Editing Loop quantum gravity (section)

=== Testing the master constraint ===
The constraints in their primitive form are rather singular, this was the reason for integrating them over test functions to obtain smeared constraints. However, it would appear that the equation for the master constraint, given above, is even more singular involving the product of two primitive constraints (although integrated over space). Squaring the constraint is dangerous as it could lead to worsened ultraviolet behaviour of the corresponding operator and hence the master constraint programme must be approached with care.

In doing so the master constraint programme has been satisfactorily tested in a number of model systems with non-trivial constraint algebras, free and interacting field theories.{{sfn|Dittrich|Thiemann|2006a|pp=1025–1066}}{{sfn|Dittrich|Thiemann|2006b|pp=1067–1088}}{{sfn|Dittrich|Thiemann|2006c|pp=1089–1120}}{{sfn|Dittrich|Thiemann|2006d|pp=1121–1142}}{{sfn|Dittrich|Thiemann|2006e|pp=1143–1162}} The master constraint for LQG was established as a genuine positive self-adjoint operator and the physical Hilbert space of LQG was shown to be non-empty,{{sfn|Thiemann|2006b|pp=2249–2265}} a consistency test LQG must pass to be a viable theory of quantum general relativity.