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 Bi : 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.
Highlights
If λ is a nonzero element of C − λ is a unit in Q and Q/( − λ)Q is trivial
For λ ∈ C, let Qλ be the set of formal elements f | =λ for all f ∈ Q
We should observe that Q/( − λ)Q = 0, since − λ is a unit in Q, and Qλ Q/( − λ)Q
Summary
A commutative C-algebra R is said to be a Poisson algebra if there exists a bilinear product {−, −} : R × R → R, called a Poisson bracket, such that (R, {−, −}) is a Lie algebra satisfying Leibniz’s rule {ab, c} = a{b, c}+{a, c}b for all a, b, c ∈ R. A quantization of R is an associative C[[ ]]-algebra R[[ ]] equipped with a star product ∗ : R[[ ]] × R[[ ]] → R[[ ]] such that for all a, b ∈ R, a ∗ b = ab + B1(a, b) + B2(a, b) 2 +. We can recover the Poisson algebra R from its quantization Q. Q is a nontrivial ideal such that Q/ Q R as a commutative C-algebra and the Poisson bracket {−, −} in R is obtained by (1).
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