice of variables that allows us to asymptotically approach a constant spacing recursion relation, because of the mixing of the $\mu$ and $\tau$ variables in the step size functions. Thus, we will make the assumption that in the large $\mu,\tau$ limit, the solution does not change much under step sizes $\delta N_{x}^{-1}$ and $\delta N_{\vartheta}^{-1}$. To see how this affects the resulting stability of the solutions, we will look at a simpler example first. If we start with the Fibonacci relation $R_{\tau}\equiv\psi_{\tau+1}-\psi_{\tau}-\psi_{\tau-1}=0$, then the two independent solutions are of the form $\psi_{\tau}=\kappa^{\tau}$, where $\kappa$ is either the golden ratio $\phi=(1+\sqrt{5})/2$ or else $-\phi^{-1}$. Only the latter solution meets the criterion for stability, since $|\phi|>1$. When we change this relation to

[TABLE]

with $n\neq 1$, the situation changes – only one of the two solutions outlined above will solve the relation asymptotically. In particular, when we examine the error ${\tilde{R}}_{\tau}$ we get when we plug $\kappa^{\tau}$ into the altered relation (32), i.e.

[TABLE]

the error is proportional to $\psi_{\tau}$ itself. As $\tau\to\infty$, therefore, the error for the $\kappa=\phi$ solution grows without bound, while that of $\kappa=-\phi^{-1}$ goes to zero. Thus, we see in this situation a relation between the stability and the asymptotic behavior of a solution.

Returning to the Schwarzschild relation, in the large $\mu,\tau$ limit the coefficient functions of the recursion relation are to leading order

[TABLE]

In turn, the relation itself becomes

[TABLE]

From this point on, we assume that we have a solution to this relation which does not vary greatly when, for example, $\mu$ is changed by $\pm 2\delta/\sqrt{\mu}$, and similarly for $\tau$. Both $N_{x}$ and $N_{\vartheta}$ are constant to first order in shifts $\mu\pm 2\delta N_{x}^{-1}$ and similarly for $\tau$, in the asymptotic limit. Thus, we assume that $\alpha=2\delta N_{x}^{-1}$ and $\beta=2\delta N_{\vartheta}^{-1}$ are constants, and use the scalings $\mu=\alpha m$ and $\tau=\beta n$. When this is done, we get an equation similar to the case when $N_{x}=\tau$ and $N_{\vartheta}=\mu$, but with constant coefficients; this is the crucial difference that allows stable solutions to the case here. Using the decomposition $\psi_{\alpha m,\beta n}=u_{n}\exp(im\theta)$, we arrive at the matrix equation

[TABLE]

The matrix here has eigenvalues $\lambda$ with $|\lambda|=1$ for all $m,n$, so the solution is stable. Using arguments as in the Fibonacci example, the non-equidistant equation of the second scheme is shown to be stable.

5 Conclusions

Following [9], we explicitly introduced loop quantum cosmological models which take into account the full lattice structure of inhomogeneous states. Such lattices are in general refined by adding new vertices when acting with the Hamiltonian constraint. Thus, also dynamical equations even in homogeneous models should respect this property. Several interesting features arose: One obtains non-equidistant difference equations which, when imposed for functions on the whole real line as in isotropic loop quantum cosmology, are more restrictive than equidistant ones due to the absence of superselected sectors. This leaves the singularity issue unchanged since for this one only needs to consider a few steps in the equation. But a stability analysis of solutions and the verification of the correct classical limit in all semiclassical regimes can be more challenging. We presented an example for such an analysis, but also introduced a procedure by which one can transform the resulting equations to equidistant ones up to quantum corrections, which is sufficient for a semiclassical analysis. Interestingly, properties of the transformation itself provide hints to the correct semiclassical behavior. As a side-result, we demonstrated that one particular version of lattice refinements naturally gives rise to Misner-type variables.

