\input phyzzx

\def\bx{{\bf x}}
\def\eqdef{\buildrel\rm def\over=}
\def\a{\hat a^{\vphantom\dagger}}
\def\ad{\hat a^\dagger}
\def\bd{\hat b^\dagger}
\def\bp{{\bf p}}

\hsize=6.8in	\hoffset=-0.2in	\vsize=8.75in \voffset=-0.25in
%\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#7.\quad
Due October 30, 2000.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider the matrix $\gamma^5\eqdef i\gamma^0\gamma^1\gamma^2\gamma^3$.

\spointbegin
Show that $\gamma^5$ anticommutes with each of the $\gamma^\mu$ matrices,
$\gamma^5\gamma^\mu=-\gamma^\mu\gamma^5$.

\spoint %b
Show that $\gamma^5$ is hermitian and that $(\gamma^5)^2=1$.

\spoint %c
Show that $\gamma^5=(-i/24)\epsilon_{\kappa\lambda\mu\nu}
\gamma^\kappa\gamma^\lambda\gamma^\mu\gamma^\nu$ and
$\gamma^{[\kappa}\gamma^\lambda\gamma^\mu\gamma^{\nu]}=
-i\epsilon^{\kappa\lambda\mu\nu}\,\gamma^5$.\brk
(Sign convention: $\epsilon^{0123}=+1$, $\epsilon_{0123}=-1$.)

\spoint %d
Show that $\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}=
i\epsilon^{\kappa\lambda\mu\nu}\,\gamma_\kappa\gamma^5$.

\spoint %e
Show that any $4\times4$ matrix $\Gamma$ is a unique linear combination
of the following 16 matrices:
1, $\gamma^\mu$, $\gamma^{[\mu}\gamma^{\nu]}$, $\gamma^5\gamma^\mu$
and $\gamma^5$.

\pointcon
Under continuous Lorentz symmetries,
Dirac spinor fields $\Psi(x)$ transform according to
$\Psi'(x')=M(L)\Psi(x=L^{-1}x')$ where
$M\bigl(L=e^\theta\bigr))=
	\exp\bigl(-{i\over2}\theta_{\alpha\beta}S^{\alpha\beta}\bigr)$.
Consider the transformation rules for the independent bilinears
$\overline\Psi\Gamma\Psi$, namely (\cf~(e))
$$
S\ =\ \overline\Psi\Psi,\quad
V^\mu\ =\ \overline\Psi\gamma^\mu\Psi,\quad
T^{\mu\nu}\ =\ \overline\Psi\gamma^{[\mu}\gamma^{\nu]}\Psi,\quad
A^\mu\ =\ \overline\Psi\gamma^5\gamma^\mu\Psi\,\quad{\rm and}\quad
P\ =\ \overline\Psi\gamma^5\Psi.
\eqn\BiLinears
$$

\spoint %f
Show that under {\sl continuous} Lorentz symmetries, the $S$ and the $P$
transform as  scalars, the $V^\mu$ and the $A^\mu$ as vectors and the
$T^{\mu\nu}$ as an antisymmetric tensor.

\bigskip
\point %2
Under the {\sl parity} symmetry ${\cal P}:(\bx,t)\mapsto(-\bx,t)$,
Dirac spinor fields transform according to
$$
\hat{\cal P}\,\hat\Psi(\bx,t)\,\hat{\cal P}\
\equiv\ \hat\Psi'(\bx,t)\ =\ \pm\gamma^0\,\hat\Psi(-\bx,t)
\eqn\Parity
$$
where the overall sign depends on the so-called {\it intrinsic parity}
of a particular Dirac field.
Note: $\hat{\cal P}$ here is a unitary operator in the fermionic Fock space;
by nature of the parity symmetry, $\hat{\cal P}^2=1$.

\spointbegin
Verify the covariance of the Dirac equation under this symmetry.

\spoint
Find the transformation rules of the bilinears~\BiLinears\ under parity
and show that while $S$ is a true scalar and $V$ is a true (polar) vector,
$P$ is a pseudoscalar and $A$ is an axial vector.

\bigskip
\point %3
In theories involving both bosons and fermions,
one often has to combine commutation and
anti-commutation relations of various operators, depending on the overall
statistics of the operators involved.
For that purpose, it is useful to define a `mixed' commutator bracket
$$
[\hat A,\hat B\}\ \buildrel \scriptscriptstyle{\rm def}\over= \
\hat A\hat B\ -\ (-1)^{AB}\hat B\hat A
\eqn\MixedDef
$$
where $(-1)^{AB}$ is $-1$ if both $\hat A$ and $\hat B$ have overall Fermi
statistics (\ie, each comprises an odd number of {\sl fermionic}
creation/annihilation operators --- the number of bosonic creation/annihilation
operators does not matter) and $+1$ in all other cases.

