In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric on the spatial slice and the metric's conjugate momentum, which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time. These are the metric canonical coordinates. In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekarvariables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU gauge field and its complementary variable.
Overview
Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity and in turn loop quantum gravity and quantum holonomy theory. Let us introduce a set of three vector fields, that are orthogonal, that is, The are called a triad or drei-bein. There are now two different types of indices, "space" indices that behave like regular indices in a curved space, and "internal" indices which behave like indices of flat-space. Define the dual drei-bein as We then have the two orthogonality relationships where is the inverse matrix of the metric . and . It is then easy to verify from the first orthogonality relation that we have obtained a formula for the inverse metric in terms of the drei-beins - the drei-beins may be thought of as the "square-root" of the metric. Actually what is really considered is which involves the densitized drei-bein instead. One recovers from the metric times a factor given by its determinant. It is clear that and contain the same information, just rearranged. Now the choice for is not unique, and in fact one can perform a local in space rotationwith respect to the internal indices without changing the metric. This is the origin of the gauge invariance. Now if one is going to operate on objects that have internal indices one needs to introduce an appropriate derivative, for example the covariant derivative for the object will be where is the usual Levi-Civita connection and is the so-called spin connection. Let us take the configuration variable to be where and. The densitized drei-bein is the conjugate momentum variable of this three-dimensional SU gauge field , in that it satisfies the Poisson bracket relation The constant is the Immirzi parameter, a factor that renormalizes Newton's constant. The densitized drei-bein can be used to re construct the metric as discussed above and the connection can be used to reconstruct the extrinsic curvature. Ashtekar variables correspond to the choice , is then called the chiral spin connection. The reason for this choice of spin connection was that Ashtekar could much simplify the most troublesome equation of canonical general relativity, namely the Hamiltonian constraint of LQG; this choice made its second, formidable, term vanish and the remaining term became polynomial in his new variables. This raised new hopes for the canonical quantum gravity programme. However it did present certain difficulties. Although Ashtekar variables had the virtue of simplifying the Hamiltonian, it has the problem that the variables become complex. When one quantizes the theory it is a difficult task to ensure that one recovers real general relativity as opposed to complex general relativity. Also the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with. There were serious difficulties in promoting this quantity to a quantum operator. It was Thomas Thiemann who was able to use the generalization of Ashtekar's formalism to real connections and in particular devised a way of simplifying the original Hamiltonian, together with the second term, in 1996. He was also able to promote this Hamiltonian constraint to a well defined quantum operator within the loop representation. For an account of these developments see John Baez's homepage entry, The Hamiltonian Constraint in the Loop Representation of Quantum Gravity. Smolin and others independently discovered that there exists in fact a Lagrangian formulation of the theory by considering the self-dual formulation of the tetradic Palatini action principle of general relativity. These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg and in terms of tetrads by Henneaux et al.
