loop quantum gravity

1. Motivation

General Relativity (GR) describes spacetime as a smooth geometry governed by Einstein’s equations:

G_{\mu\nu} = 8\pi G \, T_{\mu\nu}.

Quantum Mechanics (QM) demands that all dynamical fields be quantized.
How can spacetime itself be quantized, without introducing strings or extra dimensions?

Loop Quantum Gravity (LQG) attempts exactly this:
a background-independent, nonperturbative quantization of GR.

2. Canonical Quantization of GR

Start from the Hamiltonian formulation of GR.
Using ADM decomposition, the metric splits into 3-metric $q_{ab}$ and conjugate momentum $\pi^{ab}$ .
But these variables are cumbersome.

Ashtekar Variables

Introduce new canonical variables:

Connection: $A^i_a$ (an $SU(2)$ gauge field)
Conjugate electric field: $E^a_i$ (related to triads)

with Poisson brackets:

\{ A^i_a(x), E^b_j(y) \} = 8\pi G \, \gamma \, \delta^b_a \, \delta^i_j \, \delta^{(3)}(x,y).

Here $\gamma$ is the Barbero–Immirzi parameter.

3. Constraints

The dynamics of GR in canonical form is encoded in constraints:

Gauss constraint (internal $SU(2)$ gauge invariance):

\mathcal{G}_i = D_a E^a_i \approx 0.

Diffeomorphism constraint (spatial invariance):

\mathcal{V}_a = E^b_i F^i_{ab} \approx 0.

Hamiltonian constraint (time evolution):

\mathcal{H} = \frac{E^a_i E^b_j}{\sqrt{\det E}} \left( \epsilon^{ij}{}_{k} F^k_{ab} - 2(1+\gamma^2) K^i_{[a}K^j_{b]} \right) \approx 0.

Physical states must satisfy all three constraints.

4. Holonomies and Fluxes

Instead of quantizing $A^i_a$ directly, LQG uses:

Holonomies: parallel transports of $A^i_a$ along a curve $\ell$ :

h_\ell[A] = \mathcal{P}\exp\!\left(\int_\ell A\right).

Fluxes: smeared $E^a_i$ across surfaces.

This leads to a holonomy–flux algebra, well-suited for background-independent quantization.

5. Spin Networks

Kinematic Hilbert space of LQG is spanned by spin network states:

Graphs with edges labeled by $SU(2)$ representations (spins $j$ )
Vertices labeled by intertwiners

Spin networks diagonalize geometric operators:

Area operator:

\hat{A}(S) = 8\pi \gamma \ell_P^2 \sum_{p\in S\cap \Gamma} \sqrt{j_p(j_p+1)}.

Volume operator:

\hat{V}(R) = \sum_{v\in R\cap \Gamma} \sqrt{|\hat{q}_v|}.

Result: areas and volumes are quantized in discrete units $\sim \ell_P^2$ .

6. Loop Quantum Cosmology (LQC)

Applying LQG ideas to homogeneous cosmologies yields:

Big Bang singularity is replaced by a quantum bounce.
Modified Friedmann equation:

H^2 = \frac{8\pi G}{3}\rho \left(1 - \frac{\rho}{\rho_c}\right),

where $\rho_c \sim \rho_{\text{Planck}}$ .

At $\rho = \rho_c$ , expansion reverses, avoiding the singularity.

7. Challenges and Open Questions

Solving the Hamiltonian constraint in full LQG remains difficult.
The semiclassical limit (recovering smooth GR at large scales) must be fully established.
Connection to experimental tests (cosmic microwave background, gravitational waves) is still indirect.

Status

LQG provides a mathematically rigorous framework where spacetime geometry is quantized.
It is complementary to string theory:

String theory: unification-first, background-dependent
LQG: quantum spacetime-first, background-independent

Both aim at quantum gravity, from opposite directions.