FROM (n + 1)-LEVEL ATOM CHAINS TO n-DIMENSIONAL NOISES

Published Jan 1, 2004• 3 citations

Abstract

Stéphane ATTAL and Yan PAUTRAT This article is dedicated to the memory of Paul-André MEYER In quantum physics, the state space of a countable chain of (n+1)-level atoms becomes, in the continuous field limit, a Fock space with multiplicity n. In a more functional analytic language, the continuous tensor product space over IR + of copies of the space C n+1 is the symmetric Fock space Γs(L 2 (IR +; C n)). In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of the n-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks in IR n whose jumps take exactly n+1 different values. We show that these probabilistic approximations are carried by the convergence of the basic matrix basis a i j (p) of ⊗INC n+1 to the usual creation, annihilation and gauge processes on the Fock space. I.

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