\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}}
%\def\kl{{\bk,\lambda}}
%\def\klp{{\bk',\lambda'}}
\def\PMBOLD#1#2{\setbox0=\hbox{$#1{#2}$}\setbox2=\hbox{$#1{0}$}%
	\copy0\kern-\wd0\kern 0.1\wd2\box0 }
\def\pmbold#1{\mathpalette\PMBOLD{#1}}

\hsize=6.8in	\hoffset=-0.2in	%\vsize=8.5in	\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#6.\quad
Due October 14, 1998.
\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
$L^\mu_{\,\,\nu}=\exp\bigl(e^\theta\bigr)^\mu_{\,\,\nu}$ Dirac spinor
field $\Psi(x)$ transforms according to
$\Psi'(x)=\Lambda\Psi(L^{-1}x)$ where
$\Lambda=\exp\bigl(-{i\over2}\theta_{\mu\nu}S^{\mu\nu}\bigr)$.
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.

\medskip
\point %2
Under the {\sl parity} symmetry ${\cal P}(\bx,t)=(-\bx,t)$, 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.

\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.

\goodbreak\medskip
\point %3
The last exercise is about the two-component Weyl spinors.
\spointbegin
Let $\varphi$ be a left-handed Weyl spinor.
Show that $\sigma^2\varphi^*$ transforms under continuous Lorentz symmetries
as a right-handed Weyl spinor.
Hint: Prove and use $\sigma^2\sigma_\mu=\bar\sigma_\mu^*\sigma^2$.

\pointcon
Given (a), we can re-write any four-component Dirac spinor field $\Psi(x)$
as two left-handed Weyl spinor fields $\chi(x)$ and $\varphi(x)$,
$$
\Psi(x)\ \equiv\ \pmatrix{\psi_L(x)\cr \psi_R(x)\cr }\
=\ \pmatrix{\ \chi(x)\cr \sigma^2\varphi^*(x)\cr } .
\eqn\DiracWeyl
$$

\spoint %2
Re-express the Dirac bilinears \BiLinears\ in terms of the Weyl spinors
$\chi$ and $\varphi$.

\spoint %c
Show that in terms of $\chi$ and $\varphi$,
the Dirac Lagrangian becomes
$$
{\cal L}\ \equiv\ \overline\Psi(i\gamma^\mu\partial_\mu-m)\Psi\
=\ i\chi^\dagger\bar\sigma^\mu\partial_\mu\chi\
+\ i\varphi^\dagger\bar\sigma^\mu\partial_\mu\varphi\
-\ m\,\varphi^\top\!\sigma^2\chi\ -\ m\,\varphi^\dagger\sigma^2\chi^*
\eqn\LDirac
$$
(up to a total derivative).
Note that $\chi(x)$, $\varphi(x)$, $\chi^*(x)$ and $\varphi^*(x)$ are
{\it fermionic} fields, so in the classical limit they {\it anticommute}
with each other rather than commute.
Thus, $\varphi^\top\!\sigma^2\chi=+\chi^\top\!\sigma^2\varphi$ even though the
$\sigma^2$ matrix is antisymmetric.

\spoint %d
Verify that each term in the Lagrangian~\LDirac\ is Lorentz invariant.\brk
Hint: Use ${\rm Spin}(3,1)=SL(2,{\bf C})$, \cf\ previous homework.

\pointcon
Next Consider the $U(1)$ symmetries of a Dirac spinor field.
%
\spoint %e
The ordinary particle number symmetry acts as
$\Psi(x)\to e^{i\alpha}\Psi(x)$ or
$\chi(x)\to e^{+i\alpha}\chi(x)$, $\varphi(x)\to e^{-i\alpha}\varphi(x)$.
Show that the Noether current for this symmetry is $j^\mu=V^\mu$ and verify
explicitly that it is conserved.
This symmetry is often called {\sl vector like} or `{\sl the vector symmetry}'
after the true-vector nature of its current.

