Griffiths Introduction to Elementary Particles · Problem 7.7

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · helicity-spinors

Problem

Construct the normalized spinors

u^{(+)}

and

u^{(-)}

representing an electron of momentum

\mathbf{p}

with helicity

\pm 1

. [That is, find the

u

's that satisfy equation (7.34), with positive

E

, and are eigenspinors of the helicity operator

(\hat{p}\cdot\boldsymbol{\Sigma})

with eigenvalues

\pm 1

\left[\textit{Solution: } u^{(\pm)} = N\begin{pmatrix} u_A \\ \dfrac{\pm c|\mathbf{p}|}{(E+mc^2)}\,u_A \end{pmatrix},\right.

\text{where } u_A = \begin{pmatrix} p_z \pm |\mathbf{p}| \\ p_x + ip_y \end{pmatrix} \quad \text{and} \quad N^2 = \frac{(E+mc^2)}{2|\mathbf{p}|c(|\mathbf{p}|\pm p_z)}\left.\right]