Many Electrons and the Photon Field -- The many-body structure of nonrelativistic quantum electrodynamics

Recent experimental progress in the field of cavity quantum electrodynamics allows to study the regime of strong interaction between quantized light and complex matter systems. Due to the coherent coupling between photons and matter-degrees of freedom, polaritons -- hybrid light-matter quasiparticles -- emerge, which can significantly influence matter properties and complex processes such as chemical reactions (strong coupling). In this thesis we propose a way to overcome these problems by reformulating the coupled electron-photon problem in an exact way in a different, purpose-build Hilbert space, where no longer electrons and photons are the basic physical entities but the polaritons. Representing an N-electron-M-mode system by an N-polariton wave function with hybrid Fermi-Bose statistics, we show explicitly how to turn electronic-structure methods into polaritonic-structure methods that are accurate from the weak to the strong-coupling regime. We elucidate this paradigmatic shift by a comprehensive review of light-matter coupling, as well as by highlighting the connection between different electronic-structure methods and quantum-optical models. This extensive discussion accentuates that the polariton description is not only a mathematical trick, but it is grounded in a simple and intuitive physical argument: when the excitations of a system are hybrid entities a formulation of the theory in terms of these new entities is natural. Finally, we discuss in great detail how to adopt standard algorithms of electronic-structure methods to adhere to the new hybrid Fermi-Bose statistics. Guaranteeing the corresponding nonlinear inequality constraints in practice requires a careful development, implementation and validation of numerical algorithms. This extra numerical complexity is the price we pay for making the coupled matter-photon problem feasible for first-principle methods.