Bound States of Non-Hermitian Quantum Field Theories

TL;DR

This paper demonstrates that non-Hermitian ${ m PT}$-symmetric quantum field theories with a $-g\,\phi^4$ interaction can host two-particle bound states, unlike their Hermitian counterparts, across various dimensions.

Contribution

It reveals the existence of bound states in non-Hermitian ${\rm PT}$-symmetric $-g\phi^4$ quantum field theories, contrasting with the absence of such states in Hermitian $g\phi^4$ theories.

Findings

Bound states exist in non-Hermitian ${\rm PT}$-symmetric $-g\phi^4$ theories.

No bound states in Hermitian $g\phi^4$ theories.

Bound states persist for all dimensions $0\leq D<3$.

Abstract

The spectrum of the Hermitian Hamiltonian ${1\over2}p^2+{1\over2}m^2x^2+gx^4$ ($g>0$), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\over2}p^2+{1 \over2}m^2x^2-gx^4$, where the coupling constant $g$ is real and positive, is ${\cal PT}$-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 ${\cal PT}$-symmetric $-g\phi^4$ quantum field theory for all dimensions $0\leq D<3$ but is not present in the conventional Hermitian $g\phi^4$ field theory.