On the Triviality of ( )_(d+1) in the Nonrelativistic and Lee Approximations

摘要

We study the nonrelativistic approximation of the ()_(d+1) theory. Being trivial for d>=4, if d=3 it is either trivial in the two-particle sector or nontrivial in the two-particle sector but with a Hamiltonian unbounded from below in the three-particle sector due to the Thomas effect. If d=2 the theory is nontrivial in the two-particle sector, with a semibounded Hamiltonian in the three-particle sector, but the energy per particle is not bounded in the limit where the particle number tends to infinity, The Lee approximation, which differs from the nonrelativistic one by the fact that the kinematics is relativistic, leads, however, to results which are consistent with the folklore regarding triviality: for d=2 it is nontrivial with semibounded Hamiltonian, and for d=3 it is either trivial or the energy per particle is unbounded (and hence the theory is physically unacceptable).