Ashtekar variables
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 metricon 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.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.[2] Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity[3] and in turn loop quantum gravity and quantum holonomy theory.[4] Let us introduce a set of three vector fieldsare called a triad or drei-bein (German literal translation, "three-leg").which behave like indices of flat-space (the corresponding "metric" which raises and lowers internal indices is simplyIt is then easy to verify from the first orthogonality relation, employingthat we have obtained a formula for the inverse metric in terms of the drei-beins.Actually what is really considered is which involves the "densitized" drei-beinthe metric times a factor given by its determinant.is not unique, and in fact one can perform a local in space rotation with respect to the internal indicesThe densitized drei-bein is the conjugate momentum variable of this three-dimensional SU(2) gauge field (or connection)in that it satisfies the Poisson bracket relation The constantis the Immirzi parameter, a factor that renormalizes Newton's constantThe 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 choiceis 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 simplification raised new hopes for the canonical quantum gravity programme.[5] 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.[6] When one quantizes the theory it is a difficult task to ensure that one recovers real general relativity, as opposed to complex general relativity.There were serious difficulties in promoting this quantity to a quantum operator.In 1996 Thomas Thiemann who was able to use a generalization of Ashtekar's formalism to real connections (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.[7][8] 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.A purely tensorial proof of the new variables in terms of triads was given by Goldberg[12] and in terms of tetrads by Henneaux, Nelson, & Schomblond (1989).