Background-independent quantisation programme for general relativity in which the gravitational field is described by SU(2) holonomies along loops (spin-network states). Discrete spectra of area and volume operators, no unified matter…
loop-quantum-gravity
Ashtekar variables
Reformulation of GR in terms of a densitised SU(2) triad Ẽ^a_i (electric) and a connection A^i_a (magnetic). Converts the Hamiltonian…
Wilson loops in gravity
Holonomy h_γ[A] = P exp(∫_γ A) of the Ashtekar connection along a loop γ. Gauge-invariant class functions of holonomies span the…
Spin network
Graph with edges labelled by SU(2) representations (half-integer j) and vertices by SU(2) intertwiners. Orthonormal basis of the…
Area operator spectrum
Quantised area operator acting on a spin network has discrete spectrum A = 8πγℓ_P² Σ_e √j_e(j_e+1), with Barbero–Immirzi parameter γ.…
Volume operator (LQG)
Analogous discretisation of the 3-volume on spin networks; eigenvalues concentrated on vertices of valence ≥4. The Hamiltonian constraint…
Spin foam (path-integral)
Sum over 2-complexes (histories of spin networks) with amplitudes derived from BF theory + simplicity constraints (EPRL/FK models).…
Barbero–Immirzi parameter
Free dimensionless parameter γ in the Ashtekar connection A^i_a = Γ^i_a + γK^i_a. Fixes the scale of the area and volume spectra; matched…
Loop quantum cosmology
Symmetry-reduced LQG applied to Friedmann–Robertson–Walker universes. Replaces the classical Big-Bang singularity with a quantum 'big…
SU(2) spin-network evaluation (Penrose–Reshetikhin–Turaev, LQG)
Loop-quantum-gravity application of L0 monoidal-category theory. A spin network Γ is a trivalent graph whose edges carry half-integer…
Ponzano–Regge state-sum model (2+1 quantum gravity, LQG)
Loop-quantum-gravity application of L0 2-category and simplicial-set theory. Ponzano and Regge 1968 proposed the first state-sum model of…
Ashtekar area-spectrum A = 8 pi l_P^2 gamma sum sqrt(j(j+1))
Ashtekar-Lewandowski area-spectrum framework (Ashtekar 1986 PRL 57, 2244; Rovelli-Smolin 1995 Nucl Phys B 442, 593). Setup: in loop quantum…
Theorem: 4 j (j+1) - 3 = 0 at j = 1/2 (LQG area-spectrum minimum)
Theorem (LQG-area-min canonical): at j = 1/2, the SU(2) Casimir j(j+1) = 1/2 * 3/2 = 3/4, hence 4 j(j+1) = 3 exactly. Canonical sympy pin:…
Theorem: 1 - 1 = 0 (trivial-group orbit count)
Theorem (Burnside-trivial canonical): for the trivial group G = {e} acting on any set X, the Burnside / Cauchy-Frobenius lemma gives #…
Ashtekar variables (1986)
A Ashtekar 1986 SU(2) self-dual variables; basis of loop-quantum-gravity Hamiltonian formulation; modern Lewandowski + Sahlmann update.
Spin network (Rovelli-Smolin 1995)
Rovelli-Smolin 1995 spin-network states; discrete area + volume operators; basis of LQG kinematical Hilbert space.
EPRL spin-foam (2008)
Engle-Pereira-Rovelli-Livine 2008 spin-foam model; semi-classical limit recovers Regge-action; modern numerical + Lefschetz-thimble.
LQC (Bojowald 2001)
M Bojowald 2001 loop-quantum-cosmology; big-bounce vs big-bang; modern Ashtekar-Singh 2011 effective-equations.
Immirzi (1996)
G Immirzi 1996 quantization parameter; black-hole entropy fixes gamma=0.2375; ongoing debate of physical meaning.
Ashtekar variables (1986)
A Ashtekar 1986 self-dual SU(2)-connection formulation of GR; foundation of LQG canonical-quantization scheme.
Immirzi parameter (1996)
G Immirzi 1996 Barbero-Immirzi γ-parameter ambiguity; modern black-hole-entropy fitting + log-corrections.
Spin networks (Rovelli-Smolin 1995)
C Rovelli-L Smolin 1995 spin-network basis; modern quantum-geometry area + volume operator discrete spectra.
BH entropy (Rovelli 1996)
C Rovelli 1996 microstate-counting LQG black-hole-entropy = A/4 + log corrections; modern semi-classical recovery.
GFT (Oriti 2006)
D Oriti 2006 group-field-theory; modern emergent-spacetime as condensate; cosmological-bounce + semi-classical limit.
Loop cosmology (Bojowald 2001)
M Bojowald 2001 loop-quantum-cosmology + bounce-replaces-Big-Bang; modern observational-tests via primordial-power-spectrum.