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%%from  Rudro Rana Biswas
\usepackage[pagebackref,  % this puts links to the page numbers where refs appear
%pdftex, 
bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 215C QFT Spring 2026}\\
{\Large\bf Assignment 2  \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Monday, April 13, 2026}




\bigskip
\hrule
\begin{enumerate}

\item {\bf Brain-warmer.} [optional]
Find the constant $a$ in 
$$ a \prod_{\mu=0}^{D-1} \gamma^\mu = \gamma^5 $$ 
so that $(\gamma^5 )^2 = 1 $.  

\item{\bf Chiral anomaly in two dimensions.} 

Consider a massive relativistic Dirac fermion in 1+1 dimensions, with
$$ S = \int \dd x \dd t \bar \psi \( \ii \gamma^\mu \( \partial_\mu + e A_\mu \)-  m \) \psi.$$
By heat-kernel regularization of  
its expectation value,
show that the divergence of the axial current 
$ j^5_\mu \equiv \ii \bar \psi \gamma_\mu \gamma^5 \psi$ is
$$ \partial_\mu j^5_\mu = 2 \ii m \bar \psi \gamma^5 \psi + { e\over 2 \pi } \eps_{\mu\nu} F^{\mu\nu} . $$



\item{\bf Chiral anomaly in six dimensions.} [optional]
Find the divergence of the axial current in QED in $D=5+1$.  






\item {\bf An application of the anomaly to a theory without gauge fields.}

Consider a 1+1d theory of Dirac fermions coupled
to a background scalar field $\theta$ as follows:
$$ \CL = \bar \Psi \( \ii  \slashed{\partial} + m e^{ \ii \theta \gamma^5 } \) \Psi .$$
We wish to ask: if we subject the fermion to various configurations
of $\theta(x)$ (such as a domain wall where $\theta(x>0) = \pi + \theta(x<0)$)
what does the fermion number do in the groundstate?   

\begin{enumerate}

\item Convince yourself that when $\theta$ is constant
$$ \vev{ j^\mu} = 0 $$
where $j^\mu = \bar \Psi \gamma^\mu \Psi$ is the fermion number current.



\item 
Minimally couple the fermion to a {\it background} gauge field $A_\mu$.
Let $ e^{ \ii \Gamma[A, \theta] } = \int [ d\Psi] e^{ \ii S} $.
Convince yourself that the term linear in $A$ in $\Gamma[A, \theta] = \text{const} + \int A_\mu J^\mu + \CO(A^2) $ 
is the vacuum expectation value of the current $ \vev{ j^\mu } = J^\mu$ { (at $A=0$)}.


\item Show that by a local chiral transformation $ \Psi \to e^{ {\cor -} \ii \theta(x) \gamma^5/2}\Psi$ 
we can remove the dependence on $\theta$ from the mass term.
%action the position dependence from $\theta$.

\item Where does the theta-dependence go?  
Use the 2d chiral anomaly to relate $ \vev{j^\mu}$ to $\partial \theta$.  
Notice that the result is independent of $m$.
[This relation was found by Goldstone and Wilczek.  The associated physics is realized
in Polyacetylene.]


\item Show that a domain wall where $\theta$ jumps from $0$ to $\pi$ 
localizes {\it fractional} fermion number.


\item{} 
%[bonus problem] 
Consider the Dirac hamiltonian in the presence of such a domain wall.
Show that there is an exponentially-localized mode of zero energy.



\end{enumerate}




\end{enumerate}
\end{document}
