Seminar on low-dimensional mathematics "Moscow-Petersburg"
March 28, 2003

DEQUANTIZATION OF MATHEMATICS
G.L. Litvinov

     This is a talk on heuristic aspects of Idempotent Mathematics
in the  spirit of  current works of V.P. Maslov and his collaborators.
Idempotent Mathematics can be treated as a result of a dequantizati-
on of the traditional Mathematics  as  the Planck constant tends to
zero taking pure imaginary values.   For example, the field of real
numbers can be treated as a quantum object  whereas idempotent semi-
rings  can  be  examined  as "classical" or "semiclassical" objects
(a semiring is called  idempotent  if the semiring addition is idem-
potent, i.e. x + x = x).
      In the spirit of  N. Bohr's correspondence principle there is
a (heuristic) correspondence between important, useful, and interes-
ting constructions and results over fields and similar results over
idempotent semirings.  For example,  the superposition principle in
Quantum Mechanics (i.e. the linearity of the Schroedinger equation)
corresponds to  a linearity of the Hamilton-Jacobi and Bellman equa-
tions over idempotent semirings.  A systematic application  of this
correspondence principle  leads  to  a variety of results including
such  exotic applications  as  a methodology  to construct computer
devices (processors) for numerical calculations  and to get the cor-
responing patents. The so-called tropical algebraic geometry in the
sense of O. Viro, G. Mikhalkin and others is  an idempotent version
of the traditional algebraic geometry.

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Seminar home-page http://www.pdmi.ras.ru/~lowdimma