Isospectral local Hermitian theory for the $\mathcal{PT}$-symmetric $i\phi^3$ quantum field theory

TL;DR

This paper introduces a perturbative method to derive a local Hermitian equivalent of the $ ext{PT}$-symmetric $ioldsymbol{ extphi}^3$ quantum field theory, generalizing previous quantum mechanics results to higher dimensions.

Contribution

A new perturbative approach to compute the isospectral Hermitian theory for $ ext{PT}$-symmetric $ioldsymbol{ extphi}^3$ quantum field theory, applicable across all dimensions.

Findings

Method reproduces previous quantum mechanics results in 1D.

The Hermitian theory form is consistent across dimensions with only coefficient differences.

Previous quantum mechanics results can directly inform the quantum field theory form.

Abstract

We propose a new method to calculate perturbatively the isospectral Hermitian theory for the $\mathcal{PT}$-symmetric $i\phi^3$ quantum field theory in $d$ dimensions, whose result is local. The result of the new method in $1$ dimension reproduces our previous result in the $ix^3$ quantum mechanics, and the new method can be seen as a generalization of our previous method to quantum field theory. We also find the isospectral local Hermitian theory has the same form in all dimensions and differs in coefficients only, and our previous results in quantum mechanics can be used directly to determine the form of the isospectral local Hermitian quantum field theory.