$\cal PT$-symmetric theory in two dimensional spaces
Year: 2016
Type: preprint
Abstract: In the 1990's, C. M. Bender established the $\cal PT$-symmetric quantum theory. In this theory, the parity operator $\cal P$ and time reversal operator $\cal T$ are of great significance. In this paper, by using the concrete forms of $\cal P$ and $\cal T$ in two dimensional spaces, we show that if ${\cal T}^2= I$, then there are many $\cal P$ commuting with $\cal T$. However, if ${\cal T}^2= -I$, then $\cal P$ commuting with $\cal T$ can only be $\pm I$. We give out the geometrical interpretation of how $\cal P$ commutes with $\cal T$, and show that if $\cal T$ is given, then $\cal P$ links with the quadric surfaces; if $\cal P$ is given, then $\cal T$ links with the quadric curves. Moreover, we present the conditions when an operator is unbroken $\cal PT$-symmetric, and show that each Hermitian operator is unbroken $\cal PT$-symmetric.
Source: arXiv (Cornell University)
Authors: Minyi Huang, Yu Yang, Junde Wu
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2520900464