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\begin_body

\begin_layout Chapter
Non-perturbative Quantum Field Theory with n-Particle Irreducible Effective
 Action Techniques
\begin_inset CommandInset label
LatexCommand label
name "chap:chap2"

\end_inset


\end_layout

\begin_layout Section
Synopsis
\end_layout

\begin_layout Standard
This chapter reviews the 
\begin_inset Formula $n$
\end_inset

-particle irreducible effective action (
\begin_inset Formula $n$
\end_inset

PIEA) techniques for quantum field theory.
 For 
\begin_inset Formula $n>1$
\end_inset

 the 
\begin_inset Formula $n$
\end_inset

PIEAs are non-perturbative, effecting resummations of 
\begin_inset Formula $2$
\end_inset

- through 
\begin_inset Formula $n$
\end_inset

-th order correlation functions to infinite order in perturbation theory.
 The equations of motion derived from 
\begin_inset Formula $n$
\end_inset

PIEAs are closed, unlike the otherwise very similar Schwinger-Dyson equations
 in quantum field theory or Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY)
 equations of kinetic theory.
 Because of this and the fact the the methods can be derived from first
 principles and are manifestly self-consistent, 
\begin_inset Formula $n$
\end_inset

PIEAs have been advocated for applications in non-equlibrium field theory
 ranging from cold atom research to cosmology (see, e.g., 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2004,Berges2015"

\end_inset

).
 For 
\begin_inset Formula $n\geq3$
\end_inset

, 
\begin_inset Formula $n$
\end_inset

PIEAs can provide the basis for a self-consistent first principles derivation
 of kinetic theory (see, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Carrington2009,Smolic2012,Berges2004"

\end_inset

).
\end_layout

\begin_layout Standard
In section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-1PI-effective"

\end_inset

 the venerable (but perturbative) 1PIEA is introduced following standard
 textbook treatments.
 Then in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-2PI-effective"

\end_inset

 the non-perturbative 2PIEA is reviewed, following the reviews by Berges
 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2004,Berges2015"

\end_inset

 and the extensive book by Calzetta and Hu 
\begin_inset CommandInset citation
LatexCommand citep
key "Calzetta2008"

\end_inset

.
 Here the full expression for the 2PIEA of a quartically coupled scalar
 field theory is derived to three loop order.
 The general pattern of higher 
\begin_inset Formula $n$
\end_inset

PIEAs is then discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Higher-PI-effective"

\end_inset

, where the 3PIEA is explicitly derived for future reference.
 Finally, section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Analytic-properties-of-2PIEA-and-resummation"

\end_inset

 contains a novel discussion of the analytic properties of effective actions
 using the 2PIEA of a toy model where exact results are available, so that
 the 2PIEA results can be compared to perturbation theory and other resummation
 methods.
 This discussion is based on a paper published by the author 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015"

\end_inset

.
\end_layout

\begin_layout Section
The 1PI effective action
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:The-1PI-effective"

\end_inset


\end_layout

\begin_layout Standard
Generating functionals and effective action methods form the basis for the
 rest of this work.
 This section reviews the one particle irreducible effective action.
 Originally developed by Goldstone, Salam and Weinberg 
\begin_inset CommandInset citation
LatexCommand citep
key "Goldstone1962"

\end_inset

 and, independently, Jona-Lasinio 
\begin_inset CommandInset citation
LatexCommand citep
key "Jona-Lasinio1964"

\end_inset

, the 1PIEA is widely used, particularly in discussions of theories with
 spontaneous symmetry breaking or gauge symmetries.
 The treatment starts with the generating functional 
\begin_inset Formula $Z\left[J\right]$
\end_inset

, closely related to the partition function described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Time-contours-Green-functions-NEQFT"

\end_inset

.
 
\begin_inset Formula $Z\left[J\right]$
\end_inset

 generates all correlation functions of the field theory and can be related
 to a Feynman diagram expansion.
 From 
\begin_inset Formula $Z\left[J\right]$
\end_inset

 one derives 
\begin_inset Formula $W\left[J\right]$
\end_inset

 which contains only the 
\begin_inset Quotes eld
\end_inset

connected
\begin_inset Quotes erd
\end_inset

 correlation functions, i.e.
 those whose diagrams are made of one connected component.
 The 1PIEA 
\begin_inset Formula $\Gamma\left[\varphi\right]$
\end_inset

 is then found by Legendre transforming 
\begin_inset Formula $W\left[J\right]$
\end_inset

 and is the generating function of 
\begin_inset Quotes eld
\end_inset

proper
\begin_inset Quotes erd
\end_inset

 correlation functions, meaning those which cannot be divided by cutting
 one line of the corresponding Feynman diagram.
 The skeleton expansion of the connected generating functional provides
 an elegant and systematic means of relating connected correlation functions
 with the proper correlation functions.
 Hence 
\begin_inset Formula $\Gamma\left[\varphi\right]$
\end_inset

 contains all of the physical information of a field theory in relatively
 compact form.
 Further, terms in the derivative expansion of 
\begin_inset Formula $\Gamma\left[\varphi\right]$
\end_inset

 have a direct physical meaning.
 Most importantly, the term with no derivatives (the 
\emph on
effective potential
\emph default
) is equal to the minimum of the energy, over all states 
\begin_inset Formula $\Ket{\Psi}$
\end_inset

, subject to the constraint that the mean field is equal to 
\begin_inset Formula $\left\langle \phi\right\rangle =\varphi$
\end_inset

.
\end_layout

\begin_layout Standard
The previous chapter outlined how using various choices of time contour
 in the functional integral allow one to compute vacuum, finite temperature
 and non-equilibrium properties of a field theory.
 The differences between these formalisms are largely irrelevant to most
 of the following considerations.
 The important thing is that all observables of the theory can be computed
 from a formula formally identical to 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:vacuum-to-vacuum-transition-matrix-element"

\end_inset

, only where the range of the integration variables, boundary conditions
 and time contour are interpreted appropriately according to the particular
 physical situation.
 To that end it proves useful to generalise the generating functional 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:transition-amplitude-generating-functional"

\end_inset

 to a quantity usually called the partition function:
\begin_inset Formula 
\begin{equation}
Z\left[J\right]=\int\mathcal{D}\left[\phi\right]\exp\left[\frac{i}{\hbar}\left(S\left[\phi\right]+\int_{x}J\left(x\right)\phi\left(x\right)\right)\right],\label{eq:partition-function-for-1PIEA}
\end{equation}

\end_inset

where
\begin_inset Formula 
\begin{equation}
S\left[\phi\right]=\int_{x}\mathcal{L}\left(\phi\right),
\end{equation}

\end_inset

is the classical action.
 The only difference between 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:partition-function-for-1PIEA"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:transition-amplitude-generating-functional"

\end_inset

 is that the time contour and boundary conditions of the functional integral
 are hidden in the notation and can be re-interpreted as needed.
 Then, the sought after generalisation of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:vacuum-to-vacuum-transition-matrix-element"

\end_inset

 is
\begin_inset Formula 
\begin{equation}
G^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right)\equiv\left\langle \mathrm{T}_{C}\left[\hat{\phi}\left(x_{1}\right)\cdots\hat{\phi}\left(x_{n}\right)\right]\right\rangle =\left.\frac{1}{Z\left[J\right]}\left(-i\hbar\frac{\delta}{\delta J\left(x_{1}\right)}\right)\cdots\left(-i\hbar\frac{\delta}{\delta J\left(x_{n}\right)}\right)Z\left[J\right]\right|_{J=0},
\end{equation}

\end_inset

which defines the functions 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

, 
\begin_inset Formula $n=1,2,3,\cdots$
\end_inset

, which are the contour ordered Green functions of the theory.
 (There is no harm in defining also the trivial cases 
\begin_inset Formula $G^{\left(0\right)}\equiv\left\langle 1\right\rangle =1$
\end_inset

 and 
\begin_inset Formula $G^{\left(n\right)}=0$
\end_inset

 for 
\begin_inset Formula $n<0$
\end_inset

, which makes some formulae simpler.)
\end_layout

\begin_layout Standard
Applying the trivial identity
\begin_inset Formula 
\begin{equation}
0=\int\mathcal{D}\left[\phi\right]\frac{\delta}{\delta\phi\left(x_{1}\right)}F\left[\phi\right],
\end{equation}

\end_inset

where 
\begin_inset Formula $F\left[\phi\right]$
\end_inset

 is an arbitrary functional, to the case 
\begin_inset Formula 
\begin{equation}
F\left[\phi\right]=\phi\left(x_{2}\right)\cdots\phi\left(x_{n}\right)\exp\left[\frac{i}{\hbar}\left(\int_{x}\mathcal{L}\left(\phi\right)+J\left(x\right)\phi\left(x\right)\right)\right]
\end{equation}

\end_inset

and working out the derivative explicitly gives the 
\emph on
Schwinger-Dyson equation
\emph default

\begin_inset Formula 
\begin{multline}
\left\langle \mathrm{T}_{C}\left[\phi\left(x_{2}\right)\cdots\phi\left(x_{n}\right)\left(\frac{\delta S\left[\phi\right]}{\delta\phi\left(x_{1}\right)}+J\left(x_{1}\right)\right)\right]\right\rangle \\
=i\hbar\sum_{j=2}^{n}\delta_{C}\left(x_{1},x_{j}\right)G^{\left(n-2\right)}\left(\phi\left(x_{2}\right),\cdots\phi\left(x_{j-1}\right),\phi\left(x_{j+1}\right),\cdots,\phi\left(x_{n}\right)\right),
\end{multline}

\end_inset

which says that the classical equation of motion 
\begin_inset Formula $\frac{\delta S\left[\phi\right]}{\delta\phi\left(x\right)}+J\left(x\right)=0$
\end_inset

 is obeyed inside correlation functions up to the existence of delta function
 
\emph on
contact terms
\emph default
 on the right hand side.
 Substituting an explicit form for 
\begin_inset Formula $S\left[\phi\right]$
\end_inset

 gives a hierarchy of equations, one for each 
\begin_inset Formula $n$
\end_inset

, connecting correlation functions for different values of 
\begin_inset Formula $n$
\end_inset

.
 This is exactly analogous to the BBGKY hierarchy in statistical mechanics
 (see, e.g., 
\begin_inset CommandInset citation
LatexCommand citep
key "Born1946,Kirkwood1946"

\end_inset

).
 Solving this hierarchy of equations in practice requires truncation at
 some finite order, which necessitates invoking a closure ansatz approximating
 some higher order correlation functions in terms of lower order ones.
 This ansatz can be physically motivated but always involves some degree
 of arbitrariness.
 In contrast, while the higher 
\begin_inset Formula $n$
\end_inset

PIEA discussed later provide a superficially very similar framework (a hierarchy
 of equations of motion connecting correlation functions of different order),
 the equations of motion resulting from 
\begin_inset Formula $n$
\end_inset

PIEA are automatically self-consistent and closed, without requiring any
 arbitrary ansatz or approximation.
\end_layout

\begin_layout Standard
Usually the Lagrangian can be written
\begin_inset Formula 
\begin{equation}
\mathcal{L}\left(\phi\right)=\frac{1}{2}\phi\Delta_{0}^{-1}\phi-V_{\mathrm{int}}\left(\phi\right),
\end{equation}

\end_inset

where the first term is the quadratic part of the Lagrangian, 
\begin_inset Formula $\Delta_{0}^{-1}$
\end_inset

 is a differential operator and 
\begin_inset Formula $V_{\mathrm{int}}\left(\phi\right)$
\end_inset

 is a polynomial containing cubic and higher terms in 
\begin_inset Formula $\phi$
\end_inset

.
 In this case there is a perturbation expansion for 
\begin_inset Formula $Z$
\end_inset

 which can be derived by expanding the exponential in 
\begin_inset Formula $V_{\mathrm{int}}\left(\phi\right)$
\end_inset

:
\begin_inset Formula 
\begin{eqnarray}
Z\left[J\right] & = & \int\mathcal{D}\left[\phi\right]\exp\left[-\frac{i}{\hbar}\int_{x}V_{\mathrm{int}}\left(\phi\right)\right]\exp\left[\frac{i}{\hbar}\left(\int_{x}\frac{1}{2}\phi\Delta_{0}^{-1}\phi+J\left(x\right)\phi\left(x\right)\right)\right]\nonumber \\
 & = & \int\mathcal{D}\left[\phi\right]\exp\left[-\frac{i}{\hbar}\int_{x}V_{\mathrm{int}}\left(-i\hbar\frac{\delta}{\delta J}\right)\right]\exp\left[\frac{i}{\hbar}\left(\int_{x}\frac{1}{2}\phi\Delta_{0}^{-1}\phi+J\left(x\right)\phi\left(x\right)\right)\right].
\end{eqnarray}

\end_inset

The 
\begin_inset Formula $V_{\mathrm{int}}\left(-i\hbar\frac{\delta}{\delta J}\right)$
\end_inset

 term can now be extracted from the integral
\begin_inset Foot
status open

\begin_layout Plain Layout
It is this step that is responsible for the asymptotic nature of perturbation
 theory, c.f.
 section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Analytic-properties-of-2PIEA-and-resummation"

\end_inset

.
\end_layout

\end_inset

, giving
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
Z\left[J\right] & = & \exp\left[-\frac{i}{\hbar}\int_{x}V_{\mathrm{int}}\left(-i\hbar\frac{\delta}{\delta J}\right)\right]\int\mathcal{D}\left[\phi\right]\exp\left[\frac{i}{\hbar}\left(\int_{x}\frac{1}{2}\phi\Delta_{0}^{-1}\phi+J\left(x\right)\phi\left(x\right)\right)\right]\nonumber \\
 & = & \exp\left[-\frac{i}{\hbar}\int_{x}V_{\mathrm{int}}\left(-i\hbar\frac{\delta}{\delta J}\right)\right]\exp\left[-\frac{i}{\hbar}\frac{1}{2}J\Delta_{0}J\right]Z\left[0\right],
\end{eqnarray}

\end_inset

where the last line is obtained by completing the square in the exponent
 of the Gaussian integral.
 Thus
\begin_inset Formula 
\begin{multline}
G^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right)=\left(-i\hbar\frac{\delta}{\delta J\left(x_{1}\right)}\right)\cdots\left(-i\hbar\frac{\delta}{\delta J\left(x_{n}\right)}\right)\\
\times\left.\exp\left[-\frac{i}{\hbar}\int_{x}V_{\mathrm{int}}\left(-i\hbar\frac{\delta}{\delta J\left(x\right)}\right)\right]\exp\left[-\frac{i}{\hbar}\int_{xy}\frac{1}{2}J\left(x\right)\Delta_{0}\left(x,y\right)J\left(y\right)\right]\right|_{J=0}.
\end{multline}

\end_inset

Expanding the exponentials one finds the usual Wick expansion of the Green
 functions which is embodied in the 
\emph on
Feynman rules
\emph default
:
\end_layout

\begin_layout Enumerate
To find a contribution to 
\begin_inset Formula $G^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right)$
\end_inset

 draw 
\begin_inset Formula $n$
\end_inset

 points (
\begin_inset Quotes eld
\end_inset

external legs
\begin_inset Quotes erd
\end_inset

), associated to 
\begin_inset Formula $x_{1}$
\end_inset

 through 
\begin_inset Formula $x_{n}$
\end_inset

, and
\end_layout

\begin_layout Enumerate
Draw a 
\begin_inset Formula $k$
\end_inset

 point vertex for every interaction 
\begin_inset Formula $\frac{1}{k!}g_{k}\phi^{k}\subset V_{\mathrm{int}}$
\end_inset

 appearing in the expansion of the exponential.
 Each such vertex is associated with a factor 
\begin_inset Formula $\left(-\frac{i}{\hbar}\right)g_{k}$
\end_inset

 and an integral 
\begin_inset Formula $\int_{x}$
\end_inset

 over position of the vertex, and
\end_layout

\begin_layout Enumerate
Connect external legs and vertices with lines (
\begin_inset Quotes eld
\end_inset

propagators
\begin_inset Quotes erd
\end_inset

) in all possible ways such that: all external legs are attached to one
 
\begin_inset Quotes eld
\end_inset

line end,
\begin_inset Quotes erd
\end_inset

 each 
\begin_inset Formula $k$
\end_inset

 point vertex is attached to 
\begin_inset Formula $k$
\end_inset

 line ends (nothing forbids attaching both ends of a line to the same vertex)
 and no line ends except on a vertex or external leg.
 Each line is associated with a factor 
\begin_inset Formula $i\hbar\Delta_{0}\left(x,y\right)$
\end_inset

 where 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 are the position of the two ends.
\end_layout

\begin_layout Enumerate
Also, associate to the diagram an overall symmetry factor 
\begin_inset Formula $1/\left|\Gamma\right|$
\end_inset

, where 
\begin_inset Formula $\left|\Gamma\right|$
\end_inset

 is the order of the automorphism group of the diagram under interchanges
 of lines and vertices.
\end_layout

\begin_layout Standard
Using these rules it is possible to compute 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

 systematically, order by order, in a perturbation expansion in the coupling
 constants in 
\begin_inset Formula $V_{\mathrm{int}}$
\end_inset

.
 From 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

, one can also recover 
\begin_inset Formula $Z\left[J\right]$
\end_inset

:
\begin_inset Formula 
\begin{equation}
Z\left[J\right]=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{i}{\hbar}\right)^{n}\int_{x_{1}\cdots x_{n}}J\left(x_{1}\right)\cdots J\left(x_{n}\right)G^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right).
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The diagram expansion of 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

contains many disconnected diagrams, i.e.
 diagrams with more than one connected component.
 This leads to an undesirable duplication of computation (since a disconnected
 contribution to 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

 is also a contribution to various 
\begin_inset Formula $G^{\left(k\right)}$
\end_inset

 with 
\begin_inset Formula $k<n$
\end_inset

) and non-extensivity of 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

 since a disconnected contribution factorises 
\begin_inset Formula $\sim G^{\left(k\right)}\left(x_{1},\cdots,x_{k}\right)G^{\left(n-k\right)}\left(x_{n-k},\cdots,x_{n}\right)$
\end_inset

 and so is independent of the relative separation between points in the
 two clusters 
\begin_inset Formula $x_{1},\cdots,x_{k}$
\end_inset

 and 
\begin_inset Formula $x_{n-k},\cdots,x_{n}$
\end_inset

.
 Clearly the connected components are the basic objects which one would
 prefer to compute.
 Fortunately, though the combinatorics will not be proved here
\begin_inset Foot
status open

\begin_layout Plain Layout
It can be found in many quantum field theory books, see e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Srednicki2007"

\end_inset

.
\end_layout

\end_inset

, the connected diagrams obey the simple relationship given schematically
 as
\begin_inset Formula 
\begin{equation}
\text{all diagrams}=\exp\left(\text{connected diagrams}\right).
\end{equation}

\end_inset

This strongly motivates the introduction of the 
\emph on
connected generating functional
\emph default

\begin_inset Formula 
\begin{equation}
W\left[J\right]=-i\hbar\ln Z\left[J\right].
\end{equation}

\end_inset

The diagram expansion for 
\begin_inset Formula $W\left[J\right]$
\end_inset

 has the same Feynman rules as above, except that one only includes connected
 diagrams and, traditionally, the extra overall factor 
\begin_inset Formula $-i\hbar$
\end_inset

.
 To lowest order
\begin_inset Formula 
\begin{equation}
W\left[J\right]=W\left[0\right]+\int_{x}J\left(x\right)\left\langle \phi\left(x\right)\right\rangle -\frac{1}{2}\int_{xy}J\left(x\right)\Delta_{0}\left(x,y\right)J\left(y\right)+\mathcal{O}\left(V_{\mathrm{int}}\right).
\end{equation}

\end_inset

Note that 
\begin_inset Formula $\left\langle \phi\left(x\right)\right\rangle \sim\mathcal{O}\left(V_{\mathrm{int}}\right)$
\end_inset

 if there is no linear term in 
\begin_inset Formula $\mathcal{L}\left(\phi\right)$
\end_inset

, which will be assumed below.
 As usual, for most physical purposes the overall normalisation 
\begin_inset Formula $W\left[0\right]$
\end_inset

 is unimportant.
 The connected correlation functions are
\begin_inset Formula 
\begin{equation}
G_{C}^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right)=\left.\left(-i\hbar\frac{\delta}{\delta J\left(x_{1}\right)}\right)\cdots\left(-i\hbar\frac{\delta}{\delta J\left(x_{n}\right)}\right)W\left[J\right]\right|_{J=0},
\end{equation}

\end_inset

so that, at leading order in interactions,
\begin_inset Formula 
\begin{eqnarray}
G_{C}^{\left(2\right)}\left(x,y\right) & = & \hbar^{2}\Delta_{0}\left(x,y\right)+\mathcal{O}\left(V_{\mathrm{int}}\right),\\
G_{C}^{\left(1\right)}\left(x\right) & = & G_{C}^{\left(n\geq3\right)}\left(x_{1},\cdots,x_{n}\right)=0+\mathcal{O}\left(V_{\mathrm{int}}\right).
\end{eqnarray}

\end_inset

From these 
\begin_inset Formula $W\left[J\right]$
\end_inset

 can be recovered if need be:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
W\left[J\right]=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{i}{\hbar}\right)^{n}\int_{x_{1}\cdots x_{n}}J\left(x_{1}\right)\cdots J\left(x_{n}\right)G_{C}^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right).
\end{equation}

\end_inset

It is also convenient to define
\begin_inset Formula 
\begin{equation}
\Delta\left(x,y\right)=\hbar^{-2}G_{C}^{\left(2\right)}\left(x,y\right),
\end{equation}

\end_inset

i.e.
 simply stripping off the factor of 
\begin_inset Formula $\hbar^{2}$
\end_inset

