 #1
shinobi20
 257
 17
 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?