Griffiths Introduction to Elementary Particles · Problem 10.23

Weak Interactions — David J. Griffiths

Physics — QFT · gamma5-helicity-chiral-spinors

Problem

Show that γ5u(p)=(c(p⋅σ)E+mc200c(p⋅σ)E−mc2)u(p)\gamma^5 u(p) = \begin{pmatrix} \dfrac{c(\mathbf{p}\cdot\boldsymbol{\sigma})}{E + mc^2} & 0 \\ 0 & \dfrac{c(\mathbf{p}\cdot\boldsymbol{\sigma})}{E - mc^2} \end{pmatrix} u(p) and that for a massless particle (E=∣p∣cE = |\mathbf{p}|c), γ5u(p)=(p^⋅Σ)u(p)\gamma^5 u(p) = (\hat{p}\cdot\boldsymbol{\Sigma})u(p), where Σ=(σ00σ)\boldsymbol{\Sigma} = \begin{pmatrix} \boldsymbol{\sigma} & 0 \\ 0 & \boldsymbol{\sigma} \end{pmatrix}. Also show that for an antiparticle, γ5v(p)=−(p^⋅Σ)v(p)\gamma^5 v(p) = -(\hat{p}\cdot\boldsymbol{\Sigma})v(p) (if m=0m = 0).

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