Dual spinor and gamma matrices

I'm sorry I didn't have a chance to go back and review this thread before you found the issue. I believe we should leave the thread open for a bit in case there are any other questions about the hint.
  • #1
Basu23
Here it is a simple problem which is giving me an headache,Recall from class that in order to build an invariant out of spinors we had to introduce a somewhat
unexpected form for the dual spinor, i.e. ߰ψ = ψ⋅γ0
Then showing that ߰ is invariant depends on the result that (ei/4⋅σμν⋅ωμν) ⋅γ0 = γ0⋅e-i/4⋅σμν⋅ωμν
Prove this by expanding out the exponential for the first three terms and using the (anti)commutation relations of the gamma matrices.

2. Homework Equations
The Metric signature is (-+++)

You can read the problem (number 2) by clicking the link below.

https://inside.mines.edu/~aflourno/Particle/HW4.pdf

The Attempt at a Solution



My issue is that I can't see why there is a - sign in the exponential after transpose/conjugating and moving the gamma matrix.

Moving the gamma matrix γ0 to the left cancels the negative sign from conjugating the σ0j in the exponential hence the + sign should not change.
Which step am I missing?

You can read the solution by clicking the link below.

https://inside.mines.edu/~aflourno/Particle/HW4solutions.pdf


Thank you for reading and any replies.[/B]
 
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  • #2
The hint would seem to explain it... the [itex]\gamma^0[/itex] will commute with the rotation generators and anti-commute with the boost generators. But the conjugate transpose of (i times) the rotation generators are their negatives (they times i must be anti-hermitian to generate unitary rotations). Contrawise the boost generators (times i) are hermitian (under conjugate transpose) and thus will not change sign under conjugate transpose. Either one or the other but not both sign changes will occur in all terms.

[EDIT, added:] This makes all the generators (times i) anti-pseudo-Hermitian in the pseudo-Hilbert space wherein we represent relativistic spinnors. The [itex]\gamma^0[/itex] matrix is effectively the indefinite part of the pHilbert space's inner product expressed by the Dirac bar adjoint vs the usual conjugate transpose. The way it is expressed here is still horribly basis dependent to some extent. The [itex]\gamma^0[/itex] matrix as it is used here is really a very different object than the [itex]\gamma^0[/itex] matrix used to represent the time inversion transformation in pin(3,1). I believe the object that is in this representation equal to [itex]\gamma^0[/itex] is referred to as [itex]\beta[/itex] and will differ from [itex]\gamma^0[/itex] in other representations.

It is similar to how we use a matrix to represent a rank 2,0 tensor [itex]g^{\mu\nu}[/itex] as well as a rank 1,1 tensor (operator) [itex]R^\mu_\nu[/itex]. You might even have two which have the same matrix form in a given basis (time inversion operator vs space-time metric in the (-,+++) signature). But they are distinctly different types of objects with different behaviors under change of basis transformations.
 
Last edited:
  • #3
.The anticommutation relations were clear to me but I have just realized I was not taking into account the i before before the generators thus the -sign after the conjugate/transposition.
A silly mistake.Thank you
 
  • #4
Well its the silly mistakes that are easiest to correct.
 

1. What are dual spinors?

Dual spinors are mathematical objects used in the theory of spinors, which are mathematical representations of the fundamental building blocks of matter. Dual spinors are used to describe the transformation properties of spinors under rotations and Lorentz transformations.

2. How do dual spinors relate to gamma matrices?

Dual spinors are related to gamma matrices through the Dirac equation, which is a mathematical equation that describes the behavior of spin-1/2 particles. The Dirac equation uses gamma matrices to represent the spinor wave function and its conjugate, which are both needed to describe spinors and their transformations.

3. What is the significance of dual spinor and gamma matrices in physics?

Dual spinor and gamma matrices are crucial in theoretical physics, particularly in the theory of relativity and quantum mechanics. They are used to describe the spin and angular momentum of particles, and are essential in predicting and understanding the behavior of fundamental particles.

4. How are dual spinor and gamma matrices used in practical applications?

Dual spinor and gamma matrices are used in numerous practical applications, such as in the design of magnetic materials, particle accelerators, and quantum computing. They are also used in the development of new theories and models in physics, which can lead to new technologies and advancements in various fields.

5. Are there any challenges in working with dual spinor and gamma matrices?

Yes, there can be challenges in working with dual spinor and gamma matrices, as they involve complex mathematical concepts and calculations. Additionally, their application in certain areas of physics, such as in quantum gravity, is still an active area of research and can present theoretical challenges.

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