Graduate Understanding Dirac Adjoint Derivation & Spinor Transformations

[nigelscott]

I am trying to understand the derivation of the Dirac adjoint. I understand the derivation of the following identities involving Spinors, the Gamma matrices and Lorentz transformations:

(S^μν)^† = γ⁰S^μνγ⁰

s[Λ] = exp(Ω_μνS^μν/2)

s[Λ]^† = exp(Ω_μν(S^μν/2)^†)

The part I'm having trouble with is showing that the last line is also equal to:

γ⁰S[Λ]^-1γ⁰

Its probably simple but I'm having a mental block with it. Appreciate any help to get me going again.

 

[haushofer]

You should also dagger the Omega; it can be complex in general. Without doing the calculation myself, it seems you need the Clifford algebra and give an expression for the inverse of S. But this should be in any decent qft book, like Peskin.

 

[nigelscott]

OK. I have been looking at Equation 103 in http://isites.harvard.edu/fs/docs/icb.topic473482.files/10-spinors.pdf. I don't understand how γ⁰ get absorbed in the exponential.

 

[Dr.AbeNikIanEdL]

Well, $(\gamma^0)^2 = 1$, so for any $n$, $(\gamma^0 S_{\mu\nu}\gamma^0)^n = \gamma^0 (S_{\mu\nu})^n \gamma^0$. Using this and the series expression $\exp x = \sum \frac{x^n}{n!}$ should make this clear. But it is not clear to me how this works for general, complex $\Omega$ as haushofer said...

 

[nigelscott]

OK. So just to verify (neglecting indeces):

γ⁰(1 + iΩS)^†γ⁰

multiplying from left and right gives:

γ⁰γ⁰ - iΩγ⁰S^†γ⁰

= exp(-iΩγ⁰S^†γ⁰)

correct.

 

[Dr.AbeNikIanEdL]

Thats what I think (last equal of course up to order $\Omega^2$).