# Covariant Loop Gravity

Carlo Rovelli

► To cite this version:

Carlo Rovelli. Covariant Loop Gravity. Lecture Notes in Physics, 2013, Lecture Notes in Physics, 863, pp.57-66. 10.1007/978-3-642-33036-0_3. hal-00957717

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# Covariant Loop Gravity

Carlo Rovelli

**Abstract** I summarize and illustrate the present state of the covariant formulation of the loop quantum gravity dynamics.

## 3.1 The Definition of the Theory

A quantum theory of gravity is defined by the family of transition amplitudes [1–5].

$$W_{\mathcal{C}}(h_l) = \int_{SU(2)} dh_{vf} \prod_f \delta(h_f) \prod_v A(h_{vf}), \qquad (3.1)$$

where the vertex amplitude is given by

$$A(h_{ab}) = \sum_{j_{ab}} \int_{SL(2C)} dg_a \prod_{ab} \mathrm{tr}_{j_{ab}} [h_{ab} Y_{\gamma}^{\dagger} g_a g_b^{-1} Y_{\gamma}]. \qquad (3.2)$$

$\mathcal{C}$ is a two-complex with faces $f$, edges $e$ and vertices $v$, bounded by a graph $\Gamma = \partial\mathcal{C}$ with links $l$ and nodes $n$, and $Y_{\gamma}$ is the linear map

$$Y_{\gamma} : |j; m\rangle \mapsto |(\gamma(j+1), j); j, m\rangle \qquad (3.3)$$

from the SU(2) spin-$j$ representation space $\mathcal{H}_j$ to the unitary SL(2, C) representation space $\mathcal{H}_{p,k}$ with quantum numbers $p = \gamma(j+1)$ and $k = j$. See [2] for the full details on the notation. These transition amplitudes are ultraviolet finite and admits a $q$-deformed version [6–8] where they are finite. The theory can be coupled to fermions and Yang–Mills fields [9].

The transition amplitude depends on $L$ SU(2) elements $h_l$ associated with the $L$ links $l$ of the graph that bounds $\mathcal{C}$ and is therefore an element $W$ of the boundary Hilbert space $H_{\partial\mathcal{C}}$ [2]. Semi-classical states $\psi$ in $H_{\partial\mathcal{C}}$ describe discretized 3d geometries $q_{\psi}$ formed by glued polyhedra [10, 11]. In particular, Regge geometries.

Evidence has piled up [12–15] that (3.1) converges to the Hamilton function $S_{\Delta}[q]$ of a Regge discretization of gravity in the classical limit, if $\mathcal{C}$ is the two-skeleton of the dual of the triangulation $\Delta$. The structure of the theory is therefore

<footnote>C. Rovelli (✉)</footnote>

<footnote>Centre de Physique Théorique, Case 907, Luminy, 13288 Marseille, France</footnote>

<footnote>e-mail: rovelli@cpt.univ-mrs.fr</footnote>
<figure><img src="image_2.png" /></figure>

Thus, the LQG transition amplitudes on a two-complex define a family of approximations to the full theory, and the classical limit of each of these (for $\mathcal{C}$ the two-skeleton of $\Delta^*$) is given by Regge gravity on $\Delta$.

In the rest of this article, I illustrate properties and various aspects of these transition amplitudes. This article does not cover the quantum geometry defined by the boundary states of the theory, which is amply discussed in many reviews. (See for instance [2].)

## 3.2 Properties

The structure described above can be compared with that of lattice QCD, where $\mathcal{C}$ corresponds to a given lattice. Here Eqs. (3.1) and (3.2) can be shown to provide a discretization of the formal “sum over 4-geometries” of the exponential of the GR action [16]

$$Z \sim \int Dg e^{\frac{i}{\hbar} \int R \sqrt{g} d^4 x}, \qquad (3.4)$$

on a manifold with boundaries with fixed geometry. There is a crucial difference, however. In QCD, the continuous limit requires the refinement of the lattice as well as the tuning of a coupling constant to a critical value. In gravity, instead, there is no parameter to be tuned. The difference can be traced directly to the general covariance of the continuous theory. As shown in detail in [17], a general covariant system can always be discretized without introducing a lattice spacing, and therefore a scale. This difference is important and is the source of confusion.

