The philosophy of loops is canonical, i.e., an analysis of the evolution of variables defined classically through a foliation of spacetime by a family of space-like three- surfaces ∑t. The standard choice is the three-dimensional metric gij, and its canonical conjugate, related to the extrinsic curvature. If the system is reparametrization invariant, the total hamiltonian vanishes, and this hamiltonian constraint is usually called the Wheeler-DeWitt equation. Choosing the canonical variables is fundamental, to say the least.
Abhay Ashtekar‘s insights stems from the definition of an original set of variables stemming from Einstein-Hilbert Lagrangian written in the form,
S = ∫ea ∧ eb ∧ Rcdεabcd —– (1)
where, ea are the one-forms associated to the tetrad,
ea ≡ eaμdxμ —– (2)
The associated SO(1, 3) connection one-form ϖab is called the spin connection. Its field strength is the curvature expressed as a two form:
Rab ≡ dϖab + ϖac ∧ ϖcb —– (3)
Ashtekar’s variables are actually based on the SU(2) self-dual connection
A = ϖ − i ∗ ϖ —– (4)
Its field strength is
F ≡ dA + A ∧ A —– (5)
The dynamical variables are then (Ai, Ei ≡ F0i). The main virtue of these variables is that constraints are then linearized. One of them is exactly analogous to Gauss’ law:
DiEi = 0 —– (6)
There is another one related to three-dimensional diffeomorphisms invariance,
trFijEi = 0 —– (7)
and, finally, there is the Hamiltonian constraint,
trFijEiEj = 0 —– (8)
On a purely mathematical basis, there is no doubt that Astekhar’s variables are of a great ingenuity. As a physical tool to describe the metric of space, they are not real in general. This forces a reality condition to be imposed, which is akward. For this reason it is usually prefered to use the Barbero-Immirzi formalism in which the connection depends on a free parameter, γ
Aia + ϖia + γKia —– (9)
ϖ being the spin connection, and K the extrinsic curvature. When γ = i, Ashtekar’s formalism is recovered, for other values of γ, the explicit form of the constraints is more complicated. Even if there is a Hamiltonian constraint that seems promising, was isn’t particularly clear is if the quantum constraint algebra is isomorphic to the classical algebra.
Some states which satisfy the Astekhar constraints are given by the loop representation, which can be introduced from the construct (depending both on the gauge field A and on a parametrized loop γ)
W (γ, A) ≡ trPeφγA —– (10)
and a functional transform mapping functionals of the gauge field ψ(A) into functionals of loops, ψ(γ):
ψ(γ) ≡ ∫DAW(γ, A) ψ(A) —– (11)
When one divides by diffeomorphisms, it is found that functions of knot classes (diffeomorphisms classes of smooth, non self-intersecting loops) satisfy all the constraints. Some particular states sought to reproduce smooth spaces at coarse graining are the Weaves. It is not clear to what extent they also approach the conjugate variables (that is, the extrinsic curvature) as well.
In the presence of a cosmological constant the hamiltonian constraint reads:
εijkEaiEbj(Fkab + λ/3εabcEck) = 0 —– (12)
A particular class of solutions expounded by Lee Smolin of the constraint are self-dual solutions of the form
Fiab = -λ/3εabcEci —– (13)
Loop states in general (suitable symmetrized) can be represented as spin network states: colored lines (carrying some SU(2) representation) meeting at nodes where intertwining SU(2) operators act. There is also a path integral representation, known as spin foam, a topological theory of colored surfaces representing the evolution of a spin network. Spin foams can also be considered as an independent approach to the quantization of the gravitational field. In addition to its specific problems, the hamiltonian constraint does not say in what sense (with respect to what) the three-dimensional dynamics evolve.