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：我们提出一个概念框架，将通信理论中出现的长椭球微分算子解释为熵算子；实际上，我们将其期望值表示为若干项之和，其中每一项都可通过量子场论启示的嵌入得到熵解读。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意义，特别是对截断傅里叶变换与长椭球算子可交换的“幸运巧合”给出了新的解释。其核心是复希尔伯特空间中向量关于实线性子空间的熵概念，该概念近期由作者通过冯·诺依曼代数的Tomita-Takesaki模理论引入。我们将长椭球算子推广至高维情形，并证明其存在一个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关于全直线上存在与截断傅里叶变换可交换的自然自伴延拓的一维结论。