doi:10.21468/SciPostPhysLectNotes.50engRomán-Roche, JuanZueco, DavidEffective theory for matter in non-perturbative cavity QEDART-2022-132314Starting from a general material system of N particles coupled to a cavity, we use a coherent-state path integral formulation to produce a effective theory for the material degrees of freedom. We tackle the effects of image charges, the A^2 term and a multimode arbitrary-geometry cavity. The resulting (non-local) action has the photonic degrees of freedom replaced by an effective position-dependent interaction between the particles. In the large-NN limit, we discuss how the theory can be cast into an effective Hamiltonian where the cavity induced interactions are made explicit. The theory is applicable, beyond cavity QED, to any system where bulk material is linearly coupled to a diagonalizable bosonic bath. We highlight the differences of the theory with other well-known methods and numerically study its finite-size scaling on the Dicke model. Finally, we showcase its descriptive power with three examples: photon condensation, the 2D free electron gas in a cavity and the modification of magnetic interactions between molecular spins; recovering, condensing and extending some recent results in the literature.2022http://zaguan.unizar.es/record/12389310.21468/SciPostPhysLectNotes.50http://zaguan.unizar.es/record/123893oai:zaguan.unizar.es:123893info:eu-repo/grantAgreement/ES/CSIC/PTI-001info:eu-repo/grantAgreement/ES/DGA/E09-17R-Q-MADinfo:eu-repo/grantAgreement/EC/H2020/862893/EU/Molecular spin qudits offering new hope for quantum computing/FATMOLSThis project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No H2020 862893-FATMOLSinfo:eu-repo/grantAgreement/ES/MICINN-AEI/PID2020-115221GB-C41SciPost Physics Lecture Notes 50 (2022), [32 pp.]byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess