This question is related to <a href="http://mathoverflow.net/questions/406/how-is-tropicalization-like-taking-the-classical-limit">my earlier, even more open-ended question</a> on tropilcalization. I will give some background and ask my question at the end.
On **R**, consider the family of commutative, associative operations ⊕<sub><i>h</i></sub>, indexed by positive _h_, given by <i>x</i> ⊕<sub><i>h</i></sub> <i>y</i> = -_h_ ln( exp(-<i>x</i>/<i>h</i>) + exp(-<i>y</i>/<i>h</i>) ). For _h_>0, the semigroup (**R**,⊕<sub><i>h</i></sub>) is isomorphic to the normal additive groupsemi (**R**<sub>>0</sub>,+). But as _h_ → 0, for fixed _x_ and _y_ we have the limit <i>x</i> ⊕<sub><i>h</i></sub> <i>y</i> → min(<i>x</i>,<i>y</i>). This defines the <i>tropical addition</i>, and it's conventional to include the additive unit ∞ = -<i>h</i> ln(0).
There is a continuous/integral version of the observation that in the limit, + (in the guise ⊕<sub><i>h</i></sub>) becomes max. Indeed, let _f_ : **R**<sup><i>n</i></sup> → **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 ∫<sub><b>R</b><sup><i>n</i></sup></sub> exp(-<i>f</i>(<i>x</i>)/<i>h</i>) <i>dx</i> 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 |<i>x</i>|<sup>ε</sup> 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 ∫<sub><b>R</b><sup><i>n</i></sup></sub> exp(-<i>f</i>(<i>x</i>)/<i>h</i>) <i>dx</i> is (exp of -<i>h</i><sup>-1</sup> times) the "⊕<sub><i>h</i></sub> integral" of _f_, whereas the "⊕<sub>0</sub> 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**<sup><i>n</i></sup> → **R** continuous and growing reasonably quickly to infinity, but this time for real non-zero _h_ consider the integral ∫<sub><b>R</b><sup><i>n</i></sup></sub> exp(-<i>f</i>(<i>x</i>)/(<i>ih</i>)) <i>dx</i>, 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 |<i>x</i>|<sup>1+ε</sup>, 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 ⊕<sub><i>h</i></sub> defined above but related to the Wick-rotated integral? The most naive approach, replacing _h_ by _ih_ and so considering _x_ ⊕<sub><i>ih</i></sub> _y_ = -_ih_ ln( exp(-<i>x</i>/<i>ih</i>) + exp(-<i>y</i>/<i>ih</i>) ), 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 <i>ih</i> by <i>ih</i>+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(<i>x</i>,<i>y</i>) corresponds to {global minimum}?