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

parity_letter_L

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.

 

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About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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