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\noindent PHY--396~L.\quad
Problem set \#15.\quad
Due February 10, 2005.
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\pointbegin
First, a reading assignment:
\S7.4 of the {\sl Peskin \& Schroeder} textbook about the diagrammatic
proof of Ward--Takahashi identities for QED.

\medskip
\point %2
Next, consider ``scalar QED'', the theory of a charged scalar field
$\phi(x)\neq\phi^*(x)$ coupled to the electromagnetic field $A^\mu(x)$,
$$
{\cal L}\ =\ -\coeff14 F^2_{\mu\nu}\ +\ D^\mu\phi^* D_\mu\phi\
-\ M^2\phi^*\phi\ -\ \coeff14\lambda(\phi^*\phi)^2.
\eqno\eq
$$
Write down Ward--Takahashi identities for this theory and prove them
using the diagrammatic method.

\medskip
\point %3
Finally, consider a theory comprised of the EM field $A^\mu(x)$,
a charged scalar field $\phi(x)$, a charged Dirac field $\psi(x)$,
and a neutral Dirac field $\chi(x)$, interacting with each other
according to
$$
\eqalign{
{\cal L}\ ={}&
-\coeff14 F^2_{\mu\nu}\ +\ D^\mu\phi^* D_\mu\phi\
    -\ M^2\phi^*\phi\ -\ \coeff14\lambda(\phi^*\phi)^2\cr
&+\ \overline\psi(i\not\!\!D-m)\psi\ +\ \overline\chi(i\!\not\!\partial-m')\chi\
    -\ \gamma\phi\overline\psi\chi\ -\ \gamma\phi^*\overline\chi\psi.\cr
}\eqno\eq
$$
Write down the Ward--Takahashi identities for this theory
and sketch out a proof.


\bye
