# Signal communication and modular theory

## TL;DR

This paper interprets the prolate differential operator in communication theory as an entropy operator using quantum field theory concepts, generalizing classical results and establishing new commutation properties in higher dimensions.

## Contribution

It introduces a novel entropy-based interpretation of the prolate operator and extends its properties to higher dimensions, connecting classical Fourier analysis with quantum algebraic structures.

## Key findings

- Prolate operator can be viewed as an entropy operator in a quantum framework.
- The truncated Fourier transform commutes with a generalized prolate operator in higher dimensions.
- Extension of the one-dimensional commutation result to multi-dimensional cases.

## Abstract

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.