.
 It is useful for later to note that, in terms of the 
\begin_inset Formula $G^{\left(n\right)}$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
G_{C}^{\left(2\right)}\left(x,y\right)=-i\hbar\left[G^{\left(2\right)}\left(x,y\right)-G^{\left(1\right)}\left(x\right)G^{\left(1\right)}\left(y\right)\right],
\end{equation}

\end_inset

thus
\begin_inset Formula 
\begin{equation}
G^{\left(2\right)}\left(x,y\right)=i\hbar\Delta\left(x,y\right)+\left\langle \phi\left(x\right)\right\rangle \left\langle \phi\left(y\right)\right\rangle .
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
There is a great deal of duplication of computation, even in the connected
 correlation functions.
 This can be seen by the example of the two point function of a theory with
 
\begin_inset Formula $V_{\mathrm{int}}\left(\phi\right)=\frac{1}{4!}\lambda\phi^{4}$
\end_inset

 expanded to third order in the interaction:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{perturbation-theory-propagator}
\end_layout

\begin_layout Plain Layout


\backslash
begin{align}
\end_layout

\begin_layout Plain Layout

i
\backslash
hbar
\backslash
Delta
\backslash
left(x,y
\backslash
right)&
\end_layout

\begin_layout Plain Layout

=
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,o}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{2}
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v,v,o} 
\backslash
fmfdot{v}
\backslash
end{fmfgraph}} 
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{4}
\backslash
parbox{20mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v1,v1,v2,v2,o} 
\backslash
fmfdot{v1}
\backslash
fmfdot{v2}
\backslash
end{fmfgraph}} 
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{4}
\backslash
parbox{20mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v1,v2} 
\backslash
fmf{phantom}{v2,t}
\backslash
fmffreeze 
\backslash
fmf{plain,left}{v1,v2,t,v2,v1} 
\backslash
fmfdot{v1,v2}
\backslash
end{fmfgraph}} 
\backslash
nonumber
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

&+
\backslash
frac{1}{6}
\backslash
parbox{20mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{plain}{i,v1,v2,o} 
\backslash
fmf{plain,left,tension=0}{v1,v2,v1} 
\backslash
fmfdot{v1,v2}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{plain}{i,v1,v1,v2,v2,v3,v3,o} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{tl,t1,t2,tr} 
\backslash
fmf{plain}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v1,x1,t1}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v2,v3,t2}
\backslash
fmffreeze 
\backslash
fmf{plain,left}{v1,x1,v1} 
\backslash
fmf{plain,left}{v2,v3,v2} 
\backslash
fmf{plain,left}{v3,t2,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{tl,t2,t1,tr} 
\backslash
fmf{plain}{i,v2,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v1,x1,t1}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v2,v3,t2}
\backslash
fmffreeze 
\backslash
fmf{plain,left}{v1,x1,v1} 
\backslash
fmf{plain,left}{v2,v3,v2} 
\backslash
fmf{plain,left}{v3,t2,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\backslash
nonumber
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

&+
\backslash
frac{1}{12}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,60) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{plain}{i,v1,v1,v2,v3,o} 
\backslash
fmf{plain,left,tension=0}{v2,v3,v2} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{12}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,60) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{plain}{i,v2,v3,v1,v1,o} 
\backslash
fmf{plain,left,tension=0}{v2,v3,v2} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,60) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v1,v2,v3,t}
\backslash
fmffreeze 
\backslash
fmf{plain,left}{v1,v2,v3,t,v3,v2,v1} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{12}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{tl,t1,t2,tr} 
\backslash
fmf{plain}{i,x1,v1,x2,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{x1,v2,t1}
\backslash
fmffreeze 
\backslash
fmf{phantom}{x2,v3,t2}
\backslash
fmffreeze 
\backslash
fmf{plain,left=0.5}{v1,v2} 
\backslash
fmf{plain,right=0.5}{v1,v3} 
\backslash
fmf{plain,left=0.5}{v2,v3,v2} 
\backslash
fmf{plain}{v2,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\backslash
nonumber
\backslash

\backslash
 
\end_layout

\begin_layout Plain Layout

&+
\backslash
frac{1}{4}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain,left=0.5}{v1,v3,v2} 
\backslash
fmf{plain,right}{v1,v2} 
\backslash
fmf{phantom,tension=0.1}{v3,t} 
\backslash
fmf{plain,left}{v3,t,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{lm2,lm1,i,l1,l2}
\backslash
fmfright{rm2,rm1,o,r1,r2}
\backslash
fmftop{tl,t1,t2,t3,tr} 
\backslash
fmf{plain}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{l1,v2,v3,r1}
\backslash
fmffreeze 
\backslash
fmf{plain,left=0.6}{v1,v2} 
\backslash
fmf{plain,left=0.4}{v2,v3} 
\backslash
fmf{plain,left=0.6}{v3,v1} 
\backslash
fmf{plain,left}{v2,t1,v2}
\backslash
fmf{plain,left}{v3,t3,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{4}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,x1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{t,v3,x1}
\backslash
fmffreeze 
\backslash
fmf{plain,right=0.3}{v1,v3,v2} 
\backslash
fmf{plain,left=0.5}{v1,v3,v2} 
\backslash
fmfdot{v1,v2,v3}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
mathcal{O}
\backslash
left(
\backslash
lambda^4
\backslash
right).
\backslash
label{eq:G-perturbation-series-diagrams}
\end_layout

\begin_layout Plain Layout


\backslash
end{align}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

There are obviously a number of repeated structures in this expression,
 which can be seen to follow the pattern of a geometric series:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{dyson-series}
\end_layout

\begin_layout Plain Layout


\backslash
begin{align}
\end_layout

\begin_layout Plain Layout

i
\backslash
hbar
\backslash
Delta
\backslash
left(x,y
\backslash
right)&=
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,o}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v,o} 
\backslash
fmfblob{.15w}{v}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v1,v2,o} 
\backslash
fmfblob{.15w}{v1}
\backslash
fmfblob{.15w}{v2}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v1,v2,v3,o} 
\backslash
fmfblob{.15w}{v1}
\backslash
fmfblob{.15w}{v2}
\backslash
fmfblob{.15w}{v3}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
cdots,
\backslash
label{eq:dyson-series-for-propagator}
\end_layout

\begin_layout Plain Layout


\backslash
end{align}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

where the shaded 
\begin_inset Quotes eld
\end_inset

blobs
\begin_inset Quotes erd
\end_inset

 represent the sum of all two point 
\emph on
one particle irreducible
\emph default
 (1PI) Feynman diagrams, that is, the set of diagrams which do not fall
 apart upon the cutting of any one line.
 The first few terms of the self-energy are seen to be
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{self-energy-series}
\end_layout

\begin_layout Plain Layout


\backslash
begin{align}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v,o} 
\backslash
fmfblob{.15w}{v}
\backslash
end{fmfgraph}}&=
\end_layout

\begin_layout Plain Layout


\backslash
frac{1}{2}
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmf{plain}{i,v,v,o} 
\backslash
fmfdot{v}
\backslash
end{fmfgraph}} 
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{4}
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v1,v2} 
\backslash
fmf{phantom}{v2,t}
\backslash
fmffreeze 
\backslash
fmf{plain,left}{v1,v2,t,v2,v1} 
\backslash
fmfdot{v1,v2}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{6}
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20) 
\backslash
fmfleft{i}
\backslash
fmfright{o} 
\backslash
fmf{plain}{i,v1,v2,o} 
\backslash
fmf{plain,left,tension=0}{v1,v2,v1} 
\backslash
fmfdot{v1,v2}
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{20mm}{ 
\backslash
begin{fmfgraph}(40,60) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{v1,v2,v3,t}
\backslash
fmffreeze 
\backslash
fmf{plain,left}{v1,v2,v3,t,v3,v2,v1} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\backslash
nonumber
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

&+
\backslash
frac{1}{12}
\backslash
parbox{15mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{tl,t1,t2,tr} 
\backslash
fmf{plain}{i,x1,v1,x2,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{x1,v2,t1}
\backslash
fmffreeze 
\backslash
fmf{phantom}{x2,v3,t2}
\backslash
fmffreeze 
\backslash
fmf{plain,left=0.5}{v1,v2} 
\backslash
fmf{plain,right=0.5}{v1,v3} 
\backslash
fmf{plain,left=0.5}{v2,v3,v2} 
\backslash
fmf{plain}{v2,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{4}
\backslash
parbox{15mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{plain,left=0.5}{v1,v3,v2} 
\backslash
fmf{plain,right}{v1,v2} 
\backslash
fmf{phantom,tension=0.1}{v3,t} 
\backslash
fmf{plain,left}{v3,t,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{8}
\backslash
parbox{15mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{lm2,lm1,i,l1,l2}
\backslash
fmfright{rm2,rm1,o,r1,r2}
\backslash
fmftop{tl,t1,t2,t3,tr} 
\backslash
fmf{plain}{i,v1,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{l1,v2,v3,r1}
\backslash
fmffreeze 
\backslash
fmf{plain,left=0.6}{v1,v2} 
\backslash
fmf{plain,left=0.4}{v2,v3} 
\backslash
fmf{plain,left=0.6}{v3,v1} 
\backslash
fmf{plain,left}{v2,t1,v2}
\backslash
fmf{plain,left}{v3,t3,v3} 
\backslash
fmfdot{v1,v2,v3} 
\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
frac{1}{4}
\backslash
parbox{15mm}{ 
\backslash
begin{fmfgraph}(40,40) 
\backslash
fmfleft{i}
\backslash
fmfright{o}
\backslash
fmftop{t} 
\backslash
fmf{plain}{i,v1,x1,v2,o}
\backslash
fmffreeze 
\backslash
fmf{phantom}{t,v3,x1}
\backslash
fmffreeze 
\backslash
fmf{plain,right=0.3}{v1,v3,v2} 
\backslash
fmf{plain,left=0.5}{v1,v3,v2} 
\backslash
fmfdot{v1,v2,v3}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}}
\end_layout

\begin_layout Plain Layout

+
\backslash
mathcal{O}
\backslash
left(
\backslash
lambda^4
\backslash
right).
\end_layout

\begin_layout Plain Layout


\backslash
end{align}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The geometric series can be easily summed.
 Let 
\begin_inset Formula $-\frac{i}{\hbar}\Sigma$
\end_inset

 denote the 1PI 
\begin_inset Quotes eld
\end_inset

blob.
\begin_inset Quotes erd
\end_inset

 Then 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:dyson-series-for-propagator"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
refers to ERT 
\backslash
begin{fmffile}...
 above
\end_layout

\end_inset

 can be written
\begin_inset Formula 
\begin{align}
i\hbar\Delta & =i\hbar\Delta_{0}+\left(i\hbar\Delta_{0}\right)\left(-\frac{i}{\hbar}\Sigma\right)\left(i\hbar\Delta_{0}\right)+\left(i\hbar\Delta_{0}\right)\left(-\frac{i}{\hbar}\Sigma\right)\left(i\hbar\Delta_{0}\right)\left(-\frac{i}{\hbar}\Sigma\right)\left(i\hbar\Delta_{0}\right)+\cdots\nonumber \\
 & =i\hbar\Delta_{0}+i\hbar\Delta_{0}\Sigma\Delta.
\end{align}

\end_inset

Acting on the right with 
\begin_inset Formula $\Delta^{-1}$
\end_inset

 and on the left with 
\begin_inset Formula $\Delta_{0}^{-1}$
\end_inset

 one finds
\begin_inset Formula 
\begin{equation}
\Delta^{-1}=\Delta_{0}^{-1}-\Sigma,
\end{equation}

\end_inset

which is known as the 
\emph on
Dyson equation
\emph default
 for 
\begin_inset Formula $\Delta$
\end_inset

.
 
\begin_inset Formula $\Sigma$
\end_inset

 is called the 
\emph on
self-energy
\emph default
.
 A similar 
\emph on
skeleton
\emph default
 expansion occurs for the higher order correlation functions 
\begin_inset Formula $G^{\left(n\geq3\right)}$
\end_inset

 which can also be written in terms of 1PI subgraphs.
\end_layout

\begin_layout Standard
There is a generating function for the one particle irreducible diagrams:
 the 1PI effective action 
\begin_inset Formula $\Gamma\left[\varphi\right]$
\end_inset

 which is the Legendre transform of 
\begin_inset Formula $W\left[J\right]$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\Gamma\left[\varphi\right]=W\left[J\right]-J\frac{\delta W}{\delta J},
\end{equation}

\end_inset

where 
\begin_inset Formula $J$
\end_inset

 on the right hand side is eliminated in terms of the mean field 
\begin_inset Formula $\varphi=\left\langle \phi\right\rangle $
\end_inset

 by solving
\begin_inset Foot
status open

\begin_layout Plain Layout
There are physically relevant cases where this equation cannot be inverted.
 The most important example in this class is spontaneous symmetry breaking.
 There the saddle point method of evaluating 
\begin_inset Formula $\Gamma$
\end_inset

 derived below gives a non-convex potential which also has an imaginary
 part.
 The non-convex portion of the potential is unphysical, corresponding to
 a metastable state, and the imaginary part of the potential is related
 to the lifetime of that state.
 The quantity actually found by the saddle point method is an effective
 action which expands about a single classical configuration.
 This is equal to 
\begin_inset Formula $\Gamma$
\end_inset

 in the case that it is convex.
 However, for a non-convex effective potential the two quantities differ
 and the true form of 
\begin_inset Formula $\Gamma$
\end_inset

 in these cases is given by a Maxwell construction much like that used for
 the free energy in thermodynamics (see, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Zinn-Justin1990,Alexandre2012"

\end_inset

 and references therein).
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
\frac{\delta W}{\delta J}=\varphi.
\end{equation}

\end_inset

Then the equation of motion followed by 
\begin_inset Formula $\Gamma\left[\varphi\right]$
\end_inset

 is
\begin_inset Formula 
\begin{equation}
\frac{\delta\Gamma}{\delta\varphi}=-J.\label{eq:1piea-eom}
\end{equation}

\end_inset

A useful result is
\begin_inset Formula 
\begin{equation}
\int_{y}\frac{\delta^{2}\Gamma}{\delta\varphi\left(x\right)\delta\varphi\left(y\right)}\frac{\delta^{2}W}{\delta J\left(y\right)\delta J\left(z\right)}=\int_{y}-\frac{\delta J\left(y\right)}{\delta\varphi\left(x\right)}\frac{\delta\varphi\left(z\right)}{\delta J\left(y\right)}=-\delta\left(x-z\right),
\end{equation}

\end_inset

so that
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\frac{\delta^{2}\Gamma}{\delta\varphi\left(x\right)\delta\varphi\left(y\right)}=\Delta^{-1}\left(x,y\right).\label{eq:1PI-Delta-def-from-Gamma}
\end{equation}

\end_inset

Taking another derivative gives
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
0 & = & \frac{\delta}{\delta\varphi\left(w\right)}\int_{y}\frac{\delta^{2}\Gamma}{\delta\varphi\left(x\right)\delta\varphi\left(y\right)}\frac{\delta^{2}W}{\delta J\left(y\right)\delta J\left(z\right)}\nonumber \\
 & = & \int_{y}\frac{\delta^{3}\Gamma}{\delta\varphi\left(w\right)\delta\varphi\left(x\right)\delta\varphi\left(y\right)}\frac{\delta^{2}W}{\delta J\left(y\right)\delta J\left(z\right)}+\int_{yv}\frac{\delta^{2}\Gamma}{\delta\varphi\left(x\right)\delta\varphi\left(y\right)}\frac{\delta J\left(v\right)}{\delta\varphi\left(w\right)}\frac{\delta^{3}W}{\delta J\left(v\right)\delta J\left(y\right)\delta J\left(z\right)}\nonumber \\
 & = & \int_{y}\frac{\delta^{3}\Gamma}{\delta\varphi\left(w\right)\delta\varphi\left(x\right)\delta\varphi\left(y\right)}\frac{\delta^{2}W}{\delta J\left(y\right)\delta J\left(z\right)}\nonumber \\
 &  & -\int_{yv}\frac{\delta^{2}\Gamma}{\delta\varphi\left(x\right)\delta\varphi\left(y\right)}\frac{\delta\Gamma}{\delta\varphi\left(w\right)\delta\varphi\left(v\right)}\frac{\delta^{3}W}{\delta J\left(v\right)\delta J\left(y\right)\delta J\left(z\right)},
\end{eqnarray}

\end_inset

from which
\begin_inset Formula 
\begin{equation}
\frac{\delta^{3}\Gamma}{\delta\varphi\left(w\right)\delta\varphi\left(x\right)\delta\varphi\left(u\right)}=-\int_{yzv}\Delta^{-1}\left(w,v\right)\Delta^{-1}\left(x,y\right)\Delta^{-1}\left(u,z\right)\frac{\delta^{3}W}{\delta J\left(v\right)\delta J\left(y\right)\delta J\left(z\right)}.
\end{equation}

\end_inset

Similar results can be derived for higher order correlation functions by
 taking more derivatives.
 One eventually finds that 
\begin_inset Formula $W$
\end_inset

 is given by a sum of 
\emph on
tree
\emph default
 diagrams (i.e., no loops) where the vertices are given by the derivatives
 of 
\begin_inset Formula $\Gamma$
\end_inset

.
 This is the skeleton expansion of 
\begin_inset Formula $W$
\end_inset

, and from it one can see that 
\begin_inset Formula $\Gamma$
\end_inset

 consists of 1PI diagrams only.
 A detailed proof is not needed here and can be found in, e.g., chapter 5
 of 
\begin_inset CommandInset citation
LatexCommand citep
key "Zinn-Justin1990"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $\Gamma\left[\varphi\right]$
\end_inset

 is most often evaluated in a semi-classical or saddle point approximation.
 Here the path integral is expanded around the field configuration 
\begin_inset Formula $\phi_{\mathrm{cl.}}$
\end_inset

 of stationary phase, which is the one that satisfies the classical equation
 of motion 
\begin_inset Formula 
\begin{equation}
\left.\frac{\delta S\left[\phi\right]}{\delta\phi\left(x\right)}\right|_{\phi=\phi_{\mathrm{cl.}}\left[J\right]}+J\left(x\right)=0.
\end{equation}

\end_inset

Writing 
\begin_inset Formula $\phi=\phi_{\mathrm{cl.}}+\hbar^{1/2}\tilde{\phi}$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
S\left[\phi\right]=S\left[\phi_{\mathrm{cl.}}\right]+\hbar^{1/2}\tilde{\phi}\left.\frac{\delta S\left[\phi\right]}{\delta\phi\left(x\right)}\right|_{\phi=\phi_{\mathrm{cl.}}\left[J\right]}+\hbar\int\left(\frac{1}{2}\tilde{\phi}\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]\tilde{\phi}-V\left(\phi_{\mathrm{cl.}},\tilde{\phi}\right)\right),
\end{equation}

\end_inset

which defines 
\begin_inset Formula $\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]$
\end_inset

 as (up to a factor 
\begin_inset Formula $\hbar$
\end_inset

) the part of 
\begin_inset Formula $S\left[\phi\right]$
\end_inset

 which is quadratic in 
\begin_inset Formula $\tilde{\phi}$
\end_inset

 and 
\begin_inset Formula $V\left(\phi_{\mathrm{cl.}},\tilde{\phi}\right)$
\end_inset

 as (up to a factor 
\begin_inset Formula $\hbar$
\end_inset

) the part of 
\begin_inset Formula $V_{\mathrm{int}}$
\end_inset

 which is greater than quadratic in 
\begin_inset Formula $\tilde{\phi}$
\end_inset

.
 For example, in a theory with cubic and quartic couplings 
\begin_inset Formula $V_{\mathrm{int}}\left(\phi\right)=\frac{1}{3!}g\phi^{3}+\frac{1}{4!}\lambda\phi^{4}$
\end_inset

, 
\begin_inset Formula $V\left(\phi_{\mathrm{cl.}},\tilde{\phi}\right)$
\end_inset

 is
\begin_inset Formula 
\begin{equation}
V\left(\phi_{\mathrm{cl.}},\tilde{\phi}\right)=\frac{1}{3!}\hbar^{1/2}\left(g+\lambda\phi_{\mathrm{cl.}}\right)\tilde{\phi}^{3}+\frac{1}{4!}\hbar\lambda\tilde{\phi}^{4}.
\end{equation}

\end_inset

If one substitutes 
\begin_inset Formula $S\left[\phi_{\mathrm{cl.}}+\hbar^{1/2}\tilde{\phi}\right]$
\end_inset

 in 
\begin_inset Formula $Z\left[J\right]$
\end_inset

 the linear term in 
\begin_inset Formula $\tilde{\phi}$
\end_inset

 cancels the 
\begin_inset Formula $J\tilde{\phi}$
\end_inset

 term due to the equation of motion for 
\begin_inset Formula $\phi_{\mathrm{cl.}}$
\end_inset

 and one has
\begin_inset Formula 
\begin{eqnarray}
Z\left[J\right] & = & \exp\left[\frac{i}{\hbar}\left(S\left[\phi_{\mathrm{cl.}}\right]+J\phi_{\mathrm{cl.}}\right)\right]\int\mathcal{D}\left[\phi\right]\exp\left[i\int\left(\frac{1}{2}\tilde{\phi}\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]\tilde{\phi}-V\left(\phi_{\mathrm{cl.}},\tilde{\phi}\right)\right)\right]\nonumber \\
 & = & \exp\left[\frac{i}{\hbar}\left(S\left[\phi_{\mathrm{cl.}}\right]+J\phi_{\mathrm{cl.}}\right)\right]\int\mathcal{D}\left[\phi\right]\left(1+\mathcal{O}\left(\hbar\right)\right)\exp\left[i\int\frac{1}{2}\tilde{\phi}\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]\tilde{\phi}\right]\nonumber \\
 & = & \exp\left[\frac{i}{\hbar}\left(S\left[\phi_{\mathrm{cl.}}\right]+J\phi_{\mathrm{cl.}}\right)\right]\left(N\left(\det\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]\right)^{-1/2}+\mathcal{O}\left(\hbar\right)\right),
\end{eqnarray}