A second important difference is that the three-metric $q$ cannot be fully diagonalized in $H_{\partial \mathcal{C}}$. A maximal set of commuting operators in $H_{\partial \mathcal{C}}$ is formed by areas and volumes of the polyhedra, but these quantities do not suffice to determine the geometry of the glued polyhedra. From this perspective the metric is like angular momentum: the full set of data that determine the geometry is formed by operators that do not commute. Therefore, classical 3-metrics can only be associated with semi-classical states in $H_{\partial \mathcal{C}}$. To have semi-classical states (where fluctuations are
small with respect to expectation values) we need large quantum numbers. In par-
ticular, we need large spins, and therefore large distances compared to the Planck
scale. Physically, this means that at small scale the geometry of space cannot be
Riemannian: it is a quantized geometry, and there is a Heisenberg uncertainty pre-
venting the full 3-geometry to be sharp. In other words, the $\hbar \to 0$ limit of the theory
is also necessarily a large-distance limit. At short scale, quantum gravity does not
have a proper continuous limit taking it to classical general relativity (GR). This was
of course expected on physical grounds.

Individual amplitudes (3.1)–(3.2) can be obtained as Feynman amplitudes of a
proper quantum field theory (QFT), using the group-field-theory formalism [18].
By separating the terms with vanishing spins and reinterpreting them as defined
on sub-two-complexes, it can be possible to re-express the limit as a series [19].
This observation, and the analogy with the standard Feynman expansion provide a
second interpretation to the expansion in $\mathcal{C}$, as a perturbative expansion. States in
$\mathcal{H}_r$ can be viewed as formed by $N$ quanta, where $N$ is the number of nodes of the
graph. Each node of the graph is like a particle in QED, namely a quantum of elec-
tromagnetic field. Here, each node represents a quantum of gravitational field. But
there is a key difference. QED Fock quanta carry quantum numbers coding where
they are located in the background space-manifold. Here, since in general relativity
the gravitational field is also physical space, individual quanta of gravity are also
quanta of space. Therefore they do not carry information about their localization in
space, but only information about the *relative* location with respect to one another.
This information is coded by the graph structure. Thus, the quanta of gravity form
themselves the texture of physical space. Therefore the graph can also be seen as a
generalization of the lattices of lattice QCD.

This convergence between the perturbative-QED picture and the lattice-QCD pic-
ture follows directly from the key physics of general relativity: the fact that the
gravitational field is physical space itself. Indeed, the lattice sites of lattice QCD are
small regions of space; according to general relativity, these are excitations of the
gravitational field, therefore they are themselves quanta of a (generally covariant)
quantum field theory. An $N$-quanta state of gravity has therefore the same struc-
ture as a Yang-Mills state on a lattice with $N$ sites. This convergence between the
perturbative-QED and the lattice-QCD pictures is a beautiful feature of loop gravity.

## 3.3 The Discretization of Parametrized Systems

To understand the particular features of the transition amplitudes of loop quantum gravity, it is useful to compare them with the following definition of the amplitudes of a finite-dimensional ‘general covariant’ system. Consider a dynamical system with configuration variable $q \in \mathcal{C}$, and Lagrangian $L(q, \dot{q})$. Given an initial configuration $q$ at time $t$ and a final configuration $q'$ at time $t'$, let $q_{q,t,q',t'}: R \to \mathcal{C}$ be a solution of the equations of motion such that $q_{q,t,q',t'}(t) = q$ and $q_{q,t,q',t'}(t') = q'$.
Assume for the moment this exists and is unique. The Hamilton function is the function on $(\mathscr{C} \times \mathbb{R})^2$ defined by

$$ S q, t, q', t' \rangle = \int_{t}^{t'} dt \, L(q, t, q', t', \dot{q}, t, q', t'), \quad (3.5) $$

namely, the value of the action on the solution of the equation of motion determined by given initial and final data. This function, introduced by Hamilton in 1834 [20] codes the solution of the dynamics of the system, has remarkable properties and is a powerful tool that remains meaningful in background-independent physics. Let $H$ be the quantum Hamiltonian operator of the system and $|q\rangle$ the eigenstates of its $q$ observables. The transition amplitude

