Student Friendly Introduction to QFT · Problem 18

Vectors: Spin 1 Fields

Physics — QFT · relativistic-spin-helicity-ultrarelativistic

Problem

Show that as a Dirac particle of spinor form

u = u_1 + u_2

approaches the speed of light in the

x

direction (i.e.,

p^1 \to \infty

p^2 \ll p^1

, and

p^3 \ll p^1

), then

u

approaches a spin state in the plus

x

direction (i.e., the spinor

u = u_1 + u_2

is an eigenvector of

\Sigma_1

of (4-39) with eigenvalue

+\frac{1}{2}

(in natural units).) Then show that for

p^2 = p^1

and

p^3 = p^1

, this is not true.