\end_inset

Note that while 
\begin_inset Formula $V\left(\phi_{\mathrm{cl.}},\tilde{\phi}\right)=\mathcal{O}\left(\hbar^{1/2}\right)$
\end_inset

, 
\begin_inset Formula $Z\left[J\right]$
\end_inset

 necessarily involves an even number of cubic vertices so the leading correction
 is 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

.
 The constant 
\begin_inset Formula $N$
\end_inset

 is determined by 
\begin_inset Formula $Z\left[0\right]=1$
\end_inset

.
 Also note that 
\begin_inset Formula $\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]$
\end_inset

 depends on 
\begin_inset Formula $\phi_{\mathrm{cl.}}$
\end_inset

.
 Thus
\begin_inset Foot
status open

\begin_layout Plain Layout
Recall the identity 
\begin_inset Formula $\ln\det M=\mathrm{Tr}\ln M$
\end_inset

.
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
W\left[J\right]=S\left[\phi_{\mathrm{cl.}}\right]+J\phi_{\mathrm{cl.}}+\frac{i\hbar}{2}\left(\mathrm{Tr}\ln\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]-\mathrm{Tr}\ln\Delta_{0}^{-1}\left[0\right]\right)+\mathcal{O}\left(\hbar^{2}\right).
\end{equation}

\end_inset

To perform the Legendre transform one needs to relate 
\begin_inset Formula $\phi_{\mathrm{cl.}}$
\end_inset

 to 
\begin_inset Formula $\varphi$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\varphi=\frac{\delta W}{\delta J}=\frac{\delta\phi_{\mathrm{cl.}}}{\delta J}\frac{\delta S}{\delta\phi_{\mathrm{cl.}}}+\phi_{\mathrm{cl.}}+J\frac{\delta\phi_{\mathrm{cl.}}}{\delta J}+\frac{\delta\phi_{\mathrm{cl.}}}{\delta J}\frac{\delta}{\delta\phi_{\mathrm{cl.}}}\frac{i\hbar}{2}\mathrm{Tr}\ln\Delta_{0}^{-1}\left[\phi_{\mathrm{cl.}}\right]+\mathcal{O}\left(\hbar^{2}\right)=\phi_{\mathrm{cl.}}+\mathcal{O}\left(\hbar\right).
\end{equation}

\end_inset

Using 
\begin_inset Formula $\phi_{\mathrm{cl}.}=\varphi+\mathcal{O}\left(\hbar\right)$
\end_inset

 in 
\begin_inset Formula $S$
\end_inset

 gives
\begin_inset Formula 
\begin{equation}
S\left[\phi_{\mathrm{cl.}}\right]+J\phi_{\mathrm{cl.}}=S\left[\varphi\right]+J\varphi+\mathcal{O}\left(\hbar^{2}\right),
\end{equation}

\end_inset

because the classical equation of motion for 
\begin_inset Formula $\phi_{\mathrm{cl.}}$
\end_inset

 implies the cancellation of the 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

 term, so that
\begin_inset Formula 
\begin{equation}
\Gamma\left[\varphi\right]=S\left[\varphi\right]+\frac{i\hbar}{2}\left(\mathrm{Tr}\ln\Delta_{0}^{-1}\left[\varphi\right]-\mathrm{Tr}\ln\Delta_{0}^{-1}\left[0\right]\right)+\mathcal{O}\left(\hbar^{2}\right).
\end{equation}

\end_inset

The 
\begin_inset Formula $\Delta_{0}^{-1}\left[0\right]$
\end_inset

 term is just a shift by an overall constant which is often neglected, leading
 one to write
\begin_inset Formula 
\begin{equation}
\Gamma\left[\varphi\right]=S\left[\varphi\right]+\frac{i\hbar}{2}\mathrm{Tr}\ln\Delta_{0}^{-1}\left[\varphi\right]+\mathcal{O}\left(\hbar^{2}\right).\label{eq:1piea}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
There is a nice topological property of Feynman diagrams which makes the
 organisation of the semi-classical expansion at higher orders easy.
 A diagram contributing to 
\begin_inset Formula $\Gamma$
\end_inset

 has an overall factor of 
\begin_inset Formula $\hbar$
\end_inset

 coming from the definition of 
\begin_inset Formula $W\left[J\right]$
\end_inset

, another 
\begin_inset Formula $\hbar$
\end_inset

 for each 
\begin_inset Formula $i\hbar\Delta_{0}$
\end_inset

 propagator appearing in the diagram, and a factor of 
\begin_inset Formula $\hbar^{-1}$
\end_inset

 for each vertex.
 Thus the total power of 
\begin_inset Formula $\hbar$
\end_inset

 for a diagram is 
\begin_inset Formula $I-V+1$
\end_inset

 where 
\begin_inset Formula $I$
\end_inset

 is the number of lines and 
\begin_inset Formula $V$
\end_inset

 the number of vertices.
 However, the Euler characteristic 
\begin_inset Formula $L-I+V=1$
\end_inset

, where 
\begin_inset Formula $L$
\end_inset

 is the number of faces or 
\begin_inset Quotes eld
\end_inset

loops
\begin_inset Quotes erd
\end_inset

 in the graph, is a topological invariant for all planar graphs.
 Thus, for planar graphs the contribution to 
\begin_inset Formula $\Gamma$
\end_inset

 is 
\begin_inset Formula $\mathcal{O}\left(\hbar^{L}\right)$
\end_inset

.
 The definition of 
\begin_inset Formula $L$
\end_inset

 can be extended from planar graphs to all graphs by using the same formula.
 Thus, in the semi-classical expansion, powers of 
\begin_inset Formula $\hbar$
\end_inset

 count the number of loops in a graph.
 This agrees with 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1piea"

\end_inset

: the 
\begin_inset Formula $\mathcal{O}\left(\hbar^{0}\right)$
\end_inset

 term is just the classical action and the 
\begin_inset Formula $\mathcal{O}\left(\hbar^{1}\right)$
\end_inset

 term is the sum of all one loop graphs, as can be seen explicitly by expanding
 the logarithm in a Taylor series in 
\begin_inset Formula $V_{\mathrm{int}}$
\end_inset

.
 The terms not shown above are the 1PI diagrams with two or more loops.
\end_layout

\begin_layout Section
The 2PI effective action
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:The-2PI-effective"

\end_inset


\end_layout

\begin_layout Standard
Two-particle irreducible effective actions (2PIEA) and approximation schemes
 based on them are useful when it is necessary to go beyond the perturbative
 field theory embodied by the 1PIEA, with applications to thermal and non-equili
brium plasmas/fluids, strongly coupled quantum field theories and systems
 dominated by many-body collective effects.
 The technique was originally developed by Lee and Yang 
\begin_inset CommandInset citation
LatexCommand citep
key "Lee1960"

\end_inset

, Luttinger and Ward 
\begin_inset CommandInset citation
LatexCommand citep
key "Luttinger1960"

\end_inset

, Baym 
\begin_inset CommandInset citation
LatexCommand citep
key "Baym1962"

\end_inset

 and others in the context of many-body theory, then extended by Cornwall,
 Jackiw and Tomboulis 
\begin_inset CommandInset citation
LatexCommand citep
key "Cornwall1974"

\end_inset

 to relativistic field theory in terms of functional integrals.
 Since then a broad literature has developed surrounding 2PI effective actions
 and their generalisations (see 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2004"

\end_inset

 for a good introductory review).
\end_layout

\begin_layout Standard
The essential idea behind the 2PIEA is to define a functional 
\begin_inset Formula $\Gamma\left[\varphi,\Delta\right]$
\end_inset

 of not only the mean field 
\begin_inset Formula $\varphi$
\end_inset

, but also the (connected) two point correlation function 
\begin_inset Formula $\Delta$
\end_inset

.
 By allowing 
\begin_inset Formula $\Delta$
\end_inset

 to vary it is possible to eliminate the diagrams yielding propagator correction
s.
 This is achieved by a double Legendre transform procedure, first defining
 a generalised partition functional
\begin_inset Formula 
\begin{equation}
Z\left[J,K\right]=\int\mathcal{D}\left[\phi\right]\exp\left[\frac{i}{\hbar}\left(S\left[\phi\right]+J\phi+\frac{1}{2}\phi K\phi\right)\right],
\end{equation}

\end_inset

where spacetime integrations are hidden for brevity, i.e.
 
\begin_inset Formula $\phi K\phi\equiv\int_{xy}\phi\left(x\right)K\left(x,y\right)\phi\left(y\right)$
\end_inset

 etc.
 Then 
\begin_inset Formula $W\left[J,K\right]=-i\hbar\ln Z\left[J,K\right]$
\end_inset

 as before and
\begin_inset Formula 
\begin{equation}
\Gamma\left[\varphi,\Delta\right]=W\left[J,K\right]-J\frac{\delta W\left[J,K\right]}{\delta J}-K\frac{\delta W\left[J,K\right]}{\delta K},
\end{equation}

\end_inset

where 
\begin_inset Formula $J$
\end_inset

 and 
\begin_inset Formula $K$
\end_inset

 are eliminated by solving
\begin_inset Formula 
\begin{eqnarray}
\frac{\delta W}{\delta J\left(x\right)} & = & \varphi\left(x\right),\\
\frac{\delta W}{\delta K\left(x,y\right)} & = & \frac{1}{2}\left(i\hbar\Delta\left(x,y\right)+\varphi\left(x\right)\varphi\left(y\right)\right).
\end{eqnarray}

\end_inset

The equations of motion obeyed by the 2PIEA are then
\begin_inset Formula 
\begin{eqnarray}
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\varphi} & = & -J-K\varphi,\label{eq:2pi-eom-varphi}\\
\frac{\delta\Gamma\left[\varphi,\Delta\right]}{\delta\Delta} & = & -\frac{1}{2}i\hbar K.\label{eq:2pi-eom-Delta}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
The easiest way to compute the double Legendre transform is stepwise, using
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:1piea"

\end_inset

 with 
\begin_inset Formula $S\to S+\frac{1}{2}\phi K\phi$
\end_inset

 and 
\begin_inset Formula $\Delta_{0}^{-1}\to\Delta_{0}^{-1}+K$
\end_inset

 as an intermediate step, then performing the transform with respect to
 
\begin_inset Formula $K$
\end_inset

.
 This leads to
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Gamma\left[\varphi,\Delta\right]=S\left[\varphi\right]+\frac{i\hbar}{2}\mathrm{Tr}\ln\Delta^{-1}+\frac{i\hbar}{2}\mathrm{Tr}\left(\Delta_{0}^{-1}\Delta-1\right)+\Gamma_{2}\left[\varphi,\Delta\right],\label{eq:2piea}
\end{equation}

\end_inset

where 
\begin_inset Formula $\Gamma_{2}\left[\varphi,\Delta\right]\sim\mathcal{O}\left(\hbar^{2}\right)$
\end_inset

 on using
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
K=\Delta^{-1}-\Delta_{0}^{-1}+\mathcal{O}\left(\hbar\right).
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The equation of motion for the propagator becomes
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\Delta^{-1}=\Delta_{0}^{-1}-\Sigma,\label{eq:2PI-Dyson-equation}
\end{equation}

\end_inset

where
\begin_inset Formula 
\begin{equation}
\Sigma=\frac{2i}{\hbar}\frac{\delta\Gamma_{2}\left[\varphi,\Delta\right]}{\delta\Delta},
\end{equation}

\end_inset

which one recognises as the Dyson equation and self-energy respectively.
 Since 
\begin_inset Formula $\Sigma$
\end_inset

 consists of 1PI graphs and the effect of 
\begin_inset Formula $\delta/\delta\Delta$
\end_inset

 is to remove a propagator from any graph in 
\begin_inset Formula $\Gamma_{2}$
\end_inset

 in all possible ways, 
\begin_inset Formula $\Gamma_{2}$
\end_inset

 must consist of 
\emph on
two particle irreducible
\emph default
 (2PI) graphs, i.e.
 those which do not fall apart under any cutting of two lines.
 Note that 
\begin_inset Formula $\Delta$
\end_inset

 depends on 
\begin_inset Formula $\hbar$
\end_inset

 through 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2PI-Dyson-equation"

\end_inset

, which must be accounted for when comparing 
\begin_inset Formula $\Gamma\left[\varphi,\Delta\left[\varphi\right]\right]$
\end_inset

 to the 1PIEA order by order in 
\begin_inset Formula $\hbar$
\end_inset

.
\end_layout

\begin_layout Standard
For reference it is useful to give the 2PIEA for a quartic scalar field
 theory, generalised to a set of fields 
\begin_inset Formula $\phi_{a}$
\end_inset

, 
\begin_inset Formula $a=1,\cdots,N$
\end_inset

, up to three loop order.
 No symmetry is yet imposed, though the study of 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 symmetric theories forms the bulk of this thesis.
 The classical action is given by
\begin_inset Formula 
\begin{equation}
S\left[\phi\right]=\frac{1}{2}\phi_{a}\left(-\square\delta_{ab}-m_{ab}^{2}\right)\phi_{b}-\frac{1}{3!}g_{abc}\phi_{a}\phi_{b}\phi_{c}-\frac{1}{4!}\lambda_{abcd}\phi_{a}\phi_{b}\phi_{c}\phi_{d},\label{eq:generic-scalar-action}
\end{equation}

\end_inset

where the linear term is dropped without loss of generality
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
If a linear term is present, it can be removed by shifting 
\begin_inset Formula $\phi_{a}$
\end_inset

.
 This results in a constant term which can also be dropped without consequence.
\end_layout

\end_inset

.
 The couplings 
\begin_inset Formula $m_{ab}^{2}$
\end_inset

, 
\begin_inset Formula $g_{abc}$
\end_inset

 and 
\begin_inset Formula $\lambda_{abcd}$
\end_inset

 can always be taken to be totally symmetric under permutations of indices
 since the 
\begin_inset Formula $\phi_{a}$
\end_inset

s commute.
 There are 
\begin_inset Formula $N\left(N+1\right)/2$
\end_inset

, 
\begin_inset Formula $N\left(N+1\right)\left(N+2\right)/3!$
\end_inset

 and 
\begin_inset Formula $N\left(N+1\right)\left(N+2\right)\left(N+3\right)/4!$
\end_inset

 linearly independent quadratic, cubic and quartic couplings respectively,
 though 
\begin_inset Formula $N\left(N-1\right)/2$
\end_inset

 rotations of the fields into each other can be used to remove some of these,
 e.g.
 by diagonalising 
\begin_inset Formula $m_{ab}^{2}$
\end_inset

.
 In the case of 
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset

 symmetry there are only two constants with 
\begin_inset Formula $m_{ab}^{2}=m^{2}\delta_{ab}$
\end_inset

, 
\begin_inset Formula $g_{abc}=0$
\end_inset

 and 
\begin_inset Formula $\lambda_{abcd}\propto\lambda\left(\delta_{ab}\delta_{cd}+\text{permutations}\right)$
\end_inset

.
 One has the bare propagator
\begin_inset Formula 
\begin{align}
\Delta_{0ab}^{-1}\left[\varphi\right]\left(x,y\right) & =\left.\frac{\delta^{2}S}{\delta\phi_{a}\left(x\right)\delta\phi_{b}\left(y\right)}\right|_{\phi=\varphi}\nonumber \\
 & =-\left(\square_{x}\delta_{ab}+m_{ab}^{2}+g_{abc}\varphi_{c}\left(x\right)+\frac{1}{2}\lambda_{abcd}\varphi_{c}\left(x\right)\varphi_{d}\left(x\right)\right)\delta\left(x-y\right),
\end{align}

\end_inset

and the effective interaction Lagrangian (i.e.
 the cubic and quartic part of 
\begin_inset Formula $S$
\end_inset

 after shifting the field 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
`{a}
\end_layout

\end_inset

 la 
\begin_inset Formula $\phi=\varphi+\hbar^{1/2}\tilde{\phi}$
\end_inset

)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
-\hbar V\left(\varphi,\tilde{\phi}\right)=-\frac{1}{3!}\hbar^{3/2}\left(g_{abc}+\lambda_{abcd}\varphi_{d}\right)\tilde{\phi}_{a}\tilde{\phi}_{b}\tilde{\phi}_{c}-\frac{1}{4!}\hbar^{2}\lambda_{abcd}\tilde{\phi}_{a}\tilde{\phi}_{b}\tilde{\phi}_{c}\tilde{\phi}_{d}.
\end{equation}

\end_inset

It is convenient to introduce the shorthands
\begin_inset Formula 
\begin{align}
V_{0abc} & \equiv\left.\frac{\delta^{3}S}{\delta\phi_{a}\delta\phi_{b}\delta\phi_{c}}\right|_{\phi=\varphi}=-\left(g_{abc}+\lambda_{abcd}\varphi_{d}\right),\\
W_{abcd} & \equiv\left.\frac{\delta^{4}S}{\delta\phi_{a}\delta\phi_{b}\delta\phi_{c}\delta\phi_{d}}\right|_{\phi=\varphi}=-\lambda_{abcd}.
\end{align}

\end_inset

(This matches the notation of 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015"

\end_inset

.) 
\begin_inset Formula $\Gamma_{2}\left[\varphi,\Delta\right]$
\end_inset

 is the sum of 2PI Feynman diagrams with interactions given by the above
 and propagators given by the variational propagator 
\begin_inset Formula $i\hbar\Delta_{ab}$
\end_inset

.
 Up to three loop order this is
\begin_inset Formula 
\begin{equation}
\Gamma_{2}=\Phi_{1}+\Phi_{2}+\Phi_{3}+\Phi_{4}+\Phi_{5}+\mathcal{O}\left(\hbar^{4}\right),\label{eq:2piea-diagram-terms}
\end{equation}

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
For easy copy'n'pasting:
\begin_inset Formula 
\begin{eqnarray*}
\Phi_{1} & = & -\frac{\hbar^{2}}{8}W_{abcd}\Delta_{ab}\Delta_{cd}\\
\Phi_{2} & = & \frac{\hbar^{2}}{12}V_{0abc}V_{0def}\Delta_{ad}\Delta_{be}\Delta_{cf}\\
\Phi_{3} & = & \frac{i\hbar^{3}}{4!}V_{0abc}V_{0def}V_{0ghi}V_{0jkl}\Delta_{ad}\Delta_{bg}\Delta_{cj}\Delta_{eh}\Delta_{fk}\Delta_{il}\\
\Phi_{4} & = & -\frac{i\hbar^{3}}{8}W_{abcd}V_{0efg}V_{0hij}\Delta_{ae}\Delta_{bf}\Delta_{ch}\Delta_{di}\Delta_{gj}\\
\Phi_{5} & = & \frac{i\hbar^{3}}{48}W_{abcd}W_{efgh}\Delta_{ae}\Delta_{bf}\Delta_{cg}\Delta_{dh}
\end{eqnarray*}

\end_inset


\end_layout

\end_inset

where
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{fmffile}{twopi-diagrams}
\end_layout

\begin_layout Plain Layout


\backslash
begin{eqnarray}
\end_layout

\begin_layout Plain Layout


\backslash
Phi_1 & = 
\backslash
parbox{12mm}{
\backslash
begin{fmfgraph}(40,40)
\end_layout

\begin_layout Plain Layout


\backslash
fmftop{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfbottom{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{phantom}{i,v,o}
\backslash
fmffreeze
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,tension=0.7}{v,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=90,tension=0.7}{v,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} & = -
\backslash
frac{
\backslash
hbar^{2}}{8}W_{abcd}
\backslash
Delta_{ab}
\backslash
Delta_{cd}, 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
Phi_2 & = 
\backslash
parbox{12mm}{
\backslash
begin{fmfgraph}(40,40)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{v1}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=1}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=1}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v1}
\backslash
fmfdot{v2}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} & = 
\backslash
frac{
\backslash
hbar^{2}}{12}V_{0abc}V_{0def}
\backslash
Delta_{ad}
\backslash
Delta_{be}
\backslash
Delta_{cf}, 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
Phi_3 & = 
\backslash
parbox{12mm}{
\backslash
begin{fmfgraph}(40,40)
\end_layout

\begin_layout Plain Layout


\backslash
fmfsurroundn{v}{12}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v4}
\backslash
fmfdot{v8}
\backslash
fmfdot{v12}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=.5}{v4,v8}
\backslash
fmf{plain,right=.5}{v8,v12}
\backslash
fmf{plain,right=.5}{v12,v4}
\backslash
fmffreeze
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{i,v4}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{i,v8}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{i,v12}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{i}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} & = 
\backslash
frac{i
\backslash
hbar^{3}}{4!}V_{0abc}V_{0def}V_{0ghi}V_{0jkl}
\backslash
Delta_{ad}
\backslash
Delta_{bg}
\backslash
Delta_{cj}
\backslash
Delta_{eh}
\backslash
Delta_{fk}
\backslash
Delta_{il},
\end_layout

\begin_layout Plain Layout


\backslash
end{eqnarray}
\end_layout

\begin_layout Plain Layout


\backslash
begin{eqnarray}
\end_layout

\begin_layout Plain Layout


\backslash
Phi_4 & = 
\backslash
parbox{12mm}{
\backslash
begin{fmfgraph}(40,40)
\end_layout

