This question is related to my earlier, even more open-ended question on tropilcalization. I will give some background and ask my question at the end.
On R, consider the family of commutative, associative operations ⊕h, indexed by positive h, given by x ⊕h y = -h ln( exp(-x/h) + exp(-y/h) ). For h>0, the semigroup (R,⊕h) is isomorphic to the normal additive groupsemi (R>0,+). But as h → 0, for fixed x and y we have the limit x ⊕h y → min(x,y). This defines the tropical addition, and it's conventional to include the additive unit ∞ = -h ln(0).
There is a continuous/integral version of the observation that in the limit, + (in the guise ⊕h) becomes max. Indeed, let f : Rn → R be a continuous function bounded below, and assume that f grows to +∞ in all directions, fast enough so that for any h>0, the integral ∫Rn exp(-f(x)/h) dx converges (or anyway for h small enough; if it converges for any h then it does for all smaller h, and to converge for small h requires only very mild growth rates; as |x|ε for ε>0 is certainly good enough). Then asymptotically as h → 0, the integral is supported at the (or, rather, in a formal neighborhood of the) globally-minimal values of f. To make the correspondence explicit, note that ∫Rn exp(-f(x)/h) dx is (exp of -h-1 times) the "⊕h integral" of f, whereas the "⊕0 integral" of a function is its global minimum value.
There is another fact about asymptotic integrals, related by "Wick rotation", which is what the physicists call it any time you switch a variable from pure-real to pure-imaginary. As above, let f : Rn → R continuous and growing reasonably quickly to infinity, but this time for real non-zero h consider the integral ∫Rn exp(-f(x)/(ih)) dx, where i = √-1. The integral never converges absolutely (and so does not exist in the sense of Lebesgue), but it converges conditionally as a Riemann integral, e.g. if f is differentiable and given mild conditions on the growth of the norm of the derivative. (If f grows at least as fast as |x|1+ε, we're fine, I think.) In any case, let's assume that the integral converges conditionally for small enough (real, non-zero) h. Then the method of stationary phase shows that asymptotically, the integral is supported at (formal neighborhoods of) critical points of f.
My question is this: Is there a version of "tropical arithmetic" like the operation ⊕h defined above but related to the Wick-rotated integral? The most naive approach, replacing h by ih and so considering x ⊕ih y = -ih ln( exp(-x/ih) + exp(-y/ih) ), is not defined because of the problem of picking a branch of the logarithm. But perhaps this problem can be fixed for small h, or by approximating each pure-imaginary ih by ih+ε for some very small positive ε? Put another way: what is the operation on numbers that corresponds to {critical points} in the same way that min(x,y) corresponds to {global minimum}?
