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


\noindent
\hfill {\bf Due 11:00am {Monday, November 6, 2023}} 

\bigskip
\hrule


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


\begin{enumerate}




\item{\bf The $\ii \eps$ prescription produces time-ordered correlators.} [Bonus problem]

Check that the four-point function of a free scalar field of mass $m$
$$ Z^{-1}\int [D\phi] e^{ \ii S[\phi]} \prod_{i=1}^4 \phi(x_i) |_{m^2 \to m^2 - \ii \eps} ,$$
defined by the $\ii \eps$ prescription,
is the {\it time-ordered} vacuum expectation value 
$$=  \bra{0} \CT \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \ket{0} .$$
One way to do this is to show that they are both equal to
\be\label{eq:wicked} D_T(12)D_T(34) + D_T(13)D_T(24) + D_T(14)D_T(23) , \ee
where $D_T(ij) \equiv \bra{0} \CT \phi(x_i) \phi(x_j) \ket{0}$.  


\item {\bf The propagator is a Green's function.}

\begin{enumerate}
\item \label{parta}Consider the {\it retarded} propagator for a real, free, massive scalar field:
$$ D_R(x-y) \equiv \theta(x^0-y^0) \bra{0} [ \phi(x), \phi(y) ] \ket{0} . $$
Show that it is a Green's function for the Klein-Gordon operator,
in the sense that 
$$ \( \Box_x + m^2 \) D_R(x-y) = a \delta^{d+1} (x-y) , ~~~~~\Box_x \equiv { \partial \over \partial x^\mu} 
{\partial \over \partial x_\mu} . $$
Find $a$.



\item Use the previous result to generalize the mode 
expansion of the scalar field
to the situation with an external {\it source}, 
\ie~when we add to the lagrangian density $ \phi(x) j(x) $
for some fixed c-number function $j(x)$.


\item{} \label{problem-time-ordered-G}
Show that the {\it time-ordered} propagator for a real, free, massive scalar field
$$ D(x-y) \equiv 
\bra{0} \CT \phi(x) \phi(y) \ket{0} 
\equiv
\theta(x^0-y^0) \bra{0}  \phi(x) \phi(y)  \ket{0} 
+ \theta(y^0-x^0) \bra{0}  \phi(y) \phi(x)  \ket{0} $$
is {\it also} a Green's function for the Klein-Gordon operator.
%\item 
%Remind yourself that last quarter you derived the equation \eqref{eq:SD}
%(for a free theory)
%more arduously, from a more canonical (\ie\ Hamiltonian) point of view, 
That is, consider
what happens when you
act with the wave operator $ +\cob \Box_x + m^2 $ on 
the time-ordered two-point function.

[Hints: Use the canonical equal-time commutation relations: 
$$ [ \phi(\vec x), \phi(\vec y) ] = 0 , ~~~
[ \partial_{x^0} \phi(\vec x), \phi(\vec y) ] = -  \ii \delta^{D-1}(\vec x - \vec y) . $$
Do not neglect the fact that $ \partial_t \theta(t) = \delta(t)$: 
the time derivatives act on the time-ordering symbol!]


\item 
This is an example of something that is easier to understand from the path integral.
%Consider the path integral for a real, free, massive scalar field with a source $J(x)$, 
%continued from the Euclidean integral.  
In the next problem, we'll understand 
why the correlation functions of $\phi$ should
 solve the equations of motion, up to `contact terms'. 


\end{enumerate}

\item {\bf Schwinger-Dyson equations.}

\label{problem:SD}

Consider the path integral
$$ \int [D\phi] e^{ \ii S[\phi]}. $$
Using the fact that the integration measure is independent of 
the choice of field variable, we have 
$$0 =  \int [D\phi] { \delta \over \delta \phi(x)} \( \text{anything} \)  $$
 (as long as `anything' doesn't grow too fast at large $\phi$)\cob.
So this equation says that we can integrate by parts in the functional integral.

(Why is this true?  As always when questions about 
functional calculus arise, 
you should think of spacetime as discrete and hence the path integral measure as simply 
the product of integrals of the field value at each spacetime point, $ \int [D\phi] 
\equiv \int \prod_{x} d\phi(x)  $, 
this is just the statement that 
$$ 0 = \int d\phi_x { \partial \over \partial \phi_x} \( \text{anything} \) $$
with $ \phi_x \equiv \phi(x)$, \ie\ that we can integrate by parts in an ordinary integral
if there is no boundary of the integration region.)

