Notation and Conventions

The conventions below are used throughout the thesis. Since signs and numerical factors vary considerably from one reference to another, we fix them once and for all; whenever our conventions differ from those of the standard monographs [1–3], this is pointed out in the text.

Units and Constants

Unless stated otherwise, we set $c = k_{\mathrm B} = 1$ and keep Newton's constant $G$ and the reduced Planck constant $\hbar$ explicit. We write

so that $\hbar\kappa = 8\pi {\ell_{\mathrm{P}}}^2$; in ordinary units, ${\ell_{\mathrm{P}}} = \sqrt{\hbar G/c^3} \simeq 1.616 \times 10^{-35}\ \mathrm{m}$. Ordinary units are restored, in particular in Chapter 4, whenever orders of magnitude are given. In Chapter 4 only, the letter $\kappa$ also denotes the surface gravity of a horizon.

Indices

Indices	Range	Nature
$\mu,\nu,\rho,\ldots$	$0,1,2,3$	spacetime indices (manifold $\mathcal M$)
$a,b,c,\ldots$	$1,2,3$	spatial indices (hypersurface $\Sigma$)
$I,J,K,\ldots$	$0,1,2,3$	internal Lorentz indices
$i,j,k,\ldots$	$1,2,3$	internal ${\mathrm{SO}}(3)$ or ${\mathfrak{su}}(2)$ indices

Repeated indices are summed over. The internal Minkowski metric is $\eta_{IJ} = \mathrm{diag}(-1,1,1,1)$, and the signature of $g_{\mu\nu}$ is $(-,+,+,+)$. Internal spatial indices are raised and lowered with $\delta_{ij}$. Antisymmetrization is normalized: $X_{[ab]} = \tfrac12(X_{ab} - X_{ba})$.

Levi-Civita Symbols

The totally antisymmetric internal symbol satisfies $\epsilon_{0123} = +1$, hence $\epsilon^{0123} = -1$; in three dimensions, $\epsilon_{123} = \epsilon^{123} = 1$ and $\epsilon_{0ijk} = \epsilon_{ijk}$. The spacetime symbol $\tilde\epsilon^{\mu\nu\rho\sigma}$ and the spatial symbol $\tilde\epsilon^{abc}$ are densities taking the values $0,\pm1$; for instance,

Differential Forms and Internal Duality

Differential forms are written without spacetime indices: $e^I = e^I_\mu\, {\mathrm{d}} x^\mu$, $\omega^{IJ} = \omega^{IJ}_\mu\, {\mathrm{d}} x^\mu$. For an antisymmetric quantity $X^{IJ}$ with values in the Lorentz algebra, the internal Hodge dual is

The curvature of a connection $\omega$ is $F^{IJ} = {\mathrm{d}}\omega^{IJ} + \omega^I{}_K \wedge \omega^{KJ}$; it is denoted $R^{IJ}$ when $\omega = \omega[e]$ is the Levi-Civita connection.

The Algebra su(2)

The anti-Hermitian generators of ${\mathfrak{su}}(2)$ in the fundamental representation are $\tau_i = -\tfrac{{\mathrm{i}}}{2}\sigma_i$, where $\sigma_i$ are the Pauli matrices. They satisfy

In the spin-$j$ representation, of dimension $d_j = 2j+1$, the generators are denoted $\tau^{(j)}_i$; the Casimir operator is $\tau^{(j)}_i \tau^{(j)i} = -j(j+1)\,\mathbb 1$. We write $J_i = {\mathrm{i}} \tau_i$ for the usual Hermitian generators.

Canonical Variables

The configuration variable is the Ashtekar–Barbero connection $A^i_a = \Gamma^i_a + \gamma K^i_a$, and its conjugate momentum is the densitized triad $E^a_i$, with

The curvature of $A$ is $F^i_{ab} = \partial_a A^i_b - \partial_b A^i_a + \epsilon^i{}_{jk} A^j_a A^k_b$.

Abbreviations

Abbreviation	Meaning
% Include a list of abbreviations (a table of two columns)

Symbols

ADM | Arnowitt–Deser–Misner (decomposition, action, energy) | | EPRL | Engle–Pereira–Rovelli–Livine (spin-foam model) | | FGP | Frodden–Ghosh–Perez (local first law) | | FK | Freidel–Krasnov (spin-foam model) | | GFT | Group Field Theory | | GR | General Relativity | | LOST | Lewandowski–Okołów–Sahlmann–Thiemann (uniqueness theorem) | | LQG | Loop Quantum Gravity | | QFT | Quantum Field Theory | | TOCY | Turaev–Ooguri–Crane–Yetter (topological spin-foam models) |

Symbol	Meaning
$\mathcal M$, $\Sigma$	spacetime manifold; spatial hypersurface
$g_{\mu\nu}$, $q_{ab}$	spacetime metric; induced spatial metric
$e^I_\mu$, $\omega^{IJ}_\mu$	tetrad; spin connection
$F^{IJ}$, $R^{IJ}$	curvature of $\omega$; curvature of the Levi-Civita connection $\omega[e]$
$\Sigma^{IJ}$	the 2-form $e^I\wedge e^J$
$\kappa$	$8\pi G$ (the surface gravity in Chapter 4)
$\gamma$	Barbero–Immirzi parameter
$N$, $N^a$	lapse function; shift vector
$K_{ab}$, $K^i_a$	extrinsic curvature; its triad form
$\pi^{ab}$	momentum conjugate to $q_{ab}$
$e^i_a$, $E^a_i$	triad; densitized triad
$\Gamma^i_a$, $A^i_a$	spin connection of the triad; Ashtekar–Barbero connection
$G_i$, $C_a$, $C$	Gauss, diffeomorphism (vector) and Hamiltonian (scalar) constraints
$h_e[A]$, $E_i(S)$	holonomy along the link $e$; flux through the surface $S$
$\Gamma$, $j_e$, $\iota_n$	graph; spin of a link; intertwiner at a node
$\Psi_{\Gamma,j_e,\iota_n}$	spin-network state
$\hat A(S)$, $\hat V(R)$	area and volume operators
$\tau_i$, $J_i$	anti-Hermitian and Hermitian generators of ${\mathfrak{su}}(2)$
$T_H$, $T_U$	Hawking and Unruh temperatures
$S_{\mathrm{BH}}$	Bekenstein–Hawking entropy
$a$	proper acceleration (the Kerr parameter in Section 4.1)
$\mathcal N$, $z(\gamma)$	number of punctures; single-puncture partition function
$\sigma = (\mathcal C, j_f, \iota_e)$	spin foam on the 2-complex $\mathcal C$
$A_v$, $A_e$, $A_f$	vertex, edge and face amplitudes