A mathematical model for the Fermi weak interactions

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

Time evolution for the Pauli-Fierz operator (Markov approximation and Rabi cycle)

This article is concerned with a system of particles interacting with the quantized electromagnetic field (photons) in the non relativistic Quantum Electrodynamics (QED) framework and governed by the Pauli-Fierz Hamiltonian. We are interested not only in deriving approximations of several quantities when the coupling constant is small but also in obtaining different controls of the error terms. First, we investigate the time dynamics approximation in two situations, the Markovian (Theorem 1.4 completed by Theorem 1.16) and non Markovian (Theorem 1.6) cases. These two contexts differ in particular regarding the approximation leading terms, the error control and the initial states. Second, we examine two applications. The first application is the study of marginal transition probabilities related to those analyzed by Bethe and Salpeter in \cite{B-S}, such as proving the exponential decay in the Markovian case assuming the Fermi Golden Rule (FGR) hypothesis (Theorem 1.17 or Theorem 1.15) and obtaining a FGR type approximation in the non Markovian case (Theorem 1.5). The second application, in the non Markovian case, includes the derivation of Rabi cycles from QED (Theorem 1.7). All the results are established under the following assumptions at some steps of the proofs: an ultraviolet and an infrared regularization are imposed, the quadratic terms of the Pauli-Fierz Hamiltonian are dropped, and the dipole approximation is assumed but only to obtain optimal error controls.Comment: This version improves some results such as Theorem 1.4, which now includes an estimate without the dipolar approximation, and some proofs are then reorganize