Analytic representations with Theta functions for systems on Z(d) and on S

TL;DR

This paper develops analytic representations using Theta functions for systems on finite cyclic groups and circles, exploring their kernels and phase space functions like Wigner and Weyl functions.

Contribution

It introduces new Theta function-based analytic representations for systems on Z(d) and S, and studies their reproducing kernel formalism and phase space functions.

Findings

Reproducing kernel formalism is established for both systems.

Wigner and Weyl functions are characterized within this framework.

Analytic representations provide new insights into quantum systems on discrete and circular spaces.

Abstract

An analytic representation with Theta functions on a torus, for systems with variables in Z(d), is considered. Another analytic representation with Theta functions on a strip, for systems with positions in a circle S and momenta in Z, is also considered. The reproducing kernel formalism for these two systems is studied. Wigner and Weyl functions in this language, are also studied