\begin_layout Plain Layout


\backslash
fmfsurroundn{v}{12}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v4}
\backslash
fmfdot{v7}
\backslash
fmfdot{v1}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=.5}{v4,v7}
\backslash
fmf{plain,right=.8}{v7,v1}
\backslash
fmf{plain,right=.5}{v1,v4}
\backslash
fmffreeze
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=0.2}{v4,v7}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=0.2}{v4,v1}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} & = -
\backslash
frac{i
\backslash
hbar^{3}}{8}W_{abcd}V_{0efg}V_{0hij}
\backslash
Delta_{ae}
\backslash
Delta_{bf}
\backslash
Delta_{ch}
\backslash
Delta_{di}
\backslash
Delta_{gj}, 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
Phi_5 & = 
\backslash
parbox{12mm}{
\backslash
begin{fmfgraph}(40,40)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{v1}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=1}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=1}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=0.5}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,right=0.5}{v1,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v1}
\backslash
fmfdot{v2}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} & = 
\backslash
frac{i
\backslash
hbar^{3}}{48}W_{abcd}W_{efgh}
\backslash
Delta_{ae}
\backslash
Delta_{bf}
\backslash
Delta_{cg}
\backslash
Delta_{dh}.
\end_layout

\begin_layout Plain Layout


\backslash
end{eqnarray}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

These diagrams are called 
\begin_inset Quotes eld
\end_inset

EIGHT,
\begin_inset Quotes erd
\end_inset

 
\begin_inset Quotes eld
\end_inset

EGG,
\begin_inset Quotes erd
\end_inset

 
\begin_inset Quotes eld
\end_inset

MERCEDES,
\begin_inset Quotes erd
\end_inset

 
\begin_inset Quotes eld
\end_inset

HAIR,
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Quotes eld
\end_inset

BBALL,
\begin_inset Quotes erd
\end_inset

 respectively, in the nomenclature of 
\begin_inset CommandInset citation
LatexCommand citep
key "Carrington2011,Carrington2012"

\end_inset

.
 Keeping only 
\begin_inset Formula $\Phi_{1}$
\end_inset

 is called the 
\emph on
Hartree-Fock
\emph default
 (HF) approximation, which has special properties due to the simple form
 of the self-energy, while keeping both 
\begin_inset Formula $\Phi_{1}$
\end_inset

 and 
\begin_inset Formula $\Phi_{2}$
\end_inset

 is called the 
\emph on
sunset
\emph default
 approximation.
 Note that at four loop order there are eleven new diagrams (including the
 first non-planar diagram), so the number of diagrams increases rapidly
 with order.
 However, the growth is much slower than for 1PI or connected diagrams and
 the 2PIEA gives a relatively compact formulation of the theory.
\end_layout

\begin_layout Standard
It is also convenient to give explicitly the equations of motion in the
 sunset approximation for later reference.
 They are:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
-\left(\square\delta_{ab}+m_{ab}^{2}\right)\varphi_{b} & = & \frac{1}{2}g_{abc}\varphi_{b}\varphi_{c}+\frac{1}{3!}\lambda_{abcd}\varphi_{b}\varphi_{c}\varphi_{d}\nonumber \\
 &  & +\frac{i\hbar}{2}\left(g_{abc}+\lambda_{abcd}\varphi_{d}\right)\Delta_{cb}+\frac{\hbar^{2}}{6}\lambda_{abcd}V_{0efg}\Delta_{be}\Delta_{cf}\Delta_{dg}\nonumber \\
 &  & -J_{a}-K_{ab}\varphi_{b}+\mathcal{O}\left(\hbar^{3}\right),\\
\Delta_{ab}^{-1} & = & \Delta_{0ab}^{-1}+\frac{i\hbar}{2}W_{abcd}\Delta_{cd}-\frac{i\hbar}{2}V_{0acd}V_{0bef}\Delta_{ce}\Delta_{df}\nonumber \\
 &  & +K_{ab}+\mathcal{O}\left(\hbar^{2}\right).
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
Notice the following phenomenon: a loopwise truncation of the 2PIEA is not
 a loopwise truncation of the equations of motion.
 That is, starting from the action to order 
\begin_inset Formula $\mathcal{O}\left(\hbar^{2}\right)$
\end_inset

, one obtains the equation of motion for 
\begin_inset Formula $\varphi$
\end_inset

 to 
\begin_inset Formula $\mathcal{O}\left(\hbar^{2}\right)$
\end_inset

 but the equation of motion for 
\begin_inset Formula $\Delta$
\end_inset

 to only 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

.
 This is because taking a derivative with respect to 
\begin_inset Formula $\Delta$
\end_inset

 removes a propagator and therefore opens up a loop.
 This behaviour holds also in higher 
\begin_inset Formula $n$
\end_inset

PIEAs, and in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap4"

\end_inset

 some of the consequences of this for physics will be discussed.
\end_layout

\begin_layout Section
Higher 
\begin_inset Formula $n$
\end_inset

PI effective actions
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Higher-PI-effective"

\end_inset


\end_layout

\begin_layout Standard
The construction for 2PIEA via a two-fold Legendre transform can be immediately
 generalised to arbitrary 
\begin_inset Formula $n$
\end_inset

-fold Legendre transforms giving the 
\begin_inset Formula $n$
\end_inset

PIEA.
 Apparently de Dominicis and Martin 
\begin_inset CommandInset citation
LatexCommand citep
key "DeDominicis1964"

\end_inset

 were the first to realise this and study 
\begin_inset Formula $n\geq3$
\end_inset

.
 They were mentioned, but not used, in the pioneering 2PIEA paper 
\begin_inset CommandInset citation
LatexCommand citep
key "Cornwall1974"

\end_inset

 and developed further in 
\begin_inset CommandInset citation
LatexCommand citep
key "Norton1975"

\end_inset

.
 More early work on higher effective actions was carried on by Vasiliev
 
\begin_inset CommandInset citation
LatexCommand citep
key "Vasiliev1998"

\end_inset

, whose book was unfortunately not available in English for more than twenty
 years, though there were some reviews suggesting that the English literature
 was at least aware of these developments (e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Kleinert1982,Haymaker1991"

\end_inset

).
 The recent resurgence of interest in higher 
\begin_inset Formula $n$
\end_inset

PIEAs has largely been driven by their advantages for non-equilibrium problems
 and can likely be credited to the reviews by Berges 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2004,Berges2015"

\end_inset

 and advances in computer power.
 Explicit expressions are given here for the three loop 3PIEA.
 Higher order 
\begin_inset Formula $n$
\end_inset

PIEA can be found in the literature (e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2004,Berges2004c,Carrington2011,Carrington2012,Carrington2013,Carrington2013b"

\end_inset

).
\end_layout

\begin_layout Standard
The definition of the 
\begin_inset Formula $n$
\end_inset

PIEA is conceptually simple enough, having seen now the 1PIEA and 2PIEA
 cases
\begin_inset Foot
status open

\begin_layout Plain Layout
Note however that the 
\begin_inset Formula $n$
\end_inset

PIEA is defined by the Legendre transform, 
\emph on
not
\emph default
 by any irreducibility property of the Feynman graphs, though for low enough
 loop orders the graphs are irreducible as the name implies.
 At high enough loop order for 
\begin_inset Formula $n>2$
\end_inset

 the name becomes misleading.
 For example, the five-loop 5PIEA contains graphs that are not five-particle
 irreducible 
\begin_inset CommandInset citation
LatexCommand citep
key "Carrington2012"

\end_inset

!)
\end_layout

\end_inset

.
 One defines a generalised partition function
\begin_inset Formula 
\begin{align}
Z^{\left(n\right)}\left[J,K^{\left(2\right)},\cdots,K^{\left(n\right)}\right] & =\int\mathcal{D}\left[\phi\right]\exp\left[\frac{i}{\hbar}\left(S\left[\phi\right]+J_{x}\phi_{x}\right.\right.\nonumber \\
 & \left.\left.+\frac{1}{2}\phi_{x}K_{xy}^{\left(2\right)}\phi_{y}+\cdots+\frac{1}{n!}K_{x_{1}\cdots x_{n}}^{\left(n\right)}\phi_{x_{1}}\cdots\phi_{x_{n}}\right)\right],
\end{align}

\end_inset

and generating function 
\begin_inset Formula $W^{\left(n\right)}=-i\hbar\ln Z^{\left(n\right)}$
\end_inset

, and perform the multiple Legendre transform
\begin_inset Formula 
\begin{equation}
\Gamma^{\left(n\right)}\left[\varphi,\Delta,V,\cdots,V^{\left(n\right)}\right]=W^{\left(n\right)}-J\frac{\delta W^{\left(n\right)}}{\delta J}-K^{\left(2\right)}\frac{\delta W^{\left(n\right)}}{\delta K^{\left(2\right)}}-\cdots-K^{\left(n\right)}\frac{\delta W^{\left(n\right)}}{\delta K^{\left(n\right)}},\label{eq:nPIEA-legendre-transform-def}
\end{equation}

\end_inset

where the sources 
\begin_inset Formula $J$
\end_inset

, 
\begin_inset Formula $K^{\left(2\right)}$
\end_inset

 through 
\begin_inset Formula $K^{\left(n\right)}$
\end_inset

 are eliminated in terms of 
\begin_inset Formula $\varphi$
\end_inset

, 
\begin_inset Formula $\Delta$
\end_inset

 and the proper (1PI) three- through 
\begin_inset Formula $n$
\end_inset

-point vertex functions 
\begin_inset Formula $V$
\end_inset

 through 
\begin_inset Formula $V^{\left(n\right)}$
\end_inset

.
 The relation of the 
\begin_inset Formula $V$
\end_inset

s to the correlation functions can be worked out using the skeleton expansion
 so that, e.g.,
\begin_inset Formula 
\begin{align}
\hbar^{2}\Delta_{ad}\Delta_{be}\Delta_{cf}V_{def} & =\left\langle \mathrm{T}\left[\phi_{a}\phi_{b}\phi_{c}\right]\right\rangle -\left\langle \mathrm{T}\left[\phi_{a}\phi_{b}\right]\right\rangle \left\langle \phi_{c}\right\rangle \nonumber \\
 & -\left\langle \mathrm{T}\left[\phi_{c}\phi_{a}\right]\right\rangle \left\langle \phi_{b}\right\rangle -\left\langle \mathrm{T}\left[\phi_{b}\phi_{c}\right]\right\rangle \left\langle \phi_{a}\right\rangle \nonumber \\
 & +2\left\langle \phi_{a}\right\rangle \left\langle \phi_{b}\right\rangle \left\langle \phi_{c}\right\rangle ,
\end{align}

\end_inset

which can be connected to 
\begin_inset Formula $W^{\left(n\right)}$
\end_inset

 using
\begin_inset Formula 
\begin{align}
3!\frac{\delta W^{\left(n\right)}}{\delta K_{abc}^{\left(3\right)}} & =\left\langle \mathrm{T}\left[\phi_{a}\phi_{b}\phi_{c}\right]\right\rangle \nonumber \\
 & =\hbar^{2}\Delta_{ad}\Delta_{be}\Delta_{cf}V_{def}+i\hbar\Delta_{ab}\varphi_{c}\nonumber \\
 & +i\hbar\Delta_{ca}\varphi_{b}+i\hbar\Delta_{bc}\varphi_{a}+\varphi_{a}\varphi_{b}\varphi_{c},
\end{align}

\end_inset

and similar results for the higher order 
\begin_inset Formula $V$
\end_inset

s.
 The 
\begin_inset Formula $n$
\end_inset

PI equation of motion is then, in the absence of sources,
\begin_inset Formula 
\begin{equation}
\frac{\delta\Gamma^{\left(n\right)}}{\delta\varphi}=\frac{\delta\Gamma^{\left(n\right)}}{\delta\Delta}=\frac{\delta\Gamma^{\left(n\right)}}{\delta V}=\cdots\frac{\delta\Gamma^{\left(n\right)}}{\delta V^{\left(n\right)}}=0.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
As before, the 3PIEA can be computed using the previous result for the 2PIEA
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2piea"

\end_inset

, with 
\begin_inset Formula $S\to S+\frac{1}{3!}K_{abc}^{\left(3\right)}\phi_{a}\phi_{b}\phi_{c}$
\end_inset

 and 
\begin_inset Formula $\Delta_{0ab}^{-1}\to\Delta_{0ab}^{-1}+K_{abc}^{\left(3\right)}\varphi_{c}$
\end_inset

which results in a shifted vertex 
\begin_inset Formula $\tilde{V}=V_{0}+K^{\left(3\right)}$
\end_inset

 in 
\begin_inset Formula $\Gamma_{2}$
\end_inset

, then performing the Legendre transform with respect to 
\begin_inset Formula $K^{\left(3\right)}$
\end_inset

.
 One gets, on cancelling the disconnected contributions,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right] & =S\left[\varphi\right]+\frac{i\hbar}{2}\mathrm{Tr}\ln\Delta^{-1}+\frac{i\hbar}{2}\mathrm{Tr}\left(\Delta_{0}^{-1}\Delta-1\right)+\Gamma_{2}\left[\varphi,\Delta,\tilde{V}\right]\nonumber \\
 & -K_{abc}^{\left(3\right)}\frac{1}{3!}\hbar^{2}\Delta_{ad}\Delta_{be}\Delta_{cf}V_{def},
\end{align}

\end_inset

which can be further simplified using the fact that, by definition of the
 Legendre transform, the right hand side is stationary under variations
 of 
\begin_inset Formula $K^{\left(3\right)}$
\end_inset

 at fixed 
\begin_inset Formula $V$
\end_inset

.
 This gives to order 
\begin_inset Formula $\mathcal{O}\left(\hbar^{4}\right)$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\delta\Gamma_{2}}{\delta K_{abc}^{\left(3\right)}} & =\frac{\delta\left(\Phi_{2}+\Phi_{3}+\Phi_{4}\right)}{\delta\tilde{V}_{abc}}\nonumber \\
 & =\frac{\hbar^{2}}{3!}\tilde{V}_{def}\Delta_{ad}\Delta_{be}\Delta_{cf}+\frac{i\hbar^{3}}{3!}\tilde{V}_{def}\tilde{V}_{ghi}\tilde{V}_{jkl}\Delta_{ad}\Delta_{bg}\Delta_{cj}\Delta_{eh}\Delta_{fk}\Delta_{il}\nonumber \\
 & -\frac{i\hbar^{3}}{4}W_{hijd}\tilde{V}_{efg}\left(\frac{1}{3}\Delta_{ah}\Delta_{bi}\Delta_{ej}\Delta_{df}\Delta_{cg}+\text{cyclic permutations }\left(a\to b\to c\to a\right)\right)\nonumber \\
 & =\frac{1}{3!}\hbar^{2}\Delta_{ad}\Delta_{be}\Delta_{cf}V_{def},
\end{align}

\end_inset

where the second line comes from the definitions of the diagrams and the
 third line is from the source term in 
\begin_inset Formula $\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right]$
\end_inset

.
 Note that the permutations of the 
\begin_inset Formula $W\tilde{V}\Delta^{5}$
\end_inset

 term enter because 
\begin_inset Formula $K^{\left(3\right)}$
\end_inset

 is symmetric in its indices.
 The 
\begin_inset Formula $1/3$
\end_inset

 compensates for the would-be over-counting.
 This shows that 
\begin_inset Formula $\tilde{V}=V+\mathcal{O}\left(\hbar\right)$
\end_inset

, so that, up to higher order terms which are not written, the 
\begin_inset Formula $\tilde{V}$
\end_inset

s in the 
\begin_inset Formula $\mathcal{O}\left(\hbar^{3}\right)$
\end_inset

 terms can be replaced by 
\begin_inset Formula $V$
\end_inset

s.
 Then one has
\begin_inset Formula 
\begin{align}
\tilde{V}_{xyz} & =V_{0xyz}+K_{xyz}^{\left(3\right)}\nonumber \\
 & =V_{xyz}-i\hbar V_{xef}V_{yhi}V_{zkl}\Delta_{eh}\Delta_{fk}\Delta_{il}\nonumber \\
 & +\frac{i\hbar}{2}\left(W_{xyjd}V_{efz}\Delta_{ej}\Delta_{df}+\text{cyclic permutations }\left(x\to y\to z\to x\right)\right)\nonumber \\
 & +\mathcal{O}\left(\hbar^{2}\right).
\end{align}

\end_inset

Again, the stationarity of the Legendre transform under variations of 
\begin_inset Formula $K^{\left(3\right)}$
\end_inset

 at fixed 
\begin_inset Formula $V$
\end_inset

 implies that the 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

 correction to 
\begin_inset Formula $K^{\left(3\right)}$
\end_inset

 only contributes to 
\begin_inset Formula $\Gamma$
\end_inset

 at 
\begin_inset Formula $\mathcal{O}\left(\hbar^{4}\right)$
\end_inset

 (one order higher than the naive 
\begin_inset Formula $\mathcal{O}\left(\hbar^{3}\right)$
\end_inset

).
 Thus, to three loop order one only needs that 
\begin_inset Formula $K^{\left(3\right)}=V-V_{0}+\mathcal{O}\left(\hbar\right)$
\end_inset

.
 Substituting this into the expression for 
\begin_inset Formula $\Gamma$
\end_inset

 gives, finally
\begin_inset Formula 
\begin{equation}
\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right]=S\left[\varphi\right]+\frac{i\hbar}{2}\mathrm{Tr}\ln\Delta^{-1}+\frac{i\hbar}{2}\mathrm{Tr}\left(\Delta_{0}^{-1}\Delta-1\right)+\Gamma_{3}\left[\varphi,\Delta,V\right],\label{eq:3piea}
\end{equation}

\end_inset

where
\begin_inset Formula 
\begin{equation}
\Gamma_{3}=\Phi_{1}-\Phi_{2}+\frac{\hbar^{2}}{3!}V_{0abc}\Delta_{ad}\Delta_{be}\Delta_{cf}V_{def}+\Phi_{3}+\Phi_{4}+\Phi_{5}+\mathcal{O}\left(\hbar^{4}\right),
\end{equation}

\end_inset

where the three point vertices in the 
\begin_inset Formula $\Phi$
\end_inset

s are now the variational function 
\begin_inset Formula $V$
\end_inset

.
 From 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:nPIEA-legendre-transform-def"

\end_inset

 the 3PI equations of motion are
\begin_inset Formula 
\begin{eqnarray}
\frac{\delta\Gamma^{\left(3\right)}}{\delta\varphi_{a}} & = & -J_{a}-K_{ab}^{\left(2\right)}\varphi_{b}-\frac{1}{2}K_{abc}^{\left(3\right)}\left(i\hbar\Delta_{bc}+\varphi_{b}\varphi_{c}\right),\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{ab}} & = & -\frac{i\hbar}{2}K_{ab}^{\left(2\right)}-\frac{i\hbar}{2}K_{abc}^{\left(3\right)}\varphi_{c}\nonumber \\
 &  & -\frac{\hbar^{2}}{3!2}\left(K_{acd}^{\left(3\right)}\Delta_{ce}\Delta_{df}V_{bef}+K_{bcd}^{\left(3\right)}\Delta_{ce}\Delta_{df}V_{aef}\right)\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta V_{abc}} & = & -\frac{1}{3!}\hbar^{2}\Delta_{ad}\Delta_{be}\Delta_{cf}K_{def}^{\left(3\right)},
\end{eqnarray}

\end_inset

while from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:3piea"

\end_inset

 the left hand sides are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\frac{\delta\Gamma^{\left(3\right)}}{\delta\varphi_{a}} & = & \frac{\delta S}{\delta\varphi_{a}}-\frac{i\hbar}{2}\left(g_{abc}+\lambda_{abcd}\varphi_{d}\right)\Delta_{cb}-\frac{\hbar^{2}}{3!}\lambda_{abcd}\Delta_{be}\Delta_{cf}\Delta_{dg}V_{efg}+\mathcal{O}\left(\hbar^{3}\right),\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta\Delta_{ab}} & = & -\frac{i\hbar}{2}\left(\Delta_{ab}^{-1}+\mathrm{Tr}\Delta_{0ab}^{-1}-\Sigma_{ab}\right)+\mathcal{O}\left(\hbar^{4}\right),\\
\frac{\delta\Gamma^{\left(3\right)}}{\delta V_{abc}} & = & \frac{\hbar^{2}}{3!}\left(V_{0def}-V_{def}\right)\Delta_{ad}\Delta_{be}\Delta_{cf}+\frac{i\hbar^{3}}{3!}V_{def}V_{ghi}V_{jkl}\Delta_{ad}\Delta_{bg}\Delta_{cj}\Delta_{eh}\Delta_{fk}\Delta_{il}\nonumber \\
 &  & -\frac{i\hbar^{3}}{4}W_{hijd}V_{efg}\left(\frac{1}{3}\Delta_{ah}\Delta_{bi}\Delta_{ej}\Delta_{df}\Delta_{cg}+\text{cyclic permutations }\left(a\to b\to c\to a\right)\right)\nonumber \\
 &  & +\mathcal{O}\left(\hbar^{4}\right),
\end{eqnarray}

\end_inset

where now 
\begin_inset Formula 
\begin{equation}
\Sigma_{ab}=\frac{2i}{\hbar}\frac{\delta\Gamma_{3}}{\delta\Delta_{ab}}.
\end{equation}

\end_inset

Note that, as in the 2PI case, there are 
\begin_inset Formula $\mathcal{O}\left(\hbar^{2}\right)$
\end_inset

 corrections to the 
\begin_inset Formula $\varphi$
\end_inset

 and 
\begin_inset Formula $\Delta$
\end_inset

 equations of motion, but only 
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset

 corrections to 
\begin_inset Formula $V$
\end_inset

.
 Thus a loopwise truncation of 
\begin_inset Formula $\Gamma^{\left(3\right)}$
\end_inset

 is not a loopwise truncation of the corresponding equations of motion.
\end_layout

\begin_layout Standard
Note that if one truncates 
\begin_inset Formula $\Gamma^{\left(3\right)}$
\end_inset

 to two loop order and sets 
\begin_inset Formula $K^{\left(3\right)}=0$
\end_inset

