Divergence of Soft Photon Number and Absence of Ground State for Nelson’s Model

We treat Nelson’s Hamiltonian HN(κ) under the infrared singularity condition [Ar, (3.14)]. HN(κ) describes a system of a quantum particle moving on the 3-dimensional Euclidean space R3 under the influence of an external potential V and interacting with a massless scalar Bose field. Betz et al. showed in [BHLMS, (6.7)] that soft photon number for Nelson’s Hamiltonian with external potentials in the Kato class diverges under the infrared singularity condition. On the other hand, Lőrinczi et al. showed in [LMS, Theorem 4.5] that in 3-dimensional Euclidean space there is no ground state of Nelson’s Hamiltonian with the confining external potential under the infrared singularity condition. To prove both of the two results in [BHLMS, LMS], functional-integral manner is used. In this paper, with an operator-theoretical method we also give an explicit estimate of soft photon number divergence, which is our secondary purpose. Moreover, with this method we improve the result by Lőrinczi et al. so that our result can accept the Coulomb potential and eliminate the restriction on the coupling constant. That is our main purpose in this paper. Following physical intuition, if soft photon number for the Hamiltonian of a system diverges, we expect that the Hamiltonian has no ground state in the standard state space constructed from the Fock space. And another representation is required of us to describe the system, which is known as the Bloch-Nordsiek theorem in physics [BN]. We are interested in proving in mathematics that the divergence of soft photons implies absence of ground state in the standard state space, and we tried it in [AHH] from an operator-theoretical point of view. Because in [AHH] we treated general models in some degree, we could not entirely kick out ground states from the standard state space under the infrared singularity condition. Developing the way employed in [AHH] with some techniques in [BFS, GLL, HHS] and applying them to Nelson’s Hamiltonian, we achieve our purposes. The success in proving the finite bound of soft photon number [BFS, GLL, HHS] is in the following two steps. We first show that