$$ W q, t, q', t' \rangle = q' |e^{-\frac{i}{\hbar} H(t'-t)}|q\rangle \quad (3.6) $$

codes all the quantum dynamics. Now, consider a parametrized formulation of this same system. That is, consider a new action which depends on $n + 1$ variables $x = (q, t) \in \mathscr{C} \times \mathcal{R} \equiv \mathcal{C}_{ex}$ evolving in $\tau$

$$ I[q] = \int dt L(q, \dot{q}) \rightarrow \int d\tau i L(q, \dot{q}/i) = \int d\tau \mathcal{L}(x, \dot{x}) \equiv I[x], \quad (3.7) $$

where on the left-hand side of the arrow the dot indicates a derivative with respect to $t$, while on the right-hand side of the arrow (and from now on) it indicates a derivative with respect to $\tau$. For instance, a single-particle Lagrangian $L = m\dot{q}^2/2 - V(q)$ gives $\mathcal{L} = m\dot{q}^2/(2i) - iV(q)$. This ‘parametrized’ system describes the same physics as the original one, but in a different logic. The physical time variable $t$ is now treated on the same footing as the other dynamical variables<fnref n="1" /> and the system has a local gauge invariance under reparametrizations of $\tau$. As was recognized in the early days of the canonical analysis of general relativity [24], this is precisely the structure of general relativity. The general relativistic coordinates $x^\mu$ play the same role as $\tau$ above and reparametrization invariance is the invariance under general coordinate transformations. Above, the ‘parametrized’ form of the dynamics has been derived from an original ‘un-parametrized’ form, while GR is directly written in the parametrized form, with a local gauge-invariance. ‘De-parametrizing’ GR is possible in principle, but spoils many of its formal properties, making the formalism far more cumbersome and intractable. In GR we better live with reparametrization invariance and, in fact, take advantage of it.

What is the Hamilton function of the parametrized system? From the definition,

$$ S x, \tau, x', \tau' \rangle = \int_{\tau}^{\tau'} d\tau \mathcal{L}(x_{,x'}, \dot{x}_{,x'}), \quad (3.8) $$

where $x_{,x'}(\tau)$ is a solution of the equations of motion such that $x_{,x'}(\tau) = x$ and $x_{,x'}(\tau') = x'$. But a moment of reflection will convince the reader that

$$ S x, \tau, x', \tau' \rangle = S x, x' \rangle, \quad (3.9) $$

<footnote n="1">¹That is, dynamics is not anymore interpreted as the description of the evolution in $t$ of the $n$ configuration variables $q$, but rather as a description of the possible relations between the $n + 1$ variables $x = (q, t)$ (see Sect. 3.2.4 in [21] and [22, 23]).</footnote>
namely the Hamilton function of a parametrized system is independent from $\tau$ and $\tau'$.<fnref n="2" /> Second, because of the gauge there are many solutions of the classical equations with the same boundary data, but $S(x, x')$ is independent from the one chosen, that is, it is gauge invariant. Both facts are immediate consequence of the invariance of the action under reparametrizations of $\tau$. Furthermore, since the new action is just the old one in new variables, $S(x, x')$ is precisely nothing else than (3.5).

That is, remarkably, the Hamilton function of the parametrized system is the same object as the Hamilton function of the original system. In fact, notice that the original Hamilton function was already a function of (two copies of) the *extended* configuration space $\mathcal{C}_{\text{ex}} = \mathcal{C} \times \mathcal{R}$. The Hamilton function was already born as if it knew it had to work in a parametrized language.

Let me now give a definition of the path integral that defines the transitions amplitudes of this system, as a limit of multiple integrals, as in the original Feyman's derivation of path integral, or as in lattice QCD. This can be defined as

$$W_N(x, x') = \int \frac{dq_n dt_n}{\mu(q_n, t_n)} e^{\frac{i}{\hbar} \sum_{n=1}^{N} (t_n - t_{n-1}) L(q_n, q_{n-1}, t_n, t_{n-1})} \quad (3.11)$$

The essential point is that the discretization of the parametrized action does not involve a lattice spacing. This can be seen explicitly, for instance, by noticing that the discretized form of the general-covariant action of a one-dimensional system with kinetic and potential energy

$$S_N[q_n, t_n] = \sum_{n=1}^{N} a \left[ \frac{m \frac{(q_{n+1}-q_n)^2}{a^2}}{t_{n+1}-t_n} - \frac{t_{n+1}-t_n}{a} V(q_n) \right] \\ = \sum_{n=1}^{N} \left[ \frac{m (q_{n+1}-q_n)^2}{t_{n+1}-t_n} - (t_{n+1}-t_n) V(q_n) \right]$$

cancels exactly! Notice that the 'lattice spacing' $a$ is replaced by the variable quantity $(t_n - t_{n-1})$ [17].

