Loop Quantum Gravity

The starting point of Loop Quantum Gravity is a reformulation of general relativity using connection variables instead of the metric. In the ADM formalism, the phase space is parametrized by the spatial metric $q_{ab}$ and its conjugate momentum. Ashtekar’s key insight was to use an $SU(2)$ connection $A^i_a$ and a densitized triad $E^a_i$ as canonical variables:

Here $\Gamma^i_a$ is the spin connection compatible with the triad, $K^i_a = K_{ab} e^{bi}$ is the extrinsic curvature in triad form, and $\gamma$ is the Barbero-Immirzi parameter. The densitized triad is $E^a_i = \sqrt{\det q}\, e^a_i$, encoding both the metric and orientation. The fundamental Poisson bracket is:

The spatial metric is recovered as $q_{ab} = E^i_a E^j_b \delta_{ij} / \det(E)$. The constraints of general relativity (Gauss, diffeomorphism, Hamiltonian) take polynomial form in these variables, which is a major advantage over metric variables.

Rather than quantizing $A^i_a$ directly (which is not well-defined in the continuum limit), LQG uses holonomies along edges $e$ and fluxes through surfaces $S$ as the fundamental variables. The holonomy of the connection along an edge is:

where $\tau_i = -i\sigma_i/2$ are $\mathfrak{su}(2)$ generators and $\mathcal{P}$ denotes path ordering. The flux of the densitized triad through a surface $S$ is:

The holonomy-flux algebra captures the full content of the classical phase space. Crucially, the LOST theorem (Lewandowski, Okolow, Sahlmann, Thiemann) shows that the kinematical Hilbert space is unique under natural assumptions of diffeomorphism invariance.

The kinematical Hilbert space $\mathcal{H}_{\rm kin}$ of LQG has an orthonormal basis given by spin network states. A spin network $|\Gamma, j_e, \iota_n\rangle$ is defined by:

• A graph $\Gamma$ embedded in the spatial manifold $\Sigma$
• Spin labels $j_e \in \{0, \frac{1}{2}, 1, \frac{3}{2}, \ldots\}$ on each edge $e$
• Intertwiners $\iota_n$ at each node $n$, ensuring gauge invariance

The spin network function evaluated on a connection is:

where $D^{(j)}$ is the spin-$j$ Wigner matrix (irreducible representation of $SU(2)$). The inner product uses the Ashtekar-Lewandowski measure $d\mu_{AL}$, which is the unique diffeomorphism-invariant measure on the space of generalized connections.

The area of a surface $S$ in the classical theory is $A(S) = \int_S \sqrt{E^a_i E^b_j \delta^{ij} n_a n_b}\, d^2\sigma$. Upon quantization, this becomes a well-defined self-adjoint operator $\hat{A}(S)$ with a discrete spectrum. Acting on a spin network state where edges $e_1, \ldots, e_n$ puncture $S$:

where the sum runs over all punctures $p$ of the surface by edges carrying spin $j_p$, and $\ell_P = \sqrt{\hbar G / c^3}$ is the Planck length. The key features are:

• The spectrum is purely discrete with a nonzero minimum (area gap): $\Delta A = 4\pi\sqrt{3}\,\gamma\,\ell_P^2$
• Eigenvalues depend on the Immirzi parameter $\gamma$
• Area is quantized in units of $\ell_P^2$, as expected from dimensional analysis

The volume of a region $R$ is classically $V(R) = \int_R \sqrt{|\det E|}\,d^3x$. The volume operator acts nontrivially only at nodes of the spin network. For a node $n$ with edges carrying spins $j_1, j_2, j_3$:

where $\hat{J}^{(a)}_i$ are $SU(2)$ generators acting on the representation of edge $a$. The volume spectrum is also discrete, but more complex than the area spectrum. A node must be at least 4-valent for a nonzero volume eigenvalue. The smallest nonzero volume eigenvalue scales as:

The Barbero-Immirzi parameter $\gamma$ is a free parameter in LQG that does not affect the classical theory but enters the quantum spectrum. Its value can be fixed by requiring consistency with the Bekenstein-Hawking entropy formula for black holes. Computing the entropy of a quantum isolated horizon with area $A$:

Matching with $S = A/(4\ell_P^2)$ gives $\gamma = \gamma_0\,\ln 2$, where $\gamma_0$ depends on the counting method. Using the most refined counting:

This value of $\gamma$ then determines all area and volume eigenvalues throughout the theory, giving LQG genuine predictive power for Planck-scale geometry.

The Hamiltonian constraint generates time evolution and is the most challenging part of LQG. In terms of connection variables, the Euclidean part of the constraint is:

where $F^i_{ab}$ is the curvature of $A^i_a$. Thiemann regularized this using holonomies around small loops $\alpha$ of coordinate area $\epsilon^2$:

The action of $\hat{H}$ on a spin network state modifies the graph by adding new edges and changing spin labels, giving a combinatorial description of quantum dynamics.

We compute the area eigenvalues $A = 8\pi\gamma\ell_P^2\sum_p\sqrt{j_p(j_p+1)}$ for spin networks with up to three edges puncturing a surface, visualize the discrete spectrum, its degeneracies, and the dependence on the Immirzi parameter: