In various physical settings where relativistic effects cannot be neglected, the Dirac operator naturally replaces the Schrödinger operator, introducing additional spectral challenges. As a result, the appropriate theoretical framework for describing matter at this scale is quantum electrodynamics (QED). However, due to the limitations of its perturbative formulation, QED lacks a fully rigorous foundation, which has led to increasing interest in non-perturbative mean-field models in recent decades. This introductory talk will begin with an overview of the physical motivations and mathematical challenges of relativistic quantum mechanics. We will then discuss recent results on the relativistic quantum vacuum in the purely magnetic case at positive temperature, and, time permitting, also touch upon the issue of renormalisation.
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