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  Master Constraint in LQG: How to derive a path integral?

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If I have the Hamiltonian operator of Loop Quantum gravity $H(x)$ with $x \in \mathbb{R}^3$ then the Master constraint is expressed by the time evolution operator $\int \mathcal{D}t exp(t \int d^3x H^\dagger(x) H(x))$ 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 $I = \int d^3x H^\dagger(x) H(x)$ I can derive a path integral. Some spin network states are eigenvectors of $I$. The inner product between spin network states $e_i = |\Gamma_i>$ are defined by

$<\Gamma_i|\Gamma_j> = \delta_{ij} (*)$

up to diffeomorphism. The spectral theorem tells us that

$I = \sum_{k=1}^\infty \lambda_k e_k \otimes e_k^\dagger$

with eigenvalues $\lambda_k$. 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?

asked Mar 16, 2017 in Mathematics by PatrickLinker (40 points) [ no revision ]

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