Massive particle has a specific chirality

[Lapidus]

What does the author mean here when he says

However, a massive particle has a specific chirality. A massive left-chiral particle may have either left- or right-helicity depending on your reference frame relative to the particle. In all reference frames the particle will still be left-chiral, no matter what helicity it is.

How does a massive particle have a specific chirality? I learned that the only massive single chiral fields are the ones with Majorana mass. Dirac fields are a mix of left-chiral and right chiral fields, they do not have a specific chirality.

Is the author thus alluding to Majorana spinors here?

Or, which massive fields do have specific chirality?

And what do people mean when they say chirality is a Lorentz invariant concept, though it mixes in the Dirac spinors?

thanks

EDIT: And yes, both Dirac and Majorana spinors break chiral symmetry! Again, how can you say that massive spinors have specific chirality?

 

[DrDu]

Hm, a Dirac spinor can have definite chirality at a given time, but it won't be a solution of the time independent Dirac equation. Not a problem in principle.

 

[Lapidus]

So when I Lorentz transform a massive left-chiral state, it stays a left-chiral state.

Whereas time evolving it with respect to a equation of motion (e.g. Dirac equation), might turn it into a right-chiral state.

Correct?

 

[Geometrorigio]

So when I Lorentz transform a massive left-chiral state, it stays a left-chiral state.

Whereas time evolving it with respect to a equation of motion (e.g. Dirac equation), might turn it into a right-chiral state.

Correct?

Do you know in principle how chirality enters the Dirac equation? How is your covariant notation?

I'll take you through a small derivation of the dirac equation, a famous one.

\partial_{t} \psi + \alpha^i \partial^i \psi = \beta m \psi

Move everything to the left hand side

\partial_{t} \psi + \alpha^i \partial^i \psi - \beta m \psi = 0

Now all you do is multiply the entire equation by \psi^{*} to obtain the action

\psi^{*}(\partial_{t} \psi + \alpha^i \partial^i \psi - \beta m \psi) = \mathcal{L}

And produces the Langrangian. It is still zero, but it is a langrangian. This equation describes how to move one particle from one point to another. You might even think of it describing the Langrangian of a possible fragment of a world line.

Now we will revert to using gamma-notation which will express the covariant language. When you take \psi^{*} and multiply it by \beta you get \bar{\psi}. So another way to write this is by saying

\bar{\psi} \beta \partial_t \psi + \bar{\psi} \beta \alpha_i \partial_i\psi + m \bar{\psi}\psi

We can change the configuration of this expression in terms of new symbols.

\gamma^{0} is the gamma notation in respect to time, we can see the coefficient of beta is the derivative taken with respect to time and \beta \alpha_i as \gamma_i. We end up with

\bar{\psi} (\gamma^{\mu}\partial_{mu} + m)\psi =\{ \bar{\psi} \gamma^{0} \partial_t \psi + \bar{\psi} \gamma^i \partial_i \psi + m \bar{\psi}\psi \}

There is what is called the fifth dirac matrix from this point. I'll assume you'll know that \gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^5. It is gamma 5 which is concerned with right-handedness and left-handedness which in the technical term means, Chirality which has Eigenvalues of either +1 or -1.

 

[DrDu]

So when I Lorentz transform a massive left-chiral state, it stays a left-chiral state.

Whereas time evolving it with respect to a equation of motion (e.g. Dirac equation), might turn it into a right-chiral state.

Correct?

Exactly! With the Dirac sea interpretation in mind I think in QFT a state with definite chirality would correspond to a coherent superposition of the vacuum and an electron positron pair.

 

[Lapidus]

Thanks!

 

[naima]

the 2 components Weyl spinors are those with a defined chirality.
I had this answer in another forum:
"A (right or left) chiral fermion is an irreducible representation of the Lorentz group. There is thus no Lorentz transformation that can convert it into another fermion of opposite chirality."
Could somebody develop this?