The spectrum of the Hermitian Hamiltonian 1/2p(2) + 1/2m(2)x(2) + gx(4) (g > 0), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian H = 1/2p(2) + 1/2m(2)x(2) - gx(4), where the coupling constant g is real and positive, is PT-symmetric. As a consequence, the spectrum of H is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: when g is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian PT-symmetric -g phi (4) quantum field theory for all dimensions 0 less than or equal to D < 3 but is not present in the conventional Hermitian g phi (4) field theory. (C) 2001 Published by Elsevier Science B.V.