Williamson theorem in classical, quantum, and statistical physics

TL;DR

This paper explores the Williamson theorem's applications across classical, quantum, and statistical physics, demonstrating its utility in analyzing normal modes, quantum states, and uncertainty relations using symplectic formalism.

Contribution

It introduces the Williamson theorem's use in various physics contexts, providing accessible demonstrations and extending its application to quantum and thermodynamic systems.

Findings

Reveals normal-mode coordinates and frequencies in Hamiltonian systems.

Analyzes quantum normal modes and canonical distributions.

Explores the role of the Williamson theorem in uncertainty relations.

Abstract

In this work we present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. We demonstrate this theorem using only basic concepts of linear algebra and symplectic matrices. As an immediate application in the context of small oscillations, we show that applying this theorem reveals the normal-mode coordinates and frequencies of the system in the Hamiltonian scenario. A modest introduction of the symplectic formalism in quantum mechanics is presented, useing the theorem to study quantum normal modes and canonical distributions of thermodynamically stable systems described by quadratic Hamiltonians. As a last example, a more advanced topic concerning uncertainty relations is developed to show once more its utility in a distinct and modern perspective.