User:Martin Bojowald/Proposed/Loop Quantum Cosmology

Prof. Martin Bojowald, Penn State Department of Physics, University Park, PA, United-States of America

Loop quantum cosmology is the application of methods of loop quantum gravity to spatially homogeneous universe models, or to inhomogeneous models obtained by adding the dynamics of non-constant fields onto homogeneous models. The framework follows the tradition of Wheeler-DeWitt quantum cosmology, but is in a position to make constructions more precise thanks to a (vague) relationship with a potential full description of quantum space-time. It can also suggest new dynamical effects by incorporating characteristic features of loop quantum gravity in a setting that is easier to evaluate. However, hopes that loop quantum cosmology can provide a physically viable model of the universe, and especially its high-density behavior, could not be realized. Instead, the main results of research on loop quantum cosmology are unexpected features of quantum space-time, such as signature change, which partially invalidate some assumptions commonly made in the field. The status of loop quantum cosmology therefore remains open.

Basic features

Loop quantum cosmology is based on a canonical quantization of symmetry-reduced cosmological models, in which one applies standard quantization techniques to systems whose configuration spaces describe spatial geometries. In the simplest examples, one assumes homogeneity and isotropy, so that the geometry, at any given time, is fully determined by a single parameter: the scale factor \(a\) of Friedmann cosmology or the volume \(V\) of some region in space whose expansion is being studied. The canonical momentum of \(V\) (times \(c^2/4\pi G\)) is proportional to the Hubble parameter \(H=\dot{a}/a\), and the dynamical law to be quantized is the Friedmann equation

\[ \tag{1} H^2=\frac{8\pi G}{3c^2}\rho \]

with the energy density \(\rho\) (plus, perhaps, curvature and cosmological-constant terms).

Difference equation

Compared with the older approach of Wheeler-DeWitt quantum cosmology, which also attempts to quantize (1), the main new ingredient of loop quantum cosmology is the use of shift operators as momentum operators on the space of spatial geometries, or finite shifts of the volume \(V\): Wheeler-DeWitt models quantize \(\hat{H}=-4\pi i G\hbar c^{-2}\partial/\partial V\) and turn the energy version

\[ \frac{3c^2}{8\pi G} VH^2=E\]

(\(E=V\rho\)) of the Friedmann equation into a differential equation,

\[ -\frac{6\pi G \hbar^2}{c^2} \frac{\partial^2(V\psi)}{\partial V^2}=\hat{E}\psi \]

for states \(\psi(V)\) (DeWitt 1967). (The ordering of \(V\) and \(\partial/\partial V\) is not unique.) Loop quantum cosmology quantizes \(\widehat{\exp(-i\delta H)}= \Delta_{\delta}\) with a shift operator

\[\tag{2} (\Delta_{\delta}\psi)_V=\psi_{V+4\pi \ell_{\rm P}^2 c\delta} \]

(introducing a free parameter \(\delta\) and the Planck length \(\ell_{\rm P}=\sqrt{G\hbar/c^3}\)) and provides a difference equation

\[ \tag{3} -\frac{3c^2}{8\pi G\delta^2} \left((V+4\pi \ell_{\rm P}^2 c\delta)\psi_{V+4\pi \ell_{\rm P}^2 c\delta}-2V\psi_V+ (V-4\pi \ell_{\rm P}^2 c\delta)\psi_{V-4\pi \ell_{\rm P}^2 c\delta}\right)= \hat{E}\psi_V \]

for a lattice-supported state \(\psi_V\) (Bojowald 2002). For \(V\) sufficiently large compared to \(4\pi\ell_{\rm P}^2c\delta\) and \(\psi_V\) allowing a smooth approximation that varies only mildly on the scale of \(\delta\), the difference equation of loop quantum cosmology has the Wheeler-DeWitt equation as its continuum limit. But interesting new features can be found at small volume \(V\), where the classical Friedmann models become singular.

