\input phyzzx

\def\bx{{\bf x}}
\def\eqdef{\buildrel\rm def\over=}
\def\a{\hat a^{\vphantom\dagger}}
\def\ad{\hat a^\dagger}
\def\b{\hat b^{\vphantom\dagger}}
\def\bd{\hat b^\dagger}
\def\bp{{\bf p}}
\def\dummy{{\phantom{0\mu}}}
\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 \#8.\quad
Due November 3, 1998.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider charge-conjugation properties of Dirac bilinears
$\hat{\overline\Psi}\Gamma\hat\Psi$.

\spointbegin
Show that $\hat{\cal C}\hat{\overline\Psi}\Gamma\hat\Psi\hat{\cal C}=
\hat{\overline\Psi}\Gamma^c\hat\Psi$ where
$\Gamma^c=\gamma^0\gamma^2\Gamma^\top\gamma^0\gamma^2$.

\spoint %b
Calculate $\Gamma^c$ for all 16 independent matrices $\Gamma$
and find out which Dirac bilinears are $\cal C$--even and which
are $\cal C$--odd.

\spoint %c
Consider a bound state of a fermion and an antifermion, \eg\
a positronium state or a neutral meson.
As argued in class, the parity of such bound state is $P=-(-1)^L$.
Show that the C-parity of this state is $C=(-1)^S(-1)^L$ and use this
fact to explain why the decay mode and the lifetime of a 1S positronium
state depend on its spin.\brk
Hint: $\ad\bd=-\bd\ad$.


\medskip
\point %2
Now consider the time-reversal $\hat{\cal T}$, which acts on particle
states as
$$
\hat{\cal T}\ket{{\rm particle\ type},\bp,s}\
=\ \eta\,i^{2m_s}\,\ket{{\rm same\ particle\ type}, -\bp, -s}
\eqn\Taction
$$
where $\eta$ is an overall phase and the $i^{2m_s}$ factor makes the time-reversal
commute with the space rotations.
Because of this factor and the anti-linearity of the $\hat{\cal T}$ operator,
$\hat{\cal T}^2\ket{\bp,s}=(\eta\,i^{-2m_s})(\eta\,i^{2m_s})^*\ket{\bp,s}=
(-1)^{2m_s}\ket{\bp,s}$ \ie, $\hat{\cal T}^2={}$rotation by $2\pi$: It leaves
bosonic states invariant but changes the overall sign of the fermionic states.

\pointcon
In terms of the particle-creation operators, eq.~\Taction\ means
$$
\hat{\cal T}\ad_{\bp,s}\hat{\cal T}^{-1}\ =\ \eta\,i^{2m_s}\,\ad_{-\bp,-s}
\qquad{\rm and}\quad
\hat{\cal T}\bd_{\bp,s}\hat{\cal T}^{-1}\ =\ \eta\,i^{2m_s}\,\bd_{-\bp,-s}
\eqn\Tops
$$
and ditto for the annihilation operators $\a_{\bp,s}$ and $\b_{\bp,s}$.

\pointcon
Let us focus on the Dirac fermions.

\spointbegin
First, prove a lemma:
$$
i^{2m_s}\,u^*(\bp,s)\ =\ i\gamma^1\gamma^3\,u(-\bp,-s),\qquad
i^{2m_s}\,v^*(\bp,s)\ =\ i\gamma^1\gamma^3\,v(-\bp,-s).
$$
Note: In the eigenbasis of the $S_z$, spin-reversal acts as
$\widehat{\cal SR}\ket{m_s}=i^{2m_s}\ket{-m_s}$, but for generic polarization
states of non-relativistic spin-half particles, one has
$\widehat{\cal SR}\ket\xi=\ket{\sigma^2\xi^*}$.
%
%Note for generic spin states of spin~$\half$ particles,
%$\ket{s}\to i^{2m_s}\,\ket{-s}$ means $\xi_s\to\xi_{-s}=\sigma^2\xi^*_s$.

\spoint %b
Use the above lemma to verify that the time-reversal of the Dirac field
$\hat\Psi(x)$ yields
$$
\hat{\cal T}\hat\Psi(\bx,t)\hat{\cal T}^{-1}\
=\ i\eta\,\gamma^1\gamma^3\,\Psi(\bx,-t).
$$

\spoint %c
Next, consider the Dirac bilinears $\hat{\overline\Psi}\Gamma\hat\Psi$
and show that
$\hat{\cal T}\hat{\overline\Psi}\Gamma\hat\Psi\hat{\cal T}^{-1}=
\hat{\overline\Psi}\Gamma^t\hat\Psi$ where
$\Gamma^t=\gamma^3\gamma^1\Gamma^*\gamma^1\gamma^3$.

\spoint %d
Calculate $\Gamma^t$ for all 16 independent matrices $\Gamma$
and find out which Dirac bilinears are $\cal T$--even and which
are $\cal T$--odd.

\spoint %e
Verify the $\cal T$--invariance of the Dirac action.

\medskip
\point %3
Finally, consider the combined $\hat{\cal C}\hat{\cal P}\hat{\cal T}$
symmetry of the Dirac field and verify that for any bilinear operator
$\hat{\cal O}(x)=\hat{\overline\Psi}(x)\Gamma\hat\Psi(x)$,
$$
\hat{\cal C}\hat{\cal P}\hat{\cal T}\,\hat {\cal O}(x)\,
	[\hat{\cal C}\hat{\cal P}\hat{\cal T}]^{-1}\
=\ \hat {\cal O}^\dagger(-x)\times (-1)^{\rm\#Lorentz\ indices\ in\ \hat{\cal O}}.
$$

\bye
