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


\noindent
\hfill {\bf Due 11:00am {Monday, November 20, 2023}} 

\bigskip
\hrule


\def\aa{{\bf a}}
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\begin{enumerate}





\item {\bf Brain-warmer:  Wick example.}

For a real scalar field, verify by hand Wick's prediction for the difference
$$ \CT\( \phi(x_1)\phi(x_2) \phi(x_3) \)  - : \phi(x_1)\phi(x_2) \phi(x_3) : $$



\item {\bf Brain-warmers: Feynman rules.}

\begin{enumerate}

Consider the field theory with action
$$ S[\phi] = \int d^{d+1} x ~ \(  \half ( \partial_\mu \phi \partial^\mu \phi - m^2 \phi^2  ) - { g \over 3! } \phi^3 \) .$$

\item Briefly state the Feynman rules in position space, emphasizing the differences from the $\phi^4$ theory.  

\item Draw the {diagrams} that correct the position-space two-point function at order $g^2$.  



\item Find the symmetry factor for {these diagrams} and verify them directly. 



\end{enumerate}

\item {\bf Particle creation by an external source.}


Compare this problem with problem 6 
on HW06.


Consider the Hamiltonian
$$ H = H_0 + \int d^3 x \( - j(t, \vec x) \phi(x) \) $$
where 
$H_0$ is the free Klein-Gordon Hamiltonian,
$\phi$ is the Klein-Gordon field,
and $j$ is a c-number scalar function.  

\begin{enumerate}
\item Show that the probability that the source creates {\it no} particles
is given by
$$ P(0) = \bigl| \bra{0} \CT \{ e^{ + \ii \int d^4 x j(x) \phi_I(x) } \} \ket{0} \bigr |^2 .$$

\item Evaluate the term in $P(0)$ of order $j^2$, and show that $P(0) = 1 - \lambda + \CO(j^4) $
where
$$ 
\lambda = \int {\dbar^3 p \over 2 E_p } | \tilde j(p) |^2 . 
$$
We will show below that $ \lambda  = \vev{N} $ is the mean number of particles created by the source.
\label{part:b}


\item Represent the term computed in part \ref{part:b} as a Feynman diagram.  Now 
represent the whole perturbation series for $P(0)$ in terms of Feynman diagrams.  
(Hint: you have done this calculation already.)
Show that this 
series exponentiates, so that it can be summed exactly $P(0) = e^{ - \lambda}$.  



\item On the next problem set, after learning about the notion of final-state phase space,
we'll find the probability for the source to create any number of particles.

\end{enumerate}


\item {\bf Propagator corrections in a solvable field theory.} 

Consider a theory of a scalar field in $D$ dimensions with action
$$ S = S_0 + S_1 $$
where
$$ S_0 = \int d^Dx \half \( \partial_\mu\phi \partial^\mu \phi  - m_0^2 \phi^2  \) $$
and 
$$ S_1 = - \int d^D x \half \delta m^2 \phi^2 ~.$$
We have artificially decomposed the mass term into two parts.  
We will do perturbation theory in small $\delta m^2$,
treating $S_1$ as an `interaction' term.  
We wish to show that the organization of perturbation theory
that we've seen lecture will correctly reassemble the mass term.

\begin{enumerate}
\item
Write down all the Feynman rules for this perturbation theory.

\item Determine the 1PI two-point function in this model, 
defined by $$ {\cor -} \ii \Sigma \equiv \sum\(\text{all 1PI diagrams with two nubbins}\) .$$

\item Show that the (geometric) summation of the propagator corrections 
correctly produces the propagator that you would have used 
had we not split up $ m_0^2 +\delta m^2 $.


\end{enumerate}


\item {\bf Wick's theorem from Schwinger-Dyson equations.} [Bonus problem]
Study the derivation of Wick's theorem 
from the Schwinger-Dyson equation for the $n$-point function of a free scalar field on 
page 81 of Schwartz' book.

\item {\bf A background field.}  [This is a bonus problem.]

