Harmonic analysis on a galois field and its subfields

TL;DR

This paper explores harmonic analysis on Galois fields and their subfields, examining Fourier and symplectic transforms, and introduces Frobenius transformations that preserve functions on subfields, linking quantum mechanics and field theory.

Contribution

It introduces a formalism for harmonic analysis on Galois fields and subfields, including Frobenius transformations that preserve functions on subfields, connecting quantum mechanics and field theory.

Findings

Frobenius transformations fix functions on subfields.

Harmonic analysis on Galois fields relates to quantum mechanics.

The formalism inherits features from Galois field theory.

Abstract

Complex functions $\chi (m)$ where $m$ belongs to a Galois field $GF(p^ \ell)$, are considered. Fourier transforms, displacements in the $GF(p^ \ell) \times GF(p^ \ell)$ phase space and symplectic $Sp(2,GF(p^ \ell))$ transforms of these functions are studied. It is shown that the formalism inherits many features from the theory of Galois fields. For example, Frobenius transformations are defined which leave fixed all functions $h(n)$ where $n$ belongs to a subfield $GF(p^ d)$ of the $GF(p^ \ell)$. The relationship between harmonic analysis (or quantum mechanics) on $GF(p^ \ell)$ and harmonic analysis on its subfields, is studied.