Why the Majorana Equation Preserves Handedness

We want to understand why the equation

\[ i\gamma^\mu \partial_\mu \psi = m\psi_c \]

does not mix independent left-handed and right-handed fields, unlike the Dirac equation.

1. Chirality and projectors

Define

\[ \psi_L = \frac{1-\gamma^5}{2}\psi, \qquad \psi_R = \frac{1+\gamma^5}{2}\psi \]

with

\[ \gamma^5 \psi_L = -\psi_L, \qquad \gamma^5 \psi_R = +\psi_R \]

2. The key structural fact

The gamma matrices satisfy

\[ \{\gamma^5, \gamma^\mu\} = 0 \]

This implies that \(\gamma^\mu\) flips chirality.

3. The derivative term flips chirality

Consider

\[ i\gamma^\mu \partial_\mu \psi_L \]

Act with \(\gamma^5\):

\[ \gamma^5 (i\gamma^\mu \partial_\mu \psi_L) = i(-\gamma^\mu \gamma^5)\partial_\mu \psi_L = i(-\gamma^\mu)(-\partial_\mu \psi_L) = i\gamma^\mu \partial_\mu \psi_L \]

\[ \gamma^5 (i\gamma^\mu \partial_\mu \psi_L) = + i\gamma^\mu \partial_\mu \psi_L \]

Therefore \(i\gamma^\mu \partial_\mu \psi_L\) is right-handed.

4. Charge conjugation flips chirality

You are using

\[ \psi_c = \gamma^2 \psi^* \]

A crucial identity is

\[ \gamma^2 (\gamma^5)^* = -\gamma^5 \gamma^2 \]

Now compute

\[ \gamma^5 \psi_c = \gamma^5 \gamma^2 \psi^* = -\gamma^2 (\gamma^5 \psi)^* \]

If \(\psi = \psi_L\), then \(\gamma^5 \psi = -\psi\), so

\[ \gamma^5 \psi_c = -\gamma^2 (-\psi)^* = +\psi_c \]

Hence charge conjugation flips chirality: a left-handed field goes to a right-handed charge conjugate.

5. Compare the equations

(a) Dirac equation

\[ i\gamma^\mu \partial_\mu \psi = m\psi \]

Split into chiral parts:

\[ i\gamma^\mu \partial_\mu \psi_L = m\psi_R \]

This requires both \(\psi_L\) and \(\psi_R\). A Dirac mass therefore mixes handedness.

(b) Majorana equation

\[ i\gamma^\mu \partial_\mu \psi = m\psi_c \]

Now start with a left-handed field \(\psi_L\):

The left-hand side, \(i\gamma^\mu \partial_\mu \psi_L\), is right-handed, and the right-hand side, \(m(\psi_L)_c\), is also right-handed.

So the equation matches right-handed to right-handed.

6. The key conclusion

The equation is closed within one chiral sector. You can write the entire dynamics using only \(\psi_L\): the opposite chirality is not an independent field, because it is generated automatically by charge conjugation.

7. What "preserves handedness" really means

It does not mean chirality never flips. Rather, it means that the theory does not require an independent field of opposite chirality.

Dirac: needs \(\psi_L\) and \(\psi_R\).
Majorana: needs only \(\psi_L\), because the role of the opposite chirality is supplied by \((\psi_L)_c\).

8. One-line intuition

\(\gamma^\mu\) flips chirality, charge conjugation flips chirality, and in the Majorana equation these flips match perfectly. So everything stays self-contained within one Weyl field.

9. Final perspective

This is why Majorana fermions can be described by a single Weyl spinor, and why the mass term is consistent without introducing new degrees of freedom.

This is additional material to complement page 486 of Group Theory in a Nutshell by Anthony Zee.