Relationships between quantized algebras and their semiclassical limits

Department of Mathematics, Chungnam National University, 99 Daehak-ro, Daejeon 34134, Korea

^* e-mail: sqoh@cnu.ac.kr

Abstract

A Poisson ℂ-algebra R appears in classical mechanical system and its quantized algebra appearing in quantum mechanical system is a ℂ[[ħ]]-algebra Q = R[[ħ]] with star product * such that for any a,b Є R ⊆ Q, a*b = ab + B₁(a,b)ħ + B₂(a,b)ħ² + … subject to {a,b}= ħ^-1(a * b ‒ b * a)|_ħ=0, … (**) where B_i : R ⨯ R → R are bilinear products. The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/ħQ with Poisson bracket (**), which is called its semiclassical limit. But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big. For instance, if λ is a nonzero element of ℂ then ħ - λ is a unit in Q and thus a so-called deformation of R, Q/(ħ - λ)Q, is trivial. Hence it seems that we need an appropriate 픽-subalgebra A of Q such that A contains all generators of Q, ħ є A and A is understandable at an algebraic point of view, where 픽 is a subring of C[[ħ]].

Here we discuss how to find nontrivial deformations from quantized algebras and the natural map in [6] from a class of infinite deformations onto its semiclassical limit. The results are illustrated by examples.