We propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed, we write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a function and its Fourier transform, in particular to the ``lucky accident" that the truncated Fourier transform commutes with the prolate operator. The key is the notion of entropy of a vector of a complex Hilbert space with respect to a real linear subspace, recently introduced by the author by means of the Tomita-Takesaki modular theory of von Neumann algebras. We consider a generalization of the prolate operator to the higher dimensional case and show that it admits a natural extension commuting with the truncated Fourier transform; this partly generalizes the one-dimensional result by Connes to the effect that there exists a natural selfadjoint extension to the full line commuting with the truncated Fourier transform.

翻译：我们提出了一个概念框架，将通信理论中的长椭球微分算子解释为熵算子；具体而言，我们将其期望值写为若干项之和，每项均可通过量子场论所建议的嵌入方式赋予熵的解读。这为Slepian等人关于同时集中函数及其傅里叶变换的经典工作增添了新的意义，特别是对于截断傅里叶变换与长椭球算子可交换这一“幸运巧合”。核心在于复希尔伯特空间中向量相对于实线性子空间的熵概念（该概念由作者近期借助冯·诺依曼代数的Tomita-Takesaki模理论引入）。我们考虑将该长椭球算子推广至高维情形，并证明其存在与截断傅里叶变换可交换的自然延拓；这在一定程度上推广了Connes的一维结论，即存在定义在全直线上的自然自伴延拓，且该延拓与截断傅里叶变换可交换。