  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.
