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\def\QED{\hskip 1em $\cal Q.~E.~D.$}

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\noindent PHY--396~K.\quad
Solutions for problem 3 of set \#5.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\noindent \undertext{Problem {\bf 3}(a)}:\brk
First, let us calculate the commutator
$[\gamma^\kappa\gamma^\lambda,\gamma^\nu]$.
According to eq.~(18), any product of 2 Dirac matrices can be re-ordered as
$$
\gamma^\nu\gamma^\mu\ =\ 2g^{\mu\nu}\ -\ \gamma^\mu\gamma^\nu .
\eqn\ReOrder
$$
Applying this formula twice, we have
$$
\gamma^\kappa\gamma^\lambda\gamma^\mu\
=\ \gamma^\kappa(2g^{\lambda\mu}-\gamma^\mu\gamma^\lambda)\
=\ 2g^{\lambda\mu}\gamma^\kappa\ -\ (\gamma^\kappa\gamma^\mu)\gamma^\lambda\
=\ 2g^{\lambda\mu}\gamma^\kappa\ -\ 2g^{\kappa\mu}\gamma\lambda\
+\ \gamma^\mu\gamma^\kappa\gamma^\lambda
\eqno\eq
$$
and hence
$$
[\gamma^\kappa\gamma^\lambda,\gamma^\nu]\
=\ 2g^{\lambda\mu}\gamma^\kappa\ -\ 2g^{\kappa\mu}\gamma^\lambda.
\eqn\DoubleComm
$$

Next, according to eq.~\ReOrder,
$[\gamma^\mu,\gamma^\nu]=2\gamma^\mu\gamma^\nu-2g^{\mu\nu}$
and hence 
$$
S^{\mu\nu}\ =\ -\coeff{i}{2}(\gamma^\mu\gamma^\nu\,-\,g^{\mu\nu}).
\eqn\Sdef
$$
Consequently,
$$
[S^{\kappa\lambda},\gamma^\mu]\
=\ -\coeff{i}{2}[\gamma^\kappa\gamma^\lambda,\gamma^\mu]\
=\ -ig^{\lambda\mu}\gamma^\kappa\ +\ ig^{\kappa\mu}\gamma^\lambda
\eqno(20)
$$
or equivalently
$$
[\gamma^\mu,S^{\kappa\lambda}]\
=\ ig^{\lambda\mu}\gamma^\kappa\ -\ ig^{\kappa\mu}\gamma^\lambda .
\eqno\eq
$$

Finally, by Leibniz rule,
$$
\eqalign{
\bigl[\gamma^\kappa\gamma^\lambda,S^{\mu\nu}\bigr]\ &
=\ \gamma^\kappa\,\bigl[\gamma^\lambda,S^{\mu\nu}\bigr]\
    +\ \bigl[\gamma^\kappa,S^{\mu\nu}\bigr]\,\gamma^\lambda\cr
&=\ \gamma^\kappa\bigl( ig^{\lambda\mu}\gamma^\nu\
	 -\ ig^{\lambda\nu}\gamma^\mu\bigr)\
    +\ \bigl( ig^{\kappa\mu}\gamma^\nu\
	    -\ ig^{\kappa\nu}\gamma^\mu\bigr)\gamma^\lambda \cr
&=\ ig^{\lambda\mu}\gamma^\kappa\gamma^\nu\
    -\ ig^{\kappa\nu}\gamma^\mu\gamma^\lambda\
    -\ ig^{\lambda\nu}\gamma^\kappa\gamma^\mu\
    +\ ig^{\kappa\mu}\gamma^\nu\gamma^\lambda\ \cr
&=\ ig^{\lambda\mu}\bigl(\gamma^\kappa\gamma^\nu-g^{\kappa\nu}\bigr)\
    -\ ig^{\kappa\nu}\bigl(\gamma^\mu\gamma^\lambda-g^{\lambda\mu}\bigr)\cr
&\qquad-\ ig^{\lambda\nu}\bigl(\gamma^\kappa\gamma^\mu-g^{\kappa\mu}\bigr)\
    +\ ig^{\kappa\mu}\bigl(\gamma^\nu\gamma^\lambda-g^{\lambda\nu}\bigr)\ \cr
&=\ 2g^{\lambda\mu}S^{\kappa\nu}\
    -\ 2g^{\kappa\nu}S^{\mu\lambda}\
    -\ 2g^{\lambda\nu}S^{\kappa\mu}\
    +\ 2g^{\kappa\mu}S^{\nu\lambda} ,\cr
}\eqno\eq
$$
and therefore,
$$
\eqalign{
\bigl[S^{\kappa\lambda},S^{\mu\nu}\bigr]\ &
=\ \coeff{i}{2}\,\bigl[\gamma^\kappa\gamma^\lambda,S^{\mu\nu}\bigr]\cr
&=\ ig^{\lambda\mu}S^{\kappa\nu}\
    -\ ig^{\kappa\nu}S^{\mu\lambda}\
    -\ ig^{\lambda\nu}S^{\kappa\mu}\
    +\ ig^{\kappa\mu}S^{\nu\lambda}\ \cr
&=\ ig^{\lambda\mu}S^{\kappa\nu}\
    +\ ig^{\kappa\nu}S^{\lambda\mu}\
    -\ ig^{\lambda\nu}S^{\kappa\mu}\
    -\ ig^{\kappa\mu}S^{\lambda\nu} .\cr
}\eqno(21)
$$
\QED

