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 demonstrate 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 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 evolves for a time that undergoes jumps during intervals centered at times is given by:

So, the master equation:

is valid *iff* the Markovian approximation is faithful and valid only on time-scales longer than , hence the jump occurs during an interval centered on . Therefore, with the Hamiltonian:

where 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 acts on the output mode and 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 is given by the Lindblad master equation:

where

is the conservative part and is the time-dependent Hamiltonian of the system and the other terms refer to the bath of the interactive system and reflect the effect of measurements, and are the Kraus-operators, not necessarily hermitians and are typically explicitly dependent on time. The Kraus number depends on the bath. In the case where the system is a closed one, the Kraus operators vanish identically and the Lindblad master equation reduces to the quantum version of the Liouville equation, giving us:

with the Lindbladian superoperator acting on the density matrix and determines its dynamics. The associated space of operators is equipped with a Hilbert-Schmidt scalar product:

with the hermitian conjugate of . We now define a pair of adjoint superoperators and as follows:

Hence, we have:

with the trace-conservation property:

The solve quantum Master equation:

one typically introduces an evolution superoperator defined implicitly by:

where is the initial-time-density-matrix, and the superoperator evolution is given by:

And in this time-ordered exponential, time is monotonically increasing from left to right.

To prove:

note that it is true at since is the identity operator. Thus, from:

one finds that:

holds, and leads to:

entailing that it satisfies the Lindblad equation:

with initial condition . Now, for the evolution operator, one writes an expression for multi-time correlations for distinct observables. For:

the time-ordered correlation is:

and can be evaluated in the Heisenberg representation formalism by using the full Hamiltonian of the system plus its environment. Since the total density matrix factorizes at each observation time and the weak Lindblad Master equation coupling assumption holds in that formalism, the time-ordered two-time correlation function satisfies an evolution equation which is the dual to:

our proof is complete.

Now note that in:

the operator represents the initial density matrix of the system and the superoperator acts on all terms to its right.

Thus, we have the crucial …

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