Symbol Correspondence for Euclidean Systems

• Main Author: Natividad, Laarni
• Format: text
• Published: Archīum Ateneo 2019
• Subjects: Star-product, Euclidean motion group, unitary irreducible representations, Wigner function, Weyl quantization
• Online Access: https://archium.ateneo.edu/theses-dissertations/448
• Institution: Ateneo De Manila University

In this dissertation, we construct the star-product or ?-product of functions on the Euclidean motion group E(n), n = 2, 3 by way of the twisted convolution formulated on E(n). The construction is motivated by the theorem that connects the Weyl quanti- zation and the Wigner function given by hWσf|gi = (2π) −n/2 Z Rn Z Rn σ(x, ξ)WG(f, g)dxdξ, where Wσ is the Weyl quantization associated with σ in the Schwartz class, S (R n ), and WG(f, g) is the Wigner function of f, g ∈ S (R n ). The objects Wσ and WG(f, g) are constructed by means of the unitary irreducible representations of E(n). We will also show the properties of Wigner functions, as well as the ?-product of functions on E(n), and establish their relationship. The objects Wσ, WG and ? are three of the main objects of the formalism of quantum mechanics in phase space.