\medskip
\noindent \undertext{Problem {\bf 3}(b)}:
$$
\eqalign{
\bigl\{\gamma^\rho,\gamma^\lambda\gamma^\mu\gamma^\nu\bigr\}\ &
=\   2g^{\rho\lambda}\,\gamma^\mu\gamma^\nu\
  -\ 2g^{\rho\mu}\,\gamma^\lambda\gamma^\nu\
  +\ 2g^{\rho\nu}\,\gamma^\lambda\gamma^\mu ,\cr\
\bigl[\gamma^\rho,\gamma^\kappa\gamma^\lambda\gamma^\mu\gamma^\nu\bigr]\ &
=\   2g^{\rho\kappa}\,\gamma^\lambda\gamma^\mu\gamma^\nu\
  -\ 2g^{\rho\lambda}\,\gamma^\kappa\gamma^\mu\gamma^\nu\
  +\ 2g^{\rho\mu}\,\gamma^\kappa\gamma^\lambda\gamma^\nu\
  -\ 2g^{\rho\nu}\,\gamma^\kappa\gamma^\lambda\gamma^\mu ,\cr
\bigl[S^{\rho\sigma},\gamma^\lambda\gamma^\mu\gamma^\nu\bigr]\ &
=\   ig^{\sigma\lambda}\,\gamma^\rho\gamma^\mu\gamma^\nu\
  +\ ig^{\sigma\mu}\,\gamma^\lambda\gamma^\rho\gamma^\nu\
  +\ ig^{\sigma\nu}\,\gamma^\lambda\gamma^\mu\gamma^\rho\cr
&\qquad -\ ig^{\rho\lambda}\,\gamma^\sigma\gamma^\mu\gamma^\nu\
        -\ ig^{\rho\mu}\,\gamma^\lambda\gamma^\sigma\gamma^\nu\
        -\ ig^{\rho\nu}\,\gamma^\lambda\gamma^\mu\gamma^\sigma .\cr
}$$
The algebra is straightforward.

