Editing Loop quantum gravity (section)

=== Quantum constraints as the equations of quantum general relativity ===
The classical result of the Poisson bracket of the smeared Gauss' law <math display="inline">G(\lambda) = \int d^3x \lambda^j (D_a E^a)^j</math> with the connections is
<math display="block">\{ G(\lambda), A_a^i \} = \partial_a \lambda^i + g \epsilon^{ijk} A_a^j \lambda^k = (D_a \lambda)^i.</math>

The quantum Gauss' law reads
<math display="block">\hat{G}_j \Psi (A) = - i D_a {\delta \lambda \Psi [A] \over \delta A_a^j} = 0.</math>

If one smears the quantum Gauss' law and study its action on the quantum state one finds that the action of the constraint on the quantum state is equivalent to shifting the argument of <math>\Psi</math> by an infinitesimal (in the sense of the parameter <math>\lambda</math> small) gauge transformation,
<math display="block">\left [ 1 + \int d^3x \lambda^j (x) \hat{G}_j \right] \Psi (A) = \Psi [A + D \lambda] = \Psi [A],</math>
and the last identity comes from the fact that the constraint annihilates the state. So the constraint, as a quantum operator, is imposing the same symmetry that its vanishing imposed classically: it is telling us that the functions <math>\Psi [A]</math> have to be gauge invariant functions of the connection. The same idea is true for the other constraints.

Therefore, the two step process in the classical theory of solving the constraints <math>C_I = 0</math> (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the 'evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions <math>\Psi</math> of the quantum equations <math>\hat{C}_I \Psi = 0</math>. This is because it solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because <math>\hat{C}_I</math> is the quantum generator of gauge transformations (gauge invariant functions are constant along the gauge orbits and thus characterize them).{{sfn|Thiemann|2003|pp=41–135}} Recall that, at the classical level, solving the admissibility conditions and evolution equations was equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in canonical quantum gravity.