Why Spinors Are Irreducible if Gamma-Traceless: Explained

In summary: You can raise and lower indices of a mwtric spinor with 2-contravariant or 2-covariant with the metric tensor.
  • #1
filip97
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I read this question

https://physics.stackexchange.com/q...onditions-is-a-vector-spinor-gamma-trace-free . Also I read Sexl and Urbantke book about groups. But I don't understand why spinors is irreducible if these are gamma-tracelees. Also I read many papers about higher spin fields in which doesn't explain and proof gamma traceless(irreducibility condition) of spinors of higher rank. Can any explain this ?
 
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  • #2
Perhaps if we ping @samalkhaiat , you can get an answer. :) It may very well be the one on the SE website, though.
 
  • #4
Ok, it is clearer. But why 1/2 component carried with divergence of Dirac spinor ? (This removing ghosts ?)
 
  • #5
filip97 said:
Ok, it is clearer.
Is it? Then why are you asking this?
But why 1/2 component carried with divergence of Dirac spinor ?
First: [itex]\psi^{\mu}(x)[/itex] is a spinor-vector field. It describes spin-3/2 particle and its anti-particle. It is NOT a Dirac spinor (Dirac spinor field describes spin-1/2 particle and its anti-particle).
Second: The divergence of [itex]\psi^{\mu}(x)[/itex] IS a Dirac spinor because (as I have already explained in the other thread) [itex]\partial_{\mu}\psi^{\mu}[/itex] satisfies the Dirac equation. In other words, the field [itex]\psi (x) \equiv \partial_{\mu}\psi^{\mu}(x)[/itex] represents one of the two spin-1/2 components of the spinor-vector field [itex]\psi^{\mu}(x)[/itex]. All of this was explained in the other thread.
(This removing ghosts ?)
Which ghosts are these?
 
  • #7
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  • #8
filip97
Can we raising and lowering indices of mwtric spinor with 2-contravariant or 2-covariant with metric tensor ? I think that we can do this with sigma(mu,nu) this write in Sexl Urbantke book of group representation. I was post this question because don't clear ho we contract Dirac equation with gamma(mu). Thanks a lot !
 

FAQ: Why Spinors Are Irreducible if Gamma-Traceless: Explained

1. What are spinors?

Spinors are mathematical objects used to describe the intrinsic angular momentum (spin) of particles in quantum mechanics. They are represented by mathematical matrices and have unique transformation properties under rotations and reflections.

2. What does it mean for a spinor to be irreducible?

In the context of quantum mechanics, irreducible means that a spinor cannot be broken down into smaller components. This is because spinors have a fixed number of components and cannot be simplified further without losing important information.

3. What is the significance of spinors being gamma-traceless?

The gamma-traceless property of spinors means that the sum of the diagonal elements of the gamma matrix (used to represent spinors) is equal to zero. This property is important in quantum mechanics as it allows for the simplification of mathematical calculations and makes it easier to solve certain equations.

4. Can you explain why spinors are irreducible if gamma-traceless?

The reason why spinors are irreducible if gamma-traceless is because the gamma-traceless property imposes a constraint on the number of independent components in a spinor. This constraint ensures that the spinor cannot be reduced any further without violating the gamma-traceless property.

5. How does understanding the irreducibility of spinors and the gamma-traceless property impact quantum mechanics?

Understanding the irreducibility of spinors and the gamma-traceless property is crucial in quantum mechanics as it allows us to accurately describe the spin of particles and make predictions about their behavior. It also simplifies mathematical calculations and allows for a deeper understanding of the fundamental principles of quantum mechanics.

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