Discreteness corrections

The discreteness of the volume in loop quantum cosmology is motivated by the structure of quantum geometry in loop quantum gravity. Seen from full gravity without any symmetries, the Hubble parameter \(H\) (or extrinsic curvature) is part of a non-Abelian connection \(A_a^i\). This connection is one of the basic fields of the theory, together with the densitized triad \(E^a_i\) on space (\(E^a_i=|\det e_b^j|e^a_i\) with the triad \(e^a_i\) and its inverse \(e_b^j\)). On the Hilbert space of loop quantum gravity (Rovelli, Smolin 1990; Ashtekar et al. 1995), the connection can be quantized only in the form of parallel transport or holonomies

\[ h_{\ell}(A)={\cal P}\exp(\smallint_{\cal C} A_a^i\tau_i t^a{\rm d}\lambda) \]

along spatial curves \({\cal C}\), with tangent vector fields \(t^a\). (The \(\tau_i\) are generators of SU(2), or \(-2i\) times Pauli matrices.) Matrix elements of holonomies along straight lines in flat space, evaluated for an isotropic connection \(A_a^i=k\delta_a^i\) with \(k=\dot{a}\), take the simple form \(\exp(i\mu k)\) with some parameter \(\mu\) related to the coordinate length of the curve. Thus, in isotropic models one can expect operators quantizing the exponential \(\exp(i\mu k)\) or \(\exp(i\delta H)\) as the shift operators of loop quantum cosmology, but there is no operator \(\hat{k}\) or \(\hat{H}\) in this setting.

This basic feature has further consequences. The variable conjugate to \(H\), the volume \(V\), is quantized to an operator \(\hat{V}\) with discrete spectrum. Since zero is an eigenvalue, \(\hat{V}\) does not have a densely defined inverse. However, negative powers of \(V\) appear in the gravitational and matter Hamiltonians, for instance in the kinetic term \(p_{\phi}^2/2V\) of a homogeneous scalar field \(\phi\) with momentum \(p_{\phi}\). Such expressions can be quantized after one rewrites the negative power of \(V\), such as \(V^{-1/2}\), as a difference expression of positive powers:

\[V^{-1/2}= \frac{\sqrt{V+\epsilon}-\sqrt{V-\epsilon}}{2\epsilon}+O(\epsilon)\,.\]

As in the difference equation, one obtains the correct continuum limit for \(V\) much larger than \(\epsilon\), and discreteness corrections which are \(O(\epsilon)\) at small \(V\).

State space

Loop quantum gravity is based on the premise that holonomies of the gravitational connection but not the connection components themselves can be quantized.

Almost-periodic functions

In isotropic cosmological models, there is a single connection component \(k\), from which Abelian "holonomies" \(\exp(i\mu k)\) with real \(\mu\) can be formed. Isotropic loop quantum cosmology should then be based on a state space on which among all functions of \(k\) only linear combinations of \(\exp(i\mu k)\) for different \(\mu\) can be represented. Working in a connection representation, states should therefore be sums \(\psi(k)=\sum_{\mu} \psi_{\mu} \exp(i\mu k)\) with countably many non-zero complex-valued \(\psi_{\mu}\) (Ashtekar, Bojowald, Lewandowski 2003).

These functions form the space of almost-periodic functions. They can be shown to constitute the space of all continuous functions on the Bohr compactification of the real line, a compact space that contains the real line densely. They provide a dense set in the (non-separable) Hilbert space obtained by taking all functions \(\exp(i\mu k)\) with arbitrary real \(\mu\) as a basis. Therefore, \((\exp(i\mu_1k),\exp(i\mu_2k))=\delta_{\mu_1,\mu_2}\). In terms of functions of \(k\), the inner product can be written as \((\psi_1,\psi_2)= \lim_{K\to\infty} (2K)^{-1}\int_{-K}^K \overline{\psi_1(k)}\psi_2(k) {\rm d}k\).

The basis functions can easily be represented as multiplication operators \(\widehat{\exp(i\mu k)}\). However, these operators are not continuous in \(\mu\) (some of their matrix elements are \((\exp(i\nu k),\widehat{\exp(i\mu k)}\exp(i\nu k))=\delta_{\mu,0}\)). It is not possible to take a derivative by \(\mu\), and therefore there is no operator for the connection component \(k\) on the same space. The loop representation is not unitarily related to the Schroedinger-type representation employed in Wheeler-DeWitt quantum cosmology. (According to the Stone-von Neumann theorem, the Schroedinger representation is unique only when one assumes continuity of the exponential operators.)

