Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators

The Yukawa-like interaction of a nonrelativistic “nucleon” field with a relativistic “meson” field is studied. E. Nelson defined a self-adjoint Hamiltonian for this model, using an approximate dressing transformation to transform the standard high-momentum cutoff Hamiltonian H_κ into an operator H₀ + A_κ + E_κ, where E_κ is the divergent nucleon self-interaction part which is removed and A_κ is controled κ-uniformly as a sesquilinear form perturbation of H₀; so H_ren is defined to transform to H₀ + A_∞. In the present work it is proved that the model has some of the main properties that are familiar in the axiomatic approach to quantum fields. Vacuum expectation values are proved to exist and satisfy the axioms of Wightman, except for Lorentz covariance. In the case of a small coupling constant, “physical nucleon” one-particle states are constructed and the “one-body problem” of Haag is solved. These field theoretic properties are translated into domain and spectral stability properties for the perturbed operator H₀ + A_∞. Stability theory for perturbations by sesquilinear forms is presented in an Appendix, with a number of new techniques.