Hamiltonian constraint
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In loop quantum gravity, dynamics such as time-evolutions of fields are controlled by the Hamiltonian constraint. The identity of the Hamiltonian constraint is a major open question in quantum gravity, as is extracting of physical observables from any such specific constraint.
The Thiemann operator has been proposed as such a constraint. Although this operator defines a complete and consistent quantum theory, doubts have been raised as to the physical reality of this theory due to inconsistencies with classical general relativity, and so variants have been proposed.
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