PT-symmetric quantum field theory in D dimensions

TL;DR

This paper extends PT-symmetric quantum mechanics to quantum field theory in D dimensions, deriving Green's functions and renormalized quantities, and shows that spectral properties persist in the field-theoretic context.

Contribution

It introduces a method to compute Green's functions in PT-symmetric quantum field theory and derives explicit expressions for key quantities in various dimensions.

Findings

Exact expressions for vacuum energy density and Green's functions in 0≤D<2

Divergences in Green's functions for D≥2 with renormalization possible

Spectral properties of PT-symmetric quantum mechanics extend to quantum field theory

Abstract

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\varepsilon\geq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional spacetime, where $\phi$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\varepsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Green's functions, and the renormalized mass to order $\varepsilon$ are derived for $0\leq D<2$. For $D\geq2$ the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.