Student Friendly Introduction to QFT · Problem 19

Vectors: Spin 1 Fields

Physics — QFT · dirac-equation-nonrelativistic-limit

Problem

Take the relativistic solutions

|\psi^{(1)}\rangle

and

|\psi^{(2)}\rangle

of (4-20) for the Dirac equation in the non-relativistic limit, i.e., with

v \ll 1

(

c = 1

in natural units), and show that they reduce to approximately

(1,0,0,0)^T e^{-ipx}

and

(0,1,0,0)^T e^{-ipx}

. Do these look remarkably like the wave functions of a spin

\frac{1}{2}

particle you encountered in NRQM, i.e., spin up state

e^{-ipx}(1,0)^T

and spin down state

e^{-ipx}(0,1)^T

? Operate on each of the two low speed 4-spinor states you found with the 4D spin operator

\Sigma_3

of (4-39). Now, operate on each of the two NRQM 2-spinor states above for up and down spin with

\frac{1}{2}\hbar\sigma_3

. Do you get the same spin eigenstates and eigenvalues? Does the low speed RQM case for

C

type (i.e., having spinors

u_r

) particles parallel that of NRQM?