- #1

shinobi20

- 267

- 19

- Homework Statement
- Let ##\psi_1##, ##\psi_2## be solutions of the Dirac equation such that ,

##\Big( \gamma^\mu p_\mu - m \Big) \psi_1= 0 \hspace{1cm} \Big( \gamma^\mu p_\mu - m \Big) \psi_2= 0 \hspace{1cm} (1)##

Define,

##\bar{\psi} \overleftarrow{p_\mu} = (p_\mu \bar{\psi})##, ##\hspace{1cm}## ##\bar{\psi} = \psi^\dagger \gamma_0 \hspace{1cm} (2)##

##\hat{\sigma}_{\mu\nu} = \frac{i}{2} \Big[ \gamma_\mu, \gamma_\nu \Big] \hspace{1cm} (3)##

From the equation

##\bar{\psi_2} \Big( -\overleftarrow{p_\mu} \gamma^\mu - m \Big) {\not}a \psi_1 + \bar{\psi_2} {\not}a \Big(\gamma^\mu p_\mu - m \Big) \psi_1 = 0 \hspace{1cm} (4)##

Show that

##\bar{\psi_2} \gamma_\mu \psi_1 = \frac{1}{2m} \Big( \bar{\psi_2} p_\mu \psi_1 - (p_\mu \bar{\psi_2}) \psi_1 \Big) - \frac{i}{2m} p^\nu \Big(\bar{\psi_2} \hat{\sigma}_{\mu\nu} \psi_1 \Big) \hspace{1cm} (5)##

Hint: write ##{\not}a{\not}p## in terms of ##\hat{\sigma}_{\mu\nu}##

- Relevant Equations
- ##\Big( \gamma^\mu p_\mu - m \Big) \psi_1= 0 \hspace{1cm} \Big( \gamma^\mu p_\mu - m \Big) \psi_2= 0 \hspace{1cm} (1)##

##\bar{\psi} \overleftarrow{p_\mu} = (p_\mu \bar{\psi})##, ##\hspace{1cm}## ##\bar{\psi} = \psi^\dagger \gamma_0 \hspace{1cm} (2)##

##\hat{\sigma}_{\mu\nu} = \frac{i}{2} \Big[ \gamma_\mu, \gamma_\nu \Big] \hspace{1cm} (3)##

##\bar{\psi_2} \Big( -\overleftarrow{p_\mu} \gamma^\mu - m \Big) {\not}a \psi_1 + \bar{\psi_2} {\not}a \Big(\gamma^\mu p_\mu - m \Big) \psi_1 = 0 \hspace{1cm} (4)##

##\bar{\psi_2} \gamma_\mu \psi_1 = \frac{1}{2m} \Big( \bar{\psi_2} p_\mu \psi_1 - (p_\mu \bar{\psi_2}) \psi_1 \Big) - \frac{i}{2m} p^\nu \Big(\bar{\psi_2} \hat{\sigma}_{\mu\nu} \psi_1 \Big) \hspace{1cm} (5)##

The left side of the equality of ##(5)## is obvious from ##(4)##, however the rest of the terms are still unknown to me. I have tried adding and subtracting terms similar to the rest of the terms so as to produce a commutator and use ##(3)##, but I can't seem to figure out how to get ##(5)## correctly. Any help or guidance on how to proceed?