The momentum \(p\) of \(k\), a densitized-triad component related to the scale factor by \(|p|=a^2\) with \({\rm sgn}p\) determining the orientation of space, is quantized as a derivative operator \(-(8\pi iG\hbar/3c^2){\rm d}/{\rm d}k\) with eigenvalues \(\frac{8}{3}\pi\ell_{\rm P}^2c\mu\) and eigenstates \(\exp(i\mu k)\). Although all real numbers are eigenvalues, the spectrum of \(\hat{p}\) is discrete (or pure point) in the sense that all its eigenstates are normalizable. (Therefore, as indicated in the section on discreteness corrections, \(\hat{p}\) or \(\hat{V}=|\hat{p}|^{3/2}\) does not have a densely defined inverse.) The sign of eigenvalues, \({\rm sgn}\mu\), has the geometric meaning of the orientation of space, while \(V_{\mu}=|\frac{8}{3}\pi\ell_{\rm P}^2c\mu|^{3/2}\) are the volume eigenvalues.

Heuristic relation to loop quantum gravity

The key role attributed to exponentials \(\exp(i\mu k)\) in constructing the state space of homogeneous and isotropic loop quantum cosmology is motivated by the kinematical representation of loop quantum gravity. A full holonomy that gives rise to the function \(\exp(i\mu k)\) for an isotropic connection could be taken as one along a curve tangential to a translation generator of the symmetry group. The full holonomy takes values in SU(2), but its matrix elements can be compared with the exponential upon evaluating it in one of the irreducible representations, labeled by half-integer spins \(j\). The matrix elements are then \(\exp(i\mu k)\) where \(\mu\) is the product of the spin label \(j\) with the coordinate length of the curve.

Matrix elements of full holonomies in the state space of loop quantum gravity are orthogonal when they belong to different curves (because they probe different degrees of freedom), or (according to the Peter-Weyl theorem) when they are taken in different irreducible representations of SU(2). The state space of isotropic loop quantum cosmology agrees with these properties in the heuristic relation sketched here, because different \(\mu\) lead to orthogonal states.

In the full theory, there are flux operators that quantize the densitized triad (the field canonically conjugate to the connection) and have discrete spectra, just like \(\hat{p}\). However, in the full theory, the set of eigenvalues is discrete (proportional to sums of the spin labels \(j\)) in addition to the fact that eigenstates are normalizable. One can understand this difference as follows: The \(\hat{p}\)-eigenvalues \(\frac{8}{3}\pi\ell_{\rm P}^2c\mu\) heuristically arise from products of discrete spins with continuous length parameters. On the state space of isotropic loop quantum cosmology, the curve length and spin are degenerate parameters and cannot be completely distinguished from each other: only their product is accessible. In the full theory, however, both factors play very different roles, a fact that foreshadows problems with isotropic loop quantum cosmology (Bojowald 2013): discreteness properties of loop quantum gravity are modeled only partially in loop quantum cosmology.

Dynamics

Since only exponentials of \(k\) can be represented in isotropic loop quantum cosmology, any Hamiltonian or other operator must be built from them, in addition to the simpler \(\hat{p}\). The Friedmann equation, however, is quadratic in \(k=\dot{a}\) and cannot be represented directly. In order to proceed, one modifies the equation so that it can be expressed in terms of exponentials only, but still agrees with the classical expression in the limit of small \(k\). At this stage, it is easier to switch from functions of \(k\) to functions of \(H\propto k/\sqrt{|p|}\) (Ashtekar, Pawlowski, Singh 2006) because the latter Hubble parameter has a clear physical meaning. It is small (by quantum-gravity standards) at low curvature where classical general relativity is well established, and becomes large only in high-curvature regimes where the Friedmann equation might well receive quantum corrections.

Such extensions of the Friedmann equation are far from being unique, but the modification by an almost-periodic function (as opposed to any function) might be restrictive enough to lead to characteristic effects. As one of the simplest choices, one takes a free parameter \(\delta\) (perhaps related to the Planck length) and writes the equation as

\[\tag{4} \frac{3c^2}{8\pi G} V \frac{\sin^2(\delta H)}{\delta^2}= E\,. \]

This equation can easily be quantized using isotropic holonomy operators. In one specific ordering, one obtains (3)

One can understand the function \(\exp(i\delta H)\), as opposed to \(\exp(i\mu k)\), as a holonomy with \(p\)-dependent \(\mu=\delta/\sqrt{|p|}\). As the universe expands and \(|p|\) grows, the curve length modeled by \(\mu\) shrinks. Such a behavior is expected for any discrete theory, for expanding space would eventually blow up any constant discreteness scale to macroscopic sizes and be in conflict with continuum physics. Loop quantum cosmology incorporates this behavior of lattice refinement by a \(p\)-dependent \(\mu\), in general by some function not necessarily of the form \(\mu(p)\propto |p|^{-1/2}\), but it is difficult to derive the precise function from a full, lattice-changing Hamiltonian in loop quantum gravity. (In general, therefore, also \(\delta\) might be \(p\)-dependent.)

Problem of time

The Friedmann equation contains the matter energy \(E\), reminiscent of the energy factor that gives rise to the time-independent Schroedinger equation in quantum mechanics. In quantum cosmology, however, as in any generally covariant theory, there is no absolute time \(t\) whose derivative one could equate with \(-i\hat{E}/\hbar\) to derive a time-dependent state equation. Evolution in quantum cosmology has to be described in a more subtle way, presenting an example for the problem of time of quantum gravity (Kuchar 1992, Anderson 2009).

A common method aims to recover evolution not from the relation of observables to an absolute time parameter, but rather from relations between physical variables. Formally, one possibility is deparameterization, in which one chooses a simple and specific matter content such as different versions of dust (with energy \(\hat{E} =i\hbar \partial/\partial P\) for some dust momentum \(P\)) or a free, massless scalar \(\phi\) (with energy \(\hat{E}=-\frac{1}{2}\hbar^2 V^{-1} \partial^2/\partial\phi^2\)).

For dust, one directly obtains a Schroedinger-type equation for evolution with respect to \(P\). For a free, massless scalar, the quantized Friedmann equation is of Klein-Gordon form:

\[ -\hbar^2 \frac{\partial^2\psi}{\partial\phi^2}= \frac{3c^2}{4\pi G} (\widehat{V\Delta})^2 \psi \]

where \(\hat{\Delta}= -4\pi iG\hbar c^{-2}\partial/\partial V\) for Wheeler-DeWitt quantizations or \(\hat{\Delta}= i\hbar\delta^{-1}\Delta_{\delta}\) with \(\Delta_{\delta}\) as in (2) for loop quantizations. One can transform it to Schroedinger-type evolution by taking a square root before acting on \(\psi\), so that

\[ \tag{5} i\hbar \frac{\partial\psi}{\partial \phi}= \pm\sqrt{\frac{3c^2}{4\pi G}} |\widehat{V\Delta}| \psi\,.\]

This equation is analogous to the Schroedinger equation with a quadratic Hamiltonian, and is therefore as easy to analyze as the harmonic oscillator regarding quantum corrections. Most of the current intuition on the dynamics of loop quantum cosmology comes from detailed numerical and analytical studies of models closely related to this system. However, with any additional terms, such as spatial curvature, a cosmological constant, a scalar mass or potential, anisotropies or inhomogeneities, the Hamiltonian is non-quadratic and exhibits stronger quantum behavior.

Example: A solvable model

The harmonic system (5) can be solved exactly. The Wheeler-DeWitt version is not very different from the standard harmonic oscillator (although it is more of an upside-down oscillator). The loop version is more contrived because of the shift operator, but enjoys the same solvability features (Bojowald 2007).

If we introduce new operators \(\hat{J}:=\delta^{-1}\hat{V}\widehat{\exp(-i\delta H)}\), we can write (5) with a shift operator for \(\hat{\Delta}\) as

\[ i\hbar \frac{\partial\psi}{\partial\phi}= \pm\sqrt{\frac{3c^2}{4\pi G}} |{\rm Im}\hat{J}|\psi \]

in a specific ordering of the quantized \(V\) and \(H\) as determined by \(\hat{J}\).

