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Scattering Theory for Mathematical Models of the Weak Interaction

Abstract : In this work, we consider, first, mathematical models of the weak decay of the vector bosons W into leptons. The free quantum field Hamiltonian is perturbed by an interaction term from the standard model of particle physics. After the introduction of high energy and spatial cut-offs, the total quantum Hamiltonian defines a self-adjoint operator on a tensor product of Fock spaces. We study the scattering theory for such models. First, the masses of the neutrinos are supposed to be positive: for all values of the coupling constant, we prove asymptotic completeness of the wave operators. In a second model, neutrinos are treated as massless particles and we consider a simpler interaction Hamiltonian: for small enough values of the coupling constant, we prove again asymptotic completeness, using singular Mourre's theory, suitable propagation estimates and the conservation of the difference of some number operators. We moreover study Hamiltonian models representing an arbitrary number of spin 1/2 fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of anti-symmetric Fock spaces and we establish the existence of a ground state. Our results rely on novel interpolated Nτ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cut-offs. Finally, the removal of spatial cut-off to define translation invariant toy models will be quickly discussed in the last chapter.
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Benjamin Alvarez. Scattering Theory for Mathematical Models of the Weak Interaction. Mathematical Physics [math-ph]. Université de Lorraine, 2019. English. ⟨NNT : 2019LORR0227⟩. ⟨tel-02499865⟩



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