By coincidence I just started responding to that question a few minutes ago in the preceding post! Your question reminded me! Here is what I had written so far:
==quote post #15==
...I'd like to find a good way for a new person to find out about the Loop Gravity UNSETTLED HAMILTONIAN SITUATION.
As far as I know (AFAIK) there is no Hamiltonian at present, only several proposals.
They have not been fully worked out.
In at least one case a proposal has been worked out in 3D but not in 4D. In at least one other case only an idea has been presented.
I would like to get other people's ideas. Some may think that LQG has a definite hamiltonian (they may disagree with me.)
First, I can give some indication of the unsettled situation by linking to some technical papers but this is definitely NOT A GOOD INTRODUCTION because exploratory proposals are the complete opposite from textbook-expository style introductions. So these are just things to have heard of and realize how much in flux the situation is. Just to have heard of, not even to know anything definite about.
Laurent Freidel is certainly someone to watch and he and Valentin Bonzom have one:
The Hamiltonian constraint in 3d Riemannian loop quantum gravity
"...This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity."
Carlo Rovelli and Alesci have one:
A regularization of the hamiltonian constraint compatible with the spinfoam dynamics
"...The resulting constraint can generate the 1-4 Pachner moves and is therefore more compatible with the dynamics defined by the spinfoam formalism. We calculate its matrix elements and observe the appearence of the 15j Wigner symbol in these."
Etera Livine and Valentin Bonzom have one:
A new Hamiltonian for the Topological BF phase with spinor networks
"...We introduce a new scalar Hamiltonian, based on recent works in quantum gravity and topological models, which is different from the plaquette operator..."
It's really important that the Hamilton be graph-changing, and e.g. be capable of a 1-to-4 Pachner move. Space can expand by giving birth to new vertices. I don't understand how this deficiency persisted so long. It's a good sign that the 15j Wigner symbol shows up (basic to spinfoam dynamics). Also I just noticed that Valentin Bonzom, a young postdoc researcher, shows up in two of the three cases.
In addition to those three, there is also another Hamilton proposal from Etera Livine, Daniele Oriti, and James Ryan
Effective Hamiltonian Constraint from Group Field Theory
"...Our strategy is to expand group field theories around non-trivial classical solutions and to interpret the induced quadratic kinematical term as defining a Hamiltonian constraint on the group field and thus on spin network wave functions..."