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:

The foundational crises caused by the wave function collapse need no introduction. Here I will address the obvious need for a collapse theory that accounts simultaneously for quantum entanglement, time-symmetry and eliminability of any incorporation of a 'decohering-observer'. The reason is clear, and not simply due to the universal wave functional collapse or the meta-entanglement Nakajima–Zwanzig-Wigner regress paradoxes. Here's how to appreciate the acuteness of the problem. Look at the following collapse equation:

ensuring the positivity of the reduced density matrix:

entails that the Lindblad quantum-jump equation:

is solvable iff ${L_k}$ commutes with the position-operator, with:

and ${\rm{d}}{W_\mu }$ the Weiner-quantum-increments.

However:

commutes with the energy operator. By the Heisenberg energy-time uncertainty principle, the collapse equation cannot be integrated to get the collapse-localization double-integral in 4-D:

Now applying the wave-particle duality to:

we face the problem of decoherence-induced space-time localization, since the time-local non-Markovian master equation:

cannot be solvable consistent with the Lindblad quantum-jump equation also being solvable.

And here's the paradox: the wave-particle duality is equivalent to the generalized Heisenberg uncertainty principle, and that would imply that the above collapse-localization double-integral in 4-D cannot have a solution and by entanglement entropy relation, that would contradict the Heisenberg time-energy uncertainty principle, thus the wave-particle duality ceases to make sense in the time-symmetry group representation of the corresponding Hilbert space of the quantum system.