Parity Transformation

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 $f$ be a function and $\mathbf{P}$ the parity transformation operator

(1D): $\mathbf{P} f(x) \rightarrow f(-x)$

(2D): $\mathbf{P} f(x,y) \rightarrow f(-x,y)$   or   $f(x,-y)$

(3D): $\mathbf{P} f(x,y,z) \rightarrow f(-x,-y,-z)$

Even and odd parity is defined as follows:

even parity: $\mathbf{P} f(...) = f(...)$
odd parity: $\mathbf{P} f(...) = -f(...)$

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.