There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and Classical mechanics. I think I first learned about this years ago on *This Week's Finds*. I'm wondering if anyone can give me a precise mathematical statement of this "fact". Or to the right introduction to tropical mathematics.

I can do the beginning. In classical mechanics, roughly (lower down I will mention some ways what I am about to say is false), when a system transitions from one configuration to another, it takes the route that *minimizes* some "action" (this idea dates at least to Maupertuis in 1744, and Wikipedia gives ancient-Greek analogs). Thus, for a system to transition from state $A$ to state $C$ in two seconds, after one second it is in the state $B$ that minimizes the sum of the action to get from $A$ to $B$ plus the action to get from $B$ to $C$. For comparison, quantum mechanics assigns to the pair $A,B$ an "amplitude", and the amplitude to go from $A$ to $C$ in two seconds is the sum over all $B$ of the amplitude to go from $A$ to $B$ times the amplitude to go from $B$ to $C$. (This is the basic principle of Heisenberg's matrix mechanics.) Anyway, we can understand both situations within the same language by considering the a matrix, indexed by states, filled with either the actions or the amplitudes to transition. In the quantum case, the matrix multiplication is the one inherited from the usual $(\times, +)$ arithmetic on $\mathbb{C}$. In the classical case, it is the $(+, \min)$ arithmetic of the tropical ring $\mathbb{T}$.

Let's be more precise. To any path through the configuration space of your system, Hamilton defines an action $\operatorname{Action}(\mathrm{path})$. The classically allowed trajectories are the critical paths of the action (rel boundary values), whereas if you believe in the path integral, the quantum amplitude is $$\int \exp\left(\frac{i}{\hbar} \operatorname{Action}(\mathrm{path}) \right) \mathrm{d}(\mathrm{path}),$$ where the integral ranges over all paths with prescribed boundary values and the measure $\mathrm{d}(\mathrm{path})$ doesn't exist (I said "if"). Here $\hbar$ is the reduced Planck constant, and the stationary-phase approximation makes it clear that as $\hbar$ goes to $0$, the integral is supported along classically allowed trajectories.

Of course, the path integral doesn't exist, so I will describe one third (and more rigorous) example, this time in statistical, not quantum, mechanics. Let $X$ be the space of possible configurations of your system, and let's say that $X$ has a natural measure $\mathrm{d}x$. Let $E : X \to \mathbb{R}$ be the energy of a configuration. Then at temperature $T$, the probability that the system is in state $x$ is (unnormalized) $\exp(-(kT)^{-1} E(x))$, by which I mean if $f : X \to \mathbb{R}$ is any function, the expected value of $f$ is (ignoring convergence issues; let's say $X$ is compact, or $E$ grows quickly and $f$ does not, or...):

$$\langle f \rangle_T = \frac{\int_X \exp(-(kT)^{-1} E(x)) f(x) \mathrm{d}x}{\int_X \exp(-(kT)^{-1} E(x)) \mathrm{d}x}$$

Here $k$ is Boltzmann's constant. It's clear that as $T$ goes to $0$, the above integral is concentrated at the $x$ that minimize $E$. In tropical land, addition and hence integration is just minimization, so in the $T \to 0$ limit, the integral becomes some sort of "tropical" integral.

Here are some of the issues that I'm having:

- When you slowly cool a system, it doesn't necessarily settle into the state that globally minimizes the energy, just into a locally-minimal state. And good thing too, else there would not be chocolate bars.
- The problems are worse for mechanics. Classically allowed paths are not necessarily even local minima, just critical points of the Action function, so the analogy between quantum and warm systems isn't perfect. Is there something like $\min$ that finds critical points rather than minima?
- More generally, the path integral is very attractive, and definitely describes a "matrix", indexed by the configuration space. But for generic systems, connecting any two configurations are many classically-allowed trajectories. So whatever the classical analogue of matrix mechanics is, it is a matrix valued in sets (or sets with functions to the tropical ring), not just in tropical numbers.
- Other than "take your equations and replace every $+$ with $\min$ and every $\times$ with $+$", I don't really know how to "tropicalize" a mathematical object.