Idempotent correspondence principle

A heuristic principle in idempotent analysis. It states that there is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semi-rings (cf. Idempotent semi-ring) in the spirit of N. Bohr's correspondence principle in quantum mechanics. The field of real numbers and other fields can be treated as "quantum objects" with respect to idempotent semi-rings; so, idempotent semi-rings can be treated as "classical" or "semi-classical" objects and as results of a "dequantization" of these quantum objects. The correspondence principle is a powerful tool for applying unexpected analogies and ideas borrowed from different areas of mathematics and theoretical physics [a1], [a2].

For example, the well-known superposition principle in quantum theory corresponds to the idempotent correspondence principle in idempotent analysis; this means that the Hamilton–Jacobi equation (cf. also Hamilton–Jacobi theory) and the Bellman equation (and its generalizations) are linear over appropriate semi-rings; see, e.g., [a3]. There is a natural analogy between idempotent measures and probability measures (cf. also Probability measure). This analogy leads to a parallelism between probability theory and stochastic processes (cf. Stochastic process) on the one hand, and optimization theory and decision processes on the other hand (cf. also Decision problem). That is why it is possible to develop optimization theory at the same level of generality as probability theory and the theory of stochastic processes. In particular, the Markov causality principle corresponds to the Bellman optimality principle; so the Bellman equation is an idempotent version of the Kolmogorov–Chapman equation for Markov stochastic processes (see, e.g., [a4] and Markov process). The well-known Legendre transform is an idempotent version of the usual Fourier transform. There are many other examples of this type (see [a1]–[a3] and Idempotent analysis).

The idempotent correspondence principle is also valid for algorithms and their software and hardware implementations (cf. Idempotent algorithm).

References

[a1]	G.L. Litvinov, V.P. Maslov, "Correspondence principle for idempotent calculus and some computer applications" J. Gunawardena (ed.) , Idempotency, Publ. Isaac Newton Institute 11, Cambridge Univ. Press (1998) ISBN 0-521-55344-X pp.420-443 Zbl 0897.68050
[a2]	G.L. Litvinov, V.P. Maslov, "Idempotent mathematics: correspondence principle and applications" Russian J. Math. Phys. , 4 : 4 (1996) (In Russian)
[a3]	V.N. Kolokoltsov, V.P. Maslov, "Idempotent analysis and applications" , Kluwer Acad. Publ. (1996) (In Russian)
[a4]	P. Del Moral, "A survey of Maslov optimization theory" V.N. Kolokoltsov (ed.) V.P. Maslov (ed.) , Idempotent analysis and applications , Kluwer Acad. Publ. (1996) pp. Appendix (In Russian)