We introduce magnetic twisted actions of X = R n on general abelian C * -algebras and study the associated twisted crossed product and pseudodifferential algebras in the framework of strict deformation quantization.
We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.
Abstract. We show that Rieffel's deformation sends covariant C(T )-algebras into C(T )-algebras. We also treat the lower semi-continuity issue, proving that Rieffel's deformation transforms covariant continuous fields of C * -algebras into continuous fields of C * -algebras. Some examples are indicated, including certain quantum groups.
Groenewold-Van Hove theorem suggest that is not always possible to transform classical observables into quantum observables (a process known as quantization) in a way that, for all Hamiltonians, the constants of motion are preserved. The latter is a strong shortcoming for the ultimate goal of quantization, as one would expect that the notion of "constants of motion" is independent of the chosen physical scheme. It has been recently developed an approach to quantization that instead of mapping every classical observable into a quantum observable, it focuses on mapping the constants of motion themselves. In this article we will discuss the relations between classical and quantum theory under the light of this new form of quantization. In particular, we will examine the mapping of a class of operators that generalizes angular momentum where quantization satisfies the usual desirable properties.
We introduce a notion of smooth fields of operators following the notion of smooth fields of Hilbert spaces recently defined by L. Lempert and R. Szőoke [16]. Formally, if ∇ is the connection of a smooth field of Hilbert spaces we show that ∇ = [∇, •] defines a connection on a suitable space of fields of operators. In order to provide examples we prove that, if u is a suitable constant of motion of h(q, p) = q 2 (i.e. {u, h} = 0), then Op(u) is a smooth field of operators over the open interval (0, ∞), where Op denotes the canonical quantization (Weyl calculus). Moreover, in such case we show that we can compute derivatives using the formula ∇X 0 (Op(u)) = Op( ∇X 0 (u)), where ∇ is a Poisson connection on the Poisson algebra of constants of motion and X0 = 2λ ∂ ∂λ . We also introduce a notion of smooth field of C * -algebras and we give an example using Hilbert modules theory.
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