, then the 
\begin_inset Formula $V$
\end_inset

 equation of motion becomes simply 
\begin_inset Formula $V=V_{0}$
\end_inset

.
 Substituting this back in 
\begin_inset Formula $\Gamma^{\left(3\right)}$
\end_inset

 gives the 2PIEA to two loop order.
 That is
\begin_inset Formula 
\begin{equation}
\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right]=\Gamma^{\left(2\right)}\left[\varphi,\Delta\right],\ \text{at two loops}.
\end{equation}

\end_inset

This is a specific example of a general equivalence hierarchy
\begin_inset Formula 
\begin{alignat*}{2}
\Gamma^{\left(1\right)}\left[\varphi\right] & =\Gamma^{\left(2\right)}\left[\varphi,\Delta\right]=\cdots & \text{at one loop},\\
\Gamma^{\left(1\right)}\left[\varphi\right] & \neq\Gamma^{\left(2\right)}\left[\varphi,\Delta\right]=\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right]=\cdots & \text{at two loops},\\
\Gamma^{\left(1\right)}\left[\varphi\right] & \neq\Gamma^{\left(2\right)}\left[\varphi,\Delta\right]\neq\Gamma^{\left(3\right)}\left[\varphi,\Delta,V\right]=\Gamma^{\left(4\right)}\left[\varphi,\Delta,V,V^{\left(4\right)}\right]=\cdots & \text{ at three loops},\\
 & \vdots & \vdots
\end{alignat*}

\end_inset

where extra arguments are evaluated at the solution of their source-free
 equations of motion when comparisons are made.
 This implies that if one is willing to work at 
\begin_inset Formula $L$
\end_inset

 loop order, there is no profit in using an 
\begin_inset Formula $n$
\end_inset

PIEA with 
\begin_inset Formula $n>L$
\end_inset

.
 However, the 
\begin_inset Formula $L$
\end_inset

-loop 
\begin_inset Formula $L$
\end_inset

PIEA is 
\emph on
self-consistently complete
\emph default
 in the sense that all Feynman diagram topologies which are possible to
 resum using 
\begin_inset Formula $\leq L$
\end_inset

 loop diagrams as skeletons are being resummed fully self-consistently.
 Not all diagrams at 
\begin_inset Formula $>L$
\end_inset

 loops are included, however.
 For example, taking the 2PI equations of motion at two loop order and solving
 them by iteration, one finds that not all three loop diagrams are present.
 This is because at three loops there are primitive vertex correction diagrams
 which cannot be represented by iterated propagator corrections.
 Thus there is profit in increasing 
\begin_inset Formula $n$
\end_inset

 as long as 
\begin_inset Formula $n<L$
\end_inset

.
 Finally, it will be shown in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap4"

\end_inset

 that symmetry improvement breaks the equivalence hierarchy, e.g.
 the symmetry improved 3PIEA 
\emph on
does not
\emph default
 reduce to the symmetry improved 2PIEA at the two loop level.
\end_layout

\begin_layout Section
Analytic properties of the 2PI effective action and resummation schemes
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Analytic-properties-of-2PIEA-and-resummation"

\end_inset


\end_layout

\begin_layout Subsection
2PIEA as a resummation scheme
\end_layout

\begin_layout Standard
One of the advantages of 
\begin_inset Formula $n$
\end_inset

PIEA (
\begin_inset Formula $n>1$
\end_inset

) is the summation of perturbation theory effected by the self-consistent
 equations of motion.
 This can be illustrated simply with the example of the 2PIEA in the Hartree-Foc
k approximation.
 The propagator equation of motion reads
\begin_inset Formula 
\begin{equation}
\Delta_{ab}^{-1}\left(x,y\right)=-\left(\square_{x}\delta_{ab}+m_{ab}^{2}+g_{abc}\varphi_{c}\left(x\right)+\frac{1}{2}\lambda_{abcd}\varphi_{c}\left(x\right)\varphi_{d}\left(x\right)+\frac{i\hbar}{2}\lambda_{abcd}\Delta_{cd}\left(x,x\right)\right)\delta\left(x-y\right).\label{eq:2piea-hartree-fock}
\end{equation}

\end_inset

Consider for simplicity the case where 
\begin_inset Formula $\varphi_{a}$
\end_inset

 is independent of position and time.
 This is the case, for example, in thermal equilibrium or vacuum since by
 assumption spacetime is also homogeneous.
 Then this equation can be solved in the Fourier domain.
 Introducing
\begin_inset Formula 
\begin{equation}
\Delta_{ab}^{-1}\left(p\right)=\int_{x-y}\mathrm{e}^{ip\left(x-y\right)}\Delta_{ab}^{-1}\left(x,y\right),
\end{equation}

\end_inset

the equation of motion is
\begin_inset Formula 
\begin{equation}
\Delta_{ab}^{-1}\left(p\right)=-\left(-p^{2}\delta_{ab}+m_{ab}^{2}+g_{abc}\varphi_{c}+\frac{1}{2}\lambda_{abcd}\varphi_{c}\varphi_{d}+\frac{i\hbar}{2}\lambda_{abcd}\int_{k}\Delta_{cd}\left(k\right)\right).
\end{equation}

\end_inset

The important point is that 
\begin_inset Formula $\int_{k}\Delta_{cd}\left(k\right)$
\end_inset

 is actually independent of 
\begin_inset Formula $p$
\end_inset

.
 Thus one can write 
\begin_inset Formula $\Delta_{ab}^{-1}\left(p\right)=p^{2}\delta_{ab}-M_{ab}^{2}$
\end_inset

 where the effective mass matrix is given by the 
\emph on
gap equation
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
M_{ab}^{2}=m_{ab}^{2}+g_{abc}\varphi_{c}+\frac{1}{2}\lambda_{abcd}\varphi_{c}\varphi_{d}+\frac{i\hbar}{2}\lambda_{abcd}\int_{k}\left[k^{2}\delta_{cd}-M_{cd}^{2}\right]^{-1}.
\end{equation}

\end_inset

There is a basis of fields in which the matrix 
\begin_inset Formula $M_{ab}^{2}=M_{a}^{2}\delta_{ab}$
\end_inset

 (no sum) is diagonalised.
 In that basis
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
M_{a}^{2}=\mu_{a}^{2}+\frac{i\hbar}{2}\sum_{c}\lambda_{ac}\int_{k}\frac{1}{k^{2}-M_{c}^{2}},
\end{equation}

\end_inset

where 
\begin_inset Formula $\mu_{a}^{2}$
\end_inset

 are the eigenvalues of 
\begin_inset Formula $m_{ab}^{2}+g_{abc}\varphi_{c}+\frac{1}{2}\lambda_{abcd}\varphi_{c}\varphi_{d}$
\end_inset

 and 
\begin_inset Formula $\lambda_{ac}$
\end_inset

 are, for each 
\begin_inset Formula $c$
\end_inset

, the eigenvalues of the matrix 
\begin_inset Formula $\lambda_{abcc}$
\end_inset

 (no sum).
\end_layout

\begin_layout Standard
The integral is divergent in 
\begin_inset Formula $d\geq2$
\end_inset

 dimensions and must be regularised.
 Using dimensional regularisation in 
\begin_inset Formula $d=4-2\epsilon$
\end_inset

 dimensions,
\begin_inset Formula 
\begin{align}
\int_{k}\frac{i}{k^{2}-M_{c}^{2}} & =-\frac{M_{c}^{2}}{16\pi^{2}}\left[\frac{1}{\epsilon}-\gamma+1+\ln\left(4\pi\right)+\mathcal{O}\left(\epsilon\right)\right]\nonumber \\
 & +\frac{M_{c}^{2}}{16\pi^{2}}\ln\left(\frac{M_{c}^{2}}{\mu^{2}}\right)+\int_{\boldsymbol{k}}\frac{1}{\sqrt{\boldsymbol{k}^{2}+M_{c}^{2}}}\frac{1}{\mathrm{e}^{\beta\sqrt{\boldsymbol{k}^{2}+M_{c}^{2}}}-1},
\end{align}

\end_inset

where 
\begin_inset Formula $\mu$
\end_inset

 is the renormalisation point and 
\begin_inset Formula $\gamma\approx0.577$
\end_inset

 is the Euler-Mascheroni constant.
 The three terms are the 
\begin_inset Formula $\overline{\mathrm{MS}}$
\end_inset

 divergent contribution, the finite vacuum contribution and the finite temperatu
re Bose-Einstein contribution respectively.
 The renormalisation procedure (not carried out in full here; see section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Renormalisation-and-solution-2PI-HF"

\end_inset

 and appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"

\end_inset

 for details), effectively results in the removal of the infinite part.
 The ambiguity in the removal of a finite part is reflected in the freedom
 to choose 
\begin_inset Formula $\mu$
\end_inset

.
 For simplicity set the temperature to zero (the finite temperature contribution
 is not important for the point that follows).
 Then,
\begin_inset Formula 
\begin{equation}
M_{a}^{2}=\mu_{a}^{2}+\frac{\hbar}{2}\sum_{c}\lambda_{ac}\frac{M_{c}^{2}}{16\pi^{2}}\ln\left(\frac{M_{c}^{2}}{\mu^{2}}\right).
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Now contrast this with the perturbative, or 1PI, evaluation of the same
 quantity.
 In the 1PI self-energy only the bare 
\begin_inset Formula $\Delta_{0}$
\end_inset

 propagators appear, so the analogue of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2piea-hartree-fock"

\end_inset

 is identical except for the replacement 
\begin_inset Formula $\Delta\to\Delta_{0}$
\end_inset

 on the right hand side.
 This results eventually in the replacement 
\begin_inset Formula $M_{c}^{2}\to\mu_{c}^{2}$
\end_inset

 on the right hand side above, giving
\begin_inset Formula 
\begin{equation}
M_{a}^{2}\left(\mathrm{1PI}\right)=\mu_{a}^{2}+\frac{\hbar}{2}\sum_{c}\lambda_{ac}\frac{\mu_{c}^{2}}{16\pi^{2}}\ln\left(\frac{\mu_{c}^{2}}{\mu^{2}}\right),
\end{equation}

\end_inset

which is what is obtained by iterating the 2PI equation once and ignoring
 terms of order 
\begin_inset Formula $\mathcal{O}\left(\hbar^{2}\right)$
\end_inset

.
 The 2PI solution has higher order contributions, starting with
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
M_{a}^{2}=M_{a}^{2}\left(\mathrm{1PI}\right)+\frac{\hbar^{2}}{1024\pi^{4}}\sum_{cd}\left\{ \lambda_{ac}\lambda_{cd}\left[1+\ln\left(\frac{\mu_{c}^{2}}{\mu^{2}}\right)\right]\mu_{d}^{2}\ln\left(\frac{\mu_{d}^{2}}{\mu^{2}}\right)\right\} +\mathcal{O}\left(\hbar^{3}\right).
\end{equation}

\end_inset

This is a large correction if, parametrically, 
\begin_inset Formula $\lambda\ln\left(\mu_{a}^{2}/\mu^{2}\right)\sim16\pi^{2}$
\end_inset

.
 This can occur in theories with very strong couplings or a hierarchy of
 masses so that some 
\begin_inset Formula $\mu_{a}^{2}/\mu^{2}$
\end_inset

 is inevitably large.
 An important case is 
\begin_inset Formula $\mu_{a}^{2}=0$
\end_inset

.
 In this case the 1PI gap equation gives zero, but the 2PI gap equation
 gives a non-perturbative solution 
\begin_inset Formula $M_{a}^{2}\sim\mu^{2}\exp\left(32\pi^{2}/\hbar\lambda\right)$
\end_inset

 which may or may not be physical.
\end_layout

\begin_layout Standard
This can be seen diagrammatically as well.
 The Hartree-Fock gap equation is
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{hartree-fock-resummation}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation}
\end_layout

\begin_layout Plain Layout


\backslash
Delta^{-1} = 
\backslash
left( 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{i,o}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} 
\backslash
right)^{-1} = 
\backslash
left( 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes}{i,o}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} 
\backslash
right)^{-1} - 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{phantom}{i,v,o}
\backslash
fmffreeze
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,tension=0.5}{v,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}},
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

where the full propagator represents 
\begin_inset Formula $i\Delta$
\end_inset

 and the dashed propagator 
\begin_inset Formula $i\Delta_{0}$
\end_inset

.
 This can be rearranged into
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{hartree-fock-resummation2}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{i,o}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} = 
\backslash
quad 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes}{i,o}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} + 
\backslash
quad 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes}{i,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{v,o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,tension=0.5}{v,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}}.
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset

Iterating this equation gives
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{fmffile}{hartree-fock-resummation3}
\end_layout

\begin_layout Plain Layout


\backslash
begin{equation}
\end_layout

\begin_layout Plain Layout


\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{i,o}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} = 
\backslash
quad 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes}{i,o}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} + 
\backslash
quad 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,20)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes}{i,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{v,o}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes,tension=0.5}{v,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}} + 
\backslash
quad 
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,50)
\end_layout

\begin_layout Plain Layout


\backslash
fmfleft{i}
\end_layout

\begin_layout Plain Layout


\backslash
fmfright{o}
\end_layout

\begin_layout Plain Layout


\backslash
fmftop{t}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes}{i,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain}{v,o}
\backslash
fmffreeze
\end_layout

\begin_layout Plain Layout


\backslash
fmf{phantom}{t,v2,v}
\backslash
fmffreeze
\end_layout

\begin_layout Plain Layout


\backslash
fmf{dashes,left=1}{v,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,tension=0.5,left=1}{v2,t,v2}
\end_layout

\begin_layout Plain Layout


\backslash
fmf{plain,left=1}{v2,v}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v}
\end_layout

\begin_layout Plain Layout


\backslash
fmfdot{v2}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmfgraph}}.
\end_layout

\begin_layout Plain Layout


\backslash
end{equation}
\end_layout

\begin_layout Plain Layout


\backslash
end{fmffile}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The 1PI Dyson equation consists of the first two terms on the right hand
 side only.
 The additional term gives, upon repeated iteration, the sum of all 
\emph on
daisy
\emph default
 (or 
\emph on
ring
\emph default
) diagrams as well as all 
\emph on
superdaisy
\emph default
 (i.e., daisies within daisies in a fractal pattern) diagrams.
 The highly non-trivial fact about 2PI approximation schemes is that all
 of these diagrams appear with the correct combinatoric factors.
 (This no longer holds for all diagrams in 
\begin_inset Formula $n$
\end_inset

PIEA with 
\begin_inset Formula $n\geq3$
\end_inset

.) From the point of view of someone engineering 
\emph on
ad hoc
\emph default
 equations of motion to effect a resummation, this is a highly desirable
 property which would be difficult to prove, especially when diagrams with
 more complicated topologies are included.
 Furthermore, since perturbation theories are generically asymptotic expansions,
 resummation in general is a dangerous procedure.
 Mathematically speaking an asymptotic series can be summed to 
\emph on
any
\emph default
 value whatsoever.
 Additional non-perturbative properties are required of any resummation
 scheme that claims to faithfully represent the original theory.
 For these reasons, while 
\begin_inset Formula $n$
\end_inset

PIEA (
\begin_inset Formula $n>1$
\end_inset

) 
\emph on
do
\emph default
 effect resummations of perturbation theory, speaking of resummations reflects
 a limited understanding of why 
\begin_inset Formula $n$
\end_inset

PIEA actually work.
 This section examines this issue in the context of the 2PIEA for a toy
 model which is exactly solvable so that a deeper understanding of the analytic
 features that support 
\begin_inset Formula $n$
\end_inset

PIEAs in general and 
\begin_inset Formula $2$
\end_inset

PIEAs in particular can be found.
 These results are then contrasted with those for some other common resummation
 schemes used in the literature, particularly the Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 and Bore-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 methods.
 Much of this section is based on a publication by the author 
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015"

\end_inset

.
\end_layout

\begin_layout Standard
Unfortunately robust comparisons are difficult because the large order behaviour
 of perturbation theory is known, at best, in a sketchy form for most field
 theories of interest.
 The use of a genuinely trivial model 
\begin_inset Quotes eld
\end_inset

field theory
\begin_inset Quotes erd
\end_inset

 in zero spacetime dimensions (i.e.
 probability theory) allows for exact results to be obtained and removes
 all complications due to renormalisation, etc.
 This model nevertheless accurately represents the typical combinatoric
 structure of large order perturbation theory, at least in those cases where
 the behaviour is known in more interesting field theories.
 A 
\begin_inset Quotes eld
\end_inset

spectral function
\begin_inset Quotes erd
\end_inset

 representation of the Green function (similar to the one first introduced
 by Bender and Wu 
\begin_inset CommandInset citation
LatexCommand citep
key "Bender1971,Bender1978"

\end_inset

) is used to capture the non-analyticity of the solutions in the various
 methods.
\end_layout

\begin_layout Standard
The existence of the spectral representation is connected to the branch
 cut of physical amplitudes on the negative coupling (
\begin_inset Formula $\lambda$
\end_inset

) axis.
 This branch cut is due to the non-existence of the theory at negative couplings
: the path integral diverges due to a potential unbounded from below.
 In a higher dimensional field theory this has a simple physical interpretation:
 the vacuum is unstable and, after tunnelling through a barrier, the system
 rolls down the potential 
\begin_inset CommandInset citation
LatexCommand citep
key "Coleman1977"

\end_inset

.
 For weak coupling the semi-classical approximation is valid and the tunnelling
 is exponentially suppressed, giving an imaginary contribution 
\begin_inset Formula $\sim\exp\left(-1/\lambda\right)$
\end_inset

 to the vacuum persistence amplitude which is inherited by the Green function.
 This exponential behaviour can also be seen in the spectral function.
\end_layout

\begin_layout Standard
Dyson 
\begin_inset CommandInset citation
LatexCommand citep
key "Dyson1952"

\end_inset

 argued that a very similar phenomenon occurs in quantum electrodynamics
 (QED).
 In QED physical observables are calculated in a perturbation series of
 the form 
\begin_inset Formula $F\left(e^{2}\right)=a_{0}+a_{2}e^{2}+a_{4}e^{4}+\cdots$
\end_inset

 where 
\begin_inset Formula $e\sim0.3$
\end_inset

 is the charge of the electron.
 Now if one imagines a world where 
\begin_inset Formula $e^{2}<0$
\end_inset

, i.e.
 like charges attract, it is easy to see that the ordinary vacuum is unstable
 to the production of many electron-positron pairs which separate into clouds
 of like-charged particles.
 At weak coupling there is a large tunnelling barrier to overcome because
 one must pay for the rest mass of the pairs and separate them far enough
 for the wrong-sign Coulomb potential to compensate.
 Thus there is a finite but exponentially suppressed rate of vacuum decay.
 A Taylor series expansion in 
\begin_inset Formula $e^{2}$
\end_inset

 cannot capture this non-analyticity so the perturbation series must be
 divergent.
\end_layout

\begin_layout Standard
Similarly, the perturbation series in 
\begin_inset Formula $\lambda$
\end_inset

 diverges for the toy model considered here.
 Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants can represent physical quantities more accurately than Taylor
 series because they can develop isolated poles in the complex 
\begin_inset Formula $\lambda$
\end_inset

 plane, however they struggle to capture the strong coupling behaviour at
 any fixed order in the approximation.
 Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants are better able to capture the non-analyticities of the Borel
 transform, however, and the widely used combination Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants give a better global approximation.
 This occurs because the Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximated Borel transform has poles in the Borel plane, which lead to
 branch cuts when the Laplace transform is taken to return to physical variables.
 Similarly, the self-consistent 2PI approximations develop branch point
 non-analyticities and approximate the exact Green function rather well
 in the entire complex 
\begin_inset Formula $\lambda$
\end_inset

 plane already at the leading non-trivial truncation.
 However, the branch cuts in the 2PI case arise because the 2PI Green function
 obeys self-consistent equations of motion, and is connected to the existence
 of unphysical solution branches.
\end_layout

\begin_layout Standard
A question that naturally arises is: how do these methods compare? Both
 2PI and Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 methods have the ability to accurately represent non-analyticities of the
 exact theory and so out-perform other methods.
 However, a natural conjecture is that self-consistently derived equations
 of motion 
\begin_inset Quotes eld
\end_inset

know more
\begin_inset Quotes erd
\end_inset

 about the analytic structure of the underlying theory than do the generic
 Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants.
 This section tests the hypothesis that the 2PI methods should be more accurate
 than Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 and, indeed, one finds this to be the case, at least in certain regimes.
\end_layout

\begin_layout Standard
The theory discussed here, although admittedly a toy model, also has physical
 relevance.
 Beneke and Moch found this toy model as the theory governing the zero mode
 of scalar fields in Euclidean de Sitter space 
\begin_inset CommandInset citation
LatexCommand citep
key "Beneke2013"

\end_inset

.
 They performed an analysis very similar to this one, finding that a non-perturb
ative treatment is necessary and compared 2PI and (Borel-)Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 resummed approximations.
 However, they present this analysis very briefly as part of a larger discussion
 of scalar fields in de Sitter space.
 Further, their comparison of the 2PI and resummed techniques, while correct,
 is not very detailed.
 The analysis presented here is a detailed discussion of the interplay between
 2PI effective actions and various resummation techniques.
 Use of the spectral function to quantify the non-analyticities present
 in the Green function in aid of this comparison is, to the author's knowledge,
 a new aspect.
\end_layout

\begin_layout Subsection
A toy model and its exact solution
\end_layout

\begin_layout Standard
The toy model is a Euclidean quartic theory in zero dimensions, i.e.
 a probability theory for a single real variable 
\begin_inset Formula $q$
\end_inset

 given by the partition function 
\begin_inset Formula 
\begin{equation}
Z\left[K\right]=N\int_{-\infty}^{\infty}\mathrm{d}q\exp\left(-\frac{1}{2}m^{2}q^{2}-\frac{1}{4!}\lambda q^{4}-\frac{1}{2}Kq^{2}\right)
\end{equation}

\end_inset

with a source 
\begin_inset Formula $K$
\end_inset

 for the two point function.
 
