How is tropicalization like taking the classical limit?

The analogy I've worked out from pieces here and there goes like this:

using the logarithm and exponential, we define for two real numbers x and y the following binary operation $x §_h y := h .ln( e^{x/h} + e^{y/h} )$ which depends on some positive real parameter $h$. Then we observe that as $h \rightarrow 0$ the number $x §_h y$ tends to $max(x,y)$. (Proof: assume without loss of generality that $x>y$, so $(y-x)/h <0$. But since $h. ln( e^{x/h} + e^{y/h} ) = h. ln( e^{x/h} . (1+e^{(y-x)/h} )$ as $h \rightarrow 0$ we tend to $h. ln (e^{x/h} . (1+0) ) = x = max(x,y)$. QED.)

Now, in quantum mechanics the canonical commutation relations between positions and momenta operators read $[x_u,p_v] = i \hbar \delta_{uv}$ and in the limit $\hbar \rightarrow 0$ those commutators thus tend to $0$, which says that we recover classical mechanics where everything commutes. And in quantum mechanics what matters are wavefunctions which are superpositions of things of the form $A.e^{iS/\hbar}$ where $A$ is some amplitude and $S$ some phase (the action of the path).

Going back to $§_h$ we can rewrite $e^{(x §_h y)/h } = e^{x/h}+ e^{y/h}$, and so there is your analogy: the tropical mathematics operation max(,) is some kind of classical limit of the (thereby quantum) operation +.

I started a page on nLab - matrix mechanics - providing links to discussions of this idea.

I think as for (1) you're mixing a very small class of exactly defined physical problems with a very small numbers of degrees of freedom, which is indeed described by things like equations, with the large physical systems, like real world, most of whose properties cannot be said in terms of "that equation does this, this variable goes there, etc.".

That's why chocolate bars are tasty. Theoretically, they will all eventually disappear. Get'em while you can.

(2) Yes and no. You do have to do a summation over all classical trajectories in quantum mechanics, but the answer may be often described simpler, e.g., as in the simplest case of perturbation theory, as a diagram expansion.