\spointbegin
Verify the Leibniz rule for the mixed brackets:
$[\hat A,\hat B\hat C\} = [\hat A,\hat B\}\hat C
+ (-1)^{AB}\hat B[\hat A,\hat C\}$
and write down a similar rule for the
$[\hat A\hat B,\hat C\}$.

\spoint
Similarly, express $[\hat A\hat B,\hat C\hat D\}$ in terms of appropriate
mixed brackets of $\hat A$ or $\hat B$ with $\hat C$ or $\hat D$.

\spoint
Prove the `mixed' Jacobi identity
$$
(-1)^{CA}[\hat A,[\hat B,\hat C\}\}\ +\ (-1)^{AB}[\hat B,[\hat C,\hat A\}\}\
+\ (-1)^{BC}[\hat C,[\hat A,\hat B\}\}\ =\ 0.
\eqn\MixedJacobi
$$
In other words (and notations),
$$
\vcenter{\openup 1\jot \ialign{&$\displaystyle{{}#}$\enspace\hfil\cr
	   [\hat B_1,[\hat B_2,\hat B_3]] &
	+\ [\hat B_2,[\hat B_3,\hat B_1]] &
	+\ [\hat B_3,[\hat B_1,\hat B_2]] &
	=\ 0,\cr
	   [\hat B_1,[\hat B_2,\hat F]] &
	+\ [\hat B_2,[\hat F,\hat B_1]] &
	+\ [\hat F,[\hat B_1,\hat B_2]] &
	=\ 0,\cr
	   \{\hat F_1,[\hat F_2,\hat B]\} &
	-\ \{\hat F_2,[\hat B,\hat F_1]\} &
	+\ [\hat B,\{\hat F_1,\hat F_2\}] &
	=\ 0,\cr
	   [\hat F_1,\{\hat F_2,\hat F_3\}] &
	+\ [\hat F_2,\{\hat F_3,\hat F_1\}] &
	+\ [\hat F_3,\{\hat F_1,\hat F_2\}] &
	=\ 0,\cr
}}\eqn\MixedJacobiSpelledOut
$$
where `$B$' and `$F$' indicate the overall statistics of the operator involved.


\medskip
\point %4
Finally, an exercise in fermionic creation and annihilation operators and their
anticommutation relations,
$$
\{\a_\alpha,\a_\beta\}\ =\ \{\ad_\alpha,\ad_\beta\}\ =\ 0,\quad
\{\a_\alpha,\ad_\beta\}\ =\ \delta_{\alpha,\beta}\,.
\eqn\CCR
$$

\spointbegin
Calculate the commutators $[\ad_\alpha\a_\beta,\ad_\gamma]$, $[\ad_\alpha\a_\beta,\a_\delta]$
and $[\ad_\alpha\a_\beta,\ad_\gamma\a_\delta]$.

\spoint %b
Consider two one-body operators $\hat A_1$ and $\hat B_1$
and let $\hat C_1$ be their commutator, $\hat C_1=[\hat A_1,\hat B_1]$.
Let $\hat A$ be the second-quantized forms of $\hat A_{\rm tot}$,
$$
\hat A\ =\sum_{\alpha,\beta}\bra\alpha\hat A_1\ket\beta\,\ad_\alpha\a_\beta\, ,
\eqn\OneParticle
$$
and ditto for the second-quantized $\hat B$ and $\hat C$.
\subpar
Verify that $[\hat A,\hat B]=\hat C$.

\spoint %c
Calculate the commutator $[\ad_\mu\a_\nu,\ad_\alpha\ad_\beta\a_\gamma\a_\delta]$.

\spoint %d
The second quantized form of a two-body additive operator
$$
\hat B_{\rm tot}\ =\ \half\sum_{i\neq j}\,
\hat B_2( i{\rm th\atop{}}\ {\rm and}\ j{\rm th\atop{}}\ {\rm particles})
$$
acting on identical fermions is
$$
\hat B\ =\ \half\sum_{\alpha,\beta,\gamma,\delta}
\bra{\alpha\otimes\beta}\hat B_2\ket{\gamma\otimes\delta}\,
	\ad_\alpha\ad_\beta\a_\delta\a_\gamma\,.
\eqn\TwoParticles
$$
This expression is similar to its bosonic counterpart, but note the reversed order
of the annihilation operators $\a_\delta$ and $\a_\gamma$.
\subpar
Consider a one-body operator $\hat A_1$ and  two-body operator $\hat B_2$
and $\hat C_2$ where
$\hat C_2=\left[\bigl(\hat A_1(1\coeff{\rm st}{})
 +\hat A_1(2\coeff{\rm nd}{})\bigr),\hat B_2\right]$.
Show that the respective second-quantized operators in the fermionic Fock space
satisfy $\hat C=[\hat A,\hat B]$.




\bye
