The basic theory was outlined by Bryce DeWitt[1] in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann[2] using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac.
Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.
In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept.
These equations describe a "flow" or orbit in phase space generated by the Hamiltonian
In theories with constraints there is also the reduced phase space quantization where the constraints are solved at the classical level and the phase space variables of the reduced phase space are then promoted to quantum operators, however this approach was thought to be impossible in General relativity as it seemed to be equivalent to finding a general solution to the classical field equations.
However, with the fairly recent development of a systematic approximation scheme for calculating observables of General relativity (for the first time) by Bianca Dittrich, based on ideas introduced by Carlo Rovelli, a viable scheme for a reduced phase space quantization of Gravity has been developed by Thomas Thiemann.
Imposing these constraints classically are basically admissibility conditions on the initial data, also they generate the 'evolution' equations (really gauge transformations) via the Poisson bracket.
Importantly the Poisson bracket algebra between the constraints fully determines the classical theory – this is something that must in some way be reproduced in the semi-classical limit of canonical quantum gravity for it to be a viable theory of quantum gravity.
In Dirac's approach it turns out that the first class quantum constraints imposed on a wavefunction also generate gauge transformations.
This is because it obviously solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because
A diffeomorphism can be thought of as simultaneously 'dragging' the metric (gravitational field) and matter fields over the bare manifold while staying in the same coordinate system, and so are more radical than invariance under a mere coordinate transformation.
This symmetry arises from the subtle requirement that the laws of general relativity cannot depend on any a-priori given space-time geometry.
This diffeomorphism invariance has an important implication: canonical quantum gravity will be manifestly finite as the ability to `drag' the metric function over the bare manifold means that small and large `distances' between abstractly defined coordinate points are gauge-equivalent!
A more rigorous argument has been provided by Lee Smolin: “A background independent operator must always be finite.
This is necessary, because the scale that the regularization parameter refers to must be described in terms of a background metric or coordinate chart introduced in the construction of the regulated operator.
Conversely, if the terms that are nonvanishing in the limit the regulator is removed have no dependence on the background metric, it must be finite.” In fact, as mentioned below, Thomas Thiemann has explicitly demonstrated that loop quantum gravity (a well developed version of canonical quantum gravity) is manifestly finite even in the presence of all forms of matter!
However, in other work, Thomas Thiemann admitted the need for renormalization as a way to fix quantization ambiguities.
[1] In perturbative quantum gravity (from which the non-renormalization arguments originate), as with any perturbative scheme, one makes the reasonable assumption that the space time at large scales should be well approximated by flat space; one scatters gravitons on this approximately flat background and one finds that their scattering amplitude has divergences which cannot be absorbed into the redefinition of the Newton constant.
Canonical quantum gravity theorists do not accept this argument; however they have not so far provided an alternative calculation of the graviton scattering amplitude which could be used to understand what happens with the terms found non-renormalizable in the perturbative treatment.
This 'quantization' of geometric observables is in fact realized in loop quantum gravity (LQG).
While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque.
This is indicated in moving to the Hamiltonian formalism by the fact that their conjugate momenta, respectively
It can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint.
The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations.
and hence Schrödinger's equation reduces to a constraint: Using metric variables lead to seemingly unsurmountable mathematical difficulties when trying to promote the classical expression to a well-defined quantum operator, and as such decades went by without making progress via this approach.
Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables.
The aim of the loop representation, in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of Gauss gauge invariant states.
Within the loop representation Thiemann has provided a well defined canonical theory in the presence of all forms of matter and explicitly demonstrated it to be manifestly finite!
However, as LQG approach is well suited to describe physics at the Planck scale, there are difficulties in making contact with familiar low energy physics and establishing it has the correct semi-classical limit.
[4][5] This work has later been further developed by him and his collaborators to an approach of identifying and quantizing time amenable to a large class of scale-invariant dilaton gravity-matter theories.