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\def\bx{{\bf x}}
\def\eqdef{\buildrel\rm def\over=}
%\def\bk{{\bf k}}
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\noindent PHY--396~K.\quad
Problem set \#1.\quad
Due September 7, 2004.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip
\pointbegin
According to the Noether theorem, a translationally invariant
system of classical fields $\phi_a$
has a conserved stress-energy tensor
$$
T^{\mu\nu}_{\rm Noether}\
= \sum_a \partder{{\cal L}}{(\partial_\mu\phi_a)}\,\partial^\nu\phi^a\
-\ g^{\mu\nu}\,{\cal L} .
\eqn\GeneralT
$$
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,
$$
T^{\mu\nu}\ =\ T^{\mu\nu}_{\rm Noether}\
+\ \partial_\lambda{\cal K}^{[\lambda\mu]\nu} ,
\eqn\Tcorr
$$
where ${\cal K}^{[\lambda\mu]\nu}$ is some 3--index Lorentz tensor antisymmetric
in its first two indices.
\spointbegin
Show that regardless of the specific form of
${\cal K}^{[\lambda\mu]\nu}(\phi,\partial\phi)$,
$$
\eqalign{
\partial_\mu T^{\mu\nu}\ &
=\ \partial_\mu T^{\mu\nu}_{\rm Noether}\ = ({\rm hopefully})=\ 0\cr
P^\mu_{\rm net}\ &
\equiv\int\!\!d^3\bx\,T^{0\mu}\ =\int\!\!d^3\bx\,T^{0\mu}_{\rm Noether}\,.\cr
}\eqno\eq
$$
\pointcon
For the scalar fields, real or complex, $T^{\mu\nu}_{\rm Noether}$ is
properly symmetric and one simply has $T^{\mu\nu}=T^{\mu\nu}_{\rm Noether}$.
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
$F_{\mu\nu}\equiv\partial_\mu A_\nu-\partial_\nu A_\mu$.
\spoint %b
Write down $T^{\mu\nu}_{\rm Noether}$ for the free electromagnetic fields and
show that it is neither symmetric nor gauge invariant.
\spoint %c
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
${\cal K}^{[\lambda\mu]\nu}$.
\spoint %d
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.
\bigskip
\point %2
Now consider the electromagnetic fields coupled to the electric
current $J^\mu$ of some charged ``matter'' fields.
Because of this coupling, only the {\sl net} energy-momentum of the
whole field system should be conserved, but not the separate
$P^\mu_{\rm EM}$ and $P^\mu_{\rm mat}$.
Consequently, we should have
$$
\partial_\mu T^{\mu\nu}_{\rm net}\ =\ 0\qquad
{\rm for}\quad T^{\mu\nu}_{\rm net}\
=\ T^{\mu\nu}_{\rm EM}\ +\ T^{\mu\nu}_{\rm mat}
\eqn\Tnet
$$
but generally $\partial_\mu T^{\mu\nu}_{\rm EM}\neq 0$ and
$\partial_\mu T^{\mu\nu}_{\rm mat}\neq 0$.
\spointbegin
Use Maxwell's equations to show that
$$
\partial_\mu T_{\rm EM}^{\mu\nu}\ =\ - F^{\nu\lambda} J_\lambda
\eqn\Tdiv
$$
and therefore any system of charged matter fields should have
its stress-energy tensor related to the electric current $J_\lambda$
according to
$$
\partial_\mu T_{\rm mat}^{\mu\nu}\
=\ + F^{\nu\lambda} J_\lambda .
\eqn\Tmatdiv
$$
\pointcon
Now consider a specific example a complex scalar field
of charge $q\neq 0$ coupled to the EM fields:
$$
{\cal L}_{\rm net}\ =\ D^\mu\Phi^* D_\mu\Phi\ -\ m^2\Phi^*\Phi\
-\ \coeff14 F^{\mu\nu} F_{\mu\nu}
\eqn\LTotal
$$
where
$$
D_\mu\Phi\ =\ (\partial_\mu\,+\,iqA_\mu)\Phi
\quad{\rm and}\quad
D_\mu\Phi^*\ =\ (\partial_\mu\,-\,iqA_\mu)\Phi^*
\eqn\Covariant
$$
are the {\sl covariant} derivatives of the scalar fields.
\spoint %b
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
and their covariant derivatives \Covariant.
\spoint %c
Write down equations of motion for the scalar fields $\Phi$ and $\Phi^*$
and use them to verify the current conservation $\partial_\mu J^\mu=0$.
\spoint %d
Write down the Noether stress-energy tensor for the
whole field system and show that
$$
T^{\mu\nu}_{\rm net}\
\equiv\ T^{\mu\nu}_{\rm EM}\ +\ T^{\mu\nu}_{\rm mat}\
=\ T^{\mu\nu}_{\rm Noether}\
+\ \partial_\lambda{\cal K}^{[\lambda\mu]\nu}
\eqn\Tnet
$$
where ${\cal K}^{[\lambda\mu]\nu}$ is the same function of EM fields
as in the free EM case~({\bf1}.c),
$T^{\mu\nu}_{\rm EM}$ 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.
\spoint %e
Use the scalar fields' equations of motion and non-commutativity
of the covariant derivatives
$$
[D_\mu,D_\nu]\Phi\ =\ iqF_{|mu\nu}\Phi,\qquad
[D_\mu,D_\nu]\Phi^*\ =\ -iqF_{|mu\nu}\Phi^*
\eqn\NonCommute
$$
to verify eq.~\Tmatdiv.
\bye