Dirac eq gamma matrices question

[copernicus1]

In almost all the books on field theory I've seen, the authors list out the different types of quantities you can construct from the Dirac spinors and the gamma matrices, but I'm confused by how these work. For instance, if $$\overline\psi\gamma^5\psi$$ is a pseudoscalar, how can $$\overline\psi\gamma^\mu\psi$$ be a vector? Aren't gamma-5 and gamma-mu just different matrices? How do you get a vector out of the second operation?

Thanks!

 

[dextercioby]

Have you seen proofs for these 2 results ? It's true that most books gloss over these things without the explicit calculations. I vaguely remember that the proofs for all the so-called <Dirac bilinears> are in Müller-Kirsten and Wiedemann's book on symmetries and supersymmetries. They contain about 100+ pages of calculations with spinors in 4D.

 

[psmt]

Gamma-5 and gamma-mu are indeed all matrices, BUT the four different gamma-mu represent the Dirac algebra, whereas gamma-5 is not an element of that representation - so that's where the difference must come in.
(If your question is about the number of components and the fact that really gamma-mu is just a single matrix, then what the authors mean is that the second quantity transforms as a *component* of a four-vector.)

 

[Bill_K]

In addition to transforming the spacetime coordinates, the Lorentz transformation also transforms the spinor components: ψ → Λψ, where Λ is a 4 x 4 matrix. For an infinitesimal transformation, Λ = I + ½ε_μνΣ^μν where Σ^μν = ½γ^μγ^ν. It's their commutators with Σ^μν that determine the transformation properties of the Dirac covariants. For example, γ^μ → Λ^-1γ^μΛ = γ'^μ.

 

[andrien]

Aren't gamma-5 and gamma-mu just different matrices? How do you get a vector out of the second operation?

For that you will have to learn how dirac spinors transform under parity.Under parity transformation
ψ^-γ⁵ψ(x,t)-ψ^-γ⁰γ⁵γ⁰ψ(-x,t)=-ψ^-γ⁵ψ(-x,t)
which shows that it has a pseudoscalar character.while ψ^-γ^μψ is a lorentz vector which means
ψ^-γ^μψ-Λ^μ_vψ^-(Λ^-1x)γ^vψ(Λ^-1x)