Editing Loop quantum gravity (section)

== Constraints and their Poisson bracket algebra ==
{{main|Poisson bracket|Hamiltonian constraint}}

=== Dirac observables ===
The constraints define a constraint surface in the original phase space. The [[Gauge theory|gauge]] motions of the constraints apply to all phase space but have the feature that they leave the constraint surface where it is, and thus the orbit of a point in the hypersurface under [[gauge transformations]] will be an orbit entirely within it. [[Dirac observables]] are defined as [[phase space]] functions, <math>O</math>, that [[Poisson commutativity|Poisson commute]] with all the constraints when the constraint equations are imposed,
<math display="block">\{ G_j , O \}_{G_j=C_a=H = 0} = \{ C_a , O \}_{G_j=C_a=H = 0} = \{ H , O \}_{G_j=C_a=H = 0} = 0,</math>
that is, they are quantities defined on the constraint surface that are invariant under the gauge transformations of the theory.

Then, solving only the constraint <math>G_j = 0</math> and determining the Dirac observables with respect to it leads us back to the [[ADM formalism|Arnowitt–Deser–Misner (ADM) phase space]] with constraints <math>H, C_a</math>. The dynamics of general relativity is generated by the constraints, 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.{{sfn|Baez|de Muniain|1994|loc=Part III, chapter 4}}