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\begin{document}
\begin{center}

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

{\Large\bf  Physics 215B QFT Winter 2026}\\
{\Large\bf Assignment 10 (``Final Exam") \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Thursday, March 19, 2026}




\bigskip
\hrule
\begin{enumerate}


\item {\bf Brainwarmers on differential forms.} [Optional]

\begin{enumerate}

\item Prove that for a $k$-form $\omega$ on a $d$-dimensional manifold with a metric signature having $s$ negative eigenvalues, applying the Hodge star twice yields:$$**\omega = (-1)^{k(d-k)+s}\omega~.$$
In the "mostly-minus" convention in 4-dimensional spacetime ($d=4$, $s=3$), calculate $**F$ where $F$ is the electromagnetic field strength 2-form.

%Hint: Recall how the Levi-Civita tensor contracts with itself, $\epsilon_{\mu_1 \dots \mu_d} \epsilon^{\nu_1 \dots \nu_d}$, and keep careful track of the sign introduced by the determinant of the mostly-minus metric.


\item Consider a static point charge $q$ localized at the origin. In spherical coordinates $(t, r, \theta, \phi)$, the mostly-minus flat spacetime metric is:
$$ds^2 = dt^2 - dr^2 - r^2 d\theta^2 - r^2 \sin^2\theta d\phi^2~.$$
The gauge potential 1-form for this static charge is $A = \frac{q}{4\pi r} dt$.
Calculate the electromagnetic field strength 2-form, $F = dA$.
Compute the dual field strength 2-form, $*F$.
Evaluate the integral of $*F$ over a spatial 2-sphere $S^2$ of radius $R$ at a constant time $t$:
$\int_{S^2} *F$
Interpret your result physically.

\item Let $A = A_\mu dx^\mu$ be the gauge potential 1-form, and $F = dA = \half F_{\mu\nu} dx^\mu dx^\nu$.  
Show that the two homogeneous Maxwell equations are equivalent to the statement $dF = 0$ (the Bianchi identity).

The Maxwell action in the presence of a source current 1-form $J$ is given by:$$S = \int_M \left( -\frac{1}{2} F \wedge *F + A \wedge *J \right)~.$$
Vary the action with respect to the gauge field $A \to A + \delta A$ to derive the equations of motion (the inhomogeneous Maxwell equations): $d*F = *J$.

[Hint: the exterior derivative $d$ obeys the generalized Leibniz rule, $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$, where $\alpha$ is a $k$-form.

\item Let's think about the relation between the form notation and the usual electric and magnetic fields in E\&M.
%In Cartesian coordinates $(t, x, y, z)$, the field strength 2-form is $F = \frac{1}{2}F_{\mu\nu}dx^\mu \wedge dx^\nu$. 
Write $F$ explicitly in terms of the electric field components $E_i$ and magnetic field components $B_i$. 
%Be mindful of how the mostly-minus signature affects the raising and lowering of indices compared to the mostly-plus convention.
Compute the dual field strength $*F$.
Show that the Lorentz invariant quantity $F \wedge *F$ evaluates to proportional to $(|\mathbf{B}|^2 - |\mathbf{E}|^2)dt \wedge dx \wedge dy \wedge dz$.  What is the Lorentz invariant $F \wedge F$ in terms of $\mathbf{E}$ and $\mathbf{B}$?



\end{enumerate}


\item {\bf Gauge theory and Lie algebra brain-warmers.} 

Please do 3 of the following 6 problems.  The rest are bonus material.

\begin{enumerate}
\item Show that the {\it adjoint} representation matrices
$$ \(T^{A}\)_{{B}C} \equiv - \ii f_{ABC} $$
furnish a $\dim \gG$-dimensional representation of the Lie algebra 
$$ 
[ T^A, T^B] = \ii f_{ABC} T^C ~~.
$$
Hint: commutators satisfy the Jacobi identity
$$ [A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0 . $$


\item{}  Show that if $(T_A)_{ij}$ are generators of a Lie algebra in some unitary representation $R$, then
so are $ - (T_A)^\star_{ij}$.  Convince yourself that these are the generators of the complex conjugate representation ${\bar R}$.

\item{}  Show that in a basis of Lie algebra generators where $ \tr T^A T^B  =\lambda \delta^{AB}$, 
the structure constants $f_{ABC}$ are completely antisymmetric.