This trivial-seeming set of equations (we get to pick the `anything') 
can be quite useful and they are called Schwinger-Dyson equations (or sometimes Ward identities).
Unlike many of the other things we'll discuss, they are true non-perturbatively,
\ie~are really true, even at finite coupling.
They provide a quantum implementation of the equations of motion.

\begin{enumerate}
\item Evaluate the RHS of 
$$0 =  \int [D\phi] { \delta \over \delta \phi(x)} \(  \phi(y) e^{ \ii S[\phi]}\) $$
{(defined by the $\ii \eps$ prescription)}
to conclude that 
\be
\label{eq:SD} 
%\vev{ \CT {\delta S\over \delta \phi(x)} \phi(y) } 
{1\over Z} \int [D\phi] e^{ \ii S} {\delta S\over \delta \phi(x)} \phi(y)
= { +} \ii \delta(x-y) . 
\ee



\item These Schwinger-Dyson equations are true in interacting field theories;
to get some practice with them we consider here a free theory.
Evaluate \eqref{eq:SD} for the case of a free massive real scalar field
to show that the (two-point) time-ordered correlation functions of $\phi$
satisfy the equations of motion, most of the time.
That is: the equations of motion are satisfied
away from other operator insertions:
\be
\label{eq:SDfree}
{ \(  + \Box_x + m^2 \) } \vev{ \CT  \phi(x) \phi(y) } = - \ii \delta(x-y) ,
\ee
with $ \Box_x \equiv \partial_{x^\mu}\partial^{x^\mu}. $




\item Find the generalization of  \eqref{eq:SDfree}
satisfied by (time-ordered) three-point functions of the free field $\phi$.




\end{enumerate}


\item{\bf QFT in $0+0$ dimensions.}

In this problem we return to the simplest scalar QFT, namely a single integral 
with quartic action.  

\begin{enumerate}

\item 
By a change of integration variable 
show that
$$ Z = \int_{-\infty}^{\infty} dq ~e^{ - S(q) } $$
with $ S(q) = \half m^2 q^2 + g q^4 $
is of the form
%$$ Z = {1\over \sqrt{m^2}} {\cal Z}\( g \over m^4 \) . $$
$$ Z = {1\over g^{1/4}} {\cal Z}\( m^2/\sqrt{g} \) . $$
This means you can make your life easier by setting $g=1$, without loss of generality.



\item Convince yourself (\eg~with Mathematica) that the integral 
really is expressible as a Bessel function.


\item 
%(Exciting bonus problem.) 
It would be nice to find a better understanding for why the partition function
of $(0+0)$-dimensional $\phi^4$ theory is a Bessel function.
%First do problem \ref{problem:SD}.  
Find a Schwinger-Dyson equation for this system
which has the form of Bessel's equation for 
%$$K(y) \equiv  {1\over \sqrt{y}} e^{- a/y }  {\cal Z}\(1/y \) $$
$$K(x^2/a) \equiv  e^{-x^2/a}  (x^2)^{-1/4} {\cal Z}\(x \) $$
for some constant $a$.
(If you get stuck I can tell you what function to choose for the
`anything' in the S-D equation.)


\item Make a plot of the perturbative approximations to the `Green function' $G \equiv \vev{q^2}$ 
as a function of $g$, truncated at orders $1$ through $6$ or so.
Plot them against the exact answer.


\item (Bonus problem) Show that $c_{n+1} \sim - {2\over 3} n c_{n}$ at large $n$, where $c_n$ is the coefficient of $g^n$ in either $G$ or $Z$ (by brute force 
or by cleverness).



\end{enumerate}



\item {\bf Diagrams in 0+0d field theory.} [Bonus problem]
%$\iota$ 
%$\boldsymbol{\iota}$
%$\mathfrak{i}$
Here we will study a bit more some field theories with no dimensions at all,
that is, integrals.  


Consider the case where we put a label on the `field': $ q \to q_a, a = 1..N$.
So we are studying 
$$ Z = \int \int_{-\infty}^{\infty} \prod_a dq_a ~e^{ - S(q) } .$$
Let $$ S(q) =  \half q_a K_{ab} q_b  +  T_{abcd} q_a q_b q_c q_d $$
where $T_{abcd}$ is a collection of couplings.
Assume $K_{ab}$ is a real symmetric matrix.


\begin{enumerate}
\item Show that the propagator has the form:
$$ a ------
%{\rm \hrule.2cm}
%________}
%\hrule.2cm
%{\text{ ________}}
b  \equiv \vev{q_a q_b}_{T=0} = \( K^{-1}\)_{ab} = \sum_k  \phi_a(k)^\star {1\over k} \phi_b(k)  $$
where $\{ k \} $ are the eigenvalues of the matrix $K$
and $\phi_a(k)$ are the eigenvectors in the $a$-basis.


\item Develop a diagrammatic expansion for the propagator $\vev{q_a q_b}$.   Show that in a diagram with a loop, we must sum over the eigenvalue label $k$.
(For definiteness, consider the order-$T$ correction to the propagator $\vev{q_k q_{k'}}$, where $k, k'$ label eigenvectors of $K$, and 
$ q_k \equiv \sum_a \phi_a(k)^\star q_a $).



