New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory

TL;DR

This paper introduces a new ansatz for calculating the metric operator in Pseudo-Hermitian field theories, simplifying the process and enabling a universal, local metric operator applicable across different space-time dimensions.

Contribution

The work proposes a novel ansatz that makes metric operator calculations in Pseudo-Hermitian field theories more straightforward and dimension-independent, contrasting with previous cumbersome, non-local methods.

Findings

Calculated the metric operator for $i\phi^{3}$ scalar field theory to first order in coupling.

The new ansatz yields a local metric operator, unlike previous non-local results.

The approach facilitates future studies, including a rigorous $ ext{PT}$-symmetric Higgs mechanism.

Abstract

In this work, a new ansatz is introduced to make the calculations of the metric operator in Pseudo-Hermitian field theory simpler. The idea is to assume that the metric operator is not only a functional of the field operators $\phi$ and its conjugate field $\pi$ but also on the field gradient $\nabla\phi$. Rather than the locality of the metric operator obtained, the ansatz enables one to calculate the metric operator just once for all dimensions of the space-time. We calculated the metric operator of the $i\phi^{3}$ scalar field theory up to first order in the coupling. The higher orders can be conjectured from their corresponding operators in the quantum mechanical case available in the literature. We assert that, the calculations existing in literature for the metric operator in field theory are cumbersome and are done case by case concerning the dimension of space-time in which the theory is investigated. Moreover, while the resulted metric operator in this work is local, the existing calculations for the metric operator leads to a non-local one. Indeed, we expect that the new results introduced in this work will greatly lead to the progress of the studies in Pseudo-Hermitian field theories where there exist a lack of such kind of studies in the literature. In fact, with the aid of this work a rigorous study of a $\mathcal{PT}$-symmetric Higgs mechanism can be reached.