PT-symmetric quantum systems with positive P

TL;DR

This paper introduces a novel PT-symmetric quantum framework replacing the traditional parity operator with a positive-definite metric, demonstrated through an N-site lattice Legendre oscillator, maintaining symmetry and utility.

Contribution

It proposes a new version of PT-symmetric quantum theory using a positive-definite operator instead of the involutory parity, expanding the theoretical foundation.

Findings

The new PT-symmetric formulation remains mathematically consistent.

Application to an N-site lattice Legendre oscillator demonstrates practicality.

The positive-definite operator enhances the interpretability of the theory.

Abstract

A new version of PT-symmetric quantum theory is proposed and illustrated by an N-site-lattice Legendre oscillator. The essence of the innovation lies in the replacement of parity P (serving as an indefinite metric in an auxiliary Krein space) by its non-involutory alternative P(positive)=Q>0 playing the role of a positive-definite nontrivial metric in an auxiliary, redundant, unphysical Hilbert space. It is shown that the QT-symmetry of this form remains appealing and technically useful.