On a c-number quantum $\tau$-function

TL;DR

This paper reviews conventional KP and Toda-lattice $ au$-functions and introduces a finite-difference generalization, aiming to deepen understanding of $q$-free fields and their relation to ordinary free fields.

Contribution

It presents a straightforward finite-difference generalization of $ au$-functions without involving operator-valued functions or non-Cartanian algebras.

Findings

Finite-difference $ au$-functions generalize differential ones.

Clarifies the relation between $q$-free fields and free fields.

Provides a basis for further studies in quantum integrable systems.

Abstract

We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $\tau$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.