Consider the following action for a real scalar field $\Phi$: 
$$ S[\Phi] = \int d^{d+1}x  \half \( \partial_\mu \Phi \partial^\mu \Phi - m^2 \Phi^2  - g \phi(x) \Phi^2 \) . $$
The last term here is a cubic coupling between $\phi$ and $\Phi$.
But here we will treat $\phi(x) $ as a fixed background field 
(analogous to $j(x)$ on previous problems)
which
acts as a spacetime-dependent mass for the dynamical field $\Phi$. 

%{\cor [I moved around some factors of $\ii$ below.]}

\begin{enumerate}
\item Show that the two-point Green's function, 
$G(x,y) \equiv \vev{ \Omega| \CT \Phi(x) \Phi(y) | \Omega} $, 
satisfies the Schwinger-Dyson equation
\be\label{eq:SD}
- \ii   \delta^{d+1} (x-y) = \(  \partial^2 + m^2 + g \phi(x) \) G(x,y) . 
\ee



\item We would like to solve this differential equation.  
As a warmup, consider the case $g=0$.  Here is a trick: 
add a fictitious additional time direction $T$
\be\label{eq:extra-diffusion} \( \partial_T -  \(  \partial^2 + m^2 \) \)  G(x,y,T)  =  \ii\delta^{d+1}(x-y) \delta(T) \ee
This is just a diffusion equation (in $d+2$ dimensions and with a funny factor of $\ii$!). 
Show that given a solution to \eqref{eq:extra-diffusion}, you can find the solution of \eqref{eq:SD}
with $g=0$ 
by 
\be\label{eq:integrate-T} G(x,y) = \int_0^\infty dT G(x,y,T) .\ee


\item Show that the solution to the diffusion equation \eqref{eq:extra-diffusion}
%for an infinitesimal time step $T$ 
is 
\be\label{eq:one-step} G(x,y,T) = {\ii \over (2\pi T)^\alpha }e^{ a{ (x-y)^2 \over 2 T } + b {m^2 \over 2 } T } .\ee
Find $\alpha, a, b$.
Use this to construct 
the path integral representation  
$$ G(x,y,T) = \int_{x(0)=x}^{x(T)=y} [Dx] e^{ - \ii \int_0^T d\tau \( \dot x^\mu \dot x_\mu + m^2 \) } . $$



\item For the case of constant $m^2$, the infinitesimal solution \eqref{eq:one-step}
actually works for finite $T$.  
Show by differentiation that plugging \eqref{eq:one-step} into \eqref{eq:integrate-T}
gives an integral representation of the free Klein-Gordon propagator.

%$$  \ii \delta^{d+1} (x-y) = \(  \partial^2 - m^2 - g \Phi(x) \) G(x,y) . 

\item Now let $g\neq0$ and suppose that $\phi$ is slowly varying.  
Generalize the path integral representation to include the dependence on $\phi$.  



\item 
Consider a non-relativistic situation, where the spacetime points $x$ and $y$ are separated by a timelike distance
large compared to $1/m$. 
Justify and use stationary-phase methods
to show that the dominant contribution to the path integral
is a straight-line trajectory between the two points $x$ and $y$.  
Evaluate the resulting amplitude as a functional of $\phi(x)$.

This calculation shows that the heavy particle made by the field $\Phi$ can be treated as a 
source for $\phi$ propagating on a fixed path in spacetime.  


\item Redo the problem for a {\it charged} scalar field, $\Phi$
in the background of a vector potential $A_\mu$, with 
$$ S[\Phi] = \int d^{d+1} x \half \( D_\mu \Phi^\star D^\mu \Phi - m^2 \Phi^\star \Phi\) , ~~
D_\mu \Phi \equiv \partial_\mu \Phi - \ii A_\mu \Phi .$$
It will help to recall that the action of a classical charged particle is 
$ \int d\tau \(  \dot x^2 +\dot x^\mu A_\mu(x) \)  $.


\end{enumerate}




\end{enumerate}
\end{document}

