Lectures on Non-Perturbative Canonical Gravity

In the citing paper

“Now, in 2+1 dimensions, using the loop representation one can recast the final description in a purely combinatorial fashion (at least in the so-called 'time-like sector' of the theory).”

In the citing paper

“In 2 + 1 dimensions with Λ = 0, this approach is closely related to the first order formalism of Section 2.3, but takes as its fundamental observables the loop variables T0[γ] and T1[γ] of Equation (55).”

In the citing paper

“More precisely, loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints, written in analogy to the (3 + 1)-dimensional case [26], that take the form 63”

In the citing paper

“For the torus universe, this “loop transform” can be obtained explicitly [26, 29, 192], and written as a simple integral transform.”

In the citing paper

“The loop representation is complicated by the existence of Mandelstam identities [190] among holonomies of loops, but for the case of the torus universe, a complete, explicit description of the states is again possible [26, 29].”

In the citing paper

“In standard quantum cosmology the constraint equation is ˆ c 2 √ | ˆ p | ξ ( c ) = 0 with a solution √ | ˆ p | ξ ( c ) = δ ( c ) which is not unique.”

In the citing paper

“This is exactly what one obtains from the classical constraint H (E) = -6 c 2 √ | p | in standard quantum cosmology [17] by quantizing 3ˆ c = -iγl 2 P d / d p . In our framework, however, this is only an approximate equation valid for large scale factors.”

In the citing paper

“The Poisson bracket in (3) preserves the space of cylindrical functions and the Dirac delta is absorbed completely by the integrations involved in the definitions of the holonomy and flux. This fact was pointed out for the first time in [1, 8].”