\begin_inset Formula $N$
\end_inset

 is a normalisation factor chosen so that 
\begin_inset Formula $Z\left[0\right]=1$
\end_inset

.
 This theory has been discussed before in the context of exact and non-perturbat
ive methods in field theory (see, e.g., 
\begin_inset CommandInset citation
LatexCommand citep
key "Malbouisson1999,Crutchfield1979"

\end_inset

 and references therein) because, despite the absence of spacetime, the
 theory possesses a perturbative expansion in terms of Feynman diagrams
 with the same combinatorial structure as more realistic theories.
 It was also discussed in 
\begin_inset CommandInset citation
LatexCommand citep
key "Beneke2013"

\end_inset

 as an effective field theory for scalar field zero modes in Euclidean de
 Sitter space.
\end_layout

\begin_layout Standard
Attention is restricted to the 
\begin_inset Formula $m^{2}>0$
\end_inset

 theory since, though the 
\begin_inset Formula $m^{2}<0$
\end_inset

 theory exists and may be interesting for other purposes, it possesses no
 sensible weak coupling limit and in zero dimensions does not give a broken
 symmetry phase anyway.
 In the absence of symmetry breaking there is no need for a source term
 for 
\begin_inset Formula $q$
\end_inset

.
 The integral diverges for 
\begin_inset Formula $\mathrm{Re}\lambda<0$
\end_inset

, but can be defined for all complex 
\begin_inset Formula $\lambda$
\end_inset

 by analytic continuation.
 Then 
\begin_inset Formula $Z\left[K\right]$
\end_inset

 possesses a branch cut along the negative 
\begin_inset Formula $\lambda$
\end_inset

 axis.
 Physically, this signals the instability of the negative 
\begin_inset Formula $\lambda$
\end_inset

 vacuum due to tunnelling away from the local minimum at 
\begin_inset Formula $q\sim0$
\end_inset

 to 
\begin_inset Formula $q\sim\pm\sqrt{-6m^{2}/\lambda}$
\end_inset

 followed by rolling down the inverted quartic potential which is unbounded
 from below.
 The branch point at 
\begin_inset Formula $\lambda=0$
\end_inset

 means that the weak coupling perturbation series has zero radius of convergence
, agreeing with the general analysis of Dyson 
\begin_inset CommandInset citation
LatexCommand citep
key "Dyson1952"

\end_inset

.
\end_layout

\begin_layout Standard
Introducing the conveniently rescaled variables 
\begin_inset Formula $k=K/m^{2}$
\end_inset

 and 
\begin_inset Formula $\rho=3m^{4}/4\lambda$
\end_inset

, the integral can be performed giving 
\begin_inset Formula 
\begin{equation}
Z\left[K\right]=\frac{2N}{m}\sqrt{\rho\left(1+k\right)}\exp\left(\rho\left(1+k\right)^{2}\right)\mathrm{K}_{1/4}\left(\rho\left(1+k\right)^{2}\right),
\end{equation}

\end_inset

 where 
\begin_inset Formula $K_{1/4}\left(\cdots\right)$
\end_inset

 is a modified Bessel function of the second kind.
 This expression is valid so long as 
\begin_inset Formula 
\begin{equation}
\mathrm{Re}\left(\sqrt{\rho}\left(1+k\right)\right)>0,
\end{equation}

\end_inset

 which extends the definition to the entirety of the cut 
\begin_inset Formula $\lambda$
\end_inset

-plane.
 The normalisation factor is 
\begin_inset Formula 
\begin{equation}
N=\frac{m}{2}\frac{\exp\left(-\rho\right)}{\sqrt{\rho}\mathrm{K}_{1/4}\left(\rho\right)},
\end{equation}

\end_inset

 so finally 
\begin_inset Formula 
\begin{equation}
Z\left[K\right]=\sqrt{1+k}\exp\left(\rho\left[\left(1+k\right)^{2}-1\right]\right)\frac{\mathrm{K}_{1/4}\left(\rho\left(1+k\right)^{2}\right)}{\mathrm{K}_{1/4}\left(\rho\right)}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Also define the generating function 
\begin_inset Formula 
\begin{equation}
W\left[K\right]=-\ln Z\left[K\right]
\end{equation}

\end_inset

 and note that connected correlations are found by taking derivatives of
 
\begin_inset Formula $W$
\end_inset

.
 For example, 
\begin_inset Formula 
\begin{align}
\frac{\partial}{\partial K}W\left[K\right] & =\frac{1}{2}\left\langle q^{2}\right\rangle _{K}\equiv\frac{1}{2}\bar{G},\label{eq:g-def}\\
\frac{\partial^{2}}{\partial K^{2}}W\left[K\right] & =-\frac{1}{4}\left(\left\langle q^{4}\right\rangle -\left\langle q^{2}\right\rangle ^{2}\right)_{K}\equiv\frac{1}{4}\left(V^{\left(4\right)}\bar{G}^{4}-2\bar{G}^{2}\right),\label{eq:4pt-vertex-def}
\end{align}

\end_inset

 where the subscript 
\begin_inset Formula $K$
\end_inset

 indicates the average is taken at a fixed value of 
\begin_inset Formula $K$
\end_inset

.
 
\begin_inset Formula $\bar{G}$
\end_inset

 and 
\begin_inset Formula $V^{\left(4\right)}$
\end_inset

 are the proper two and four point functions respectively.
 To lowest order in perturbation theory and with 
\begin_inset Formula $K=0$
\end_inset

, 
\begin_inset Formula $\bar{G}=1/m^{2}+\mathcal{O}\left(\lambda\right)$
\end_inset

 and 
\begin_inset Formula $V^{\left(4\right)}=\lambda+\mathcal{O}\left(\lambda^{2}\right)$
\end_inset

.
\end_layout

\begin_layout Standard
The exact value of 
\begin_inset Formula $W\left[K\right]$
\end_inset

 is easily obtained directly from the definition, giving 
\begin_inset Formula 
\begin{align}
W\left[K\right] & =-\frac{1}{2}\ln\left(1+k\right)-\rho\left[\left(1+k\right)^{2}-1\right]-\ln\mathrm{K}_{1/4}\left(\rho\left(1+k\right)^{2}\right)+\ln\mathrm{K}_{1/4}\left(\rho\right).
\end{align}

\end_inset

 By direct differentiation the exact two point correlation function is 
\begin_inset Formula 
\begin{align}
m^{2}\left\langle q^{2}\right\rangle _{K} & =4\rho\left(1+k\right)\left[\frac{\mathrm{K}_{3/4}\left(\rho\left(1+k\right)^{2}\right)}{\mathrm{K}_{1/4}\left(\rho\left(1+k\right)^{2}\right)}-1\right],
\end{align}

\end_inset

 or for the original (
\begin_inset Formula $K=0$
\end_inset

) theory,
\begin_inset Formula 
\begin{align}
m^{2}\bar{G} & =4\rho\left(\frac{\mathrm{K}_{3/4}\left(\rho\right)}{\mathrm{K}_{1/4}\left(\rho\right)}-1\right).\label{eq:exact-G-soln}
\end{align}

\end_inset

Like 
\begin_inset Formula $Z$
\end_inset

, 
\begin_inset Formula $\bar{G}$
\end_inset

 possesses a branch cut discontinuity from 
\begin_inset Formula $\lambda=0$
\end_inset

 to 
\begin_inset Formula $\lambda=-\infty$
\end_inset

.
 At 
\begin_inset Formula $\lambda=0$
\end_inset

 one obtains the usual free (Gaussian) theory result 
\begin_inset Formula $\bar{G}=G_{0}=m^{-2}$
\end_inset

.
 In the strong coupling limit, 
\begin_inset Formula $\lambda\to\infty$
\end_inset

, 
\begin_inset Formula $\bar{G}\sim\left[2\sqrt{6}\Gamma\left(\frac{3}{4}\right)/\Gamma\left(\frac{1}{4}\right)\right]\lambda^{-1/2}+\mathcal{O}\left(\lambda^{-1}\right)$
\end_inset

.
 
\begin_inset Formula $\bar{G}$
\end_inset

 is shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gexact"

\end_inset

, from which the branch cut is obvious.
 This can also be seen in more detail in the complex 
\begin_inset Formula $\lambda$
\end_inset

 plane as shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gexact-analytic-behaviour"

\end_inset

.
 One sees that not only does 
\begin_inset Formula $\bar{G}$
\end_inset

 possess a branch cut, it is analytic in the cut plane and is in fact a
 Herglotz-Nevanlinna function (i.e.
 
\begin_inset Formula $\bar{G}\left(\lambda\right)^{\star}=\bar{G}\left(\lambda^{\star}\right)$
\end_inset

 where 
\begin_inset Formula $\star$
\end_inset

 is complex conjugation).
 This gives rise to an integral representation in terms of a spectral function
 which can be used in order to quantify the branch cut.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/Gexact.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gexact"

\end_inset


\begin_inset Formula $\bar{G}$
\end_inset

 from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:exact-G-soln"

\end_inset

 as a function of 
\begin_inset Formula $\lambda$
\end_inset

 for 
\begin_inset Formula $m=1$
\end_inset

.
 The sign of the imaginary part depends on whether one approaches the 
\begin_inset Formula $\lambda<0$
\end_inset

 cut from above or below.
 Note the imaginary part is exponentially suppressed near the origin because
 the vacuum decay process is non-perturbative.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/Gexact-abs.pdf
	height 40theight%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gexact-abs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/Gexact-arg.pdf
	height 40theight%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gexact-arg"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gexact-analytic-behaviour"

\end_inset

Modulus 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gexact-abs"

\end_inset

 and phase 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gexact-arg"

\end_inset

 of 
\begin_inset Formula $\bar{G}$
\end_inset

 in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:exact-G-soln"

\end_inset

 for 
\begin_inset Formula $m^{2}=1$
\end_inset

 in the complex 
\begin_inset Formula $\lambda$
\end_inset

 plane.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Start with the integral representation for 
\begin_inset Formula $\bar{G}$
\end_inset

 using the Cauchy formula 
\begin_inset Formula 
\begin{equation}
\bar{G}\left(\lambda\right)=\frac{1}{2\pi i}\oint_{C}\frac{\bar{G}\left(\lambda'\right)}{\lambda'-\lambda}\mathrm{d}\lambda',
\end{equation}

\end_inset

where the contour 
\begin_inset Formula $C$
\end_inset

 circles 
\begin_inset Formula $\lambda$
\end_inset

 in the counter-clockwise direction and avoids the cut.
 Deforming the contour to run on the circle at infinity and around the cut
 and using 
\begin_inset Formula $\bar{G}\left(\lambda\right)\to0$
\end_inset

 as 
\begin_inset Formula $\left|\lambda\right|\to\infty$
\end_inset

, the integral can be written in terms of a spectral function 
\begin_inset Formula 
\begin{equation}
\sigma\left(s\right)=\frac{\bar{G}\left(-s-i\epsilon\right)-\bar{G}\left(-s+i\epsilon\right)}{2\pi i}\Theta\left(s\right)=\frac{\mathrm{Im}\left[\bar{G}\left(-s-i\epsilon\right)\right]}{\pi}\Theta\left(s\right),
\end{equation}

\end_inset

such that 
\begin_inset Formula 
\begin{equation}
\bar{G}\left(\lambda\right)=\int_{0}^{\infty}\mathrm{d}s\frac{\sigma\left(s\right)}{s+\lambda}.\label{eq:spectral-function-def}
\end{equation}

\end_inset

Then, 
\begin_inset Formula 
\begin{equation}
\sigma\left(\lambda\right)=-\frac{4\sqrt{2}}{m^{2}\pi^{2}}\frac{1}{\mathrm{Im}\left[I_{-\frac{1}{4}}\left(-\rho\right)^{2}-I_{\frac{1}{4}}\left(-\rho\right)^{2}\right]}\Theta\left(\lambda\right),\label{eq:spectral-function-exact}
\end{equation}

\end_inset

 where 
\begin_inset Formula $I_{\pm\frac{1}{4}}\left(\rho\right)$
\end_inset

 are modified Bessel functions of the first kind
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Note that the physical interpretation of this spectral function is unrelated
 to the usual one in field theory since, for one thing, there is no such
 thing as energy in zero dimensions.
 One can consider 
\begin_inset Formula $\sigma\left(s\right)$
\end_inset

 a purely formal device that gives information about the analytic structure
 of 
\begin_inset Formula $\bar{G}$
\end_inset

.
\end_layout

\end_inset

.
 
\begin_inset Formula $\sigma\left(\lambda\right)$
\end_inset

 is shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sigma-exact"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/sigma-exact.pdf
	width 80col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:sigma-exact"

\end_inset

Spectral function 
\begin_inset Formula $\sigma\left(\lambda\right)$
\end_inset

 of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:spectral-function-exact"

\end_inset

 for 
\begin_inset Formula $m=1$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Expanding 
\begin_inset Formula $\bar{G}$
\end_inset

 in a Taylor series in 
\begin_inset Formula $\lambda$
\end_inset

 gives 
\begin_inset Formula 
\begin{align}
\bar{G} & =\frac{1}{m^{2}}+\frac{4}{m^{2}}\frac{\sum_{n=1}^{\infty}\frac{1}{\left(n-1\right)!}\left(-\frac{1}{4!}\lambda\right)^{n}2^{2n}\Gamma\left(2n+\frac{1}{2}\right)\frac{1}{m^{4n}}}{\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{1}{4!}\lambda\right)^{n}2^{2n}\Gamma\left(2n+\frac{1}{2}\right)\frac{1}{m^{4n}}}\nonumber \\
 & =\frac{1}{m^{2}}-\frac{\lambda}{2m^{6}}+\frac{2\lambda^{2}}{3m^{10}}-\frac{11\lambda^{3}}{8m^{14}}+\frac{34\lambda^{4}}{9m^{18}}+\cdots.\label{eq:G-perturbation-series}
\end{align}

\end_inset

The perturbative approximations 
\begin_inset Formula $G_{n}$
\end_inset

 are the 
\begin_inset Formula $\mathcal{O}\left(\lambda^{n}\right)$
\end_inset

 truncations of this series.
 The first few are shown compared to the exact 
\begin_inset Formula $\bar{G}$
\end_inset

 in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:G-pert-series"

\end_inset

.
 These approximations apparently converge poorly to the exact solution,
 and in fact the series diverges.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/G-pert-series.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:G-pert-series"

\end_inset

Perturbative approximations to 
\begin_inset Formula $\bar{G}$
\end_inset

 up to 
\begin_inset Formula $\mathcal{O}\left(\lambda^{5}\right)$
\end_inset

 for 
\begin_inset Formula $m=1$
\end_inset

, compared to the exact solution.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The series for 
\begin_inset Formula $\bar{G}$
\end_inset

 can also be described in terms of Feynman diagrams by the following rules:
 
\end_layout

\begin_layout Enumerate
Draw all connected graphs with two external lines constructed from lines
 and four point vertices.
\end_layout

\begin_layout Enumerate
Associate to each line a factor 
\begin_inset Formula $G_{0}=1/m^{2}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Associate to each vertex a factor 
\begin_inset Formula $-\lambda$
\end_inset

.
\end_layout

\begin_layout Enumerate
Divide by an overall symmetry factor being the order of the symmetry group
 of the diagram under permutations of lines and vertices.
\end_layout

\begin_layout Standard
The diagrams of this series are exactly those seen before in the 
\begin_inset Formula $\lambda\phi^{4}$
\end_inset

 theory in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:G-perturbation-series-diagrams"

\end_inset

.
 The series 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:G-perturbation-series"

\end_inset

 has the form 
\begin_inset Formula $\bar{G}=m^{-2}\sum_{n=0}^{\infty}c_{n}\left(\lambda/m^{4}\right)^{n}$
\end_inset

 where the coefficients asymptotically obey 
\begin_inset Formula $c_{n+1}\sim-\frac{2}{3}nc_{n}$
\end_inset

 as 
\begin_inset Formula $n\to\infty$
\end_inset

, thus the radius of convergence of the series is zero.
 This is consistent with the fact that one is perturbing around a branch
 point of the exact solution: no approximation of 
\begin_inset Formula $\bar{G}$
\end_inset

 in terms of analytic functions can converge at 
\begin_inset Formula $\lambda=0$
\end_inset

 because 
\begin_inset Formula $Z\left[K\right]$
\end_inset

 is itself undefined for 
\begin_inset Formula $\mathrm{Re}\lambda<0$
\end_inset

.
 The terms of the series start to increase when 
\begin_inset Formula $c_{n+1}\left(\lambda/m^{4}\right)\sim c_{n}$
\end_inset

, i.e.
 
\begin_inset Formula $n\sim3m^{4}/2\lambda$
\end_inset

, meaning the series is useful for 
\begin_inset Formula $\lambda\ll m^{4}$
\end_inset

 but fails immediately for a moderately strong coupling 
\begin_inset Formula $\lambda\approx m^{4}$
\end_inset

.
 This is typical asymptotic series behaviour as shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gpert-asymptotic-series"

\end_inset

.
 Extrapolating perturbation theory to strong coupling 
\begin_inset Formula $\lambda\gg m^{4}$
\end_inset

 is simply impossible, although the exact solution is well behaved there.
 (In fact 
\begin_inset Formula $\bar{G}$
\end_inset

 can be expanded as 
\begin_inset Formula $\bar{G}=m^{-2}\sum_{n=1}^{\infty}\tilde{c}_{n}\left(\lambda/m^{4}\right)^{-n/2}$
\end_inset

, displaying explicitly the branch point at 
\begin_inset Formula $\lambda=\infty$
\end_inset

.)
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/G-pert-asympt-series.pdf
	width 75col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gpert-asymptotic-series"

\end_inset

Relative error 
\begin_inset Formula $\left|\left(G_{n}-\bar{G}\right)/\bar{G}\right|$
\end_inset

of the 
\begin_inset Formula $n$
\end_inset

-th perturbative approximation to 
\begin_inset Formula $\bar{G}$
\end_inset

 for 
\begin_inset Formula $m=1$
\end_inset

 and 
\begin_inset Formula $\lambda=1/4$
\end_inset

, showing the decreasing then increasing behaviour typical of asymptotic
 series
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Borel summation
\end_layout

\begin_layout Standard
The series expansion for 
\begin_inset Formula $\bar{G}$
\end_inset

 diverges for all 
\begin_inset Formula $\lambda\ne0$
\end_inset

.
 This is typical of perturbation series and usually signals some singularity
 of the exact solution for unphysical values of 
\begin_inset Formula $\lambda$
\end_inset

.
 Indeed, the toy model does not exist for 
\begin_inset Formula $\mathrm{Re}\lambda<0$
\end_inset

 and the exact solution possesses a branch cut on the negative 
\begin_inset Formula $\lambda$
\end_inset

 axis, a feature which cannot be reproduced in any order of perturbation
 theory.
 However, the perturbation series is asymptotic and does contain true informatio
n about the exact solution, even if 
\begin_inset Formula $\lambda$
\end_inset

 is large enough that the series is not useful practically.
 Because of the ubiquity of this phenomenon, mathematicians have invented
 a number of series summation techniques which assign a finite value to
 certain types of divergent series and which obey certain consistency properties
 (e.g.
 the value assigned to a convergent series is just its na
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
"{i}
\end_layout

\end_inset

ve sum).
 This section discusses Borel summation, which is capable of summing factorially
 divergent series like 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:G-perturbation-series"

\end_inset

.
\end_layout

\begin_layout Standard
Suppose that 
\begin_inset Formula $\sum_{n=0}^{\infty}a_{n}$
\end_inset

 is a divergent series but that the 
\emph on
Borel transform
\emph default
 of the series, defined as 
\begin_inset Formula 
\begin{equation}
\phi\left(x\right)=\sum_{n=0}^{\infty}\frac{a_{n}x^{n}}{n!},
\end{equation}

\end_inset

 converges for sufficiently small 
\begin_inset Formula $x$
\end_inset

.
 Then, if the integral 
\begin_inset Formula 
\begin{equation}
B\left(x\right)=\int_{0}^{\infty}\mathrm{e}^{-t}\phi\left(xt\right)\mathrm{d}t\label{eq:borel-sum}
\end{equation}

\end_inset

exists the 
\emph on
Borel sum
\emph default
 
\begin_inset CommandInset citation
LatexCommand citep
key "Bender1999,Ellis1996,Dorigoni2014"

\end_inset

 of the divergent series is defined as 
\begin_inset Formula $\mathcal{B}\left[\sum_{n=0}^{\infty}a_{n}\right]\equiv B\left(1\right)$
\end_inset

.
 This definition is justified by substituting the series for 
\begin_inset Formula $\phi\left(xt\right)$
\end_inset

 into the integral and evaluating term-wise and noting that 
\begin_inset Formula $B\left(x\right)\sim\sum_{n=0}^{\infty}a_{n}x^{n}$
\end_inset

.
 The main drawback of Borel summation is that one must know the precise
 form of 
\begin_inset Formula $a_{n}$
\end_inset

 for all 
\begin_inset Formula $n$
\end_inset

 to compute 
\begin_inset Formula $\phi\left(x\right)$
\end_inset

, which is rarely the case in field theory.
 For this reason Borel summation cannot be usefully applied directly in
 practice.
 However, one may use Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants as discussed in the next section to recast the Borel transform
 in a useful way.
 In the case of the toy model, though, one can verify that the perturbation
 series is Borel summable and the Borel sum is equal to the exact solution
 (the computation is unenlightening and best done by a computer algebra
 package).
\end_layout

\begin_layout Standard
Note that key to Borel-summability of the perturbation series is the alternating
 sign 
\begin_inset Formula $\left(-1\right)^{n}$
\end_inset

 of the 
\begin_inset Formula $n$
\end_inset

-th order term.
 To see this consider the two series 
\begin_inset Formula 
\begin{align}
S_{1} & =\sum_{n=0}^{\infty}\left(-\lambda\right)^{n}n!,\\
S_{2} & =\sum_{n=0}^{\infty}\lambda^{n}n!,
\end{align}

