Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the Intermediate Vector BosonsW±

J.-M Barbaroux,J.-C Guillot

doi:10.1155/2009/978903

Abstract

We consider a Hamiltonian with cutoffs describing the weak decay of spin 1 massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove the absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.

Highlights

We consider a mathematical model of the weak interaction as patterned according to the Standard Model in Quantum Field Theory see 1, 2
The total Hamiltonian, which is the sum of the free energy of the particles and antiparticles and of the interaction, is a self-adjoint operator in the Fock space for the leptons and the vector bosons and it has an unique ground state in the Fock space for a sufficiently small coupling constant
We are expecting that the spectrum of the Hamiltonian associated with every model of weak decays is absolutely continuous above the energy of the ground state, and this article is a first step towards a proof of such a statement

Summary

Introduction

We consider a mathematical model of the weak interaction as patterned according to the Standard Model in Quantum Field Theory see 1, 2. The total Hamiltonian, which is the sum of the free energy of the particles and antiparticles and of the interaction, is a self-adjoint operator in the Fock space for the leptons and the vector bosons and it has an unique ground state in the Fock space for a sufficiently small coupling constant. We are expecting that the spectrum of the Hamiltonian associated with every model of weak decays is absolutely continuous above the energy of the ground state, and this article is a first step towards a proof of such a statement. For related results about models in Quantum Field Theory see 20, in the case of the Quantum Electrodynamics and in the case of the weak interaction.

The Model

Main Results

Proof of the Mourre Inequality

En zσn df z