Exact Results in Quantum Theory

In this talk, we will consider mathematical models describing the processes of the Fermi theory of weak interactions. The formal QFT Hamiltonian arising from the Standard Model of particle physics in ill-defined. To obtain a self-adjoint operator on a tensor product of Fock spaces, spatial and high-energy cut-offs are introduced into the interaction term. We will review some results concerning the spectral and scattering theories of such models. First, we will consider models describing the weak decay of massive vector bosons into the full family of leptons. Neutrinos may be treated as massive or massless particles. Next, we will study abstract effective models of interacting fermion fields describing processes such as the decay of a muon into an electron, a muon neutrino, and an electron antineutrino. This talk is based on joint works with B. Alvarez and J.-C. Guillot.