We can therefore recover the structure illustrated in the diagram in Sect. 3.1. The discretized quantum transition amplitude (3.11) yields the full quantum transition amplitude in the $N \to \infty$ limit and the Hamilton function of a discretization of the classical parametrized theory in the $\hbar \to 0$ limit. A concrete example is illustrated in detail in [17].

$$\frac{\partial S}{\partial \tau} = H(x, \partial S/\partial x). \quad (3.10)$$

But the canonical Hamiltonian $H(x, p_x)$ vanishes because of the gauge invariance and therefore $\partial S/\partial \tau = 0$.

<footnote n="2">²S is a solution of the Hamilton-Jacobi equation</footnote>
## 3.4 The Discretization of Classical General Relativity

Consider a finite 4d region $\mathcal{B}$ surrounded by a 3d surface $\Sigma$.<fnref n="3" /> For simplicity, I begin with metric variables—later on I will use other variables to describe the gravitational field. We can fix pure gravity boundary data on $\Sigma$ by giving the three-metric $q$ of $\Sigma$. Let $g_q$ be a 4d metric on $\mathcal{B}$ which satisfies the Einstein's equations and induces the 3-metric $q$ on $\Sigma$. The possibility that such $g_q$ might be non-existent or non-unique does not concern us here, as discussed in the previous section. Consider the action of GR, including the boundary term

$$I[g] = \frac{1}{2} \int_{\mathcal{B}} d^4 x \sqrt{g} R[g] + \int_{\Sigma} d^3 x \sqrt{q} k, \qquad (3.12)$$

where $k = k^{ab}q_{ab}$ is the trace of the extrinsic curvature $k^{ab}$ on the boundary and $q$ is the induced 3-metric. If $g$ is a solution of the equations of motion, the first term vanishes. Thus, the Hamilton function of $q$ is the action of $g_q$, that is

$$S[q] = I[g_q] = \int_{\Sigma} d^3 x \sqrt{q} k^{ab}[q](x) q_{ab}(x). \qquad (3.13)$$

The non-trivial part of this expression is the dependence of the extrinsic curvature $k^{ab}[q](x)$ on the 3-metric. This dependence is non-local: in general the extrinsic curvature in a point $x$ depends on the value of the metric on the entire surface $\Sigma$.<fnref n="4" />

Now, consider a discretization of GR. As a warm-up, consider the best-known discretization, which is Regge calculus. Fix a triangulation $\Delta$ of $\mathcal{B}$ and consider Regge geometries on $\mathcal{B}$. A Regge geometry is a geometry which is everywhere flat except on the triangles $t$ of $\Delta$, where the curvature is distributional and fully determined by the deficit angle $\theta_t$ at the triangle, determined by the sum of the dihedral angles of the flat 4-simplices meeting at $t$. This geometry is uniquely determined by giving the lengths $l_i$ of the segments $i$ of $\Delta$, which determine the deficit angles $\theta_t = \theta_t(l_i)$ and the areas $A_t = A_t(l_i)$ of these triangles. The Regge action is a local function of these lengths and reads

$$I_{\Delta}[l_i] = \sum_{t \in \Delta} \theta_t(l_i) A_t(l_i) + \sum_{t \in \partial \Delta} \theta_t(l_{ib}) A_t(l_{ib}). \qquad (3.14)$$

The triangulation of $\mathcal{B}$ indices a boundary triangulation $\partial \Delta$ on the boundary surface $\Sigma$. The boundary 3-metric is fully determined by the lengths of the boundary

<footnote n="3">³In the applications, we are particularly interested in the cases where $\mathcal{B}$ is a ball or a segment of a cylinder, and therefore $\Sigma$ has the topology of $S_3$, or $S_3 \times S_3$, which are respectively relevant to discuss scattering and cosmology.</footnote>

