Massive particle has a specific chirality

In summary, the author is discussing the concept of chirality and how it is a Lorentz invariant concept. He is saying that massive spinors have specific chirality and that this is something that is not always possible to change.
  • #1
Lapidus
344
11
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?
 
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  • #2


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.
 
  • #3


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?
 
  • #4


Lapidus said:
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.

[tex]\partial_{t} \psi + \alpha^i \partial^i \psi = \beta m \psi[/tex]

Move everything to the left hand side

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

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

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

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 [tex]\psi^{*}[/tex] and multiply it by [tex]\beta[/tex] you get [tex]\bar{\psi}[/tex]. So another way to write this is by saying

[tex]\bar{\psi} \beta \partial_t \psi + \bar{\psi} \beta \alpha_i \partial_i\psi + m \bar{\psi}\psi[/tex]

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

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

[tex]\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 \}[/tex]

There is what is called the fifth dirac matrix from this point. I'll assume you'll know that [tex]\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^5[/tex]. 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.
 
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  • #5


Lapidus said:
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.
 
  • #6


Thanks!
 
  • #7


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?
 

1. What is a massive particle?

A massive particle is a subatomic particle that has mass, meaning it has a physical weight and takes up space. Examples of massive particles include protons, neutrons, and electrons.

2. What is chirality?

Chirality is a property of particles that refers to their handedness or the direction in which they spin or rotate. A particle with a specific chirality means that it only spins or rotates in one direction.

3. How are chirality and mass related?

Chirality and mass are related because the mass of a particle can determine its chirality. For example, all massless particles have a specific chirality, while massive particles can have either a specific chirality or a mixture of both chiralities.

4. Why is chirality important in particle physics?

Chirality is important in particle physics because it affects how particles interact with each other and with the fundamental forces of nature. It can also determine the stability and decay of particles.

5. What are some applications of studying particles with specific chirality?

Studying particles with specific chirality can have practical applications in fields such as medicine, where chirality plays a role in the effectiveness and safety of drugs. It can also help scientists understand the fundamental laws of nature and the origins of the universe.

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