Hermiticity Breaking and Restoration in the $(g\phi^4 + h\phi^6)_{1+1}$ Field Theoretic Model

TL;DR

This paper explores how quantum corrections can break and restore hermiticity in a 1+1 dimensional scalar field model, revealing a physically acceptable broken hermiticity phase with PT symmetry that was previously overlooked.

Contribution

It introduces hermiticity as a symmetry and demonstrates the existence of a physically acceptable broken hermiticity phase in a scalar field theory, challenging prior assumptions.

Findings

Quantum corrections can break hermiticity while maintaining physical acceptability.

The broken hermiticity phase possesses PT symmetry and is relevant for universality.

Ignoring the broken hermiticity phase leads to incorrect theoretical predictions.

Abstract

We introduce hermiticity as a new symmetry and show that when starting with a model which is Hermitian in the classical level, quantum corrections can break hermiticity while the theory stay physically acceptable. To show this, we calculated the effective potential of the ($g\phi^{4}+h\phi^{6}$)$_{1+1}$ model up to first order in $g$ and $h$ couplings which is sufficient as the region of interest has finite correlation length for which mean field calculation may suffice. We show that, in the literature, there is a skipped phase of the theory due to the wrong believe that the theory in the broken hermiticity phase is unphysical. However, in view of recent discoveries of the reality of the spectrum of the non-Hermitian but $PT$ symmetric theories, in the broken hermiticity phase the theory possesses $PT$ symmetry and thus physically acceptable. In fact, ignoring this phase will lead to violation of universality when comparing this model predictions with other models in the same class of universality.