Links. Relating different physical systems through the common QFT algebraic structure

TL;DR

This paper explores the algebraic structure of quantum field theory (QFT), focusing on how the doubling of degrees of freedom and q-deformed Hopf algebra link various physical systems like oscillators, thermal fields, and quantum states.

Contribution

It reveals the common algebraic framework underlying diverse physical systems through the q-deformed Hopf algebra, highlighting the unifying role of algebraic structures in QFT.

Findings

Doubling of degrees of freedom relates to inequivalent representations in QFT

Q-deformed Hopf algebra describes the algebraic structure of various systems

Links between systems like oscillators and thermal fields are established through algebraic analogy

Abstract

In this report I review some aspects of the algebraic structure of QFT related with the doubling of the degrees of freedom of the system under study. I show how such a doubling is related to the characterizing feature of QFT consisting in the existence of infinitely many unitarily inequivalent representations of the canonical (anti-)commutation relations and how this is described by the q-deformed Hopf algebra. I consider several examples, such as the damped harmonic oscillator, the quantum Brownian motion, thermal field theories, squeezed states, classical-to-quantum relation, and show the analogies, or links, among them arising from the common algebraic structure of the q-deformed Hopf algebra.