It is our understanding that this general procedure of defining lattice refining models mostly agrees with the intuition used specifically in isotropic models in [25], and adapted to anisotropic ones in [34].555We thank A. Ashtekar for discussions of this point. However, there are some departures from what is assumed in [25]. First, we do not see indications to refer to the area operator while the area spectrum was not only used in [25] to fix the constant $\delta$ and the volume dependence of the step size but in fact provided the main motivation. Secondly, due to this motivation [25] presents a more narrow focus which from our viewpoint corresponds to only one single refinement model. It has a vertex number proportional to volume, which is a limiting case not realized by known full Hamiltonian constraints, and puts special emphasis on geometrical areas to determine the vertex number. Finally, commutators for inverse volume operators are to be treated differently from [25], taking into account a lattice refining model which would not be possible in a purely homogeneous formulation. As shown in the appendix, this enlarges expected quantum corrections to the classical functions.

We have discussed similar cases for illustration here, but keep a more general viewpoint on the refinement as a function of volume. A preliminary stability analysis for the Schwarzschild interior, consistent with [35] indeed suggests that a behavior different from what is suggested in [25] is preferred, which indicates that models can provide tight conditions for the general analysis of quantum dynamics. We emphasize that stability arguments as used here are independent of physical inner product issues since they refer to properties of general solutions. A general analysis as started here allows detailed tests of the full dynamics in manageable settings, which can verify the self-consistency of the framework of loop quantum gravity — or possibly point to limitations which need to be better understood.

Acknowledgements

We thank Kevin Vandersloot for discussions. This work was supported in part by NSF grant PHY0554771. GK is grateful for research support from the University of Massachusetts and Glaser Trust of New York.

Appendix A Inverse volume terms in homogeneous models and lattice

refinement

We have seen in this paper, following [9], that Hamiltonian constraint operators with triad-dependent parameters in holonomies allow one to model lattice refinements faithfully, with interesting results and some improvements over the original non-refining models. However, as always there are also some features of inhomogeneous states and operators which are not present and difficult to mimic in homogeneous models. Thus, even models generalized in this way by allowing for lattice refinement effects have to be interpreted with great care. While qualitative effects can be investigated fruitfully to test the full framework, there is no basis for drawing quantitative conclusions. The prime example is that of commutator terms which appear ubiquitously in composite operators of loop quantum gravity, such as the coefficients of difference equations or also matter Hamiltonians.

In the main construction of this paper we used holonomies associated with links of a lattice, rather than edges of a fixed coordinate length. This allows us, effectively, to take into account lattice refinements which change the number of vertices. It applies to holonomies (16) along a closed loop used to quantize curvature components which determine the step size of difference equations, but also to the link holonomies used in commutators to quantize inverse triad components based on (14). What is not modeled in homogeneous models is the fact that a lattice operator makes use of the local volume $\hat{V}_{v}$ at a given vertex $v$ where the commutator is acting, rather than the total volume $\hat{V}=\sum_{v}\hat{V}_{v}$ of the whole box in which the lattice is embedded. In a fully inhomogeneous setting the difference does not matter since volume contributions from vertices not touched by the edge used in a commutator drop out in the end, $[h_{e},\hat{V}]=\sum_{v\in e}[h_{e},\hat{V}_{v}]$. But in homogeneous models there is a difference since volume contributions from different vertices, in an exactly homogeneous setting, are all identical. Thus, the total volume $V={\cal N}V_{v}$ is the number of vertices multiplied with the local volume $V_{v}$. Then, $[h,\hat{V}_{v}]$ rather than $[h,\hat{V}]$ is expected as the contribution to constraint operators from the inhomogeneous perspective. In homogeneous models as in [25], on the other hand, $[h,\hat{V}]$ is more straightforward to use. We now show that without corrections this would imply crucial deviations from the inhomogeneous behavior.

It is easy to see that commutators differ depending on whether the local or total volume is used. For simplicity of the argument, we proceed with an isotropic situation where $V=|p|^{3/2}$ in terms of the basic isotropic densitized triad component $p$. A local lattice flux, for a surface $S$ intersecting only a single link, would be $\rho=\int_{S}{\mathrm{d}}^{2}y\tilde{p}=\ell_{0}^{2}p/L_{0}^{2}=p/{\cal N}^{2/3}$ for links of coordinate length $\ell_{0}$, such that $V_{v}=|\rho|^{3/2}=|p|^{3/2}/{\cal N}$ is the local volume. (We again use a coordinate box of size $L_{0}^{2}$ and introduce rescaled flux variables $p=L_{0}^{2}\tilde{p}$.) Isotropic states are spanned by $e^{i\mu c/2}$ where $\mu\in{\mathbb{R}}$ is related to the triad eigenvalues by $p_{\mu}=4\pi\gamma\ell_{\mathrm{P}}^{2}\mu/3$. Using a link holonomy $h\sim e^{i\ell_{0}\tilde{c}/2}=e^{ic/2{\cal N}^{1/3}}$ which as a multiplication operator increases $\mu$ by $1/{\cal N}^{1/3}$, a commutator with the total volume will have eigenvalues of the form

