Hamiltonian constraint of LQG information

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In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, ${\displaystyle q_{ab}(x)}$, on the spatial slice (the distance function induced on the spatial slice by the spacetime metric), and its conjugate momentum variable related to the extrinsic curvature, ${\displaystyle K^{ab}(x)}$, (this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time).^[1] These are the metric canonical coordinates.

Dynamics such as time-evolutions of fields are controlled by the Hamiltonian constraint.

The identity of the Hamiltonian constraint is a major open question in quantum gravity, as is extracting of physical observables from any such specific constraint.

In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable.^[2] The Hamiltonian was much simplified in this reformulation. This led to the loop representation of quantum general relativity^[3] and in turn loop quantum gravity.

Within the loop quantum gravity representation Thomas Thiemann was able to formulate a mathematically rigorous operator as a proposal as such a constraint.^[4] Although this operator defines a complete and consistent quantum theory, doubts have been raised as to the physical reality of this theory due to inconsistencies with classical general relativity (the quantum constraint algebra closes, but it is not isomorphic to the classical constraint algebra of GR, which is seen as circumstantial evidence of inconsistencies definitely not a proof of inconsistencies), and so variants have been proposed.