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\begin{document}
\begin{center}

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

{\Large\bf  Physics 215A QFT Fall 2023}\\
{\Large\bf Assignment 4 \answer{--~~~ Solutions} }
\end{center}


\noindent
\hfill {\bf Due 11:00am {Monday, October 30, 2023}} 



\bigskip
\hrule


\def\aa{{\bf a}}
\def\nnn{{\bf n}}


\begin{enumerate}

\item{\bf Scale transformations.}

\begin{enumerate}

\item Check that the following action 
\be S[\phi] = \int d^D x \( \half \partial_\mu \phi \partial^\mu \phi - g \phi^p \) \ee
is invariant under the scale transformation 
$ \phi(x^\mu) \mapsto  \lambda^{d_\phi} \phi(\lambda x^\mu) $
for some choice of $p$ and some choice of $d_\phi$ that depend on $D$.  
Check that your answer is the same as the mass dimension of $\phi$.


\item Find the infinitesimal form of the transformation of $\phi$ under a scale transformation.


\item 
Find the associated conserved current $j^\mu_s$ and conserved charge $ {\bf s } \equiv \int d^d x j^0_s $  (the dilatation generator).  


\item Check using the canonical commutation that the dilatation generator  ${\bf s}$ indeed generates dilatations.  

\item For a general field theory with Lagrangian density that depends on the fields and their first derivatives, relate the dilatation current $j_s^\mu$ to the stress-energy tensor.



\end{enumerate}





\item {\bf Gaussian integrals are your friend.}



\begin{enumerate}

\item Show that 
$$\int_{-\infty}^\infty dx e^{ - \half a x^2 + j x } = \sqrt{ 2\pi \over a } e^{  j^2 \over 2 a } . $$
[Hint: square the integral and use polar coordinates.]

\item Consider a collection of variables $ x_i, i=1..N$
and a { real, symmetric} matrix $a_{ij}$.  Show that 
$$ \int \prod_{i=1}^N dx_i e^{ - \half x_i a_{ij} x_j + J^i x_i } 
= {(2\pi)^{N/2}\over \sqrt{\det a }} e^{ \half \cob J^i a^{-1}_{ij} J^j } . $$
(Summation convention in effect, as always.)

[Hint: change integration variables to diagonalize $a$.  $\det a = \prod a_i $,
where $a_i$ are the eigenvalues of $a$.]




\item I include this problem partly because it might be helpful for a future problem.
In that regard, for any function of the $N$ variables, $f(x)$, 
let 
$$\vev{f(x)} \equiv 
{ \int \prod_{i=1}^N dx_i e^{ - \half x_i a_{ij} x_j } f(x) \over Z[J{ =0}]} , ~~~
Z[J] = \int \prod_{i=1}^N dx_i e^{ - \half x_i a_{ij} x_j { + J^i x_i}}.$$
Show that
$$ \vev{ x_i x_j } 
%\int \prod_{i=1}^N dx_i e^{ - \half x_i a_{ij} x_j } } 
= \partial_{J_i} \partial_{J_j} \log Z[J]|_{J=0} = a^{-1}_{ij} $$
Also, { convince yourself} that 
$$ \vev{ e^{ J_i x_i } } = {Z[J] \over  Z[J=0]}. $$

\item Note that the number $N$ in the previous parts may be infinite.  
This is really the only path integral we know how to do.


%\item Consider a Gaussian field $X$, governed by the (quadratic)  euclidean \cob action
%$$ S[x] = \int dt \half \( \dot X^2 \cor + \cob \Omega^2 X^2 \).$$
%Show that 
%$$ 
%\vev{e^{ - \int ds J(s) X(s) } }_X 
%= \cor \CN \cob e^{ +{1 \over 2 } \int ds dt J(s) G(s, t) J(t) } $$
%where $G$ is the (Feynman) Green's function for $X$, satisfying: 
%$$\( - \partial_s^2 + \Omega^2 \) G(s, t) = \delta(s-t). $$
%\cor Here $ \CN $ is a normalization factor which is independent of $J$. \cob
%Note the similarity with the previous problem, under the replacement
%$$ a = - \partial_s^2 + \Omega^2, ~~a^{-1} = G .$$


