\input phyzzx

\def\pd{{\vphantom\dagger}}
\def\a{\hat a^\pd}
\def\ad{\hat a^\dagger}
\def\bk{{\bf k}}
\def\kl{{\bk,\lambda}}
\def\bx{{\bf x}}
\def\by{{\bf y}}
\def\eqdef{\buildrel\rm def\over=}

\hsize=7in	\hoffset=-0.35in 	\vsize=8.5in	%\voffset=-0.25in
\quad\noindent PHY--396~K.\qquad
Problem set \#4.\qquad
Due October 7, 2003.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
In homework\#2 we developed Hamiltonian formalism for a massive
vector field $A^\mu(x)$.
Upon quantization, the 3--vector field ${\bf A}(x)$ and its canonical
conjugate $-{\bf E}(x)$ become quantum fields subject to equal-time
commutation relations
$$
[\hat A^i(\bx),\hat A^j(\by)]\ =\ 0,\quad
[\hat E^i(\bx),\hat E^j(\by)]\ =\ 0,\quad
[\hat A^i(\bx),\hat E^j(\by)]\ =\ -i\delta^{ij}\delta^{(3)}(\bx-\by)
\eqn\CCR
$$
($\hbar=1,c=1$ units)
governed by the free Hamiltonian
$$
\hat H\ = \int\!\!d^3\bx\left(
	\half{\bf\hat E}^2\ +\ {(\nabla\cdot {\bf\hat E})^2\over 2m^2}\
	+\ \half(\nabla\times{\bf\hat A})^2\ +\ \half m^2{\bf\hat A}^2
	\right)
\eqn\VectorHam
$$
(we assume $J^\mu=0$).
For the non-dynamical $A^0$ field,
its time-independent equation of motion becomes an operatorial
identity
$$
\hat A^0(x)\ =\ -{\nabla\cdot{\bf\hat E}(x)\over m^2}\,.
\eqn\Azero
$$

\pointcon
The purpose of the present exercise is to expand fields in terms
of creation and annihilation operators $\ad_\kl$ and $\a_\kl$
where $\lambda$ labels three different polarization states of
a vector particle ($\rm spin=1$).
Generally, bases for polarization states correspond to $\bf k$--dependent
complex bases ${\bf e}_\lambda({\bf k})$ for ordinary 3--vectors,
$$
{\bf e}^{}_\lambda({\bf k}) \cdot {\bf e}^*_{\lambda'}({\bf k})\
=\ \delta_{\lambda,\lambda'}
\eqn\ComplexBasis
$$
Of particular convenience is the helicity basis of eigenvectors
of the vector product $i{\bf k}\times{}$, namely
$$
i{\bf k}\times {\bf e}^{}_\lambda({\bf k})\
=\ \lambda|{\bf k}|{\bf e}^{}_\lambda({\bf k}),\qquad
\lambda=-1,0,+1.
\eqn\Helicity
$$
By convention, the overall phases of the helicity eigenvectors
are chosen such that
$$
{\bf e}^{}_0({\bf k})\ =\ {{\bf k}\over |{\bf k}|}\,\quad
{\bf e}^*_\lambda({\bf k})\ =\ (-1)^\lambda {\bf e}^{}_{-\lambda}({\bf k}),\quad
{\bf e}^{}_\lambda(-{\bf k})\ =\ -{\bf e}^*_\lambda(+{\bf k}).
\eqn\Phases
$$

\pointcon
Combining Fourier transform with a basis decomposition, we have
$$
{\bf\hat A}(\bx)\ = \int\!\!{d^3{\bf k}\over(2\pi)^3}\sum_\lambda
e^{i\bf kx}{\bf e}^{}_\lambda({\bf k})\,\hat A_\kl\,,\qquad
\hat A_\kl\ = \int\!\!d^3\bx\,e^{-i\bf kx}
	 {\bf e}^*_\lambda({\bf k})\cdot{\bf\hat A}(\bx)
\eqn\Fourier
$$
and ditto for the ${\bf\hat E}(\bx)$ fields and its $\hat E_\kl$ modes.

