The Hamiltonian operator of loop quantum gravity is a totally constrained system $$H = \int_\Sigma \mathrm{d}^3x\ (N\mathcal{H}+N^a V_a+G).$$
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 $N,N^a$ the corresponding constraint generators.
In research literature, this Hamiltonian was criticized to be not Hermitian 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 $$M:=\int_\Sigma \mathrm{d}^3x\ \frac{\mathcal{H}^2}{\sqrt{\det q}}$$ with 3-d-metric $q_{ab}$ was introduced that solves these issues. Loop quantum gravity can be treated canonically, but according to this paper, 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 if I treat Loop Quantum Gravity with the path integral with action $$S = \int \mathrm{d}^4x\ (E_a^i \dot{A_i^a}-N\mathcal{H}+N^a V_a+G) \tag{$\star$}$$ 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 $$\int \mathrm{d}[E_a^i] \mathrm{d}[A_i^a] \mathrm{d}[N_{\text{Master}}] \exp\left(i E_a^i \dot{A_i^a} - i\int \mathrm{d}t N_{\text{Master}} M\right) \\ = \int \mathrm{d}[E_a^i] \mathrm{d}[A_i^a] \exp\left(i E_a^i \dot{A_i^a}\right) \delta(M)$$ a better version of the path integral induced by the action $(\star)$?
