Dequantization, tropical and idempotent mathematics
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.