[TABLE]

If the local volume is used, on the other hand, we have to refer to local edge labels $\mu_{e}=3\rho/4\pi\gamma\ell_{\mathrm{P}}^{2}$, rather than using the total $p$. Thus,

[TABLE]

For large volume, $p\gg{\cal N}$, both expressions give the correct classical limit $\frac{3}{2}{\cal N}^{-1/3}\sqrt{|p|}$ expected from $\{e^{ic/2N^{1/3}},|p|^{3/2}\}$. However, quantum corrections, i.e. deviations from this classical limit for finite $p$, are much larger for the second version using the local volume as it would occur in an inhomogeneous quantization. The smooth classical function ${\mathrm{d}}V/{\mathrm{d}}p$ in a Poisson bracket appears in discretized form by the large step-size $N^{2/3}$ in (35) rather than the small one $N^{-1/3}$ in (34). Perturbative corrections, derived by Taylor expanding the difference terms and keeping higher order corrections to the classical expression, are thus larger. (This can have cosmological implications [37, 38, 39].) Non-perturbative effects as observed for the inverse scale factor operator in isotropic loop quantum cosmology which has an upper bound at finite volume [40], start to arise for $p\sim{\cal N}^{2/3}$ when the local volume is used but only at the much smaller $p\sim{\cal N}^{-1/3}$ for the total volume. Since only the local volume is relevant for inhomogeneous quantizations, quantum corrections from inverse volume operators can be large.

Unfortunately, this effect is more difficult to mimic in exact homogeneous models unlike the behavior of holonomies under lattice refinements and has therefore been overlooked in [25]. The connection components appearing in holonomies can simply be divided by a function ${\cal N}$ of triad components to implement shrinking edges due to subdivision. This is not possible for the volume itself to use a local version in a homogeneous model since, if we would divide the total volume by the appropriate function of triad components, only a constant would remain for an ${\cal N}$ proportional to volume and the commutator would be zero. The only way to have this effect faithfully implemented in a homogeneous model is to use higher SU(2) representations for holonomies in commutators but not for holonomies used in the loop to quantize curvature components. (This is not possible if one writes the constraint as a single trace, $\mathop{\mathrm{tr}}(h_{\alpha}h_{e}[h_{e}^{-1},\hat{V}])$ but can easily be done using the equivalent form $\mathop{\mathrm{tr}}(\tau_{i}h_{\alpha})\mathop{\mathrm{tr}}(\tau_{i}h_{e}[h_{e}^{-1},\hat{V}])$. We emphasize that higher representations for commutators are advocated here only in exactly homogeneous models to mimic inhomogeneous effects. Fully inhomogeneous operators usually need not refer to higher representations.) In a representation of spin $j$, matrix elements of holonomies contain exponentials $\exp(imc/2{\cal N}^{1/3})$ with $-j\leq m\leq j$, which increases the shifts in volume labels resulting from commutators. Resulting expressions for commutators can be found in [41, 42]. If the representation label $j$ is of the order ${\cal N}$, effects as they result from lattice refinements and using the local volume are correctly implemented. Accordingly, corrections from inverse triad components quantized through commutators are much larger than they would otherwise be.666This is the reason why inverse triad corrections did not play a large role in recent numerical studies of the so-called $\bar{\mu}$-quantization, as repeatedly emphasized in that context [43, 25, 44, 45]. This is only the case because this quantization is a hybrid version of lattice refinements which are taken into account for holonomies but not for the local volume used in commutators. Inverse triad corrections are thus artificially, though subtly, suppressed in those quantizations, departing from quantizations of inhomogeneous models.