Editing Loop quantum gravity (section)

=== The master constraint ===
Thiemann's Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Hamiltonian constraint equations in terms of a single master constraint <math>M</math>, which involves the square of the constraints in question. An initial objection to the use of the master constraint was that on first sight it did not seem to encode information about the observables; because the Master constraint is quadratic in the constraint, when one computes its Poisson bracket with any quantity, the result is proportional to the constraint, therefore it vanishes when the constraints are imposed and as such does not select out particular phase space functions. However, it was realized that the condition 
<math display="block">\{ O , \{ O , M \} \}_{M = 0} = 0,</math>
is where <math>O</math> is at least a twice differentiable function on phase space is equivalent to <math>O</math> being a weak Dirac observable with respect to the constraints in question. So the master constraint does capture information about the observables. Because of its significance this is known as the master equation.{{sfn|Thiemann|2006a|pp=2211–2247}}

That the master constraint Poisson algebra is an honest Lie algebra opens the possibility of using a method, known as group averaging, in order to construct solutions of the infinite number of Hamiltonian constraints, a physical inner product thereon and [[Dirac observables]] via what is known as [[refined algebraic quantization]], or RAQ.<ref>{{cite arXiv |last=Thiemann |first=Thomas |title=Introduction to Modern Canonical Quantum General Relativity |date=2001-10-05 |eprint=gr-qc/0110034}}</ref>