The Nelson model on static Lorentzian manifolds

We are concerned with the Nelson model defined on static Lorentzian manifolds. Static Lorentzian manifold is defined by a Lorentzian manifold with a metric depending on position but independent of time. The Nelson model is a simple but non-trivial model describing the strong interaction in quantum field theory. It is however assumed that fermions are governed by Schrödinger operator. Then it is the so called non-relativistic quantum field theory. From mathematical point of view the model is defined as a self-adjoint operator acting on some tensor product of Hilbert spaces, and we are interested in studying the spectrum of the self-adjoint operator rigorously. In particular the existence and the absence of ground state, property of continuous spectrum and spectral scattering theory are the main topics. For the Nelson model some physical folklore has been estab-lished rigorously. E.g., the absence of ground state of the Nelson model under infrared singular condition and the existence of ground state under the infrared regular condition are established. In this note we extend the Nelson model to the model defined on static Lorentzian manifold and study its spectrum. The Hilbert space of the state vectors is defined by H = L2(R3)⊗F, (1.1) where F = n=0 L 2 sym(R3n) denotes the boson Fock space over L2(R3). Then the stan-dard Nelson model is defined by a self-adjoint operator of the form