Up to the absolute value, the Hamiltonian is linear in \(\hat{J}\). Heisenberg's equations are linear as well, as follows from the \({\rm sl}(2,{\mathbb R})\)-commutators

\[[\hat{V},\hat{J}]= -\frac{4\pi G\hbar}{c^2}\hat{J}\quad,\quad{} [\hat{V},\hat{J}^{\dagger}]= \frac{4\pi G\hbar}{c^2}\hat{J}^{\dagger}\quad,\quad{} [\hat{J},\hat{J}^{\dagger}]=\frac{8\pi G\hbar}{c^2}\hat{V}\,.\]

They can easily be solved, or evaluated as Ehrenfest-type equations

\[ \frac{\partial\langle\hat{V}\rangle}{\partial\phi}= \pm\frac{\sqrt{12\pi G}\hbar}{c} {\rm Re}\langle\hat{J}\rangle\quad,\quad \frac{\partial {\rm Re}\langle\hat{J}\rangle}{\partial\phi}= \pm\frac{\sqrt{12\pi G}\hbar}{c} \langle\hat{V}\rangle\]

or \(\partial^2\langle\hat{V}\rangle/\partial\phi^2= (12\pi G \hbar^2/c^2) \langle\hat{V}\rangle\). Irrespective of the precise state, the volume expectation value changes as \(\langle\hat{V}\rangle(\phi)\propto \cosh(\phi-\phi_0)\). (The \(\sinh\) solution is eliminated by the reality condition \(\delta^2\hat{J}\hat{J}^{\dagger}=\hat{V}^2\), which implies that \(\langle\hat{V}\rangle^2-\delta^2({\rm Re}\langle\hat{J}\rangle)^2= \delta^2({\rm Im}\langle\hat{J}\rangle)^2+O(\hbar)\) in states that are semiclassical at low curvature. Only the \(\cosh\)-solution for \(\langle\hat{V}\rangle\), and the \(\sinh\)-solution for \({\rm Re}\langle\hat{J}\rangle\), is consistent with the right-hand side being positive.) The expectation value never becomes zero and instead "bounces" at some minimum value. In these models, there is no complete collapse, as realized classically at the big-bang singularity.

The scalar \(\phi\) is not identical to time measured by observers co-moving in a Friedmann universe. One can relate \(\phi\)-changes to proper-time evolution by using \({\rm d}\phi/{\rm d}\tau= (8\pi G/3c^2) p_{\phi}/V\) with the constant \(\phi\)-momentum \(p_{\phi}\). The changes of \(\langle\hat{V}\rangle\) with respect to \(\phi\) then translate into a modified Friedmann equation

\[ \tag{6} H^2=\frac{8\pi G}{3c^2}\rho\left(1-\frac{\rho}{\rho_{\delta}}\right) \]

for an effective scale factor \(a(\tau)\), where \(\rho_{\delta}=3c^2/8\pi G \delta^2\). This derivation confirms preceding numerical tests of this equation obtained by comparing its solutions with the evolution of wave functions (Vandersloot 2005; Ashtekar, Pawlowski, Singh 2006). When the average density reaches \(\rho_{\delta}\) (\(3/8\pi\) times the Planck density for \(c\delta\) being the Planck length), \(H=0\) and the scale factor has an extremum. Computing the modified Raychaudhuri equation confirms that the extremum is a minimum.

Extensions

Loop quantum cosmology has shown interesting effects, such as a potential of singularity resolution at the big bang. It therefore appears to be promising as a universe model. At the same time, however, it is highly oversimplified compared to full loop quantum gravity and all its results must be taken with caution. Attempts to extend the results of loop quantum cosmology to a theory of generic quantum space-time have not been successful, but instead they have led to a critical view on loop quantum cosmology itself.

There are three main steps to be completed before loop quantum cosmology and its results could be seen as reliable:

Quantum back-reaction

Even for isotropic models, the general dynamical behavior remains poorly understood. Most investigations focus on consequences of the modification (4) of the Friedmann equation, or of inverse-triad corrections. But as in all interacting quantum systems, there is also quantum back-reaction of fluctuations and other moments of a state on expectation values. These quantum corrections depend on the evolving state, which in turn depends on the initial state. In contrast to familiar systems, quantum cosmology, lacking a Hamiltonian bounded from below, does not offer a natural initial state such as a ground state or vacuum. In order to estimate quantum corrections, a large state space must therefore be controlled. While the harmonic dynamics of the solvable model (5), which is insensitive to the precise state used, provides indications about consequences of the holonomy modification, it does not address generic features of quantum back-reaction.

