The characterization of positivity properties of Weyl operators is a notoriously difficult problem, and not much progress has been made since the pioneering work of Kastler, Loupias, and Miracle-Sole (KLM). In this paper we begin by reviewing and giving simpler proofs of some known results for trace-class Weyl operators; the latter play an essential role in quantum mechanics. We then apply time-frequency analysis techniques to prove a phase space version of the KLM condition; the main tools are Gabor frames and the Wigner formalism. Finally, discrete approximations of the KLM condition, which are tractable numerically, are provided.

中文翻译：

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关于跟踪类运算符的积极性

外尔算子的正性质的表征是一个众所周知的难题，自从 Kastler、Loupias 和 Miracle-Sole (KLM) 的开创性工作以来，没有取得太大进展。在本文中，我们首先回顾并给出迹类 Weyl 算子的一些已知结果的更简单的证明；后者在量子力学中起着至关重要的作用。然后我们应用时频分析技术来证明 KLM 条件的相空间版本；主要工具是 Gabor 框架和 Wigner 形式主义。最后，提供了在数值上易于处理的 KLM 条件的离散近似值。