The geometry properties of parity and time reversal operators in two dimensional spaces

TL;DR

This paper explores the geometric properties of parity and time reversal operators in two-dimensional spaces, revealing their connections to quadric surfaces and curves, and discusses conditions for unbroken PT-symmetry.

Contribution

It provides a detailed geometric analysis of parity and time reversal operators in 2D, including their relations to quadric surfaces and curves, and introduces generalized unbroken PT-symmetric conditions.

Findings

Parity operators relate to quadric surfaces when time reversal is fixed.

Time reversal operators relate to quadric curves when parity is fixed.

Conditions for unbroken PT-symmetry are generalized and applied to Hermitian operators.

Abstract

The parity operator $\cal P$ and time reversal operator $\cal T$ are two important operators in the quantum theory, in particular, in the $\cal PT$-symmetric quantum theory. By using the concrete forms of $\cal P$ and $\cal T$, we discuss their geometrical properties in two dimensional spaces. It is showed that if $\cal T$ is given, then all $\cal P$ links with the quadric surfaces; if $\cal P$ is given, then all $\cal T$ links with the quadric curves. Moreover, we give out the generalized unbroken $\cal PT$-symmetric condition of an operator. The unbroken $\cal PT$-symmetry of a Hermitian operator is also showed in this way.