# Bachelor Thesis With Prof. Czakon in 2023

Evolution of coherence in Quantum Mechanics

Quarks and gluons possess an internal degree of freedom conventionally called color. In high-energy collisions color exchange is described with the help of diagrammatic Feynman rules, which are easy to apply if the process only involves a small number of quark and gluons. An important part of modelling of high-energy processes concerns multiple radiation resulting in high-multiplicity final states. This is usually achieved with the help of parton shower Monte Carlo algorithms, which are based on the leading color approximation, which effectively neglects the evolution of color and hence also quantum coherence. Recently, several methods have been proposed to describe radiation including some coherence effects. The purpose of the thesis is to explore the available methods and develop a simplified code for the evolution of a quantum state due to radiation.

**The student will learn:**

- how is color described through representations of the SU(3) group
- how does color enter physical transition probabilities in Feynman diagrams
- what is a parton shower

**Requirements:**

- good understanding of basics of Quantum Mechanics
- programming skills

Unitary physics of non-hermitian Hamilton operators

As part of the basic postulates of Quantum Mechanics, one requires the Hamilton operator to be hermitian (self-adjoint), which yields a unitary evolution of the quantum state of a given system. Non-hermitian Hamilton operators are used as an approximation for unstable systems, where they are generated through the neglect of the decay products in the Hilbert space. It turns out, however, that there is a large class of Hamilton operators that are not hermitian and yet yield unitary evolution. This class of operators is of interest in Quatum Field Theory, since it provides simple solvable models with Asymptotic Freedom, a property that has been previously thought to be stricly restriced to non-abelian gauge theories. The purpose of the thesis is to study the fundamental properties of non-hermitian Hamilton operators, in particular their spectra, and the demonstrate the unitary evolution with numerical simulations.

**The student will learn:**

- how to construct a Hilbert space and Hamilton operator with wave functions on the complex plane
- what symmetries guarantee unitary evolution despite non-hermiticity
- how to numerically obtain spectra of Hamilton operators and how to numerically model time evolution

**Requirements:**

- good understanding of basics of Quantum Mechanics
- programming skills