The Hamiltonian operator of Loop quantum gravity is a totally constraint system

$$H = \int_\Sigma d^3x\ (N\mathcal{H}+N^a V_a+G)$$

Here, $\Sigma$ is a 3-dimensional hypersurface; a slice of spacetime. Moreover, $\mathcal{H}$ is the Hamiltonian constraint, $V_a$ the diffeomorphism constraint, $G$ the Gauss law term and $N,N^a$ 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 $A_a^i$ and the triad $E_a^i$. Therefore the Master constraint

$$M:=\int_\Sigma d^3x\ \mathcal{H}^2/\sqrt{\det q}$$

with 3-d-metric $q_{ab}$ 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

$$S = \int d^4x\ (E_a^i \dot{A_i^a}-N\mathcal{H}+N^a V_a+G) \tag{$\star$}$$ 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 $$\int d[E_a^i] d[A_i^a] d[N_{Master}] exp(i E_a^i \dot{A_i^a} - i\int dt N_{Master} M) $$ $$= \int d[E_a^i] d[A_i^a] exp(i E_a^i \dot{A_i^a}) \delta(M)$$ a better version than the path integral induced by the action $(\star)$?

This post imported from StackExchange Physics at 2016-12-24 22:42 (UTC), posted by SE-user kryomaxim