\input phyzzx

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

\hsize=6.8in	\hoffset=-0.2in
\vsize=8.75in	\voffset=-0.25in
%\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#8.\quad
Due November 13, 2000.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
First, a little exercise about Weyl spinors.
As discussed in class, a  Dirac spinor field $\Psi(x)$ is
physically equivalent to two left-handed Weyl spinor fields $\chi(x)$ and
$\varphi(x)$.
In Weyl basis,
$$
\Psi(x)\ \equiv\ \pmatrix{\psi_L(x)\cr \psi_R(x)\cr }\
=\ \pmatrix{\ \chi(x)\cr -\sigma^2\varphi^*(x)\cr } .
\eqn\DiracWeyl
$$
\spointbegin
Show that in terms of the Weyl spinor fields,
the Dirac Lagrangian becomes
$$
{\cal L}\ \equiv\ \overline\Psi(i\gamma^\mu\partial_\mu-m)\Psi\
=\ i\chi^\dagger\bar\sigma^\mu\partial_\mu\chi\
+\ i\varphi^\dagger\bar\sigma^\mu\partial_\mu\varphi\
+\ m\left(\varphi^\top\!\sigma^2\chi\,+\,\varphi^\dagger\sigma^2\chi^*\right)
\eqn\LDirac
$$
(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 %b
Verify that each term in the Lagrangian~\LDirac\ is separately invariant
(up to a total derivative) under the combined Parity + Charge Conjugation
symmetry $\widehat{\cal CP}$.


\bigskip
\point %2
The rest of this homework concerns Dirac spinor fields and their
discrete symmetries.
We begin with the 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
$\Gamma^c=\gamma^0\gamma^2\Gamma^\top\gamma^0\gamma^2$.\brk
Hint: Mind the anticommutativity of the fermionic fields.

\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: $\ad\bd=-\bd\ad$.


\bigskip
\point %3
The time-reversal symmetry involves an {\sl anti-linear, anti-unitary}
operator $\hat{\cal T}$ that invert directions of all particle momenta
and spins,\foot{%
    Please see J.~J.~Sakurai {\it Modern Quantum Mechanics}, \S4.4 for a discussion
    of time reversal in general and spin reversal in particular.
    }
%
$$
\hat{\cal T}\ket{{\rm particle\ type},\bp,s}\
=\ ({\rm phase)}\,\ket{{\rm same\ particle\ type}, -\bp, -s} .
\eqn\Taction
$$
The phase factor here here combines an arbitrary but fixed overall phase
$\eta$ with a phase factor inherent in spin reversal:
$\widehat{\cal SR}\ket{m_s}=i^{2m_s}\ket{-m_s}$
in the $\hat S_z$ eigenbasis, or more generally,
$$
\widehat{\cal SR}\ket\xi\
=\ i^{2S}e^{-\pi i \hat S_y}\ket\xi^* .
\eqn\SpinReversal
$$
In particular, for spin~$\half$ particles,
$\widehat{\cal SR}\ket\xi=\sigma_2\ket\xi^*$,
in agreement with the rule $\xi_{-s}=\sigma_2\xi_s^*$ we used in class.
Note however that reversing the spin twice results in a rotation by $2\pi$,
$\widehat{\cal SR}^2=e^{-2\pi i\hat S_y}=(-1)^{2S}$, which is trivial
for integral spins but changes the overall sign of the spin state for
half-integral spins.
Consequently, the time-reversal operator in the Fock space satisfies
$$
\hat{\cal T}^2\ =\ (-1)^F\ \equiv\ \rm rotation\ by\ 2\pi.
\eqn\Tsquare
$$

\pointcon
In this exercise, we consider time-reversal of the Dirac spinor field.
First, we need a lemma:
$$
i^{2m_s}u^*(-\bp,-m_s)\ =\ -i\gamma^1\gamma^3 u(+\bp,+m_s),\qquad
i^{2m_s}v^*(-\bp,-m_s)\ =\ -i\gamma^1\gamma^3 v(+\bp,+m_s).
\eqn\Lemma
$$
\spointbegin
Prove this lemma.

\pointcon
In terms of electronic and positronic creation and annihilation operators,
eq.~\Taction\ means
$$
\eqalign{
\hat{\cal T}\ad(\bp,m_s)\hat{\cal T}^{-1}\ &
=\ (\mp i)\,i^{2m_s}\,\ad(-\bp,-m_s),\cr
\hat{\cal T}\bd(\bp,m_s)\hat{\cal T}^{-1}\ &
=\ (\pm i)\,i^{2m_s}\,\bd(-\bp,-m_s),\cr
\hat{\cal T}\a(\bp,m_s)\hat{\cal T}^{-1}\ &
=\ (\pm i)\,i^{2m_s}\,\a(-\bp,-m_s),\cr
\hat{\cal T}\b(\bp,m_s)\hat{\cal T}^{-1}\ &
=\ (\mp i)\,i^{2m_s}\,\b(-\bp,-m_s),\cr
}\eqn\Tops
$$
where the spin-independent phase factors $\pm i$ make for a consistent
time-reversal of the Dirac spinor field.


\spoint %b
Use lemma~\Lemma\ and eqs.~\Tops\ to show that
$$
\hat{\cal T}\hat\Psi(\bx,t)\hat{\cal T}^{-1}\
=\ \pm\gamma^1\gamma^3\,\Psi(\bx,-t).
\eqno\eq
$$

\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.

\bigskip
\point %4
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
$\hat{\cal O}(x)=\hat{\overline\Psi}(x)\Gamma\hat\Psi(x)$,
$$
\hat{\cal C}\hat{\cal P}\hat{\cal T}\,\hat {\cal O}(x)\,
	[\hat{\cal C}\hat{\cal P}\hat{\cal T}]^{-1}\
=\ \hat {\cal O}^\dagger(-x)\times (-1)^{\rm\#Lorentz\ indices\ in\ \hat{\cal O}}.
\eqn\CPT
$$
Actually, eq.~\CPT\ holds for any physically measurable operator $\hat{\cal O}(x)$
in any legitimate quantum field theory --- this is the famous CPT theorem ---
but the exercise is limited to the Dirac bilinear operators only.

\bye
