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

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