\item Consider the case where $K_{ab} = t\(  \delta_{a, b+1} + \delta_{a+1, b}\) $,
with periodic boundary conditions: $ a+N \equiv a$.
Find the eigenvalues.
Show that in this case if 
$$ T_{abcd} q_a q_b q_c q_d = \sum_a g q_a^4 $$
the $k$-label is conserved at vertices, \ie~the vertex is accompanied by a 
delta function on the sum of the incoming eigenvalues.
 \label{prob:discrete-circle} 
 
 \answer{This matrix $K$ is the hamiltonian for a particle hopping on a discrete circle, 
 let's write this as  
 $$ K \ket{n} = \ket{n+1} - \ket{n-1} ,$$
 where $ \ket{n+N} = \ket{n}$.
This has translation invariance, so $[ K, T]$ 
where $ T \ket{n} = \ket{n+1} $.  
More specifically, $ K = T + T^{-1}$.  
The eigenvectors of translation are  $ \ket{\theta} 
= \sum_{n=1}^n e^{ \ii n \theta} \ket{n} $ since
$$ T \ket{\theta} = e^{ -\ii \theta } \ket{\theta}. $$
Hence the eigenvalues of $K$ are of the form $ 2 \cos \theta$.  
Since $n \in \IZ$, $ \theta \equiv \theta + 2 \pi$.  
Since $ n \equiv n+N$, only $ \theta = 2 \pi m /N , m \in \IZ$ 
are allowed.
Hence the eigenvalues of $K$ are $ 2 \cos 2 \pi m / N , m = 1.. N$.  
 }
 
 

\item (Bonus question) What is the more general condition on $T_{abcd}$ in order that the $k$-label is
conserved at vertices?



\item (Bonus question) Study the physics of the model described in \ref{prob:discrete-circle}.


\end{enumerate}









\item{\bf The vacuum is a fluid with $p = - \rho$.} [Bonus problem]

We said in lecture that the vacuum energy density $\rho$ gravitates and that, when positive, its effect is to cause space to inflate -- to expand exponentially in time.  
An important aspect of this phenomenon is that the vacuum fluctuations produce not only an energy density, but a {\it pressure}, $p = T^i_i$ (no sum on $i$), of the form $p = - \rho$, which is negative for $\rho>0$.  
The vacuum therefore acts as a perfect fluid with $P = - \rho$.  
(The stress tensor for a perfect fluid in terms of its velocity field $u^\mu$ takes the form 
$T^{\mu\nu} = (p + \rho) u^\mu u^\nu + p g^{\mu\nu}$, 
so in a frame with $u^\mu = (1,\vec 0^\mu)$, 
$T^0_0 = \rho, T^i_i = P$.)
Solving Einstein's equations with such a source produces an inflating universe.
In this problem we show that this is is the case.

\begin{enumerate}
\item Show that the energy-momentum tensor for a free relativistic scalar field 
($S[\phi] = \int d^D x \sqrt{g} \CL, \CL = \half g^{\mu\nu} \partial_\mu \phi \partial_\nu\phi - {m^2\over 2}  \phi^2 $)
takes the form
$$ T_{\mu\nu} =  a \partial_\mu \phi \partial_\nu \phi - b g_{\mu\nu} \CL  $$
with some constants $a, b$.  

\item 
Reproduce the formal expression for the vacuum energy
$$ \vev{0| \HH |0} = V \int \dbar^d k \half \hbar \omega_{\vec k} $$
using the two point function
$$ \bra{0} \phi(x)^2 \ket{0} = \vev{0| \phi(0) \phi(0) |0} 
= \lim_{\vec x, t \to 0} \bra{0} \phi(x) \phi(0) \ket{0}  $$
and its derivatives.
($V$ is the volume of space.)


(We will learn to draw this amplitude as a Feynman diagram 
which is a circle (a line connecting a point to itself).)


\item 
Show that the vacuum expectation value of the pressure
$$   \vev{0 | T_{ii}  | 0} $$ 
(no sum on $i$)
gives the same answer, up to a sign.  

[Hints: You'll find a quite different looking integral from the vacuum energy.  
Use rotation invariance of the vacuum to simplify the answer.
The claim is that however you regulate the integral for the vacuum pressure and $\half \int \dbar^d k \omega_k$, you'll get the same answer.  A convenient regulator is {\it dimensional regularization}: treat the dimension $d$ as an arbitrary complex number.]


\item Show that the resulting vacuum energy momentum tensor ($T_{00} = \rho, T_{ii} = - \rho$ (no sum on $i$)) is the same as the contribution to the energy-momentum tensor from an action of the form
$$ S_\text{cc} =  \int d^D x \sqrt{g} \Lambda $$
where $\Lambda$ is a constant (the cosmological constant).

If you wish, plug in the FRW ansatz for the metric $ds^2 = - dt^2 + a(t)^2 d\vec x^2 $ and show that Einstein's equations in the presence of a positive cosmological constant 
\be  { \delta S[g] \over \delta g_{\mu\nu}(x) } = 0 , \text{ with } S[g]  = { 1\over 16 \pi G_N } \int d^D x \sqrt{g} R + S_{\text{cc}}  \ee
have the solution 
$ a(t) = e^{ H t } $ for some $H$ determined by $\Lambda$ and $G_N$.   

\item Argue that $p= - \rho$ is required in order that the vacuum energy does not specify a preferred rest frame.

\end{enumerate}







\end{enumerate}
\end{document}

