Bound States of Non-Hermitian Quantum Field Theories

The spectrum of the Hermitian Hamiltonian 1 2p 2 + 12m 2x2 + gx4 (g > 0), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian H = 1 2p 2 + 12m 2x2 − gx4, where the coupling constant g is real and positive, is PT -symmetric. As a consequence, the spectrum ofH 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 −gφ4 quantum field theory for all dimensions 0 ≤ D < 3 but is not present in the conventional Hermitian gφ4 field theory. In this Letter we show that the spectrum of the non-Hermitian PT -symmetric quartic Hamiltonian 1Contrary to appearances, this Hamiltonian is not Hermitian because its eigenfunctions are required to obey boundary conditions in the complex plane [see (4)].