In this post, I shall discuss certain relations between Weyl-integrable geometry, General Relativity, and the scalar-tensor duality. The principal motivation is that the gauge group up-lift from the Weyl fiber bundle onto the renormalization Lie algebra can handle central infrared paradoxes that GR faces given that the scalar-tensor duality gives a description of the gravitational field in terms of a space-time metric and a scalar field with an analytically closed Einstein-Riemann frame that geometrically embeds standard inflationary cosmological models as well as relativistic quantum geometric ones. We start with the following action for a scalar-tensor theory of gravity in the Jordan vacuum-frame:

$\Re$ the Ricci scalar, ${\tilde \omega \left( \Phi \right)}$ a function of the scalar field $\Phi$ and ${\tilde V\left( \Phi \right)}$ is a scalar potential. The action in terms of the redefined field $\varphi = - {\rm{In}}\left( {G\Phi } \right)$ reduces to:

Recalling that the Weyl-integrable spacetime action in this frame is given by:

with the Weylian connection and Riemannian curvature satisfying:

as well as the Palatini variational properties of $\omega \left( \varphi \right)$ and $V\left( \varphi \right)$:

thus yielding the affine connection:

which is the characteristic non-metricity condition for a Weyl-integrable geometry

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: