Relativity and gravitational theory Definition

Loop quantum gravity

Builds on Ashtekar variables Spin network Holonomy Hilbert space Canonical quantization Spin foam

Definition

Let $\Sigma$ be a three-manifold. Describe general relativity on $\R \times \Sigma$ by the Ashtekar–Barbero variables: an $SU(2)$ connection $A$ and a densitized triad $E$ on $\Sigma$ . These are subject to the Gauss, diffeomorphism and Hamiltonian constraints. Loop quantum gravity is the canonical quantization of this system in which:

the elementary variables are the holonomies $h_e(A) \in SU(2)$ of $A$ along finite curves $e$ in $\Sigma$ , and the fluxes of $E$ through surfaces;
the kinematical Hilbert space is the completion of the functions of finitely many holonomies in the Ashtekar–Lewandowski inner product, built from Haar measure;
the Gauss constraint is solved by passing to $SU(2)$ -gauge-invariant states, which are spanned by spin network states;
the diffeomorphism constraint is solved by passing to diffeomorphism classes of spin networks (s-knots), and the Hamiltonian constraint acts as an operator.

Remarks

Let $S$ be a surface, and let a spin network meet $S$ transversally at points of edges with spins $j_1, \dots, j_n$ . Then its state is an eigenvector of the area operator of $S$ , with eigenvalue $A = 8 \pi \gamma \, \ell_P^2 \sum_{k=1}^{n} \sqrt{j_k (j_k + 1)},$ where $\gamma > 0$ is the Barbero–Immirzi parameter and $\ell_P$ is the Planck length. Example: a single edge of spin $\frac12$ crossing $S$ gives the smallest nonzero area $4 \sqrt{3} \pi \gamma \ell_P^2$ . Non-example: a Fock space of gravitons on flat space is not a representation of this kind, because it uses a background metric. In loop quantum gravity no background metric is chosen. The construction of the Hamiltonian constraint operator, due to Thiemann, has ambiguities, and the covariant dynamics is studied through spin foams.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Ashtekar variables concept
Canonical quantization concept
Hilbert space concept
Holonomy concept
Spin foam concept
Spin network concept
Loop quantum gravity definition

Tag	`VFQBFV9VT5`
URI	`m2://VFQBFV9VT5`
Kind	definition
Writing	W0 stub
MSC	83C45, 81S08
Defines	Loop quantum gravity
Uses	Ashtekar variables, Spin network, Holonomy, Hilbert space, Canonical quantization, Spin foam
Changed	3 November 2026

Loop quantum gravity

Definition

Remarks

Reading path

Further reading

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