\input phyzzx

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

\hsize=6.5in	\hoffset=0in	\vsize=8.5in	\voffset=0in
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\noindent PHY--396~K.\quad
Problem set \#1.\quad
Due September 9, 2003.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

According to the Noether theorem, a translationally invariant
system of classical fields $\phi_a$
has a conserved stress-energy tensor
$$
T^{\mu\nu}_{\rm Noeter}\
= \sum_a \partder{{\cal L}}{(\partial_\mu\phi_a)}\,\partial^\nu\phi^a\
-\ g^{\mu\nu}\,{\cal L} .
\eqn\GeneralT
$$
Actually, to assure the symmetry of the stress-energy tensor,
$T^{\mu\nu}=T^{\nu\mu}$
(which is necessary for the angular momentum conservation), one sometimes
has to add a total divergence,
$$
T^{\mu\nu}\ =\ T^{\mu\nu}_{\rm Noeter}\
+\ \partial_\lambda{\cal K}^{[\lambda\mu]\nu} ,
\eqn\Tcorr
$$
where ${\cal K}^{[\lambda\mu]\nu}$ is some 3--index Lorentz tensor antisymmetric
in its first two indices.

\pointbegin
Show that regardless of the specific form of
${\cal K}^{[\lambda\mu]\nu}(\phi,\partial\phi)$,
$$
\eqalign{
\partial_\mu T^{\mu\nu}\ &
=\ \partial_\mu T^{\mu\nu}_{\rm Noether}\ = ({\rm hopefully})=\ 0\cr
P^\mu_{\rm net}\ &
\equiv\int\!\!d^3\bx\,T^{0\mu}\ =\int\!\!d^3\bx\,T^{0\mu}_{\rm Noether}\,.\cr
}\eqno\eq
$$

For the scalar fields, real or complex, $T^{\mu\nu}_{\rm Noeter}$ is
properly symmetric and one simply has $T^{\mu\nu}=T^{\mu\nu}_{\rm Noeter}$.
Unfortunately, the situation is more complicated for the vector, tensor or spinor
fields.
To illustrate the problem, consider the free electromagnetic fields
described by the Lagrangian
$$
{\cal L}(A_\mu,\partial_\nu A_\mu)\ =\ -\coeff14\,F_{\mu\nu} F^{\mu\nu}
\eqn\LEM
$$
where $A_\mu$ is a real vector field and
$F_{\mu\nu}\equiv\partial_\mu A_\nu-\partial_\nu A_\mu$.

\point %2
Write down $T^{\mu\nu}_{\rm Noeter}$ for the free electromagnetic fields and
show that it is neither symmetric nor gauge invariant.

\point %3
The properly symmetric --- and also gauge invariant ---
 stress-energy tensor for the free electromagnetism is
$$
T_{\rm EM}^{\mu\nu}\ =\ -F^{\mu\lambda} F^\nu_{\,\,\lambda}\
+\ \coeff14\,g^{\mu\nu}\,F_{\kappa\lambda} F^{\kappa\lambda} .
\eqn\TEM
$$
Show that this expression indeed has form \Tcorr\ for some
${\cal K}^{[\lambda\mu]\nu}$.

\point %4
Write down the components of the stress-energy tensor~\TEM\ in
non-relativistic notations and make sure you have the familiar
electromagnetic energy density, momentum density and pressure.


Now consider the electromagnetic fields coupled to a conserved electric
current $J^\mu$.
\point %5
Use Maxwell's equations to show that
$$
\partial_\mu T_{\rm EM}^{\mu\nu}\ =\ - F^{\nu\lambda} J_\lambda\,.
\eqn\Tdiv
$$
\par
Eq.~\Tdiv\ suggests that any system of charged ``matter'' fields
which carries the current~$J^\mu$ should have its
stress-energy tensor $T_{\rm mat}^{\mu\nu}$ obeying
$$
\partial_\mu T_{\rm mat}^{\mu\nu}\ =\ + F^\nu_{\,\,\lambda} J^\lambda_{\rm EM}\,.
\eqn\Tmatdiv
$$
Consequently, the combined stress-energy tensor
$T_{\rm EM}^{\mu\nu}+T_{\rm mat}^{\mu\nu}$ should be divergence-less and thus
lead to a conserved total energy and momentum.

Generally, testing eq.~\Tmatdiv\ for any particular system
of charged ``matter'' fields
makes use of fields' equations of `motion' and also of the fact that
{\it the covariant derivatives $D_\mu$ do not commute with each other}.
Instead, when acting upon a field $\Phi_q$ of charge $q$, one has
$$
(D_\mu D_\nu-D_\nu D_\mu)\Phi_q\ =\ iq\,F_{\mu\nu}\,\Phi_q
\eqn\NonCommute
$$
(in $c=\hbar=1$ units).
\point %6
Verify eq.~\NonCommute.

Consider a specific example of the EM coupled to a complex scalar field
of electric charge $q\neq 0$.
The net Lagrangian density of the $\rm EM+scalar$ fields is
$$
{\cal L}\ =\ D^\mu\Phi^* D_\mu\Phi\ -\ m^2\Phi^*\Phi\
-\ \coeff14 F^{\mu\nu} F_{\mu\nu}\,.
\eqn\LTotal
$$
\point %7
Verify the gauge invariance of the electric current
$$
J^\mu\ =\ -\partder{{\cal L}}{A_\mu}
\eqn\Jdef
$$
by expressing it in terms of the scalar fields % $\Phi$ and $\Phi^*$
and their {\sl covariant} derivatives $D^\mu\Phi$ and $D^\mu\Phi^*$.

\point %8
Write down equations of motion for the scalar fields $\Phi$ and $\Phi^*$
and use them to verify the current conservation $\partial_\mu J^\mu=0$.

\point %9
Write down the Noether stress-energy tensor for this field system and show
that
$$
T^{\mu\nu}_{\rm net}\
\equiv\ T^{\mu\nu}_{\rm EM}\ +\ T^{\mu\nu}_{\rm mat}\
=\ T^{\mu\nu}_{\rm Noeter}\
+\ \partial_\lambda{\cal K}^{[\lambda\mu]\nu}
\eqn\Tnet
$$
where ${\cal K}^{[\lambda\mu]\nu}$ is the same function of EM fields
as in the free EM case~[{\bf3.}],
$T^{\mu\nu}_{\rm EM}$ is exactly as in eq.~\TEM,
and
$$
T^{\mu\nu}_{\rm mat}\ =\ D^\mu\Phi^*\,D^\nu\Phi\ +\ D^\nu\Phi^*\,D^\mu\Phi\
-\ g^{\mu\nu}\bigl( D_\lambda\Phi^*\,D^\lambda\Phi\,-\,m^2\Phi^*\Phi \bigr)
\eqn\Tmat
$$
Hint: In the presence of an electric current $J^\mu$, the
$\partial_\lambda{\cal K}^{[\lambda\mu]\nu}$ correction to the electromagnetic
stress-energy tensor contains an extra $J^\mu A^\nu$ term.
This term is important for obtaining a gauge-invariant stress-energy tensor~\Tmat\
for the scalar field.

\point %10
Use the scalar field's equations of motion and eq.~\NonCommute\
to verify eq.~\Tmatdiv.


\bye
