quantum mechanics - Does the concept of both helicity and chirality make sense for a massive Dirac spinor?

Does the concept of both helicity and chirality make sense for a massive Dirac spinor?

A massive electron in the chiral basis is written as a column made up of $\psi_L$ and $\psi_R$. What is the significance of this?

These cannot be decoupled from each other due to the presence of the mass term in the Dirac equation, and we know electrons will look either left handed or right handed depending upon from which frame we are looking at them. But what is then the meaning of $\psi_L$ and $\psi_R$ separately?

Answer

I think its really important to differentiate between helicity and chirality. Helicity is the spin angular momentum of a particle projected onto its direction of motion. For a massive particle this quantity is frame dependent. Furthermore, since angular momentum is conserved, as a particle propagates helicity is conserved.

On the other hand, chirality is an innate property of a particle and doesn't change with frame. However, the mass term for a Dirac particle is,

$$\begin{equation} -m(\psi_L^\dagger \psi_R + \psi_R^\dagger\psi_L) \end{equation}$$ (in this notation the Dirac spinor is $\Psi= (\psi_L , \psi_R) ^T$). This term can be thought of as an interaction term in the Lagrangian which switches the chirality of a particle (e.g. a left chiral particle can spontaneously turn into a right chiral particle)

For a massless particle, chirality is equal to helicity.

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With that background we can finally address your questions.

1. Both helicity and chirality definitely make sense for a massive Dirac spinor. However, that doesn't mean that a Dirac spinor is a helicity and chirality eigenstate. In the same sense that energy makes sense for a particle, but it may not be an energy eigenstate.


2. As you mention the left chiral and right chiral fields can't be decoupled from each other due to the mass term. The mass term can always switch a right handed field to a left handed field and vice versa.


3. As I said above, the helicity of an electron is indeed frame dependent. So it may look like a left or right helicity electron depending on the frame, however its chirality is not frame dependent.


4. If we write the Dirac Lagrangian in terms of chirality eignestates then we have, $$\begin{equation} {\cal L} _D = i \psi _L ^\dagger \sigma ^\mu \partial _\mu \psi _L + i \psi _R ^\dagger \bar{\sigma} ^\mu \partial _\mu \psi _R - m \psi _L ^\dagger \psi _R - m \psi _R ^\dagger \psi _L \end{equation}$$ Then we can think of $\psi_L$ (left chiral particle) and $\psi_R$ (right chiral particle) as two different particles that can turn into each other spontaneously through a mass term. Putting them together, into a Dirac spinor masks this property. However, they are still well defined separately.