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, 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, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU(2) 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.

that are orthogonal, that is,

are called a triad or drei-bein). Define the dual drei-bein as

We then have the two orthogonality relationships

(this comes from substituting the formula for the dual drei-bein in terms of the drei-bein and using the orthogonality of the drei-beins).

and

and using the linear independence that

we have obtained a formula for the inverse metric in terms of the drei-beins. The drei-beins is locally flat). Actually what is really considered is

which involves the "densitized" drei-bein instead densitized is not unique, and in fact one can perform a local-in-space rotation gauge invariance. Now if one is going to operate on objects that have internal indices, one needs to introduce an appropriate derivative (covariant derivative will be

is the usual Levi-Civita connection is the so-called spin connection. Let us take the configuration variable to be

The densitized drei-bein in that it satisfies the Poisson bracket relation

is the Immirzi parameter, a factor that renormalizes Newton's constant The densitized drei-bein (the negative of the imaginary number 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 formidable second term vanish, and the remaining term became polynomial in his new variables. This simplification 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

There were serious difficulties in promoting this quantity to a quantum operator. In 1996 Thomas Thiemann takes real values) and in particular devised a way of simplifying the original Hamiltonian, together with the second term. He was also able to promote this Hamiltonian constraint to a well defined quantum operator within the loop representation.

Lee Smolin & Ted Jacobson, and Joseph Samuel, 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, Nelson, & Schomblond (1989).

References

Further reading