Hamiltonian models of interacting fermion fields in Quantum Field Theory

TL;DR

This paper proves the self-adjointness and existence of a ground state for Hamiltonian models of interacting fermion fields, accommodating arbitrary interaction strengths and regularity conditions, relevant to Fermi theory of weak interactions.

Contribution

It establishes rigorous mathematical results for fermionic quantum field models with general interactions, including existence of ground states without restrictions on interaction strength.

Findings

Hamiltonian is self-adjoint on a tensor product of Fock spaces.

Existence of a ground state is proven for the models.

Results apply to models with ultraviolet and spatial cut-offs.

Abstract

We consider 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 new interpolated $N_\tau$ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cut-offs.