\item From the transformation law for $A$, show that the non-abelian field strength
transforms in the adjoint representation of the gauge group.



\item Show that 
$$ \tr F \wedge F =d \tr \( A \wedge dA + {2\over 3 } A \wedge A \wedge A \) .$$
Write out all the indices I've suppressed.


\item{} [Bonus] If you are feeling under-employed, find $\omega_{2n-1} $  such that 
$ \tr F^n = d \omega_{2n-1}$.


\end{enumerate}


\item {\bf The field of a magnetic monopole.} [Bonus problem]

We saw that $F = dA $ implies (when $A$ is a smooth, globally well-defined differential form) that $dF=0$, which means no magnetic charge.  
If $A$ is singular, $dF$ can be nonzero.  Moreover, by a gauge transformation we can move the singularity around and hide it, so that the field is everywhere non-singular.  

A magnetic monopole of magnetic charge $g$ is defined by the condition that $ \int_{S^2} F = g $,
where $S^2$ is any sphere surrounding the monopole.
If the system is spherically symmetric, 
we can write 
$$ F = {g \over 4 \pi} d \cos\theta d \varphi .$$
(In this problem, we'll work on a sphere at fixed distance from the monopole.)
\begin{enumerate}
\item Show that the vector potential 
$$A_N = {g \over 4 \pi} \(\cos \theta-1\) d\varphi $$
gives the correct $F= dA$.
Show that it is a well-defined one-form on the sphere except at the south pole $ \theta = \pi$.


\item Show that the one-form
$$A_S = {g \over 4 \pi} \(\cos \theta+1\) d\varphi $$
also gives the correct $F= dA$.
Show that it is well-defined except at the north pole $ \theta = 0$.

\item Near the equator both $A_{N,S}$ are well-defined.
Show that 
{\it as long as $ e g \in 2 \pi \IZ$}, 
these two one-forms differ by a gauge transformation
$$ A_S - A_N = {1\over \ii e} g^{-1}(\theta,\varphi) d g(\theta,\varphi) $$
for $g(\theta,\varphi)$ a $\gU(1)$-valued function on the sphere, well-defined away from the poles.


\end{enumerate}


\item {\bf When is the QCD interaction attractive?} [Bonus problem]

Write the amplitude for {\it tree-level} scattering of a quark and antiquark 
of different flavors (say $u$ and $\bar d$)
in the $t$-channel
(in Feynman $\xi=1$ gauge).
Compare to the expression for $ e \bar \mu$ scattering in QED.

First fix the initial colors of the quarks to be different 
-- say the incoming $u$ is red and the incoming $\bar d$ is anti-green.
Show that the potential is repulsive.

Now fix the initial colors to be opposite 
-- say the incoming $u$ is red and the incoming $\bar d$ is anti-red --
so that they may form a color singlet.  Show that the potential is attractive.

Alternatively or in addition, 
describe these results in a more gauge invariant way,
by characterizing the potential in the color-singlet
and color-octet channels.

You can do this problem either by choosing a specific basis 
for the generators of $\gSU(3)$ in the fundamental 
(a common one is the Gell-Mann matrices),
or using more abstract group theory methods.





\item {\bf Spinors in other dimensions.}  The remaining two problems take place in $2+1$ and $1+1$ dimensions respectively and involve Dirac spinors.  The answers to following questions will be useful.

\begin{enumerate}
\item Show that for both $D=2$ or $D=3$ we can find a {\it two} dimensional representation of the Clifford algebra 
$\{ \gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} $.  
Hint: use the Pauli matrices with some appropriate factors of $\ii$.

\item What is $\tr \gamma^\mu \gamma^\nu$ in these two cases?

\item What is $ \tr \gamma^\mu \gamma^\nu \gamma^\rho $ in $D=2$ and $D=3$ respectively?  

\item What is $ \tr \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma$?  

