How is tropicalization like taking the classical limit?

Question

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:

1. 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.
2. 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?
3. 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.
4. 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.

Answer 1

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 +.

Answer 2

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

Answer 3

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.

Answer 4

I think the math and physics questions here are sufficiently different.

I'm not familiar with tropic geometry, but I searched and found very readable introduction at math.CO/0408099.