<footnote n="4">⁴For a simple example of this dependence consider the Euclidean theory, and say that the metric $q$ is that of a metric 3-sphere with radius $a$. Such a sphere can be imbedded in many curved 4d manifolds, and the extrinsic curvature of the imbedding is not determined by $q$. But if the 4d Riemannian manifold is a solution of the Euclidean Einstein equations, then this freedom is drastically reduced. A solution of the Einstein equations is flat space. A metric 3-sphere can be imbedded in a flat 4d space as the surface of the ball with radius $a$ and this fixes the extrinsic curvature to be $k^{ab} = a^{-1}q^{ab}$, so that $H[q(a)] = 6\pi^2 a^2$.</footnote>
links $i_b$, which I denote as $q = \{l_{ib}\}$. The Hamilton function of the Regge theory is the discrete analog of (3.13):

$$S_{\Delta}[q] = \sum_{t \in \partial \Delta} \theta_t[l_{ib}] A_t(l_{ib}), \qquad (3.15)$$

where the sum is here over the triangles in the boundary and $\theta_t$ is the discrete extrinsic curvature at the boundary triangles, namely, the angle between the 4-normals of the two boundary tetrahedra separated by the triangle $t$. While the area of a boundary triangle depends only on the length of its three sides, the extrinsic curvature $\theta_t[l_{ib}]$ is a non-trivial function of all the boundary lengths, determined by solving the bulk Regge equations of motion. (I have used the square-brackets notation to emphasize this non-local character of the dependence.)

Regge theory approximates GR in the following sense. First, say that a sequence of Regge geometries $g_n$ converges to a Riemannian geometry $g$ if there is a map $f$ from one to the other such that for any two points $x$ and $y$, $|d_g(f(x), f(y)) - d_{g_n}(x, y)| < \epsilon$, where $d_g(x, y)$ is the distance between $x$ and $y$ in the geometry $g$. Then consider a sequence $\Delta_n$ of refinements of a triangulations. Let a Regge geometry $q_n$ be on the boundary of $\Delta_n$, such that the sequence $q_n$ converges to $q$. A discretization is good if

$$\lim_{n \to \infty} S_{\Delta_n}[q_n] = S[q]. \qquad (3.16)$$

The truncated transition amplitudes given at the opening of this article gave therefore two relevant limits. The $\hbar \to 0$ limit gives the Regge Hamilton function.

$$\lim_{\hbar \to 0} (-i\hbar) \log W_{\Delta}[q] = S_{\Delta}[q]. \qquad (3.17)$$

If we restore physical units, we must add the Newton constant $1/8\pi G$ in front of the action and (3.17) reads

$$\lim_{l_P \to 0} (-i 8\pi l_P^2) \log W_{\Delta}[q_i] = S_{\Delta}[q_{\partial \Delta}], \qquad (3.18)$$

where $l_P = \sqrt{\hbar G}$ is the Planck length. Then, taking (3.16) into account, it is reasonable to define the quantum gravity transition amplitudes

$$W[q] = \lim_{n \to \infty} W_{\Delta_n}[q_n] \qquad (3.19)$$

if $\lim_{n \to \infty} q_n = q$. (The choice of a particular sequence $\Delta_n$ can be avoided by defining a stricter limit, for $\Delta \to \infty$ in the sense of nets [19].) The limit itself, however, is not of great significance from the perspective of a physicist, since the quantum theory is already defined by its family of approximations (3.1).

It is reasonable to expect these approximation to be good in the regime where the corresponding classical approximations are good, namely where (3.16) converges fast. This is the regime where the bulk deficit angles are small, namely the regime around flat space.<fnref n="5" />

