PT-symmetric Models with O(N) Symmetry

TL;DR

This paper develops PT-symmetric quantum models with O(N) symmetry, finds their Hermitian equivalents, and suggests that related scalar field theories are asymptotically free in four dimensions.

Contribution

It constructs and analyzes PT-symmetric O(N)-symmetric models, deriving their Hermitian counterparts and exploring their large-N behavior and asymptotic freedom.

Findings

Hermitian equivalent models have unusual features

Large-N effective potential matches original PT-symmetric form

Scalar field theories with similar interactions are asymptotically free

Abstract

We construct PT-symmetric quantum mechanical models with an O(N)-symmetric interaction term of the form $-g(\vec{x}^{2})^{2}/N$. Using functional integral methods, we find the equivalent Hermitian model, which has several unusual features. The effective potential obtained in the large-N limit of the Hermitian form is shown to be identical to the form obtained from the original PT-symmetric model using familiar constraint field methods. The analogous constraint field prescription in four dimensions suggests that $-g(\vec{\phi}^{2})^{2}/N$ PT-symmetric scalar field theories are asymptotically free.