\spoint %f
For {\sl massless} spinor fields there is also
{\sl the axial symmetry}
$\chi(x)\to e^{+i\alpha}\chi(x)$, $\varphi(x)\to e^{+i\alpha}\varphi(x)$.
Show that in Dirac terms this symmetry acts
as $\Psi(x)\to\exp(-i\alpha\gamma^5)\Psi(x)$,
that the corresponding Noether current is the {\sl axial} vector
$j^{5\mu}=A^\mu$ (hence the name),
which is conserved if and only if $m=0$.

\pointcon
It is often convenient to recombine the vector symmetry~(e) and the axial
symmetry~(f) into {\sl chiral symmetries}
$$
\chi(x)\ \to\ e^{+i\alpha_L}\,\chi(x),\quad \varphi(x)\ \to\ \varphi(x)
\qquad {\rm and}\quad
\chi(x)\ \to\ \chi(x),\quad \varphi(x)\ \to\ e^{-i\alpha_R}\,\varphi(x)\,.
\eqn\ChiralS
$$
The corresponding {\sl chiral currents} are
$$
j^\mu_L\ =\ \half\bigl(V^\mu-A^\mu\bigr)\
=\ i\chi^\dagger\bar\sigma^\mu\chi\qquad
{\rm and}\quad  j^\mu_R\ =\ \half\bigl(V^\mu+A^\mu\bigr)\
=\ -i\varphi^\dagger\bar\sigma^\mu\varphi\,.
\eqn\ChiralJ
$$

\pointcon
The two Weyl spinors $\chi$ and $\varphi$ --- usually called {\sl chiral spinors}
or chiral fermions after chiral symmetries \ChiralS\ ---
are related to each other via parity $\chi\leftrightarrow\sigma^2\varphi^*$.
In field theories without parity (\eg, the theory of weak interactions
or various Grand Unified theories) it is often convenient to treat all the Weyl
spinors as unrelated and dispence with Dirac spinors altogether.
The most general Lagrangian for several free Weyl spinors $\chi_k$ is
$$
{\cal L}\
= \sum_n\,i\chi_k^\dagger\bar\sigma^\mu\partial_\mu\chi_k^{\phantom\dagger}\
- \sum_{k\ell}\,\half M^{k\ell}\,\chi_k^\top\!\sigma^2\chi_\ell^{\phantom\top}\
- \sum_{k\ell}\,\half M^{*k\ell}\,\chi_k^\dagger\sigma^2\chi_\ell^*
\eqn\LGeneric
$$
where $M^{k\ell}$ is a complex symmetric mass matrix.
The Dirac Lagrangian~\LDirac\ is a special case of~\LGeneric\ for two
Weyl spinors and $M={\,0\ m\choose m\ 0\,}$.
Another special case of interested is the Majorana fermion, which is equivalent
to a single Weyl fermion $\chi$, thus
$$
{\cal L}\
=\ i\chi^\dagger\bar\sigma^\mu\partial_\mu\chi\
-\ \half m\,\chi^\top\!\sigma^2\chi\ -\ \half m^* \chi^\dagger\sigma^2\chi^* .
\eqn\LMajorana
$$

\spoint %g
Show that the Lagrangian~\LMajorana\ is invariant under `parity' acting as
${\cal CP}\chi=\pm\sigma^2\chi^*$ if and only if the Majorana mass $M$ is real.
\subpar
Actually, this $\cal CP$ symmetry combines parity with charge conjugation,
but the Majorana fermions are neutral and thus do not mind the charge conjugation.

\spoint %h
Write down the field equation for the Majorana fermion and verify that it
implies the Klein--Gordon equation $(\partial^2+|m|^2)\chi=0$.
\spoint %i
Finally, write down a generic Lagrangian for several Weyl fermions with various
electric charges in a non-trivial EM background.
What is the most general mass matrix $M^{k\ell}$ for the charged fermions? 

\bye
