Numerical and Computational Challenges
in Science and Engineering Program
Lecture
June 21, 10:00 am
Gregory Litvinov, International Sophus Lie Centre
Dequantization of Mathematics: An Introduction to Idempotent Mathematics
G.L. Litvinov and V.P. Maslov
Idempotent Mathematics can be treated as a result of a dequantization
of the traditional mathematics over numerical fields as the Planck constant
tends to zero taking pure imaginary values. There exists a correspondence
between important, interesting, and useful constructions and results
over the fields of real and complex numbers and similar constructions
and results over semirings with idempotent addition (this means that
x+x = x) in the spirit of N. Bohr's correspondence principle in Quantum
Mechanics. A systematic and consistent application of the "idempotent"
correspondence principle leads to a variety of results, often quite
unexpected. For instance, the well-known Legendre transform is nothing
more that an idempotent version of the traditional Fourier transform.
The Hamilton- Jacobi equation (which is the basic equation in Classical
Mechanics) is an idempotent version of the Schroedinger equation. The
least action principle in Classical Mechanics can be considered as an
idempotent version of the R. Feynman's approach to Quantum Mechanics
via path integrals. Some problems that are nonlinear in the traditional
sense turn out to be linear over suitable idempotent semirings (the
idempotent superposition principle). For example, the Hamilton-Jacobi
equation and different versions of the Bellman equation (which is the
basic equation in the optimization theory) are linear over suitable
idempotent semirings.
Back to Thematic
Year Index
