The Hamiltonian operator of Loop quantum gravity is a totally constraint system
H=∫Σd3x (NH+NaVa+G)
Here, Σ is a 3-dimensional hypersurface; a slice of spacetime. Moreover, H is the Hamiltonian constraint, Va the diffeomorphism constraint, G the Gauss law term and N,Na corresponding constraint generators. In research literature this Hamiltonian was criticized to be not hermitean and would not form a Lie algebra from its generators. The variables of the theory are Ashtekar's variable Aia and the triad Eia. Therefore the Master constraint
M:=∫Σd3x H2/√detq
with 3-d-metric qab was introduced that solves these issues. Loop quantum gravity can be treated canonically, but according to this paper:
http://arxiv.org/abs/0911.3432
one can derive a path integral from the Master constraint. I can't understand the derivation of it (especially with the measure factor). Question: Is there a plausible path integral in 4-d-spacetime that computes spin foam amplitudes?
What is if I treat Loop Quantum Gravity with the path integral with action
S=∫d4x (Eia˙Aai−NH+NaVa+G) is it plausible (this action is mentioned in one of my introductory textbooks) despite the non-hermiticity of the Hamiltonian? Or would this action lead to significant errors?
P.S:: is the path integral ∫d[Eia]d[Aai]d[NMaster]exp(iEia˙Aai−i∫dtNMasterM) =∫d[Eia]d[Aai]exp(iEia˙Aai)δ(M) a better version than the path integral induced by the action (⋆)?
This post imported from StackExchange Physics at 2016-12-24 22:42 (UTC), posted by SE-user kryomaxim