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\noindent PHY--396~L.\qquad
Problem set \#21.\quad
Due April 11, 2001.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
In class, I used functional formalism (path integrals) to
prove the Ward--Takahashi identities of QED.
Alternatively, one can follow Drs.\ Ward and Takahashi and derive
the same identities from the QED Feynman rules.
This derivation is explained in detail in section~\S7.4 of the
{\sl Peskin \& Schroeder} textbook.

\spointbegin
Read the textbook, then give a similar diagramatic proof of the
Ward--Takahashi identities for the {\sl scalar} QED.

\spoint %b
Prove that the renormalized scalar QED has $Z_1^{1\gamma}=Z_1^{2\gamma}=Z_2^{}$.

\bigskip
\point %2
In any {\it even} spacetime dimension $d=2n$, a {\sl massless} Dirac
spinor field $\Psi(x)$ has an axial symmetry
$$
\Psi(x)\ \to\ e^{+i\beta\Gamma}\,\Psi(x),\qquad
\overline\Psi(x)\ \to\ \overline\Psi(x) e^{+i\beta\Gamma},
\eqno\eq
$$
where $\Gamma=i^{n-1}\gamma^0\gamma^1\cdots\gamma^{(d-1)}$ generalizes the $\gamma^5$
to $d=2n$.
{\sl Classically}, this symmetry leads to the conserved axial current
$$
J_A^\mu\ =\ \overline\Psi\Gamma\gamma^\mu\Psi,\qquad
\partial_\mu J_A^\mu(x)\ =\ 0.
\eqno\eq
$$
In the quantum field theory however, the axial symmetry becomes anomalous when $\Psi$
couples to a vector gauge field $A^\mu(x)$.

\pointcon
Show that in the $d=2n$ dimensional QED,
$$
\partial_\mu J_A^\mu(x)\
=\ -{2\over n!}\left({e\over 4\pi}\right)^n\,
\epsilon^{\alpha_1\beta_1\alpha_2\beta_2\cdots\alpha_n\beta_n}
F_{\alpha_1\beta_2} F_{\alpha_2\beta_2}\cdots F_{\alpha_n\beta_n}\,.
\eqno\eq
$$

\bye
