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%%from  Rudro Rana Biswas
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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 1  \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Monday, April 6, 2025}




\bigskip
\hrule
\begin{enumerate}

\item {\bf An application of effective field theory in quantum mechanics.}  


Consider a model of two canonical quantum variables 
($[ \xx, \pp_x] = \ii = [ \yy, \pp_y], 0 = [ \xx, \pp_y] = [ \xx, \yy]$, etc)
with Hamiltonian 
%$$ S[X,Y] = \int dt \( \dot X^2 + \dot Y^2 + \lambda X^2 Y^2 \)  $$
$$ \HH = \pp_x^2 + \pp_y^2 + \lambda \xx^2 \yy^2 .$$
(This is similar to the degenerate limit of the model we studied in lecture with two QM variables
where both natural frequencies are taken to zero.)

\begin{enumerate}
\item 
Based on a semiclassical analysis, would you think that the spectrum is discrete or continuous?
%(note the flat directions $X=0, Y=0$.)


\item Study large, fixed $x$ near $y=0$.
We will treat $x$ as the slow (= low-energy) variable, 
while $y$ gets a large restoring force from the background $x$ value.  
Solve the $y$ dynamics, and find the groundstate energy as a function of $x$:
$$ V_\text{eff}(x) = E_\text{g.s. of y }(x)  .$$

%= |x|.


\item{} Presumably you did the previous part using your knowledge of the spectrum of the harmonic oscillator Hamiltonian.   Redo the previous part using path integral methods.  


\item The result for $V_\text{eff}(x)$  is not analytic in $x$ at $x=0$.  Why did that happen?  



\item Is the spectrum of the resulting 1d model 
with 
$$ \HH_\text{eff} = \pp_x^2 +  V_\text{eff}(\xx) $$
discrete?
Is this description valid in the regime that matters for the semiclassical analysis?

[Bonus: determine the spectrum of $\HH_\text{eff}$.]



\end{enumerate}


\item{\bf Entanglement and EFT.} [Bonus problem]

In a given state $\rho_{AB}$, one measure of the entanglement between two parts of a bipartite Hilbert space ${\cal H} = A \otimes B$ is the entanglement entropy, 
the von Neumann entropy  $S_A \equiv  - \tr_A \rho_A \log \rho_A $ of the reduced density matrix of one of the parts, $\rho_A = \tr_B \rho_{AB}$.  
When the whole system is in a pure state $\rho_{AB} = \ketbra{\psi}{\psi}$, this is a useful measure of their entanglement.

%In our problem of two oscillators, the Hilbert space is indeed 
Consider the problem of two oscillators we discussed in lecture as a parable about integrating out high-energy modes.  
Compute the entanglement entropy $S_q$ of the light mode $q$ in the groundstate of the combined system, by whatever means necessary.  

If $S_q\neq 0$, it means that $q$ does not have its own wavefunction, only a probability distribution on wavefunctions.   
Show that $S_q$ is very small in the limit $ \omega_0 \ll \Omega$, so that it is not a terrible idea to approximate the low-energy physics of $q$ as being described by a pure state.  

\item{\bf Emergence of the Dirac equation.}
Consider a chain of free fermions
with 
$$ H = - t \sum_n c_n^\dagger c^\nd_{n+1} + h.c. $$

%\begin{enumerate}
%
%\item 
Show that the low-energy excitations at a generic value 
of the filling 
are described the the massless Dirac lagrangian in 1+1 dimensions.
[Hint: the problem has (discrete) translation-invariance and is linear.]

Find an explicit choice of $1+1$-d gamma matrices that matches
the answer from the lattice model.
Show that the right-movers are right-handed $\gamma^5 \equiv \gamma^0 \gamma^1 = 1$ 
and the left-movers are left-handed. 


\end{enumerate}
\end{document}
