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* : **R**^{n} → **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* : **R**^{n} → **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}?