\input phyzzx

\def\a{\hat a^{\vphantom\dagger}}
\def\ad{\hat a^\dagger}
\def\bx{{\bf x}}
\def\by{{\bf y}}
\def\hilb{{\cal H}^B}
\def\eqdef{\buildrel\rm def\over=}

\hsize=7in	\hoffset=-0.35in	\vsize=9in	\voffset=-0.25in
\quad\noindent PHY--396~K.\qquad
Problem set \#2.\qquad
Due September 25, 2000.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
%The first problem of this set concerns classical fields and the
%equations that govern them.
Consider a {\sl massive} relativistic vector field $A^\mu(x)$ with the
Lagrangian density
$$
{\cal L}\ =\ -\coeff14\,F_{\mu\nu}F^{\mu\nu}\
	+\ \half m^2\,A_\mu A^\mu\ -\ A^\mu J_\mu
\eqn\Lagran
$$
(in $c=1$ units) where $F_{\mu\nu}\eqdef \partial_\mu A_\nu-\partial_\nu A_\mu$
and the current $J^\mu(x)$ is a fixed source for the $A^\mu(x)$ field.
Note that because of the mass term, the Lagrangian~\Lagran\ is {\sl not}
gauge invariant.

\spointbegin
Derive the Euler--Lagrange field equations for the massive vector field $A^\mu(x)$.

\spoint %b
Show that this field equation {\sl does not require} current conservation;
however, if the current happens to satisfy $\partial_\mu J^\mu=0$, then the
field $A^\mu(x)$ satisfies
$$
\partial_\mu A^\mu\ =\ 0\qquad {\rm and}\qquad
(\partial^2+m^2)A^\mu=J^\mu.
\eqn\TwoEqs
$$

\pointcon
Next, consider the Hamiltonian formalism for the massive vector field.
Our first step in deriving this formalism is to identify the canonically
conjugate ``momentum'' fields.
\spoint %c
Show that $\partial{\cal L}/\partial{\bf\dot A}=-{\bf E}$
but $\partial{\cal L}/\partial\dot A_0\equiv0$.

\pointcon
In other words, the canonically conjugate field to ${\bf A}(\bx)$
is $-{\bf E}(\bx)$ but the $A_0(\bx)$ does not have a canonical
conjugate!
Consequently,
$$
H\ =\ -\int\!\!d^3\bx\,{\bf\dot A}(\bx)\cdot{\bf E}(\bx)\ -\ L.
\eqn\HamDef
$$
\spoint %d
Show that in terms of the $\bf A$, $\bf E$ and $A_0$ fields and their
{\sl space} derivatives,
$$
H\ =\int\!\!d^3\bx\left\{ \half{\bf E}^2\,
	+\,A_0\left(J_0-\nabla\cdot{\bf E}\right)\,-\,\half m^2 A_0^2\
	+\ \half\left(\nabla\times{\bf A}\right)^2\,
	+\,\half m^2{\bf A}^2\,-\,{\bf J}\cdot{\bf A} \right\} .
\eqn\HamiltAAE
$$

\pointcon
Because the $A_0$ field does not have a canonical conjugate, the Hamiltonian
formalism does not produce an equation for the time-dependence of this field.
Instead, it gives us a time-independent equation relating the $A_0(\bx,t)$
to the values of other fields {\sl at the same time} $t$.
Specifically, we have
$$
{\delta H\over\delta A_0(\bx)}\ \equiv\
\left.\partder{{\cal H}}{A_0}\right|_\bx\
-\ \nabla\cdot\left.\partder{{\cal H}}{\nabla A_0}\right|_\bx\
=\ 0.\eqn\Constraint
$$
At the same time, the vector fields $\bf A$ and $\bf E$ satisfy the Hamiltonian
equations of motion,
$$
\partder{}{t}{\bf A}(\bx,t)\ =\ -\left.{\delta H\over \delta {\bf E}(\bx)}\right|_t
\,,\qquad
\partder{}{t}{\bf E}(\bx,t)\ =\ +\left.{\delta H\over \delta {\bf A}(\bx)}\right|_t
\,.\eqn\HamiltEqs
$$

\spoint %e
Write down the explicit form of all these equations.

\spoint %f
Finally, verify that the equations you have just written down are equivalent
to the Euler--Lagrange equations you derived in question~(a).

\bigskip
\point
Later in this class, we shall learn how to constuct the quantum electromagnetic
fields ${\bf\hat E}(\bx,t)$ and ${\bf\hat B}(\bx,t)$ out of creation
and annihilation operators in the photonic Fock space.
For the moment, let us simply take it for granted that they obey the
time-independent Maxwell eqs.
$$
\nabla\cdot{\bf\hat E}(\bx,t)\ =\nabla\cdot{\bf\hat B}(\bx,t)\ =\ 0
\eqn\MaxwellConstraints
$$
(we assume free EM fields, \ie\ no electric charges or currents).
In the Heisenberg picture, the quantum EM fields also obey the
time-dependent Maxwell equations
$$
\eqalign{
\partder{\bf\hat B}{t}\ &= -\nabla\times{\bf\hat E}\,,\cr
\partder{\bf\hat E}{t}\ &= +\nabla\times{\bf\hat B}\,,\cr
}\eqn\MaxwellDynamics
$$
which follow from the free electromagnetic Hamiltonian
$$
\hat H_{\it EM}\ = \int\!\!d^3\bx\,\left(
        \half{\bf\hat E}^2 + \half{\bf\hat B}^2 \right)
\eqn\FreeEMHam
$$
{\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, so {\bf write them down}.
Make sure your answer is consistent with the transversality of the
fields, \ie, with the time-independent Maxwell
equations~\MaxwellConstraints.





\bye

