A mathematical model for the Fermi weak interactions

Abstract

We consider a mathematical model of the Fermi theory of weak interactions as patterned according to the well-known current-current coupling of quantum electrodynamics. We focuss on the example of the decay of the muons into electrons, positrons and neutrinos but other examples are considered in the same way. We prove that the Hamiltonian describing this model has a ground state in the fermionic Fock space for a sufficiently small coupling constant. Furthermore we determine the absolutely continuous spectrum of the Hamiltonian and by commutator estimates we prove that the spectrum is absolutely continuous away from a small neighborhood of the thresholds of the free Hamiltonian. For all these results we do not use any infrared cutoff or infrared regularization even if fermions with zero mass are involved.