RG flows on the phase spaces and the $\tau$ functions for the generic Hamiltonian systems

TL;DR

This paper explores the definition of the tau function in generic Hamiltonian systems, linking it to classical trajectories, coherent states, and the Toda lattice, revealing insights into quantization and integrability.

Contribution

It introduces a general framework for tau functions in Hamiltonian systems and connects them to coherent states and Toda lattice structures.

Findings

Tau function relates to filling classical trajectories with coherent states.

Identification of Baker-Akhiezer function with coherent wave function.

Transition to dispersionful Toda lattice corresponds to quantization.

Abstract

We discuss the generic definition of the $\tau$ function for the arbitrary Hamiltonian system. The different approaches concerning the deformations of the curves and surfaces are compared. It is shown that the Baker-Akhiezer function for the secondary integrable system of the Toda lattice type can be identified with the coherent wave function of the initial dynamical system. The $\tau$ function appears to be related to the filling of the interior of the classical trajectory by coherent states. Transition from dispersionless to dispersionful Toda lattice corresponds to the quantization of the initial dynamical system.