\end{enumerate}


%\vfill\eject
\item {\bf Gaussian identity.}  
\label{prob:gaussian}
Show that for a gaussian quantum system
$$ \vev{ e^{ \ii K \qq }} = e^{ - A(K) \vev{\qq ^2 } } ~$$
and determine $A(K)$.
Here $ \vev{ ...} \equiv \bra{0} ... \ket{0}$, vacuum expectation value.
Here by `gaussian' 
%$$ \vev{...} \equiv { 
I mean that $\HH$ contains only quadratic and linear terms in 
both $\qq$ and its conjugate variable $\pp$
(but for the formula to be exactly correct as stated you must assume 
$\HH$ contains only terms quadratic in $\qq$ and $\pp$; 
for further entertainment fix the formula for the case with linear terms in $\HH$)\cob.

I recommend using the path integral representation (with hints from the previous problem).  Alternatively, you can use
the harmonic oscillator operator algebra.  Or, even better, do it both ways.


%{\cor In stating this problem, I should have made it clear that you should show the result for an arbitrary collection of modes $q_n$, with a Hamiltonian such as $ \sum_n {p_n^2 \over 2m } + \half  q_n A_{nm} q_m $.  }
%This result is useful for the following problem
%and in many other places.



\item {\bf Zero-phonon processes.} 

We wish to understand the probability for a 
photon to hit (our crude model of) a crystalline solid 
without exciting any vibrational excitations.  It is a nice simultaneous application of {\it many} of the things we've learned so far (quantization of the radiation field, solution of the harmonic chain, gaussian integrals, the relationship between path integrals and transition amplitudes).  

When a photon hits a lattice of atoms, Fermi's golden rule says that the (leading approximation to the) probability for a transition 
from one state of the lattice $\ket{L_i}$ to another $\ket{L_f}$
is proportional to 
$$ 
W(L_i \to L_f) = | \bra{L_f} \HH_\text{L} \ket{L_i} |^2 . $$
Here $\HH_\text{L}$
is the hamiltonian describing the interaction between the photon
and an atom in the lattice.
For the first parts of the problem, use the following form 
(to be justified in the last part of the problem):
\be\label{eq:photon} \HH_\text{L} = A e^{ \ii K \xx } + h.c. \ee
where $\xx$ is the (center of mass) position operator of the atom being struck;
$K$ is a constant (the photon wavenumber), and 
(for the purposes of the first parts of the problem) $A$ is a constant.
$+h.c.$ means `plus the hermitian conjugate of the preceding stuff'.

\begin{enumerate}

\item 
Recalling that $\xx$ (up to an additive constant) 
is part of a collection of coupled harmonic oscillators: 
$$ \xx = n x + \qq_n  $$
evaluate the ``vacuum persistence amplitude" 
$ \bra{0} \HH_\text{L} \ket{0} $.  
%A useful general fact is that for harmonic oscillators 
You will find the results of the previous problem set useful.




\item From the previous calculation,
you will find an expression that requires you to sum over wavenumbers.
Show that in one spatial dimension, the probability for
a zero-phonon transition is of the form
$$P_\text{M\"ossbauer} \propto e^{- \Gamma \ln L } $$
where $L$ is the length of the chain and $\Gamma$ is a function of other variables.
Show that this infrared divergence is missing for 
the analogous model of crystalline solids with more than one spatial dimension.
(Cultural remark: these amplitudes are called `Debye-Waller factors').




\item{} 
%[more optional] 
Convince yourself that a coupling $\HH_\text{L}$ 
of the form \eqref{eq:photon}
arises 
from the minimal coupling of the electromagnetic field 
to the constituent charges of the atom,
after accounting for the transition made by the radiation
field when the photon is absorbed by the atom.
`Minimal coupling' means replacing 
the momentum operator of the atom $\pp$, 
with the gauge-invariant combination 
$\pp \to \pp + \AA$.
You will also need to recall the form of the quantized 
electromagnetic field in terms 
creation and annihilation operators for a photon of definite momentum $K$.




\end{enumerate}


\end{enumerate}


\end{document}