\end_inset

 which differ only by the alternating sign.
 The Borel transforms 
\begin_inset Formula $\phi_{1,2}\left(x\right)$
\end_inset

 are 
\begin_inset Formula 
\begin{align}
\phi_{1}\left(x\right) & =\sum_{n=0}^{\infty}\left(-\lambda x\right)^{n}=\frac{1}{1+\lambda x},\\
\phi_{2}\left(x\right) & =\sum_{n=0}^{\infty}\left(\lambda x\right)^{n}=\frac{1}{1-\lambda x},
\end{align}

\end_inset

 and the Borel sums are 
\begin_inset Formula 
\begin{align}
\mathcal{B}\left[S_{1}\right]=B_{1}\left(1\right) & =\int_{0}^{\infty}\frac{\mathrm{e}^{-t}}{1+\lambda t}\mathrm{d}t,\\
\mathcal{B}\left[S_{2}\right]=B_{2}\left(1\right) & =\int_{0}^{\infty}\frac{\mathrm{e}^{-t}}{1-\lambda t}\mathrm{d}t.
\end{align}

\end_inset

In the first case the integral exists and 
\begin_inset Formula $\mathcal{B}\left[S_{1}\right]=\lambda^{-1}\mathrm{e}^{1/\lambda}\Gamma\left(0,\frac{1}{\lambda}\right)$
\end_inset

 where 
\begin_inset Formula $\Gamma\left(a,b\right)=\int_{b}^{\infty}t^{a-1}\mathrm{e}^{-t}\mathrm{d}t$
\end_inset

 is the incomplete gamma function.
 However, the second integral hits a pole at 
\begin_inset Formula $t=1/\lambda$
\end_inset

.
 There is no natural prescription for avoiding the pole, which leads to
 an ambiguity in the sum of 
\begin_inset Formula $\pm\pi i\lambda^{-1}\mathrm{e}^{-1/\lambda}$
\end_inset

.
 This is a non-perturbative ambiguity called a 
\emph on
renormalon
\emph default
 
\begin_inset CommandInset citation
LatexCommand citep
key "Beneke1999"

\end_inset

.
 In every known case where this arises in field theory the renormalon is
 connected to a non-perturbative finite action solution of the field equations,
 i.e.
 an instanton or soliton, and a correct evaluation of the path integral
 which sums over 
\emph on
all
\emph default
 saddle points (not just perturbative ones) removes the ambiguity.
 Key to the practical application of Borel summation is the location and
 classification of all renormalons in a given theory 
\begin_inset CommandInset citation
LatexCommand citep
key "Crutchfield1979"

\end_inset

.
 Sophisticated techniques have been developed to deal with this situation
 which are beyond the scope of this thesis 
\begin_inset CommandInset citation
LatexCommand citep
key "Dorigoni2014,Dunne2014,Basar2013"

\end_inset

.
\end_layout

\begin_layout Subsection
Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximation and Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 resummation
\end_layout

\begin_layout Standard
Borel summation is limited in its usefulness because one often only knows
 a few low order terms of perturbation theory (as well as the possibility
 of renormalons).
 Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximation is a technique which often improves perturbation series and
 is far more useful in practice, and can be combined with Borel summation.
 Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants are rational polynomials which generally converge rapidly
 to the function being fitted, are useful for numerical computation, and
 help to estimate the location of singularities of the function in the complex
 plane.
 Many software packages include standard routines for evaluating Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants, for instance the 
\noun on
PadeApproximant
\noun default
 function in 
\noun on
Mathematica
\noun default
 
\begin_inset CommandInset citation
LatexCommand citep
key "mathematica"

\end_inset

.
 This section applies Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants to both the Green function and its Borel transform, finding
 that the latter approach is clearly the better one.
\end_layout

\begin_layout Standard
The 
\begin_inset Formula $\left(N,M\right)$
\end_inset

-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximant of a function 
\begin_inset Formula $\sum_{n=0}^{\infty}a_{n}x^{n}$
\end_inset

 is given by 
\begin_inset CommandInset citation
LatexCommand citep
key "Bender1999"

\end_inset

 
\begin_inset Formula 
\begin{equation}
P_{M}^{N}\left(x\right)=\frac{\sum_{n=0}^{N}A_{n}x^{n}}{\sum_{n=0}^{M}B_{n}x^{n}},
\end{equation}

\end_inset

where, without loss of generality, one takes 
\begin_inset Formula $B_{0}=1$
\end_inset

.
 The remaining 
\begin_inset Formula $N+M+1$
\end_inset

 coefficients are chosen so that the Taylor series of 
\begin_inset Formula $P_{M}^{N}\left(x\right)$
\end_inset

 matches the perturbation series up to 
\begin_inset Formula $\mathcal{O}\left(x^{N+M}\right)$
\end_inset

.
 Due to the denominator, Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants develop poles in the complex 
\begin_inset Formula $x$
\end_inset

-plane, allowing the close approximation of more singular functions than
 Taylor series are capable of.
 Examples of the use of Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants in field theory can be found in 
\begin_inset CommandInset citation
LatexCommand citep
key "Ellis1996,Kraemmer2004"

\end_inset

 and references therein.
\end_layout

\begin_layout Standard
To find the approximants to 
\begin_inset Formula $m^{2}\bar{G}$
\end_inset

, with 
\begin_inset Formula $x=\lambda/m^{4}$
\end_inset

, one must restrict attention to the approximants where 
\begin_inset Formula $M=N+1$
\end_inset

.
 This guarantees that 
\begin_inset Formula $P_{M}^{N}\to0$
\end_inset

 as 
\begin_inset Formula $\lambda\to\infty$
\end_inset

.
 The usual diagonal approximants (
\begin_inset Formula $N=M$
\end_inset

) would give an unphysical constant 
\begin_inset Formula $P_{N}^{N}\to A_{N}/B_{N}\neq0$
\end_inset

 as 
\begin_inset Formula $\lambda\to\infty$
\end_inset

.
 Note that it is impossible to match the true 
\begin_inset Formula $1/\sqrt{\lambda}$
\end_inset

 behaviour of 
\begin_inset Formula $\bar{G}$
\end_inset

 as 
\begin_inset Formula $\lambda\to\infty$
\end_inset

 using Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants centred on the origin.
 The best that is possible in this limit is 
\begin_inset Formula $\sim\lambda^{-1}$
\end_inset

.
 (As it happens, using 
\begin_inset Formula $x\propto\sqrt{\lambda}$
\end_inset

 does not allow one to resolve this issue: the same approximants are found
 only with 
\begin_inset Formula $x^{2}$
\end_inset

 everywhere in place of 
\begin_inset Formula $x$
\end_inset

.
 This is because the 
\begin_inset Formula $1/\sqrt{\lambda}$
\end_inset

 behaviour is due to the branch point at infinity, which is infinitely far
 from the origin where the Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants are matched to perturbation theory.
 Low order Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants can extract information about the branch point near the origin,
 but evidently not the one at infinity.) The first five approximants for
 
\begin_inset Formula $m^{2}\bar{G}$
\end_inset

 are shown in Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:pade-approximants"

\end_inset

 and the first three are plotted in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Pade-approx"

\end_inset

 with comparison to the exact 
\begin_inset Formula $\bar{G}$
\end_inset

.
 Note that the existence of the integral representation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:spectral-function-def"

\end_inset

 for 
\begin_inset Formula $\bar{G}$
\end_inset

 implies that 
\begin_inset Formula $\bar{G}$
\end_inset

 is a 
\emph on
Stieltjes
\emph default
 function, meaning one can prove convergence properties for the Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants as 
\begin_inset Formula $N,M\to\infty$
\end_inset

, though this analysis is not presented here (see 
\begin_inset CommandInset citation
LatexCommand citep
key "Bender1999"

\end_inset

 for examples in other contexts).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/Pade-approx.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Pade-approx"

\end_inset

First three Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants to 
\begin_inset Formula $\bar{G}$
\end_inset

 with 
\begin_inset Formula $m=1$
\end_inset

.
 Note the simple poles developed for 
\begin_inset Formula $\lambda<0$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:pade-approximants"

\end_inset

First few Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants for 
\begin_inset Formula $m^{2}\bar{G}$
\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset space \hfill{}
\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="2">
<features booktabs="true" tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="left" valignment="top" width="0pt">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $N$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{P_{N+1}^{N}}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topspace="default" interlinespace="default">
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\begin{aligned}[t]\frac{1}{1+\frac{\lambda}{2m^{4}}}\end{aligned}
}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topspace="default" interlinespace="default">
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\begin{aligned}[t]\frac{1+\frac{2\lambda}{m^{4}}}{1+\frac{5\lambda}{2m^{4}}+\frac{7\lambda^{2}}{12m^{8}}}\end{aligned}
}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topspace="default" interlinespace="default">
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\begin{aligned}[t]\frac{1+\frac{16\lambda}{3m^{4}}+\frac{59\lambda^{2}}{12m^{8}}}{1+\frac{35\lambda}{6m^{4}}+\frac{43\lambda^{2}}{6m^{8}}+\frac{77\lambda^{3}}{72m^{12}}}\end{aligned}
}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topspace="default" interlinespace="default">
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\begin{aligned}[t]\frac{1+\frac{10\lambda}{m^{4}}+\frac{155\lambda^{2}}{6m^{8}}+\frac{15\lambda^{3}}{m^{12}}}{1+\frac{21\lambda}{2m^{4}}+\frac{365\lambda^{2}}{12m^{8}}+\frac{295\lambda^{3}}{12m^{12}}+\frac{385\lambda^{4}}{144m^{16}}}\end{aligned}
}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row topspace="default" interlinespace="default">
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\begin{aligned}[t]\frac{1+\frac{16\lambda}{m^{4}}+\frac{315\lambda^{2}}{4m^{8}}+\frac{1190\lambda^{3}}{9m^{12}}+\frac{7945\lambda^{4}}{144m^{16}}}{1+\frac{33\lambda}{2m^{4}}+\frac{259\lambda^{2}}{3m^{8}}+\frac{11935\lambda^{3}}{72m^{12}}+\frac{14315\lambda^{4}}{144m^{16}}+\frac{7315\lambda^{5}}{864m^{20}}}\end{aligned}
}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset space \hfill{}
\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The 
\begin_inset Formula $\left(N,N+1\right)$
\end_inset

-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximant can also be written as 
\begin_inset Formula 
\begin{equation}
P_{N+1}^{N}=\sum_{i=0}^{N}\frac{r_{i}}{\lambda-p_{i}},
\end{equation}

\end_inset

where 
\begin_inset Formula $r_{i}$
\end_inset

 and 
\begin_inset Formula $p_{i}$
\end_inset

 are the 
\begin_inset Formula $i$
\end_inset

-th residue and pole respectively.
 Note that since all of the coefficients in the denominators of Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:pade-approximants"

\end_inset

 are positive and real, all of the poles must either be on the negative
 real axis or they must be complex and come in complex conjugate pairs.
 Numerical experiments suggest that all the poles lie on the negative 
\begin_inset Formula $\lambda$
\end_inset

 axis, though the author is not aware of a proof of this for all 
\begin_inset Formula $N$
\end_inset

.
 Assuming this is generally true, Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximants give a representation of 
\begin_inset Formula $\bar{G}$
\end_inset

 which approximates the continuous spectral function by a sum of delta functions
 
\begin_inset Formula 
\begin{equation}
\sigma\left(s\right)\approx\sigma_{N}\left(s\right)\equiv\sum_{i=0}^{N}r_{i}\delta\left(s+p_{i}\right).
\end{equation}

\end_inset

 As 
\begin_inset Formula $N\to\infty$
\end_inset

 the poles become denser and fill the negative 
\begin_inset Formula $\lambda$
\end_inset

 axis, eventually merging into a continuous branch cut.
 Similarly the spectral function turns into a dense sum of delta functions
 which, when considered acting on any sufficiently smooth test function,
 smooths into a continuous function.
 The first few 
\begin_inset Formula $\sigma_{N}$
\end_inset

 are shown next to the exact spectral function in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Pade-spectral-function"

\end_inset

 for comparison.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/sigma-Pade.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Pade-spectral-function"

\end_inset

Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximations to the spectral function 
\begin_inset Formula $\sigma\left(\lambda\right)$
\end_inset

 (for 
\begin_inset Formula $m=1$
\end_inset

) consist of an increasingly dense set of delta functions.
 (Note the delta functions have been smoothed for visual purposes only.)
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Now consider Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximation of the Borel transform of 
\begin_inset Formula $\bar{G}$
\end_inset

.
 This is aided by noting the following connections between the Borel transform
 
\begin_inset Formula $\phi$
\end_inset

 and 
\begin_inset Formula $\bar{G}$
\end_inset

 and 
\begin_inset Formula $\sigma$
\end_inset

: 
\begin_inset Formula 
\begin{equation}
v^{-1}\sigma\left(\frac{\lambda}{v}\right)\xleftarrow[x\to v]{\mathcal{L}^{-1}}\phi\left(x\right)\xrightarrow[x\to s]{\mathcal{L}}s^{-1}\bar{G}\left(\frac{\lambda}{s}\right).
\end{equation}

\end_inset

That is, the Green function is related to the Laplace transform of the Borel
 transform, while the spectral function is related to the inverse Laplace
 transform of the Borel transform.
 These relations can be shown using the definitions of 
\begin_inset Formula $\phi$
\end_inset

 and 
\begin_inset Formula $\sigma$
\end_inset

 and the integral representation of the (inverse) Laplace transform.
 This allows one to extract the spectral function directly from the Borel
 transform.
 Note that each pole of 
\begin_inset Formula $\phi$
\end_inset

 yields by the inverse Laplace transform a term of the form 
\begin_inset Formula $\lambda^{-1}\exp\left(-k/\lambda\right)$
\end_inset

 in 
\begin_inset Formula $\sigma$
\end_inset

, where 
\begin_inset Formula $k$
\end_inset

 is controlled by the location of the pole.
 The general Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximation for 
\begin_inset Formula $\sigma$
\end_inset

 is a superposition of terms of this form.
\end_layout

\begin_layout Standard
The first few approximants to 
\begin_inset Formula $\bar{G}$
\end_inset

 and 
\begin_inset Formula $\sigma$
\end_inset

 are shown in Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gbar-PadeBorel"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sigma-PadeBorel"

\end_inset

 respectively.
 The low order Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 Green functions are reasonably accurate (within a few percent for 
\begin_inset Formula $\lambda\leq5m^{4}$
\end_inset

) but the spectral functions are not approximated particularly well.
 Certain approximants to 
\begin_inset Formula $\sigma$
\end_inset

 oscillate erratically and even become negative for certain values of 
\begin_inset Formula $\lambda$
\end_inset

.
 Except for the fact that the best approximant is the highest order one
 plotted, there is no clear sense in which the Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximations appear to converge to 
\begin_inset Formula $\sigma$
\end_inset

.
 However, even this bad approximation is at least a continuous function,
 as opposed to a sum of delta functions.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/Gbar-PadeBorel.pdf
	width 100col%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gbar-PadeBorel"

\end_inset

Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximations to the Green function 
\begin_inset Formula $\bar{G}\left(\lambda\right)$
\end_inset

 (ratio of approximant / exact) for 
\begin_inset Formula $m=1$
\end_inset

.
 Approximants are calculated by numerical integration of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:borel-sum"

\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/sigma-PadeBorel.pdf
	width 100col%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:sigma-PadeBorel"

\end_inset

Borel-Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
'e
\end_layout

\end_inset

 approximations to the spectral function 
\begin_inset Formula $\sigma\left(\lambda\right)$
\end_inset

 (ratio of approximant / exact) for 
\begin_inset Formula $m=1$
\end_inset

.
 Note that the (2,2) approximant is off scale.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
2PI results
\end_layout

\begin_layout Standard
The 2PI effective action in the toy model is
\begin_inset Formula 
\begin{equation}
\Gamma\left[G\right]=W\left[K\right]-K\partial_{K}W\left[K\right],
\end{equation}

\end_inset

where 
\begin_inset Formula $K$
\end_inset

 is solved for in terms of 
\begin_inset Formula $G$
\end_inset

.
 
\begin_inset Formula $\Gamma\left[G\right]$
\end_inset

 obeys the equation of motion 
\begin_inset Formula 
\begin{equation}
\partial_{G}\Gamma\left[G\right]=-\frac{1}{2}K.
\end{equation}

\end_inset

Explicitly, 
\begin_inset Formula 
\begin{equation}
\Gamma\left[G\right]=\frac{1}{2}\ln\left(G^{-1}\right)+\frac{1}{2}m^{2}G+\gamma_{2\mathrm{PI}}+\mathrm{const.},
\end{equation}

\end_inset

 where 
\begin_inset Formula $-\gamma_{2\mathrm{PI}}$
\end_inset

 is (minus due to Euclidean conventions) the sum of two particle irreducible
 vacuum graphs, i.e.
 those graphs which do not fall apart when any two lines are cut, where
 the lines are given by 
\begin_inset Formula $G$
\end_inset

 and vertices by 
\begin_inset Formula $-\lambda$
\end_inset

.
 The equation of motion in the absence of sources is 
\begin_inset Formula 
\begin{equation}
G^{-1}=m^{2}+2\partial_{G}\gamma_{2\mathrm{PI}},\label{eq:2pi-G-eom}
\end{equation}

\end_inset

which is Dyson's equation where the second term on the right hand side,
 
\begin_inset Formula $-\Sigma=2\partial_{G}\gamma_{2\mathrm{PI}}$
\end_inset

, represents the exact one-particle irreducible self-energy of the propagator
 
\begin_inset Formula $G$
\end_inset

.
\end_layout

\begin_layout Standard
By power counting (or counting line ends in the corresponding diagrams),
 
\begin_inset Formula $\gamma_{2\mathrm{PI}}=\sum_{n=0}^{\infty}\gamma_{n}\left(\lambda G^{2}\right)^{n}$
\end_inset

.
 It is possible to derive the 
\begin_inset Formula $\gamma_{n}$
\end_inset

 by considering the symmetry factors of the two particle irreducible Feynman
 diagrams, but they can also be obtained using knowledge of the exact solution
 
\begin_inset Formula $G=\bar{G}$
\end_inset

.
 Substituting 
\begin_inset Formula $\bar{G}$
\end_inset

 into the equation of motion and the expansion for 
\begin_inset Formula $\gamma_{2\text{PI}}$
\end_inset

, expanding about 
\begin_inset Formula $\lambda=0$
\end_inset

 and matching powers of 
\begin_inset Formula $\lambda$
\end_inset

, one finds 
\begin_inset Formula 
\begin{align}
\gamma_{2\mathrm{PI}} & =\frac{G^{2}\lambda}{8}-\frac{G^{4}\lambda^{2}}{48}+\frac{G^{6}\lambda^{3}}{48}-\frac{5G^{8}\lambda^{4}}{128}+\frac{101G^{10}\lambda^{5}}{960}-\frac{93G^{12}\lambda^{6}}{256}\nonumber \\
 & +\frac{8143G^{14}\lambda^{7}}{5376}-\frac{271217G^{16}\lambda^{8}}{36864}+\frac{374755G^{18}\lambda^{9}}{9216}-\frac{5151939G^{20}\lambda^{10}}{20480}\nonumber \\
 & +\frac{697775057G^{22}\lambda^{11}}{405504}-\frac{3802117511G^{24}\lambda^{12}}{294912}+\frac{201268707239G^{26}\lambda^{13}}{1916928}\nonumber \\
 & -\frac{11440081763125G^{28}\lambda^{14}}{12386304}+\frac{5148422676667G^{30}\lambda^{15}}{589824}-\frac{1665014342007385G^{32}\lambda^{16}}{18874368}\nonumber \\
 & +\frac{4231429245358235G^{34}\lambda^{17}}{4456448}-\frac{921138067678697395G^{36}\lambda^{18}}{84934656}+\mathcal{O}\left(\lambda^{19}G^{38}\right).\label{eq:gamma2pi-series}
\end{align}

\end_inset

The author does not know of any closed form expression for either the coefficien
ts of this series or its sum (implicit analytic expressions can be derived;
 however, these require the inversion of 
\begin_inset Formula $G=\bar{G}\left(\lambda\right)$
\end_inset

 for 
\begin_inset Formula $\lambda\left(G\right)$
\end_inset

, which is not known in closed form).
 There is a recursion formula for the coefficients 
\begin_inset CommandInset citation
LatexCommand citep
key "Kleinert1998"

\end_inset

 which, however, will not be needed here.
 After the first few terms the coefficients are well approximated by 
\begin_inset Formula $\gamma_{i+1}\sim-\frac{2}{3}i\gamma_{i}$
\end_inset

, the same as for the perturbation series.
 This has the hallmark of an asymptotic series and, like perturbation theory,
 the 2PI series does not converge.
\end_layout

\begin_layout Standard
The first non-trivial contribution to the equation of motion gives 
\begin_inset Formula 
\begin{equation}
G_{\left(1\right)}^{-1}=m^{2}+\frac{\lambda}{2}G_{\left(1\right)},
\end{equation}

\end_inset

 where the subscript 
\begin_inset Formula $\left(1\right)$
\end_inset

 indicates terms of order 
\begin_inset Formula $\mathcal{O}\left(\lambda^{1}\right)$
\end_inset

 have been kept.
 This has two solutions 
\begin_inset Formula 
\begin{equation}
G_{\left(1\right)}=\frac{-m^{2}\pm\sqrt{m^{4}+2\lambda}}{\lambda}.
\end{equation}

\end_inset

 One of these solutions is unphysical and the 
\begin_inset Formula $+$
\end_inset

 sign must be chosen.
 As 
\begin_inset Formula $\lambda\to0$
\end_inset

, 
\begin_inset Formula 
\begin{equation}
G_{\left(1\right)}\to\frac{1}{m^{2}}-\frac{\lambda}{2m^{6}}+\frac{\lambda^{2}}{2m^{10}}+\mathcal{O}\left(\lambda^{3}\right),
\end{equation}