Non-harmonic evolution has been studied numerically, for instance in (Ashtekar et al. 2007; Bentivegna, Pawlowski 2008; Brizuela, Mena-Marugan, Pawlowski 2010). In most cases, however, one assumed sharply peaked Gaussians as the simplest choice of initial states. While evolved states deviate from the initial Gaussian, the restricted initial choice makes it unclear whether a generic-enough state is being considered at high curvature. Such investigations are often used to claim that equations such as (6) are good effective equations of loop quantum cosmology also beyond the harmonic model. (Indeed, most of the intuition for effective space-time pictures and bounces in this setting comes from a direct application of (4) or (6) even to models which are far from being harmonic or kinetic-dominated.) That such a procedure cannot be correct can be seen from the fact that general insights about effective actions, generically, show the presence of higher-curvature corrections in any gravitational model (Donoghue 1994; Burgess 2004). In (6), however, there are no higher curvature terms because such corrections would come with higher time derivatives. The magnitude of the corrections in (6), on the other hand, is comparable to the one of higher-curvature terms, both given by the ratio of the average density over the Planck density.

For a reliable effective description, one must derive properties of an evolving state (or, in the language of low-energy effective actions, an interacting vacuum) along with correction terms to classical equations. Such a derivation, owing to the complexity of the large class of states to be considered, remains to be done in loop quantum cosmology.

Non-Abelian features

Isotropic models in loop quantum cosmology have been extended to anisotropic (but still homogeneous) Bianchi models (Bojowald 2003; Bojowald, Date, Vandersloot 2004; Bojowald, Cartin, Khanna 2007; Ashtekar, Wilson-Ewing 2009). However, this has been done only for diagonal metrics, which incorporate anisotropy by working with three independent scale factors \((a_1,a_2,a_3)\) for the three spatial dimensions. General anisotropic models are described by a constant matrix (the densitized triad \(E^a_i\)) with non-trivial non-Abelian structures after quantization. It remains to be shown how the Bohr compactification used in isotropic models or diagonal anisotropic ones can be realized in the non-Abelian setting. Possible alternatives (Bojowald 2013) show that the problem of how to incorporate non-Abelian features is closely related to the degeneracy of spin label \(j\) and edge length in one single quantum number \(\mu\), mentioned in the section on heuristics.

Inhomogeneity and quantum space-time

Inhomogeneity must be included on the same footing as the homogeneous variables, at two different levels: For a consistent framework of cosmological structure formation, one must extend equations such as (4) or (6) to allow for perturbative inhomogeneity. And for a potential derivation of loop quantum cosmology from loop quantum gravity, the fundamental representations as well as dynamical operators of these settings must be related to each other.

Effective models of inhomogeneous space-time:

Quantum corrections (such as holonomy modifications) must be consistently implemented in cosmological perturbation equations before any conclusions about potential observations can be drawn. This problem is highly non-trivial owing to strong consistency conditions on the inhomogeneous equations so that generally covariant dynamics results.

While there is no complete extension of modified background equations of loop quantum cosmology to inhomogeneous space-time, even at a perturbative level, there are several consistent examples. In all cases, generically, not just the dynamics but even the structure of space-time receives quantum corrections (Bojowald, Hossain, Kagan, Shankaranarayanan 2008): It is no longer possible to express quantum effects by an effective metric of Riemannian geometry. (This consequence is already indicated by the fact that simple effective equations such as (6) differ from expectations for covariant effective actions: There are corrections of the same magnitude but, lacking higher time derivatives, not the same form as higher-curvature terms.) An interesting consequence is the potential of signature change at high density (Bojowald, Paily 2012; Reyes 2009; Cailleteau, Mielczarek, Barrau, Grain 2012): Just in the regime where naive background models based on (4) or (6) would indicate a cosmic bounce, consistent extensions to inhomogeneity show that space-time turns into a quantum version of 4-dimensional Euclidean space. No structure or information can propagate through high density, preventing causal access to the collapse phase of bounce models in loop quantum cosmology.

