The Maslov dequantization, idempotent and tropical mathematics: a brief introduction.

*(Russian, English)*Zbl 1102.46049
Zap. Nauchn. Semin. POMI 326, 145-182 (2005); translation in J. Math. Sci., New York 140, No. 3, 426-444 (2007).

The author considers heuristic aspects of Idempotent and Tropical Mathematics in the spirit of works of V. P. Maslov and his collaborators. Idempotent and Tropical Mathematics can be treated as a result of a dequantization of traditional mathematics as the Planck constant tends to zero taking purely imaginary values. For example, the field of real numbers can be treated as a quantum object whereas idempotent semirings can be examined as “classical” or “semiclassical” objects (a semiring is called idempotent if the semiring addition is idempotent, i.e. \(x + x = x\)).

In the spirit of N. Bohr’s correspondence principle there is a (heuristic) correspondence between important, useful, and interesting 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 equations 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 corresponing patents. The so-called tropical algebraic geometry in the sense of O. Viro, G. Mikhalkin and others is an idempotent version of traditional algebraic geometry.

In the spirit of N. Bohr’s correspondence principle there is a (heuristic) correspondence between important, useful, and interesting 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 equations 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 corresponing patents. The so-called tropical algebraic geometry in the sense of O. Viro, G. Mikhalkin and others is an idempotent version of traditional algebraic geometry.

Reviewer: Nikolai I. Osetinski (Moskva)

##### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

00A05 | Mathematics in general |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |

14P99 | Real algebraic and real-analytic geometry |

16Y60 | Semirings |

68M07 | Mathematical problems of computer architecture |