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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 4  \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Monnesday, April 27, 2026}


\bigskip
\hrule
\begin{enumerate}
\item{ \bf All possible terms.}

Perturb the gaussian fixed point of the $\gO(N)$ model 
in $D \geq 3$
by a term
$$\delta S_6 = \int d^Dx  ~g_6 (\phi^a\phi^a)^3 ~.$$
This term is a marginal perturbation of the gaussian fixed point in $D=3$.  
Should we worry that we left it out of our study of the Wilson-Fisher fixed point?
At leading order in $\eps = 4-D$, what does it do to the Wilson-Fisher fixed point?  

%Only if you find {\it new} relevant perturbations 
%(this does not include terms that may be absorbed in changes in the bare values of 
%perturbations that we have already studied)
%must you keep track of the numerical coefficients.
%
%What would you find if you included also $\delta S_{2n}$ with $ n> 3$?  


\item {\bf An example of the power of the RG logic.}

Consider quantum mechanics of a single particle in $d$ dimensions, with 
Hamiltonian
$$ H = {p^2 \over 2m} + V(q),  ~~~[q,p]=i.$$
Consider the (say, euclidean) path integral for this problem,
$$ Z = \int [dq] e^{- S[q]}, ~~S[q] = \int dt \({m \over 2 }  \dot q^2  - V(q) \) .$$
To be more precise, with periodic boundary conditions, 
$Z(\beta) = \int_{q(t+ \beta) = q(t)} [dq] e^{- S[q]} = \tr e^{- \beta H} $
is the thermal partition function.
Alternatively, instead of $Z$, we could consider the Green's function 
$G(q_1,t_1; q_2, t_2) = \int_{q(t_1) = q_1}^{q(t_2)=q_2} [dq] e^{- S[q]}$.
% =G(q_1, q_2, t_1 - t_2).$

Working by analogy with 
our treatment of 
field theory, show that any {\color{red} smooth}\footnote{Some singular potentials are also 
relevant perturbations.  If $V(q) \sim q^{-\alpha}$, how big can $\alpha$ be for 
my statement to remain true?  Thanks to Brian Vermilyea for reminding me that a singular enough potential will cause trouble.} potential $V$ is a {\it relevant} perturbation of the 
free particle, \ie~the Gaussian fixed point with $H= {p^2 \over 2m} $.

Hint: change variables to $\phi(t)\equiv \sqrt{m} q(t)$.

Use this to explain in words why the 
%short-time behavior of the Green's function 
high energy asymptotics of the density of states 
$$ N(E) \equiv \{ \text{$\#$ of eigenvalues of $H$ less than $E$}\} $$
 is given by the {\it Weyl formula} (even for $V(q) \neq 0$):
 $$ N(E) = E^{d/2} K_d L^d + ... $$
 where $K_d  = {\Omega_{d-1}\over (2\pi)^d } $ as usual,
 and $L$ is the linear size of the box in which we put the particle (an IR cutoff).
 
Hint: we can represent the density of states by a path integral 
 using an inverse Laplace transform:
$$\tr  {1\over \omega -H } = \int d\beta ~e^{ \beta \omega } Z(\beta) $$
and the relation
$$ \Im {1\over \omega + \ii \eps - H } = \pi \delta(\omega - H ) . $$






\end{enumerate}

In a series of problems starting with the next one, we will study free massless bosons in two dimensions.
This system is solvable and 
has many physical applications -- \eg~in string theory, and at the edge of quantum Hall systems.
It is an example of a {\it conformal field theory}.
It is a field theory where the excitations are {\it not particles}.
And it's an example where we can understand all the composite operators exactly.


\begin{enumerate}[resume]

\item {\bf There is no continuous symmetry-breaking in two dimensions.}


\begin{enumerate}
\item{}
Consider a massless scalar $X$ in 2d, with action
\be\label{eq:action} S[X] = - { 1\over 4 \pi} \int d^2 \sigma \partial_a X \partial^a X .\ee
Show that the euclidean Green function 
$G_2$ satisfies 
\be\label{eq:green} \nabla^2 G_2(z, z') = - 2\pi \delta^2(z- z') \ee
($z = \sigma_1^E + \ii \sigma_2^E$)\footnote{Hint: this is a Schwinger-Dyson equation.}
and is given by 
$$ G_2(z, z') = 
{ -} 
%{1 \over 2\pi} 
\ln |z - z'| ,$$
for example by Fourier transform.
%[DEF OF sigma, nabla]


\item{} [Perhaps this part is more of a diatribe than a problem.]
The long-distance behavior of $G_2$ has important implications for the 
physics of massless scalars in two dimensions.
Thinking of $G_2$ as the two point function of the massless scalar
$$ G_2(z, z') = \vev{ X(z) X(z') } $$
let's ask the following question:

There is no potential energy for the field $X$ in \eqref{eq:action}.
Someone used to doing physics in 3+1 dimensions might think
that this means that there is a vacuum for every value of $X$.
Let's try to fix the expectation value of the scalar $ \vev{X} = x$ 
and see what happens.
Perturb the putative vacuum $\ket{x}$ a little bit
at the position $z$ by inserting the operator $X$ there.
To measure what happens, insert the operator $X$ at $z'$.
The correlator $G_2$ can thus be interpreted as a measurement
of how the effects of our perturbation fall off with distance at large $|z-z'|$.
What happens?  Contrast this with the behavior you would see for a scalar 
field with a flat potential in more than two dimensions.
%(recall problem 1 of problem set 2).
Note that the case of (0+1) dimensional QFT (\ie\ quantum mechanics)
is even more problematic in the infrared.

One way to arrive at an action like \eqref{eq:action}
is if the field $X$ arises as a Goldstone boson associated with a symmetry
$ X \to X + a $, which would be spontaneously broken by fixing the vacuum $\ket{x}$.
Then it is guaranteed by Goldstone's theorem that the action can only 
depend on derivatives of $X$.  
%We will say more about Goldstone's theorem later, 
Note that the Goldstone-ness of the massless bosons
(\ie\ whether they are massless because of a broken symmetry) is not crucial for this discussion.
One might expect massless bosons
whose masslessness is not protected by a symmetry to be lifted (to acquire a mass)
quantumly, but 
there are special cases (such as in supersymmetric theories)
where different points on the space of minima of the potential need not be related by a symmetry.


This result is called the Hohenberg-Coleman-Mermin-Wagner Theorem. Coleman's paper on the subject is S. Coleman, ``There are no Goldstone bosons in two-dimensions,'' {\it Commun.~Math.~Phys.}~{\bf 31}:259-264 (1973).  It should be called ``There is no continuous symmetry-breaking in two dimensions."

\end{enumerate}


Free massless scalars in $D=1+1$ are important for many reasons and we'll study them some more.

\end{enumerate}
\end{document}
