\input phyzzx

\def\bx{{\bf x}}
\def\eqdef{\buildrel\rm def\over=}

\hsize=6.8in	\hoffset=-0.2in	%\vsize=8.5in	\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#2.\quad
Due September 17, 1998.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider two {\sl hermitian} quantum fields $\hat\Phi(\bx,t)$ and $\hat\Pi(\bx,t)$
with the following equal-time commutation relations:
$$
\eqalign{
\bigl[\hat\Phi(\bx,t),\hat\Phi(\bx',t'=t)\bigr]\ & =\ 0,\cr
\bigl[\hat\Pi (\bx,t),\hat\Pi (\bx',t'=t)\bigr]\ &=\ 0,\cr
\bigl[\hat\Phi(\bx,t),\hat\Pi (\bx',t'=t)\bigr]\ &=\ i\delta^{(3)}(\bx-\bx').
}\eqn\ScalarCR
$$
Let the Hamiltonian for these fields be
$$
\hat H\ = \int\!\!d^3\bx\left( \half\,{\hat\Pi}^2\
	+\ \half\,(\nabla\hat\Phi)^2\ +\ \half M^2\,{\hat\Phi}^2\
	+\ \coeff{1}{24}\lambda\,{\hat\Phi}^4 \right)
\eqn\ScalarHam
$$
(in $c=\hbar=1$ units).\brk
Show that the $\hat\Phi(x)$ field satisfies a Lorentz-invariant field equation,
namely
$$
\left(\partial^\mu\partial_\mu + M^2\right)\hat\Phi(x)\
+\ \coeff16\lambda\,{\hat\Phi}^3(x)\ =\ 0 .
\eqn\ScalarEOM
$$

\bigskip
\point
Next, consider a pair of hermitian {\sl vector} fields, namely the
free electromagnetic fields ${\bf\hat E}(\bx,t)$ and ${\bf\hat B}(\bx,t)$.
These are {\sl transverse} vector fields
$$
\nabla\cdot{\bf\hat E}\ =\nabla\cdot{\bf\hat B}\ =\ 0
\eqn\MaxwellConstraints
$$
whose time-dependence
$$
\eqalign{
\partder{\bf\hat B}{t}\ &= -\nabla\times{\bf\hat E}\,,\cr
\partder{\bf\hat E}{t}\ &= +\nabla\times{\bf\hat B}\,,\cr
}\eqn\MaxwellDynamics
$$
follows from the free electromagnetic Hamiltonian
$$
\hat H_{\it EM}\ = \int\!\!d^3\bx\,\left(
	\half{\bf\hat E}^2 + \half{\bf\hat B}^2 \right) .
\eqn\FreeEMHam
$$
(Note {\sl ``rationalized''} $c=1$ units.)

\pointcon
Actually, the time-dependent Maxwell equations \MaxwellDynamics\
follow from the Hamiltonian \FreeEMHam\ {\sl and the equal-time
commutation relations}
$$
\eqalign{
\bigl[\hat E_i(\bx,t),\hat E_j(\bx',t'=t)\bigr]\ & =\ {???},\cr
\bigl[\hat B_i(\bx,t),\hat B_j(\bx',t'=t)\bigr]\ & =\ {???},\cr
\bigl[\hat E_i(\bx,t),\hat B_j(\bx',t'=t)\bigr]\ & =\ {???}.\cr
}$$
Such commutation relations for the electromagnetic fields are
completely determined by the consistency of eqs.~\MaxwellDynamics\
with the Hamiltonian~\FreeEMHam; {\bf write them down}.
Make sure your answer is consistent with the transversality of the
fields, \ie, with the time-independent Maxwell's 
equations~\MaxwellConstraints.

\bigskip
\point
Finally, consider the non-relativistic {\sl charged} scalar fields
$\hat\Phi(\bx,t)$ and $\hat\Phi^\dagger(\bx,t)$.
The electric charge and current density operators for such fields
are
$$
\hat\rho\ =\ q\hat\Phi^\dagger\hat\Phi ,\qquad
{\bf\hat J}\ =\ {q\hbar\over 2iM}\left(
	\hat\Phi^\dagger\bigl({\bf D}\hat\Phi\bigr)
	-\bigl({\bf D}\hat\Phi^\dagger\bigr)\hat\Phi \right),
\eqn\rhoJ
$$
where $q$ is the electric charge of a particle (created by the
$\hat\Phi^\dagger$ and annihilated by the $\hat\Phi$) and
$$
{\bf D}\hat\Phi\
\eqdef\ \left(\nabla - \coeff{iq}{\hbar c}{\bf A}\right)\hat\Phi
\qquad{\rm and}\quad
{\bf D}\hat\Phi^\dagger\
\eqdef\ \left(\nabla + \coeff{iq}{\hbar c}{\bf A}\right)\hat\Phi^\dagger
\eqn\covariant
$$
are the gauge-covariant gradients of the fields.

\pointcon
The charge and current densities \rhoJ\ are obviously gauge invariant.
It is less obvious --- but nevertheless true --- that they satisfy
the continuity equation
$$
\partial_\mu\hat J^\mu\
\equiv\ \partder{\hat\rho}{t}\ +\ \nabla\cdot{\bf\hat J}\ =\ 0.
\eqn\continuity
$$
Prove this.

\pointcon
Hint: Prove and use
$$
\nabla\cdot\left(\hat\Phi^\dagger\bigl({\bf D}\hat\Phi\bigr)
	-\bigl({\bf D}\hat\Phi^\dagger\bigr)\hat\Phi \right)\
=\ \hat\Phi^\dagger\bigl({\bf D}^2\hat\Phi\bigr)\
-\ \bigl({\bf D}^2\hat\Phi^\dagger\bigr)\hat\Phi ,
\eqno\eq
$$
then use the non-relativistic equations of `motion' for
the $\hat\Phi^\dagger$ and $\hat\Phi$ fields.

\bye