\end{enumerate}

\item {\bf Where to find a Chern-Simons term.}

Consider a field theory in $D=2+1$ of a 
massive Dirac fermion, coupled
to a {\it background} $\gU(1)$ gauge field $A$ with action:
$$ S[\psi, A] = \int d^3 x \bar \psi \( \ii \slashed{D} - m \) \psi $$
where $ D_\mu = \partial_\mu - \ii A_\mu $.
$A_\mu$ is not dynamical for the purposes of this problem.
Use the two-component spinors from the previous problem.

\begin{enumerate}
\item
Convince yourself that the mass term for the Dirac fermion in $D=2+1$ 
breaks parity symmetry.   By parity symmetry I mean a transformation $ \psi(x) \to \Gamma \psi(Ox )$ where $\det O = -1$,
and $\Gamma$ is a matrix acting on the spin indices, chosen so that this operation preserves $ \bar \psi \slashed{\partial} \psi$.  
%That is, parity takes $m \to -m $.



\item We would like to study 
the effective action for the gauge field
that results from integrating out the fermion field
$$ e^{ - S_{eff}[A]} = \int [D\psi D\bar \psi] e^{ - S[\psi, A]} .$$
Focus on the term quadratic in $A$:
$$ S_{eff}[A] = { \half }\int { \dbar^D q} A_\mu(q) \Pi^{\mu\nu}(q) A_\nu(q) + ... $$
We can compute $\Pi^{\mu\nu}$ by Feynman diagrams.
Convince yourself that $\Pi$ comes from a single loop of $\psi$ with two $A$ insertions.





\item Evaluate this diagram using dim reg near $D=3$.  
Show that, in the low-energy limit $ q \ll m$ (where we can't make on-shell fermions), 
$$ \Pi^{\mu\nu}(q) = a { m \over |m| }\eps^{\mu\nu\rho} q_\rho + ...$$
for some constant $a$.  Find $a$.
Convince yourself that in position space this is a Chern-Simons term, 
$S[A] = {k \over 4 \pi } \int A \wedge dA $.  
Find the level $k$.





\item{} Redo this calculation by doing the Gaussian path integral over $\psi$.


\end{enumerate}


\item {\bf A bit about the Schwinger model.}

Consider a version of QED in $D=1+1$: 
\be S[A, \psi] = \int d^2 x \( \sum_{\alpha=1}^{N_f} \bar \psi_\alpha \( \ii \slashed{D} - m_\alpha \) \psi_\alpha- { 1\over 4 g^2 } F_{\mu\nu} F^{\mu\nu}    - {\theta \over 4\pi } \epsilon_{\mu\nu} F^{\mu\nu} \)\ee
where each $\psi_\alpha$ is a 2-component spinor, and $\slashed{D} = \gamma^\mu \( \partial_ \mu + \ii A_\mu \) $.
Use the two-component spinors from the problem above.


\begin{enumerate}
\item
What are the mass dimensions of the coupling $g$?

\end{enumerate}

The model with $N_f=1$ was solved by Schwinger in 1962.   It is a model of confinement of charge, essentially because, even classically,  the Coulomb potential grows linearly in $D=1+1$, just like a confining potential.
When $\theta =\pi$, there is a line of first-order phase transitions as a function of $m_1$, terminating at a critical point at $m_1= m_\text{cr} \approx 0.3335 g$.  
(When $\theta =\pi$ things are less interesting.)
We'll set $\theta = \pi$ for the whole problem.  

If $m_2$ is very heavy, the model should reduce the $N_f=1$ model at very low energies, and so should still have a critical point at some value of $m_1$.  
However, we can expect that the critical value of $m_1$ will be modified by some dependence on $m_2$ that goes away when $m_2 \to \infty$.  

Predict the shape of the critical curve $m_{1 \text{cr}}(m_2)$ in the regime $m_2 \gg g $.  

If I were feeling mean I would end the question here.  Since I am feeling friendly, I will say a few more words.  
The heavy fermion with mass $m_2$ will contribute to the vacuum polarization for the gauge field.  

\begin{enumerate}[resume]
\item Compute the contribution to the vacuum polarization of the gauge field from the heavy fermion field, $\Pi^{\mu\nu}(q)$, in the limit $m_2 \gg g$, as a function of $m_2$ and $g$, for $q^2 \ll m_2^2$.  





\item Interpret your previous result as a correction to $g$, i.e. to the coefficient ${1\over g_\text{eff}^2}$ of the Maxwell term.  


\item Using the known formula for the critical value of $m$ in the $N_f=1$ case, predict the critical curve as a function of $m_2$.  

\end{enumerate}



\end{enumerate}

I got this problem from a talk I saw recently at the KITP.


\end{document}
