One can argue that on flat space \({\mathbb {R}}^d\), the Weyl quantization is the most natural choice and that it has the best properties (e.g., symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization—we call it the balanced geodesic Weyl quantization. Among other things, we prove that it maps square-integrable symbols to Hilbert–Schmidt operators, and that even (resp. odd) polynomials are mapped to even (resp. odd) differential operators. We also present a formula for the corresponding star product and give its asymptotic expansion up to the fourth order in Planck’s constant.

中文翻译：

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（伪）黎曼流形上的伪微分Weyl演算

有人可以说，在平坦空间\（{\ mathbb {R}} ^ d \）上，Weyl量化是最自然的选择，并且具有最佳属性（例如，辛协方差，实符号对应于Hermitian算符）。在通用流形上，没有可分辨的量化，并且量化通常以图表方式定义。在这里，我们介绍一种量化，我们相信该量化对于研究伪黎曼流形上的自然算子具有最佳的性能。这是Weyl量化的一般化，我们称其为平衡测地Weyl量化。除其他外，我们证明其将平方可积符号映射到Hilbert-Schmidt运算符，并且偶数（分别为奇数）多项式被映射为偶数（分别为奇数）。我们还给出了相应星积的公式，并给出其渐近展开式直至普朗克常数的四阶。