In physics “parity transformation” is a special kind of reflection. Parity transformation cannot be expressed as a rotation.
Do not confuse the parity transformation in physics with the parity concept in mathematics.
Let be a function and the parity transformation operator
Even and odd parity is defined as follows:
In 3D the parity transformation is equivalent to a point reflection.
In 2D the following example can be used to explain the parity transformation
The parity operation transforms the letter L either to the image on the left or to the image below the horizontal axis.
The image on the left is the reflection of the letter L in the vertical axis.
The image below the horizontal axis is the reflection of the letter L in the horizontal axis.
In a previous post I mentioned that “a rotation equals two reflections.” The reverse is not true. A reflection cannot be expressed in terms of rotations. For, example the reflected images of the letter L shown above cannot be obtained by rotating the letter L. Remember, we are restricted to 2D in this example, we are not allowed to rotate through the non-existing 3rd dimension. In 3D, the definition of the parity transformation changes (it is equivalent to a point reflection). The point reflection is not a rotation.
If the parity transformation is represented by a transformation matrix, then the determinant of that matrix must be -1. Transformations characterized by determinant=-1 cannot be expressed as rotation.
Follow-up: Parity transformation changes helicity.