\medskip
\noindent \undertext{Problem {\bf 3}(c)}:
$$
\eqalign{
\gamma^\alpha\,\gamma_\alpha\ &
=\ \half\{\gamma^\alpha,\gamma^\beta\}g_{\alpha\beta}\
    =\ g^{\alpha\beta}g_{\alpha\beta}\ =\ 4;\cr
\gamma^\alpha\gamma^\nu\gamma_\alpha\ &
=\ 2g^{\alpha\nu}\gamma_\alpha\ -\ \gamma^\nu\,\gamma^\alpha\gamma_\alpha\
    =\ 2\gamma^\nu\ -\ \gamma^\nu (4)\ =\ -2\gamma^\nu\,;\cr
\gamma^\alpha\gamma^\mu\gamma^\nu\gamma_\alpha\ &
=\ 2g^{\alpha\mu}\gamma^\nu\gamma_\alpha\
	-\ \gamma^\mu\,\gamma^\alpha\gamma^\nu\gamma_\alpha\
    =\ 2\gamma^\nu\gamma^\mu\ -\ \gamma^\mu\,(-2\gamma^\nu)\
    =\ 2\{\gamma^\nu,\gamma^\mu\}\ =\ 4g^{\mu\nu}\,;\cr
\gamma^\alpha\gamma^\lambda\gamma^\mu\gamma^\nu\gamma_\alpha\ &
=\ 2g^{\alpha\lambda}\gamma^\mu\gamma^\nu\gamma_\alpha\
	-\ \gamma^\lambda\,\gamma^\alpha\gamma^\mu\gamma^\nu\gamma_\alpha\
    =\ 2\gamma^\mu\gamma^\nu\gamma^\lambda\ -\ \gamma^\lambda\,(4g^{\mu\nu})\cr
&=\ 2\bigl(\gamma^\mu\gamma^\nu\ -\ 2g^{\mu\nu}\bigr)\,\gamma^\lambda\
    =\ -2\gamma^\nu\gamma^\mu\gamma^\lambda .\cr
}\eqno\eq
$$

\medskip
\noindent \undertext{Problem {\bf 3}(d)}:
Let $F=-{i\over2}\Theta_{\alpha\beta}S^{\alpha\beta}$,
thus $M=e^F$ and $M^{-1}=e^{-F}$.
We shall use the multiple-commutator formula for the $e^{-F}\gamma^\mu e^{+F}$,
so we begin by evaluating the single commutator
$$
\bigl[\gamma^\mu,F\bigr]\
=\ -\coeff{i}{2}\Theta_{\alpha\beta}\,\bigl[\gamma^\mu,S^{\alpha\beta}\bigr]\
=\ \half\Theta_{\alpha\beta}\bigl(g^{\mu\alpha}\gamma^\beta\,
	-\,g^{\mu\beta}\gamma^\alpha\bigr)\
=\ \Theta_{\alpha\beta}\,g^{\mu\alpha}\gamma^\beta\
=\ \Theta^\mu_{\,\,\beta}\gamma^\beta .
\eqn\GammaComm
$$
The multiple commutators follow immediately from this formula,
$$
\eqalign{
\bigl[\bigl[\gamma^\mu,F\bigr],F\bigr]\ &
=\ \Theta^\mu_{\,\,\lambda}\Theta^\lambda_{\,\,\nu}\gamma^\nu,\cr
\bigl[\bigl[\bigl[\gamma^\mu,F\bigr],F\bigr],F\bigr]\ &
=\ \Theta^\mu_{\,\,\lambda}\Theta^\lambda_{\,\,\rho}
	\Theta^\rho_{\,\,\nu}\gamma^\nu,\cr
\noalign{\vskip -1\jot}
\omit\span\omit \dotfill
}\eqno\eq
$$
Therefore, by the multiple commutator formula,
$$
\eqalign{
M^{-1}\gamma^\mu M\ &
=\ e^{-F}\gamma^\mu e^{+F}\cr
&=\ \gamma^\mu\ +\ \bigl[\gamma^\mu,F\bigr]\
	+\ \half\,\bigl[\bigl[\gamma^\mu,F\bigr],F\bigr]\
	+\ \coeff16\,\bigl[\bigl[\bigl[\gamma^\mu,F\bigr],F\bigr],F\bigr]\
	+\ \cdots\cr
&=\ \gamma^\mu\ +\ \Theta^\mu_{\,\,\nu}\gamma^\nu\
	+\ \half\,\Theta^\mu_{\,\,\lambda}\Theta^\lambda_{\,\,\nu}\gamma^\nu\
	+\ \coeff16\,\Theta^\mu_{\,\,\lambda}\Theta^\lambda_{\,\,\rho}
		\Theta^\rho_{\,\,\nu}\gamma^\nu\
	+\ \cdots\cr
&=\ L^\mu_{\,\,\nu}\gamma^\nu .\cr
}\eqno\eq
$$
\QED

\bye
