On asymptotic expansions of twisted products

Abstract

The series development of the quantum-mechanical twisted product is studied. The series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product, either pointwise or in the distributional sense depending on the nature of the factors. A condition is given that ensures convergence and is stronger than previously known results. Possible applications are examined.

Quantum Symmetry Groups of Noncommutative Spheres

“…This may be interpreted as an oscillatory integral for suitable classes of functions and distributions on R 2m , and yields the familiar series as an asymptotic expansion in powers of [14,31]. It is, therefore, a better starting point than the said series for a C * -algebraic theory of deformations.…”

DFR Perturbative Quantum Field Theory on Quantum Space Time, and Wick Reduction

“…While for large classes of functions the asymptotic expansion truncated at order n agrees with the twisted product up to terms of order λ 2(n+1) P , the issue of convergence is rather delicate, and there are very few general results (see [23], and references therein). Certainly, the asymptotic expansion converges to (the Antonet extension of) F 1 ⋆ F 2 ifF 1 ,F 2 have compact supports; in this case, F 1 , F 2 are real-analytic, i.e.…”

“…The twisted product in position space first appeared (in the form of an asymptotic expansion) in a paper by Grönewold; the integral form was first used by Baker and explicitly written down by Pool. The first rigorous results on asymptotic expansions of twisted products are probably due to Antonet, and a comprehensive investigation can be found in [23], to which we also refer for the bibliographical coordinates missing in this footnote. The seminal work of Weyl and von Neumann inspired Wigner to define the so called Wigner transform; Wigner's work in turn led Moyal to define the so called Moyal bracket or sine-bracket {f, g}⋆ = f ⋆ g − g ⋆ f ; the Moyal bracket then plaid a fundamental role in a seminal paper by Bayen et al about geometric quantization of phase manifolds.…”

Скрученная Свертка И Звездочное Произведение Мойала Обобщенных Функций

“…Максимальное расширение по дуальности было предложено Антонецем [4]- [6] и состоит в постро-ении алгебры мультипликаторов для алгебры ( , ⋆ ), или, что равносильно, для алгебры ( , ). Затем оно изучалось во многих работах, наиболее глубоко в [7]- [9] (подробный обзор и ссылки имеются в [10]). …”