CUBO A Mathematical Journal
Vol. 9, N° 2, August 2007.
A mathematical model for the Fermi weak
Interactions
Laurent Amour
Laboratoire de Mathématiques EDPPVI, UMR-CNRS 6056, Université de Reims,
Moulin de la Housse - BP 1039, 51687 REIMS Codex 2, France.
laurent.amour@univ-reims.fr
Benoit Grébert
Laboratoire de Mathematiques Jean LERAY, UMR-CNRS 6629, Université de
Nantes,
2, rue de la Houssinière, 44072 NANTES Cedex 03, France.
benoit.grebert@univ-nantes.fr
Jean-Claude Guillot
CMAP, Ecole polyteclinique, CNRS,
Route de Saclay 91128 Palaiseau, lance.
guillot@crnapx.polytechnique.fr
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 unions into electrons, positrons and neutrinos but other
examples are considered in the same way. We prove that the
Ilainiltonian describing this model lias a ground state in the ferrnioriic
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.