Extension of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric quantum mechanics to quantum field theory with cubic interaction

Abstract

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken $\mathcal{PT}$ 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 $\mathcal{C},$ which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate $\mathcal{C}$ is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating $\mathcal{C}$ directly in terms of the operator dynamical variables of the quantum theory. This 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 $\mathcal{C}$ operator in quantum field theory. The $\mathcal{C}$ operator is a time-independent observable in $\mathcal{PT}$-symmetric quantum field theory.