Any adequate account of how micro-causality and quantum coherence can explain the emergent-property of spacetime and how the Wheeler-DeWitt problem of time can be solved must incorporate a theory of how the Lindblad master equation solves the Markov quantum fluctuation problem as well as showing how the quantum Jarzynski-Hatano-Sasa relation can be homologically defined globally for both, Minkowski space and Friedmann-Robertson-Walker generalized Cartan space-times. This is a step towards those goals. Consider a wave-function $\left| {{\psi _t}^{S,m,c}} \right\rangle$ and the entropic quantum entanglement relation of the total system consisting of 'S', 'm' and the quantum-time measuring clock 'c' subject to Heisenberg's UP. It follows then that the probability that any given initial state $\left| {\psi _t^{S,m,c}} \right\rangle$ evolves for a time $T$ that undergoes $N$ jumps during intervals $\delta t$ centered at times ${t_1},...,{t_N}$ is given by:

So, the master equation:

is valid iff the Markovian approximation is faithful and valid only on time-scales longer than $1/{\Gamma _1}$, hence the jump occurs during an interval $\delta t \sim 1/{\Gamma _1}$ centered on ${t_i}$. Therefore, with the Hamiltonian:

where $\left( {\hat a,\hat b} \right);\left( {{{\hat a}^\dagger },{{\hat b}^\dagger }} \right)$ are the lowering/raising operators for the system and output mode respectively, it follows that the total system satisfies the master equation:

where the Pauli operator ${\sigma _z}$ acts on the output mode and $L_s^L$ is the Liouville superoperator. Given that it is a linear equation, it has a solution given as:

and so the evolution of the density matrix ${\rho _t}$ is given by the Lindblad master equation:

I will show that the Weyl-Wigner-Groenewold formalism of the cosmological quantum state is described by the Moyal-Wheeler-DeWitt deformation quantization equation with symplectic solutions in the Moyal-Wigner phase space: this is philosophically of foundational importance since quantum gravity necessitates that spacetime is quantized in a way described by the Moyal-Wheeler-DeWitt equation:

where in the deformation quantization Hilbert space formalism, the deformed operator in the scalar product relative to $\tilde H$ is given by:

with $\Theta$ a skew-symmetric matrix on ${\mathbb{R}^d}$, and $\chi \in f_{DQ}^{ps}\left( {{\mathbb{R}^d} \times {\mathbb{R}^d}} \right)$, $\chi \left( {0,0} \right) = 1$, and the deformation quantization differential operator is given by:

The central property of the cosmological quantum state is that it must entail the emergence of a classical universe satisfying all of the observable properties induced by the Friedmann-Robertson-Walker space-time, where such a FRW-space-time flat line-element, curved by quantum deformation, is:

with curved space-time metric: