Free coherent states and $p$-adic numbers

TL;DR

This paper introduces free coherent states for a two-degree-of-freedom system, demonstrating a homeomorphism with 2-adic integers and showing the 2-adic topology's emergence in free Fock space.

Contribution

It establishes a novel connection between free coherent states and 2-adic number topology, expanding the mathematical framework of quantum states.

Findings

Homeomorphism between 2-adic integers and coherent states

The 2-adic topology is induced by the free Fock space metric

Coherent states correspond to eigenvalues of the annihilation operator

Abstract

Free coherent states for a system with two degrees of freedom is defined. Existence of the homeomorphism of the ring of integer 2-adic numbers to the set of coherent states corresponding to an eigenvalue of the operator of annihilation is proved. It is shown that the metric of free Fock space induces the 2-adic topology on the set of coherent states.