# Questions tagged [dirac-matrices]

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### How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$\Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
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### Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
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### What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
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### Identities of Pauli matrices in two-component spinor formalism

I'm reading the review by H. K. Dreiner, H. E. Haber and S. P. Martin (arXiv:0812.1594) about the two-component spinor formalism. There are some identities and notational conventions which lead to ...
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### Mapping from spinor to tetrad

I am reading the journel by Patrick l. Nash: mapping from tetrad to Dirac spinor. While reading this ,I came across the term concrete real 4*4 irreducible representation of SO(3,3). I know SO(3) is ...
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### Almost complex structure $J_i^j$ from covariantly constant spinor $\eta$

In the context of superstring compactification on a 6-manifold which admits a covariantly conserved spinor $\eta$, which we normalize so that $\eta^{\dagger} \eta = 1$, I am trying to show that the ...
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### Supergravity and Gamma Matrices

On page 101 of Freedman and Van Proeyen's book on Supergravity they find the propagator of the gravitino, however I'm not sure how to work through the steps in (5.30), and hints or answers would be ...
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### Spin for Dirac fermions proof $\Sigma_{i}=\frac{1}{2} \gamma_{5} \gamma_{0} \gamma_{i}$

The spin for Dirac fermions is defined as: $$\Sigma_{i} \equiv \frac{1}{4}\epsilon_{ijk}\sigma_{jk}$$ Without using an explicit representation for the matrices, I would like to show that the spin ...
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### Sakurai Quantum Mechanics problems

I just began studying QM on Sakurai's "Modern Quantum Mechanics" and just finished chapter 1. I am now approaching the exercises. On exercise 2 there is a notation I can't understand: "A 2x2 square ...
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