EFFECT · Effective Equations for Fermionic Systems

Effective Equations for Fermionic Systems

Quantum mechanics is a fundamental theory in physics that accurately predicts the behaviour of nature at small scales in numerous situations. Within the framework of quantum mechanics, particles are classified based on their spin. Fermions, particles with half-odd-integer spin, are prohibited from occupying the same quantum state by the Pauli-exclusion principle, whereas this constraint does not apply to bosons, particles with integer spin, resulting in distinct physical phenomena. The time evolution of a quantum system is governed by the Schrödinger equation, whose analysis becomes intricate in the presence of many particles. Consequently, simpler equations, although less precise, are employed to approximate the system's evolution for ease of analysis. In physics, such effective theories are heuristically derived, guided by physical intuition. A mathematical study of the emergence of effective theories aids in quantifying the approximation errors and enhances the understanding of the mechanisms facilitating a simplified system description. The project aims to enhance comprehension regarding the non-equilibrium dynamics of large fermionic systems and their interaction with the quantized electromagnetic field through the mathematical rigorous derivation of effective equations. The primary focus of the project has been on analysing the semiclassical limits of two fermionic mean-field equations. These equations heuristically emerge from the many-body Schrödinger equation when the number of fermions is large. The first one is the semi-relativistic Hartree-Fock equations, describing the time evolution of fermions with relativistic dispersion law in a many-fermion mean-field limit, coupled to a semiclassical limit. As the particle number increases, quantum effects gradually diminish, allowing the approximation of the system's state by a phase space function satisfying the relativistic Vlasov equation. Considering a similar scaling limit for non-relativistic fermions interacting with the quantized electromagnetic field, the Maxwell-Schrödinger equations for extended charges heuristically emerge as a mean-field description. The project proves that these equations can be approximated by the Vlasov-Maxwell equations when the number of fermions is large. While the derivation of the Maxwell-Schrödinger equations from non-relativistic quantum electrodynamics has been investigated, its completion remains work in progress. In addition to the main results, several insights into bosonic systems have been obtained.

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