In this case, the Hamiltonian takes the form where and is the momentum conjugate to . Einstein's equations may be recovered by taking Poisson brackets with the Hamiltonian. Additional on-shell constraints, called secondary constraints by Dirac, arise from the consistency of the Poisson bracket algebra. These are and . This is the theory which is being quantized in approaches to canonical quantum gravity. 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. That is, we have: Spatial diffeomorphisms constraints of which there are an infinite number – one for value of , can be smeared by the so-called shift functions to give an equivalent set of smeared spatial diffeomorphism constraints, These generate spatial diffeomorphisms along orbits defined by the shift function . Hamiltonian constraints of which there are an infinite number, can be smeared by the so-called lapse functions to give an equivalent set of smeared Hamiltonian constraints, as mentioned above, the Poisson bracket structure between the (smeared) constraints is important because they fully determine the classical theory, and must be reproduced in the semi-classical limit of any theory of quantum gravity. The Wheeler–DeWitt equation The Wheeler–DeWitt equation (sometimes called the Hamiltonian constraint, sometimes the Einstein–Schrödinger equation) is rather central as it encodes the dynamics at the quantum level. It is analogous to Schrödinger's equation, except as the time coordinate, , is unphysical, a physical wavefunction can't depend on 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. This problem was circumvented and the formulation of a well-defined Wheeler–De-Witt equation was first accomplished with the introduction of Ashtekar–Barbero variables and the loop representation, this well defined operator formulated by Thomas Thiemann. Before this development the Wheeler–De-Witt equation had only been formulated in symmetry-reduced models, such as quantum cosmology. Canonical quantization in Ashtekar–Barbero variables and LQG Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but