<footnote n="5">⁵The expansion in $n$ should not be confused with the standard perturbative expansion used in QFT. The latter is an expansion in the amplitude of the field, keeping all its modes. The former is an expansion in the number of modes.</footnote>
But should we expect the limit (3.19) to actually converge, and converge fast? A rigorous answer to this equation is missing, but there are circumstantial arguments that indicate that this could be the case in some regimes. The Regge approximation becomes exact on flat space. That is, the Regge action and the continuous Einstein-Hilbert action are equal for a flat geometry. It follows that if the geometry is flat, the Regge discretization is ‘topological’ in the sense that it is invariant under a refinement of the triangulation. This very peculiar property was called Ditt-invariance in [17] from Bianca Dittrich, who has emphasized it in her work with Benjamin Bahr [25–27]. When the boundary data are near flatness, in the sense mentioned, a refinement of the triangulation becomes therefore irrelevant. More precisely, let $q_n$ be a sequence of boundary Regge metrics converging to a 3-geometry $q$ such that $g_q$ is close to a flat geometry. Then one may expect that in this regime $W_{\Delta_n}[q_n]$ converges fast, and therefore a small $n$ is sufficient to give a good approximation to the physical amplitudes.

## 3.5 Conclusion

Quantum gravity is often confusing. Field operator insertions in the path integral, which are the main tool for analyzing conventional QFT, are uninteresting in quantum gravity, due to diff-invariance. The discretization used to define boundary amplitudes is often confused with the quantum discreteness of space. The general structure of a background-independent quantum theory is different from a conventional QFT. What are good observables in quantum gravity, and how do we describe evolution? See for example [23, 28–41] and [42] for an overview.

I have given a tentative overall picture of the structure of the theory, the observables, and the form of the continuous and classical limits. The truncated boundary transition amplitudes (3.1)–(3.2) are the tool for extracting physics from the theory. From these quantities one can derive standard observables, for instance for analyzing cosmological evolution [43], or particle scattering [44], where the old idea that the gravitons live on the non-perturbative quantum states (e.g., [45]) is realized concretely by having the quantum excitations on the nodes of the boundary state.

These quantities admit two distinct limits. In the classical limit where quantum effects are disregarded, they converge to the Hamilton function of a truncation of GR. In the continuum limit, we expect them to converge to the transition amplitudes of the full theory. If the present indications are confirmed, there should be a regime in $q$ where the convergence is rapid and therefore the truncation can work as an effective expansion. The expansion parameter is $g_q$’s deviation from flatness.

The truncation introduced by $C$ (or $\Delta$) should not be confused with the physical quantum discreteness of the geometry. The quantum discreteness of the geometry is the fact that the geometrical size of the cells of the complex takes discrete values. It disappears in the semi-classical limit, where the theory is studied at distances large with respect to the Planck scale, while it persists in the continuum limit, with an arbitrary large two-complex. In other words, no refinement of the cellular complex
can make the size of the cells go smoothly to zero, because geometry is physically
discrete at the Planck scale. This is the most characteristic aspect of quantum gravity.

Finally, not much is known about the effect of the radiative corrections on this
structure (for partial results, see [18, 46–48]). These are finite in the deformed ver-
sion of (3.1) [6, 7] but this does not make them irrelevant. The main open problem in
quantum gravity, I think, is to study their effect on the convergence of the continuous
limit.

References

1. C. Rovelli, arXiv:1004.1780

2. C. Rovelli, arXiv:1102.3660

3. J. Engle, E. Livine, R. Pereira, C. Rovelli, Nucl. Phys. B **799**, 136 (2008). arXiv:0711.0146

4. L. Freidel, K. Krasnov, Class. Quantum Gravity **25**, 125018 (2008). arXiv:0708.1595

5. W. Kaminski, M. Kisielowski, J. Lewandowski, Class. Quantum Gravity **27**, 095006 (2010).
arXiv:0909.0939

6. M. Han, J. Math. Phys. **52**, 072501 (2011). arXiv:1012.4216

7. W.J. Fairbairn, C. Meusburger, J. Math. Phys. **53**, 022501 (2012). arXiv:1012.4784

8. M. Han, Phys. Rev. D **84**, 064010 (2011). arXiv:1105.2212

9. E. Bianchi, M. Han, E. Magliaro, C. Perini, W. Wieland, arXiv:1012.4719

10. L. Freidel, S. Speziale, Phys. Rev. D **82**, 084040 (2010). arXiv:1001.2748

11. E. Bianchi, P. Donà, S. Speziale, Phys. Rev. D **83**, 044035 (2011). arXiv:1009.3402

12. F. Conrady, L. Freidel, Class. Quantum Gravity **25**, 245010 (2008). arXiv:0806.4640

13. J.W. Barrett, R.J. Dowdall, W.J. Fairbairn, F. Hellmann, R. Pereira, Class. Quantum Gravity **27**, 165009 (2010). arXiv:0907.2440