\spointbegin
Show that $\hat A_\kl^\dagger=-\hat A_{-\kl}$,
$\hat E_\kl^\dagger=-\hat E_{-\kl}$, and derive the equal-time
commutation relations for the $\hat A_\kl$ and $\hat E_\kl$ operators.

\spoint %b
Show that
$$
\hat H\ =\int\!\!{d^3{\bf k}\over(2\pi)^3}\sum_\lambda\left(
	{C_\kl\over 2}\,\hat E_\kl^\dagger\hat E^\pd_\kl\
	+\ {\omega_{\bf k}^2\over 2C_\kl}\hat A_\kl^\dagger\hat A^\pd_\kl
	\right)
\eqn\FourierHam
$$
where $\omega_{\bf k}=\sqrt{{\bf k}^2+m^2}$ and 
$C_\kl=1+\delta_{\lambda,0}({\bf k}^2/m^2)$.

\spoint %c
Define creation and annihilation operators according to
$$
\a_\kl\
=\ {\omega_\bk\hat A^\pd_\kl\ -\ iC_\kl\hat E^\pd_\kl \over\sqrt{C_\kl}}\,,
\qquad\ad_\kl\
=\ {\omega_\bk\hat A^\dagger_\kl\ +\ iC_\kl\hat E^\dagger_\kl \over\sqrt{C_\kl}}
\eqn\BCR
$$
and verify that they satisfy (relativistically-normalized)
equal-time bosonic commutation relations.

\spoint %d
Show that
$$
\hat H\ = \int\!\!{d^3{\bf k}\over(2\pi)^3\,2\omega_\bk}\sum_\lambda
	\omega_{\bf k}\,\ad_\kl\a_\kl\
	+\ \rm const.
\eqn\Quanta
$$

\spoint %e
Next, consider the time dependence of the free vector field
and show that
$$
{\bf\hat A}(\bx,t)\
=\int\!\!{d^3{\bf k}\over(2\pi)^3\,2\omega_\bk}
\sum_\lambda\sqrt{C_\kl}\left(
	e^{-ikx}{\bf e}^{}_\lambda({\bf k})\,\a_\kl(0)\
	+\ e^{+ikx}{\bf e}^*_\lambda({\bf k})\,\ad_\kl(0)
	\right)_{k^0=+\omega_{\bf k}}\, .
\eqn\TimeDep
$$

\spoint %f
Write down a similar formula for the $\hat A^0(\bx,t)$
(use eq.~\Azero).
Together with the previous result, you should get
$$
\hat A_\mu(x)\ 
=\int\!\!{d^3{\bf k}\over(2\pi)^32\omega_\bk}\sum_\lambda\left(
	e^{-ikx}f^{}_\mu(\kl)\,\a_\kl(0)\
	+\ e^{+ikx}f^*_\mu(\kl)\,\ad_\kl(0)
	\right)_{k^0=+\omega_{\bf k}}
\eqn\Relavector
$$
where
$$
f^\mu(\kl)\ =\cases{
\bigl(0,{\bf e}^{}_\lambda({\bf k})\bigr) &
for $\lambda=\pm1$,\crr
\left({|{\bf k}|\over m},{\omega_{\bf k}\over m}\,{{\bf k}\over|{\bf k}|}\right)&
for $\lambda=0$.\cr }
\eqn\Fdef
$$
Please note that the 4--vectors $f^\mu(\kl)$ are nothing but
purely-spatial vectors ${\bf e}_\lambda({\bf k})$ Lorentz-boosted
into the moving particle's frame.
In particular, for all $(\kl)$, $f^\mu f^*_\mu=-1$ and $f^\mu k_\mu=0$.

\spoint %g
Finally, verify that the vector field~\Relavector\ satisfies the free
equations of motion $\partial_\mu \hat A^\mu(x)=0$ and
$(\partial^2+m^2)\hat A^\mu(x)=0$.


