Parity transformation changes helicity

This post is the continuation of the Parity Transformation tutorial where I demonstrated the parity transformation in 2D by using the shape of letter L.

parity_letter_L

Figure (1)

The image to the left of the vertical axis 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. Remember, we are restricted to 2D in this example, we are not allowed to rotate through the non-existing 3rd dimension.

In today’s discussion I will use the L shape to demonstrate the parity transformation as well but this time the L shape has an arrow.

 

parity (2)Figure (2)

Again, notice that in Figure (2) the image to the left of the vertical axis 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 the Parity Transformation post I mentioned that the parity operation cannot be represented as a rotation. Now, let’s look at the rotations of the L shape in 2D.

rotation_clockwise_90_degrees_each

Figure (3)

In Figure (3) we see the rotations of the L shape in 2D. None of the rotated shapes is equal to the reflected images in Figure (2).

By the way, it does not matter whether you rotate the L shape clockwise or counterclockwise, the “rotational sense indicated by the arrow” does not change. Contrast this to what happens under the parity transformation as shown in Figure (2). The “rotational sense indicated by the arrow” changes under the parity transformation.

Helicity is the formal term to use for the rotational sense of the inner flow. In our examples (red colored L shapes) the inner flow is indicated by the arrow. In these examples, the shape L combined with the arrow indicates whether the inner flow is clockwise or counterclockwise.

Don’t forget helicity is about the rotational sense of the inner flow.

In physics, helicity is more narrowly defined as the sign of the projection of the spin vector onto the momentum vector. The helicity of an elementary particle is right-handed if the direction of its spin is the same as the direction of its motion. Helicity is left-handed if the directions of spin and motion are opposite. By convention for rotation, a standard clock, with its spin vector defined by the rotation of its hands, tossed with its face directed forwards, has left-handed helicity.

  • The original L shape (red L shape in the upper-right corner of Figure (2)) has a clockwise (left-handed) rotational sense of the inner flow.
  • All shapes in Figure (3)  have a clockwise (left-handed) rotational sense of the inner flow.
  • The reflected L shapes in Figure (2) (shapes in the upper-left and the lower-right) have a counter-clockwise (right-handed) rotational sense of the inner flow.

As demonstrated in Figure(2) the 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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