14. J.W. Barrett, R.J. Dowdall, W.J. Fairbairn, H. Gomes, F. Hellmann, J. Math. Phys. **50**, 112504 (2009). arXiv:0902.1170

15. E. Magliaro, C. Perini, arXiv:1105.0216

16. C.W. Misner, Rev. Mod. Phys. **29**, 497 (1957)

17. C. Rovelli, arXiv:1107.2310

18. J. Ben Geloun, R. Gurau, V. Rivasseau, Europhys. Lett. **92**, 60008 (2010). arXiv:1008.0354

19. C. Rovelli, M. Smerlak, Class. Quantum Gravity **29**, 055004 (2012). arXiv:1010.5437

20. W. Hamilton, British Association Report, 1834, 513

21. C. Rovelli, *Quantum Gravity* (Cambridge University Press, Cambridge, 2004)

22. C. Rovelli, Found. Phys. **41**, 1475 (2011). arXiv:0903.3832

23. C. Rovelli, Phys. Rev. D **65**, 124013 (2002). arXiv:gr-qc/0110035

24. S. Deser, R. Arnowitt, C. Misner, Nuovo Cimento **19**, 668 (1961)

25. B. Dittrich, Adv. Sci. Lett. **2**, 151 (2009). arXiv:0810.3594

26. B. Bahr, B. Dittrich, S. Steinhaus, Phys. Rev. D **83**, 105026 (2011). arXiv:1101.4775

27. B. Bahr, B. Dittrich, Phys. Rev. D **80**, 124030 (2009). arXiv:0907.4323

28. S.B. Giddings, D. Marolf, J.B. Hartle, Phys. Rev. D **74**, 064018 (2006). hep-th/0512200

29. N. Tsamis, R. Woodard, Ann. Phys. **215**, 96 (1992)

30. J.B. Hartle, in *Gravitation and Quantizations: Proceedings of the 1992 Les Houches Summer School*, ed. by B. Julia, J. Zinn-Justin (Elsevier, Amsterdam, 1995), pp. 285–480. gr-qc/9304006

31. E. Witten, hep-th/0106109

32. C.E. Dolby, gr-qc/0406034

33. T. Banks, W. Fischler, S. Paban, J. High Energy Phys. **0212**, 062 (2002). hep-th/0210160

34. C. Rovelli, Class. Quantum Gravity **8**, 297 (1991)

35. H.W. Hamber, Phys. Rev. D **50**, 3932 (1994). hep-th/9311024
36. A. Ashtekar, R. Tate, C. Uggla, Int. J. Mod. Phys. D 2, 15 (1993). gr-qc/9302027

37. D.N. Page, W.K. Wootters, Phys. Rev. D 27, 2885 (1983)

38. A. Strominger, J. High Energy Phys. 0110, 034 (2001). hep-th/0106113

39. D. Marolf, Class. Quantum Gravity 12, 2469 (1995). gr-qc/9412016

40. R.M. Wald, Phys. Rev. D 48, 2377 (1993). gr-qc/9305024

41. K. Kuchař, Int. J. Mod. Phys. Proc. Suppl. D 20, 3 (2011)

42. E. Anderson, arXiv:1009.2157

43. E. Bianchi, C. Rovelli, F. Vidotto, Phys. Rev. D 82, 084035 (2010). arXiv:1003.3483

44. E. Bianchi, L. Modesto, C. Rovelli, S. Speziale, Class. Quantum Gravity 23, 6989 (2006). gr-qc/0604044

45. J. Iwasaki, C. Rovelli, Int. J. Mod. Phys. D 1, 533 (1993)

46. C. Perini, C. Rovelli, S. Speziale, Phys. Lett. B 682, 78 (2009). arXiv:0810.1714

47. T. Krajewski, J. Magnen, V. Rivasseau, A. Tanasa, P. Vitale, Phys. Rev. D 82, 124069 (2010). arXiv:1007.3150

48. V. Bonzom, M. Smerlak, Ann. Henri Poincaré 13, 185 (2012). arXiv:1103.3961