Flow of observables

Suppose a classical Hamiltonian system $(M, ω)$ with canonical coordinates. A function $f \in C^{\infty} (M)$ has associated a Hamiltonian vector field $X_{f}$ with a flow $ϕ_{s}$ . Now, we can define a family of transformations in the algebra of smooth functions (i.e., observables)

{\tilde{ϕ}}_{s} : C^{\infty} (M) ⟼ C^{\infty} (M)

by means of

g ⟼ {\tilde{ϕ}}_{s} (g) = g \circ ϕ_{s}

This is completely related to the Heisenberg vs Schrodinger picture discussion. The flow $ϕ_{s}$ in $M$ is some kind of evolution of states, like in the Schrodinger picture. And the flow ${\tilde{ϕ}}_{s}$ corresponds to Heisenberg picture.

Observe that

\frac{d}{d s} {\tilde{ϕ}}_{s} (g) = \frac{d}{d s} (g \circ ϕ_{s}) = d g_{ϕ_{s} (-)} \cdot X_{f} = X_{f} (g \circ ϕ_{s}) = {{\tilde{ϕ}}_{s} (g), f}

and

{\tilde{ϕ}}_{0} (g) = g .

And also, ${\tilde{ϕ}}_{s}$ respect the Poisson algebra structure.

Reciprocally, if we start with a function $f$ and we are able to associate a family of transformations ${{\tilde{ϕ}}_{s}^{f}}$ from $C^{\infty} (M)$ to $C^{\infty} (M)$ that preserve the structure of Poisson algebra and such that

$\frac{d}{d s} {\tilde{ϕ}}_{s}^{f} (g) = {{\tilde{ϕ}}_{s}^{f} (g), f}$
${\tilde{ϕ}}_{0}^{f} (g) = g$
we can define a flow $ϕ_{s}$ in $M$ such that

{\tilde{ϕ}}_{s}^{f} (g) = {\tilde{ϕ}}_{s} (g)

We proceed by taking the vector field ${-, f}$ and the associated flow $ϕ_{s}$ , and using the two required properties, above. Since

\frac{d}{d s} {\tilde{ϕ}}_{s} (g) = \frac{d}{d s} {\tilde{ϕ}}_{s}^{f} (g)

and

{\tilde{ϕ}}_{0} (g) = {\tilde{ϕ}}_{0}^{f} (g)

we can show that ${\tilde{ϕ}}_{s} (g) = {\tilde{ϕ}}_{s}^{f} (g)$ by fixing a point $P \in M$ .
We indeed have a duality, between transformations in $M$ and transformation in $C^{\infty} (M)$ . Why do we pass from the flow in the phase space $M$ to the "flow of observables"? Because this way, I guess, we can formulate everything in terms of algebras, and we get a deeper insight in the connection with Quantum Mechanics. Compare this with the discussion of derivations and 1-parameter group of automorphisms of the matrix algebra.