- #1
orentago
- 27
- 0
Homework Statement
From Mandl and Shaw (exercise 4.5):
Deduce the equations of motion for the fields:
[tex]\psi_L(x)\equiv{1 \over 2} (1-\gamma_5)\psi(x)[/tex]
[tex]\psi_R(x)\equiv{1 \over 2} (1+\gamma_5)\psi(x)[/tex]
for non-vanishing mass, and show that they decouple in the limit m=0. Hence show that the Lagrangian density
[tex]L(x)=\mathrm{i} \hbar c \overline{\psi}_L(x) \gamma^\mu \partial_\mu \psi_L(x)[/tex]
describes zero-mass fermions with negative helicity only, and zero-mass antifermions with positive helicity only.
Homework Equations
Lagrangian density for Dirac field:
[tex]L=c\overline{\psi}\left[ \mathrm{i}\hbar\gamma^\mu \partial_\mu -mc\right] \psi(x)[/tex]
Equations of motion:
[tex] {\partial L \over \partial \psi} - {\partial \over \partial x^\mu} \left(\partial L \over \partial \psi_{,\mu} \right)=0[/tex]
The Attempt at a Solution
I'm not exactly sure where to begin, partly because I don't understand the wording of the question. Do I simply swap [tex]\psi[/tex] for [tex]\psi_L[/tex] and [tex]\psi_R[/tex] into the Lagrangian above and sub this into the equations of motion as per normal, or should I swap [tex]\psi(x)[/tex] for [tex]\psi_L(x)+\psi_R(x)[/tex]?
For the second part I'll have to use the helicity operator I expect, but I'll cross that bridge when i come to it.