Research

Two undergraduate theses about the same parameter, seen from two places. The Physics one is deposited. The Mathematics one was defended on 23 September 2026.

Undergraduate thesis in Physics · 2025

Applications of the Koopman formalism to the dynamics of hybrid classical-quantum statistical systems

Supervisor: Jesús Clemente Gallardo · Department of Theoretical Physics · University of Zaragoza · July 2025 · written in Spanish

In short

In classical mechanics a particle has a position and a momentum at the same time. In quantum mechanics it cannot. The problem appears when you have to describe a system where both live together, for instance a molecule whose nuclei are treated as classical and whose electrons are treated as quantum: putting the two theories side by side is not enough, because they do not speak the same language.

The Koopman-von Neumann formalism lets you write classical mechanics with quantum tools, operators on a Hilbert space, which gives a common language. On that basis, Operational Dynamic Modeling introduces a continuous parameter $\kappa$ which is $\kappa = 0$ at the classical end and $\kappa = 1$ at the quantum one. The thesis walks along that transition using a numerical model of coupled harmonic oscillators, and looks at what happens on the way.

Abstract

This undergraduate thesis addresses the fundamental problem of dynamics in hybrid classical-quantum systems, an intersection of importance to several branches of physics and chemistry. Taking the Koopman-von Neumann (KvN) formalism and its generalisation through Operational Dynamic Modeling (ODM) as a basis, it explores a unified theoretical framework allowing a consistent description of the interaction between classical and quantum degrees of freedom. Through the implementation of a numerical model of coupled harmonic oscillators, it investigates the transition between the classical and quantum regimes, controlled by a continuous deformation parameter, $\kappa$ .

First page of the Physics undergraduate thesis — Fig. 1 First page of the thesis as deposited.

Zaguán · record 166671 Code for the numerical model

Undergraduate thesis in Mathematics · 2026

ODM formalism in hybrid quantum-classical systems

Supervisor: Jesús Clemente Gallardo · University of Zaragoza · defended 23 September 2026 · written in Spanish

The same parameter $\kappa$ , from its geometric side: as a continuous family of real polarizations on the doubled phase space, built through geometric quantization.

Fig. 2 The family of polarizations that κ runs through, from the Koopman-von Neumann vertical one to Wigner's. The position operator is multiplicative only at the two ends.

Defended on 23 September 2026. I will link it here once it is in Zaguán.

Short talk · ENEM

Computational complexity through the Professor Layton universe

A twenty-minute talk for mathematics undergraduates of all years, built on a simple idea: the puzzles of a video game series let you climb the complexity tower one floor at a time. Sliding blocks to get a car out is PSPACE-complete. Completing a board of queens is NP-complete. So is the longest path between two points.

The catch, which the talk says out loud, is that any particular board is always solvable in constant time with a lookup table. What is hard is the family generalised to an $n \times n$ board. And there is an ending I did not expect when I started: no puzzle in the series lives above PSPACE, and that is not a coincidence, because they are single-player perfect-information puzzles on a polynomially sized board.

Fig. 3 The floors the talk climbs, in the order it climbs them, and the ceiling it ends on: nothing in the series lives above PSPACE, and that is not a coincidence.