Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction

TL;DR

This paper extends PT-symmetric quantum mechanics to quantum field theory with cubic interactions, introducing a new method to calculate the C operator directly from dynamical variables, ensuring unitarity and positive norm.

Contribution

It develops a general method for calculating the C operator directly in quantum field theory, advancing the understanding of PT-symmetric models with cubic interactions.

Findings

C operator can be calculated directly from dynamical variables in quantum field theory.

The method confirms the unitarity of PT-symmetric quantum field theories.

C operator is identified as a new time-independent observable.

Abstract

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate C is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.