As an alternative to a consistent extension of effective equations to inhomogeneity, there are different constructions that couple non-constant fields to the background dynamics of loop quantum cosmology. In order to circumvent complicated issues posed by the consistency of the gauge system of gravity, these constructions are based on gauge fixing, deparameterization, or a combination the two. In general, while careful gauge fixing and deparameterization can lead to the right results, the correctness is not guaranteed and can be tested only by comparing models based on different gauge fixings or deparameterizations, or ideally with gauge-invariant and deparameterization-independent results. Such comparisons are complicated in quantum gravity and have not been performed yet. However, it is clear that gauge-fixing and deparameterization methods cannot be fully correct in this setting because they miss the crucial feature of signature change (which is a result found independently of gauge fixing or deparameterization). In fact, the possibility of signature change casts doubt on the general validity of deparameterization: When one deparameterizes, as in the section on the problem of time, one assumes that there is always a time variable along which a matter (or other) degree of freedom evolves. This basic assumption is violated when the space-time signature turns Euclidean. (Some aspects of signature change, in particular an instability of initial-value formulations, can sometimes be seen even in deparameterized models (Brizuela, Mena-Marugan, Pawlowski 2010). But the general space-time structure remains obscure.)

Relating loop quantum cosmology to loop quantum gravity:

Some relation between homogeneous models and full states of loop quantum gravity must be realized, especially at a dynamical level. So far, one can deduce certain features of loop quantum cosmology by applying simple operators of loop quantum gravity to special classes of lattice states (Bojowald 2004; Bojowald 2006; Bojowald 2013), making the constructions described in the section on heuristics more precise. The choice of lattice states (with edges along translation generators of the homogeneity group) can be justified as an adaptation to the desired symmetry, just as one would use adapted coordinates to express a symmetric metric in a simple form. Nevertheless, the question remains how more general states could be reduced to symmetry, and whether their reduction would lead to the same states and degrees of freedom as the reduction of lattice states. There are partial results in this direction, some of which indicate that the reduced degrees of freedom obtained from different classes of full states do not match (Engle 2007; Brunnemann, Fleischhack 2007; Engle 2013; Hanusch 2013).

Moreover, potential reductions of complicated operators such as Hamiltonians are not under control. They do not preserve symmetric states that could be used to relate the kinematical representations, even if one restricts full states to regular lattices. A complicated projection is then necessary in order to derive an isotropic operator from the full one, which would require one to address complicated issues such as the averaging problem of classical cosmology (Ellis, Buchert 2005).

Status

Most of the recent results found in loop quantum cosmology have led to more caution about the whole framework. A popular result is the bounce behavior found in equations such as (6). However, while it is easy to modify the Friedmann equation of exactly homogeneous models in a consistent way and produce an upper bound on the energy density by using a bounded function \(\delta^{-2}\sin^2(\delta H)\) instead of \(H^2\) in (4), a consistent extension to inhomogeneous geometries and a full model of space-time structure is highly non-trivial.

Research on loop quantum cosmology has led to progress in finding consistent space-time structures, and while they remain incomplete, they show that generic consistent extensions of (6) to inhomogeneity eliminate the bounce as a dynamical phenomenon because signature change makes space-time at high density take the form of a quantum version of 4-dimensional Euclidean space. There is no well-posed initial-value problem in this regime, and no evolution through high density. With such extensions, however, one is already moving beyond loop quantum cosmology because the correct nature of time can only be seen in relation to the nature of space, which is inaccessible in homogeneous models or in models in which inhomogeneous modes are added on by hand. While loop quantum cosmology has led to interesting developments and important insights into the quantum nature of space-time, as a stand-alone model it appears to be invalidated by its own results.

Acknowledgements

This work was supported in part by NSF grant PHY-1307408. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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User:Martin Bojowald/Proposed/Loop Quantum Cosmology

Contents

Basic features

Difference equation

Discreteness corrections

State space

Almost-periodic functions

Heuristic relation to loop quantum gravity

Dynamics

Problem of time

Example: A solvable model

Extensions

Quantum back-reaction

Non-Abelian features

Inhomogeneity and quantum space-time

Effective models of inhomogeneous space-time:

Relating loop quantum cosmology to loop quantum gravity:

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Acknowledgements

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