\bigskip
\point %2
Now consider the Feynman propagator for the massive vector field.
\spointbegin
First, a lemma:  Show that
$$
\sum_\lambda f^\mu(\kl)f^{\nu *}(\kl)\
=\ -g^{\mu\nu}\ +\ {k^\mu k^\nu\over m^2}\,.
\eqn\Lemma
$$

\spoint %b
Next, show that
$$
\eqalign{
\bra{0}\hat A^\mu(x)\hat A^\nu(y) \ket{0}\ &
=\int\!\!{d^3{\bf k}\over(2\pi)^3}\,{1\over{2\omega_{\bf k}}}
    \left[\left(-g^{\mu\nu}+{k^\mu k^\nu\over m^2}\right)e^{-ik(x-y)}\right]
	_{k^0=+\omega_{\bf k}}\cr
&=\ \left(-g^{\mu\nu}-{\partial^\mu \partial^\nu\over m^2}\right) D(x-y) .\cr
}\eqno\eq
$$

\spoint %c
Finally, the Feynman propagator: Show that
$$
\eqalign{
G_F^{\mu\nu}\ \equiv\ \bra{0}{\bf T^*}\hat A^\mu(x)\hat A^\nu(y) \ket{0}\ &
=\ \left(-g^{\mu\nu}-{\partial^\mu \partial^\nu\over m^2}\right) G_F(x-y)\cr
&= \int\!\!{d^4{\bf k}\over(2\pi)^4}\,
	\left(-g^{\mu\nu}+{k^\mu k^\nu\over m^2}\right)\,
	{ie^{-ik(x-y)}\over k^2-m^2+i0}\cr
}\eqn\VectorProp
$$
where
$$
{\bf T^*}\hat A^\mu(x)\hat A^\nu(y) \
=\ {\bf T}\hat A^\mu(x)\hat A^\nu(y) \
+\ i\delta^{\mu0}\delta^{\nu0}\delta^{(4)}(x-y).
\eqn\Tstar
$$
For the explanation of the $\bf T^*$ modification of the time-ordered product
of vector fields,
please see {\it Quantum Field Theory} by Claude Itzykson and
Jean--Bernard Zuber.

\bigskip
\point %3
Finally, an exercise in Dirac's $\gamma$ matrices.
\spointbegin
Verify
$\bigl[S^{\kappa\lambda},S^{\mu\nu}\bigr]=i\bigl(
g^{\lambda\mu}S^{\kappa\nu}-g^{\lambda\nu}S^{\kappa\mu}
-g^{\kappa\mu}S^{\lambda\nu}+g^{\kappa\nu}S^{\lambda\mu}\bigr)$.

\spoint %b
Verify $M^{-1}(L)\gamma^\mu M(L)= L^\mu_{\,\,\nu}\gamma^\nu$
for $L=\exp(\theta)$ (\ie, $L^\mu_{\,\,\nu}=\delta^\mu_\nu+
 \theta^\mu_{\,\,\nu}+\half\theta^\mu_{\,\,\lambda}\theta^\lambda_{\,\,\nu}+\cdots$)
and $M(L)=\exp\bigl(-\coeff{i}{2}\theta_{\alpha\beta}S^{\alpha\beta}\bigr)$

\spoint %c
Calculate $\{\gamma^\rho,\gamma^\lambda\gamma^\mu\gamma^\nu\}$,
$[\gamma^\rho,\gamma^\kappa\gamma^\lambda\gamma^\mu\gamma^\nu]$ and
$[S^{\rho\sigma},\gamma^\lambda\gamma^\mu\gamma^\nu]$.

\spoint %d
Show that
$\gamma^\alpha\gamma_\alpha=4$,
$\gamma^\alpha\gamma^\nu\gamma_\alpha=-2\gamma^\nu$,
$\gamma^\alpha\gamma^\mu\gamma^\nu\gamma_\alpha=4g^{\mu\nu}$ and
$\gamma^\alpha\gamma^\lambda\gamma^\mu\gamma^\nu\gamma_\alpha=
	-2\gamma^\nu\gamma^\mu\gamma^\lambda$.\brk
Hint: use $\gamma^\alpha\gamma^\nu=2g^{\nu\alpha}-\gamma^\nu\gamma^\alpha$
repeatedly.

\spoint %e
Consider the electron's spinor field $\Psi(x)$
in an electromagnetic background.
Show that the gauge-covariant Dirac equation
$\bigl(i\gamma^\mu D_\mu+m\bigr)\Psi(x)=0$ implies\brk
$\bigl( m^2+D^2+qF_{\mu\nu}S^{\mu\nu}\bigr)\Psi(x)=0$.


\bye
