If I have the Hamiltonian operator of Loop Quantum gravity H(x) with x∈R3 then the Master constraint is expressed by the time evolution operator ∫Dtexp(t∫d3xH†(x)H(x)) can be turned into a path integral. First I do a gauge fixing and I let t=const. such that integration over this variable can be omitted.
Now if I know the eigenstates and the corresponding eigenvalues of the operator I=∫d3xH†(x)H(x) I can derive a path integral. Some spin network states are eigenvectors of I. The inner product between spin network states ei=|Γi> are defined by
<Γi|Γj>=δij(∗)
up to diffeomorphism. The spectral theorem tells us that
I=∑∞k=1λkek⊗e†k
with eigenvalues λk. But if I use the orthogonality condition (*) then I will get a very simple statement that spin network states will not be changed under the action of I. Should I use a Jordan Block matrix for the spectral theorem where off-diagonal contributions occur on multiple eigenvalues in the diagonalized matrix? Or is the error in the condition (*)? Spin networks must change if time elapses. How a path integral can be defined here where I sum over all spin network states and not over Ashtekar variables and tedrads?