Dequantization, tropical and idempotent mathematics

Abstract:
Idempotent mathematics  is the mathematics over semifiels and semirings with idempotent addition (this means that x+x=x). One of the most important idempotent semifields  is the well-known max-plus algebra. Modern tropical mathematics is the mathematics over the max-plus algebra.

In a sense, the traditional mathematics over numerical fields can be treated as a quantum theory, while the idempotent mathematics can be treated as a `classical shadow (or counterpart)' of the traditional one. There exists the corresponding procedure of an idempotent dequantization. A special case of this dequantization is
is the so-called  Maslov dequantization based on  logarithmic transforms used by E. Schroedinger (1926) and E. Hopf (1950). In this case the parameter of the dequantization coincides with the Planck constant taking pure imaginary values. The Maslov dequantization generates "tropicalization" and leads to tropical mathematics.

There exists a correspondence between interesting, useful and important constructions and results in the traditional mathematics and similar constructions and results in idempotent mathematics. This heuristic correspondence can be formulated in the spirit of the well-known N. Bohr's  correspondence principle in quantum mechanics; in fact, the two principles are intimately connected. For example, the Hamilton--Jacobi equation is an idempotent version of the Schroedinger equation, the variational principles of classical mechanics can be treated as an idempotent version of the Feynman path integral approach to quantum mechanics.  The Legendre transform turns out to be an idempotent version of the Fourier transform etc. A systematic and consistent application of the idempotent correspondence principle leads to a variety of results (often quite unexpected) in different areas including algebra, geometry, mathematical physics, differential equations, optimization, analysis and numerical analysis, stochastic problems, computer applications. There is an idempotent version of  the representation theory, so the concept of symmetry works in the framework of idempotent mathematics.