Editing Loop quantum gravity (section)

=== Hamiltonian constraint of LQG ===
{{main| Hamiltonian constraint of LQG}}

In the long history of canonical quantum gravity formulating the Hamiltonian constraint as a quantum operator ([[Wheeler–DeWitt equation]]) in a mathematically rigorous manner has been a formidable problem. It was in the loop representation that a mathematically well defined Hamiltonian constraint was finally formulated in 1996.{{sfn|Thiemann|1996|pp=257–264}} We leave more details of its construction to the article [[Hamiltonian constraint of LQG]]. This together with the quantum versions of the Gauss law and spatial diffeomorphism constrains written in the loop representation are the central equations of LQG (modern canonical quantum General relativity).

Finding the states that are annihilated by these constraints (the physical states), and finding the corresponding physical inner product, and observables is the main goal of the technical side of LQG.

An important aspect of the Hamiltonian operator is that it only acts at vertices (a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning). More precisely, its action is non-zero on at least vertices of valence three and greater and results in a linear combination of new spin networks where the original graph has been modified by the addition of lines at each vertex together and a change in the labels of the adjacent links of the vertex.{{Citation needed|date=July 2021}}