\end_inset

 which matches perturbation theory up to 
\begin_inset Formula $\mathcal{O}\left(\lambda^{2}\right)$
\end_inset

 terms as expected.
 However, unlike perturbation theory, the strong coupling limit 
\begin_inset Formula $\lambda\to\infty$
\end_inset

 exists and gives 
\begin_inset Formula 
\begin{equation}
G_{\left(1\right)}\to\sqrt{\frac{2}{\lambda}}-\frac{m^{2}}{\lambda}+\frac{m^{4}}{\left(2\lambda\right)^{3/2}}+\mathcal{O}\left(\frac{1}{\lambda^{5/2}}\right).
\end{equation}

\end_inset

 This series has the correct form in powers of 
\begin_inset Formula $\lambda^{-1/2}$
\end_inset

, though the leading coefficient is incorrect by 
\begin_inset Formula $\approx15\%$
\end_inset

, and the sub-leading coefficients are incorrect by 
\begin_inset Formula $\approx40\%$
\end_inset

, 
\begin_inset Formula $70\%$
\end_inset

 and 
\begin_inset Formula $100\%$
\end_inset

 etc.
 This level of accuracy is remarkable considering the simple nature of the
 approximation and the fact that 
\begin_inset Formula $\gamma_{2\text{PI}}$
\end_inset

 was truncated at leading order in 
\begin_inset Formula $\lambda$
\end_inset

! 
\begin_inset Formula $G_{\left(1\right)}$
\end_inset

 is a much more uniform approximation to 
\begin_inset Formula $\bar{G}$
\end_inset

 than the perturbative approximation 
\begin_inset Formula $m^{-2}-\lambda/2m^{6}$
\end_inset

.
\end_layout

\begin_layout Standard
This uniformity is possible because of the branch cut 
\begin_inset Formula $G_{\left(1\right)}$
\end_inset

 possesses on the negative 
\begin_inset Formula $\lambda$
\end_inset

 axis.
 The discontinuity across the cut 
\begin_inset Formula 
\begin{equation}
G_{\left(1\right)}\left(\lambda+i\epsilon\right)-G_{\left(1\right)}\left(\lambda-i\epsilon\right)=2i\frac{\sqrt{-m^{4}-2\lambda}}{\lambda}\Theta\left(-\lambda-\frac{m^{4}}{2}\right),
\end{equation}

\end_inset

 gives the spectral function 
\begin_inset Formula 
\begin{equation}
\sigma_{\left(1\right)}\left(\lambda\right)=\frac{\sqrt{-m^{4}+2\lambda}}{\pi\lambda}\Theta\left(\lambda-\frac{m^{4}}{2}\right),\label{eq:sigma-2loop-2pi-approx}
\end{equation}

\end_inset

which is a far better approximation to the exact spectral function than
 obtained from any of the other techniques, as can be seen from Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sigma-2pi-2loop"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/sigma-2pi-2loop.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:sigma-2pi-2loop"

\end_inset

Comparison of exact spectral function 
\begin_inset Formula $\sigma\left(\lambda\right)$
\end_inset

 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:spectral-function-exact"

\end_inset

 with the two loop 2PI approximation 
\begin_inset Formula $\sigma_{\left(1\right)}\left(\lambda\right)$
\end_inset

 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:sigma-2loop-2pi-approx"

\end_inset

 for 
\begin_inset Formula $m=1$
\end_inset

.
 Already, the simplest non-trivial 2PI truncation gives a much better approximat
ion than perturbation theory or Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

 approximants to arbitrary order.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
All of the first order approximations are shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gbar-all-first-order-approxs"

\end_inset

.
 Note that for 
\begin_inset Formula $\lambda/m^{4}\geq0$
\end_inset

 the best approximation is the 2PI, followed by the Borel-Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

, then by Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

, then perturbation theory last of all.
 The situation for negative 
\begin_inset Formula $\lambda$
\end_inset

 is complicated.
 The best approximation overall is the 2PI, though it has an unphysical
 cusp where the branch cut starts (
\begin_inset Formula $\lambda/m^{4}=-1/2$
\end_inset

 in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Gbar-all-first-order-approxs"

\end_inset

).
 It appears the 2PI approximation trades sensitivity to the exponentially
 small portion of 
\begin_inset Formula $\sigma$
\end_inset

 in exchange for a better global approximation.
 The Pad
\begin_inset ERT
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\end_layout

\end_inset

 approximation is good at small negative 
\begin_inset Formula $\lambda$
\end_inset

 but hits a pole at 
\begin_inset Formula $\lambda/m^{4}=-2$
\end_inset

 and there loses all validity.
 The Borel-Pad
\begin_inset ERT
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\end_layout

\end_inset

 approximation is smooth and more accurate than the 2PI near the peak of
 
\begin_inset Formula $\bar{G}$
\end_inset

, but eventually becomes invalid, even negative, at sufficiently large negative
 
\begin_inset Formula $\lambda$
\end_inset

 (
\begin_inset Formula $\lambda/m^{4}\lesssim-5.37$
\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/Gbar-all-first-order-approxs.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Gbar-all-first-order-approxs"

\end_inset

Comparison of (the real part of) the exact Green function 
\begin_inset Formula $\bar{G}$
\end_inset

 to each of the approximations discussed to first non-trivial order in each
 case, for 
\begin_inset Formula $m=1$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
At 
\begin_inset Formula $n$
\end_inset

-th order, 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2pi-G-eom"

\end_inset

 is a degree 
\begin_inset Formula $2n$
\end_inset

 polynomial in 
\begin_inset Formula $G$
\end_inset

 which has 
\begin_inset Formula $2n$
\end_inset

 roots, only one of which is physical.
 For 
\begin_inset Formula $n=2$
\end_inset

 there are analytical expressions for the roots, though they are very bulky,
 and for 
\begin_inset Formula $n\geq3$
\end_inset

 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2pi-G-eom"

\end_inset

 must be solved numerically.
 Picking out the correct root for a given value of 
\begin_inset Formula $\lambda$
\end_inset

 is tricky in general and not pursued further here.
 However, one can see on general grounds that truncations of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2pi-G-eom"

\end_inset

 give a singular perturbation theory in 
\begin_inset Formula $\lambda$
\end_inset

: at 
\begin_inset Formula $\lambda=0$
\end_inset

 the physical solution starts at 
\begin_inset Formula $G_{0}$
\end_inset

 and the spurious solutions flow in from infinity as inverse powers of 
\begin_inset Formula $\lambda$
\end_inset

 as 
\begin_inset Formula $\lambda$
\end_inset

 increases to finite values.
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Also note that at least one of the spurious solutions (and always an odd
 number of them) are real since the coefficients in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2pi-G-eom"

\end_inset

 are real and complex solutions must occur in conjugate pairs.
\end_layout

\end_inset

 At strong coupling the roots generically approach each other and one must
 carefully track them through the complex plane.
 The possibility that a resummation of the 2PI series would remove some
 or all of these spurious solutions is examined using two potential methods
 in the next section with mixed results.
 
\end_layout

\begin_layout Subsection
Hybrid 2PI-Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_inset


\end_layout

\begin_layout Standard
Since the series of 2PI diagrams is asymptotic, one may use a series summation
 method to improve convergence.
 Note that this is logically independent of the resummations embodied in
 the 2PI approximation itself.
 This section studies the use of Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

 summation of the action term 
\begin_inset Formula $\gamma_{2\text{PI}}$
\end_inset

 or the equation of motion term 
\begin_inset Formula $\partial_{G}\gamma_{2\text{PI}}$
\end_inset

.
 First, consider Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


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\end_layout

\end_inset

 summation of the action which matches the series expansion up to order
 
\begin_inset Formula $N+M$
\end_inset

: 
\begin_inset Formula 
\begin{equation}
\Gamma\left[G\right]=\frac{1}{2}\ln\left(G^{-1}\right)+\frac{1}{2}m^{2}G+\frac{\sum_{n=0}^{N}A_{n}^{\left(\gamma\right)}\left(\lambda G^{2}\right)^{n}}{\sum_{n=0}^{M}B_{n}^{\left(\gamma\right)}\left(\lambda G^{2}\right)^{n}}+\mathrm{const.}
\end{equation}

\end_inset

The equation of motion becomes 
\begin_inset Formula 
\begin{align}
G^{-1} & =m^{2}+2\frac{\mathrm{d}}{\mathrm{d}G}\frac{\sum_{n=0}^{N}A_{n}^{\left(\gamma\right)}\left(\lambda G^{2}\right)^{n}}{\sum_{n=0}^{M}B_{n}^{\left(\gamma\right)}\left(\lambda G^{2}\right)^{n}}\nonumber \\
 & =m^{2}+\frac{4\sum_{n=0}^{N}\sum_{k=0}^{M}\left(n-k\right)A_{n}^{\left(\gamma\right)}B_{k}^{\left(\gamma\right)}\lambda^{n+k}G^{2\left(n+k\right)-1}}{\left[\sum_{n=0}^{M}B_{n}^{\left(\gamma\right)}\left(\lambda G^{2}\right)^{n}\right]^{2}}.\label{eq:hybrid-attempt-1}
\end{align}

\end_inset

Multiplying by the denominator this becomes a polynomial equation of degree
 
\begin_inset Formula $2\left(N+M\right)$
\end_inset

 as expected.
 This equation will have 
\begin_inset Formula $2\left(N+M\right)-1$
\end_inset

 spurious solutions as does the ordinary 2PI approximation of matching order.
 To take a specific example, consider 
\begin_inset Formula $\gamma_{2\text{PI}}$
\end_inset

 up to 
\begin_inset Formula $\mathcal{O}\left(\lambda^{4}\right)$
\end_inset

 and the matching 
\begin_inset Formula $\left(2,2\right)$
\end_inset

- and 
\begin_inset Formula $\left(1,3\right)$
\end_inset

-Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

 approximants: 
\begin_inset Formula 
\begin{align}
\gamma_{2\text{PI}} & =\frac{G^{2}\lambda}{8}-\frac{G^{4}\lambda^{2}}{48}+\frac{G^{6}\lambda^{3}}{48}-\frac{5G^{8}\lambda^{4}}{128}+\cdots\\
 & \approx\frac{\frac{G^{2}\lambda}{8}+\frac{113G^{4}\lambda^{2}}{480}}{1+\frac{41G^{2}\lambda}{20}+\frac{7G^{4}\lambda^{2}}{40}}\\
 & \approx\frac{G^{2}\lambda}{8\left(1+\frac{G^{2}\lambda}{6}-\frac{5G^{4}\lambda^{2}}{36}+\frac{113G^{6}\lambda^{3}}{432}\right)}.
\end{align}

\end_inset

The solution resulting from any of these versions of 
\begin_inset Formula $\gamma_{2\text{PI}}$
\end_inset

 cannot be written in closed form, however numerical solutions are shown
 in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hybrid-solution"

\end_inset

 for moderately small 
\begin_inset Formula $\lambda$
\end_inset

 (for 
\begin_inset Formula $\lambda$
\end_inset

 outside of this range, different roots become relevant).
 The hybrid solutions are more accurate than the fourth order 2PI solution,
 although the level of improvement depends strongly on the type of Pad
\begin_inset ERT
status open

\begin_layout Plain Layout


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\end_layout

\end_inset

 approximant employed.
 The trade-off is a loss of accuracy at large 
\begin_inset Formula $\lambda$
\end_inset

, as shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:hybrid-solution-large-lambda"

\end_inset

 for the best roots found.
 Each approximation decays with the correct leading 
\begin_inset Formula $\lambda^{-1/2}$
\end_inset

 power, however they differ by 
\begin_inset Formula $\mathcal{O}\left(1\right)$
\end_inset

 factors which are not negligible.
 Remarkably, the best approximation of those shown at large 
\begin_inset Formula $\lambda$
\end_inset

 is the 
\begin_inset Formula $\mathcal{O}\left(\lambda\right)$
\end_inset

 2PI approximation.
 For the greater computational expense the 
\begin_inset Formula $\left(2,2\right)$
\end_inset

-hybrid approximation is not noticeably an improvement at large coupling.
 The apparent clustering of the 
\begin_inset Formula $\mathcal{O}\left(\lambda^{4}\right)$
\end_inset

 2PI and 
\begin_inset Formula $\left(1,3\right)$
\end_inset

-hybrid approximations away from the exact value as 
\begin_inset Formula $\lambda\to\infty$
\end_inset

 stems from the minus sign of the leading term in 
\begin_inset Formula $\partial_{G}\gamma_{2\text{PI}}$
\end_inset

 in these approximations, suggesting that one should not use approximations
 with this property.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/hybrid-2pi-soln.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:hybrid-solution"

\end_inset

Comparison of exact 
\begin_inset Formula $\bar{G}$
\end_inset

, first and fourth order 2PI and 
\begin_inset Formula $\left(2,2\right)$
\end_inset

- and 
\begin_inset Formula $\left(1,3\right)$
\end_inset

-2PI-Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

 hybrid solutions for 
\begin_inset Formula $m=1$
\end_inset

 at small coupling.
 The 
\begin_inset Formula $\left(2,2\right)$
\end_inset

-hybrid solution lies almost on top of the exact solution.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename images/ch2/hybrid-2pi-soln-large-lambda.pdf
	width 100col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:hybrid-solution-large-lambda"

\end_inset

Comparison of exact 
\begin_inset Formula $\bar{G}$
\end_inset

, first and fourth order 2PI and 
\begin_inset Formula $\left(2,2\right)$
\end_inset

- and 
\begin_inset Formula $\left(1,3\right)$
\end_inset

-2PI-Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

 hybrid solutions at large coupling for 
\begin_inset Formula $m=1$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Next try Pad
\begin_inset ERT
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\begin_layout Plain Layout


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\end_layout

\end_inset

 summing the equation of motion: 
\begin_inset Formula 
\begin{equation}
G^{-1}=m^{2}+2\frac{G^{-1}\sum_{n=1}^{N}A_{n}^{\left(\partial\gamma\right)}\left(\lambda G^{2}\right)^{n}}{\sum_{n=0}^{M}B_{n}^{\left(\partial\gamma\right)}\left(\lambda G^{2}\right)^{n}},\label{eq:hybrid-attempt-2}
\end{equation}

\end_inset

where the explicit factor of 
\begin_inset Formula $G^{-1}$
\end_inset

 on the right hand side simply ensures the self-energy is an odd function
 of 
\begin_inset Formula $G$
\end_inset

 as it must be.
 This equation of motion is of degree 
\begin_inset Formula $\max\left(2N,2M+1\right)$
\end_inset

, which for typical choices of 
\begin_inset Formula $N$
\end_inset

 and 
\begin_inset Formula $M$
\end_inset

 results in a rough halving of the number of spurious solutions.
 To obtain an analytic result consider the first non-trivial approximant,
 i.e.
 
\begin_inset Formula $N=M=1$
\end_inset

.
 The resulting equation of motion is 
\begin_inset Formula 
\begin{equation}
G^{-1}=m^{2}+\frac{1}{2}\frac{\lambda G}{1+\frac{1}{3}\lambda G^{2}},
\end{equation}

\end_inset

which agrees with the usual 2PI equation of motion up to terms of order
 
\begin_inset Formula $\mathcal{O}\left(\lambda^{3}\right)$
\end_inset

, however it only has two spurious solutions instead of three.
 The physical solution obtained by matching at small 
\begin_inset Formula $\lambda$
\end_inset

 is 
\begin_inset Formula 
\begin{equation}
G_{\text{hy}}=\frac{1}{6m^{2}}\left(-1+\frac{36m^{4}-\lambda}{X^{1/3}}-\frac{X^{1/3}}{\lambda}\right),\label{eq:hybrid-soln}
\end{equation}

\end_inset

 where
\begin_inset Formula 
\[
X=\lambda^{3}-378\lambda^{2}m^{4}+18\lambda m^{2}\sqrt{\lambda\left(-2\lambda^{2}+144m^{8}+429\lambda m^{4}\right)}.
\]

\end_inset

Indeed, this matches perturbation theory up to 
\begin_inset Formula $\mathcal{O}\left(\lambda^{3}\right)$
\end_inset

 terms.
 However, the large 
\begin_inset Formula $\lambda$
\end_inset

 behaviour in this approximation is pathological: 
\begin_inset Formula 
\begin{equation}
G_{\text{hy}}\to-\frac{1}{2m^{2}}+\frac{18m^{2}}{\lambda}+\mathcal{O}\left(\lambda^{-3/2}\right),
\end{equation}

\end_inset

as 
\begin_inset Formula $\lambda\to\infty$
\end_inset

.
 This reflects the existence of an unphysical branch cut on the positive
 
\begin_inset Formula $\lambda$
\end_inset

 axis starting at 
\begin_inset Formula $\lambda/m^{4}=\frac{3}{4}\left(143+19\sqrt{57}\right)\approx214$
\end_inset

, which 
\begin_inset Formula $G_{\text{hy}}$
\end_inset

 has in addition to the expected cuts on the negative axis.
 The existence of this cut also renders the derivation of the spectral function
 invalid, meaning that 
\begin_inset Formula $G_{\text{hy}}$
\end_inset

 cannot be written in the form 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:spectral-function-def"

\end_inset

.
 It is not known at this stage whether other forms of Pad
\begin_inset ERT
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\begin_layout Plain Layout


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'e
\end_layout

\end_inset

 summed equations of motion lack these pathologies.
 Further development of hybrid approximation schemes is beside the main
 line of this thesis but may be a worthy topic for future work.
\end_layout

\begin_layout Section
Summary
\begin_inset ERT
status open

\begin_layout Plain Layout

}{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:ch2-Summary"

\end_inset


\end_layout

\begin_layout Standard
This chapter has investigated 
\begin_inset Formula $n$
\end_inset

-particle irreducible effective actions (
\begin_inset Formula $n$
\end_inset

PIEA) in quantum field theory.
 Green functions can be computed in a perturbation expansion in terms of
 Feynman diagrams or can be efficiently computed using the 
\begin_inset Formula $n$
\end_inset

PIEA, which were investigated and explicit derivations given for 
\begin_inset Formula $n=1,2$
\end_inset

 and 
\begin_inset Formula $3$
\end_inset

 in the case of a generic quartic scalar field theory.
 Finally, to illustrate the advantages of the 
\begin_inset Formula $n$
\end_inset

PIEA method over resummation methods, the 2PIEA was applied to an exactly
 solvable toy model and contrasted with Borel, Pad
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 and Borel-Pad
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 summation.
\end_layout

\begin_layout Standard
The study of the toy model revealed several interesting features.
 The perturbation theory has zero radius of convergence due to a branch
 cut on the negative coupling (
\begin_inset Formula $\lambda$
\end_inset

) axis, a fact which is invisible to perturbation theory.
 The theory is Borel summable, with the Borel sum giving the exact answer.
 However, Borel summation alone is not usually very helpful in practice.
 Pad
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 approximants well describe the Green function at weak coupling, though
 not at strong coupling.
 However, the combination of Borel and Pad
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 approximation yields an effective global approximation scheme for the Green
 function.
 The two point Green function of this theory admits a nice integral representati
on in terms of the spectral function (i.e.
 discontinuity of the branch cut) which shows that the Pad
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 approximants improve perturbation theory by allowing the spectral function
 to be approximated as a sum of delta functions and the Borel-Pad
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 method gives a continuous, albeit erratic and inaccurate, approximation
 to the spectral function.
 The 2PI approximation scheme surpasses perturbation theory, Pad
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 and Borel-Pad
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 approximants already at the leading non-trivial truncation.
 Like the Borel-Pad
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 method, 2PI approximations can develop branch points and represent the
 spectral function by a continuous distribution.
 However, the 2PI approximation is quantitatively superior at the leading
 truncation.
 These results are not entirely surprising because while, e.g., the Borel-Pad
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 method is a widely applicable general 
\begin_inset Quotes eld
\end_inset

black box
\begin_inset Quotes erd
\end_inset

 method, the self-consistent 2PI equations of motion are derived within
 a 
\emph on
particular
\emph default
 field theory of interest.
 This gives the 2PI method 
\begin_inset Quotes eld
\end_inset

insider information,
\begin_inset Quotes erd
\end_inset

 from which it should be able to construct a better approximation.
 This comes at the cost of spurious solutions which must be eliminated and
 the added difficulty of finding the 2PI effective action in the first place.
 Finally, a hybrid 2PI-Pad
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 scheme was introduced using Pad
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 approximants to partially resum the 2PI diagram series.
 The quality of the result depends strongly on the type of Pad
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 approximant used, with the best result found for the diagonal approximant.
 This hybrid approximation performs considerably better than the comparable
 2PI approximation at weak coupling, though not noticeably better at strong
 coupling.
\end_layout

\begin_layout Standard
\begin_inset Formula $n$
\end_inset

PIEAs are the subject of a rich literature and have found applications in
 diverse areas from early universe cosmology to nano-electronics (e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2015,Calzetta2008"

\end_inset

).
 The virtues of approximation schemes built on 
\begin_inset Formula $n$
\end_inset

PI effective actions are often explained in terms of a resummation of an
 infinite series of perturbative Feynman diagrams which are encapsulated
 in the non-perturbative Green function 
\begin_inset Formula $\Delta$
\end_inset

 and proper vertex functions 
\begin_inset Formula $V,\cdots,V^{\left(n\right)}$
\end_inset

, from which the 
\begin_inset Formula $n$
\end_inset

PI diagram series is built.
 However, 
\begin_inset Formula $n$
\end_inset

PIEAs are not resummation schemes: they are a self-consistent variational
 principle.
 The definition of the 
\begin_inset Formula $n$
\end_inset

PIEA in terms of the Legendre transform is crucial for the self-consistency
 of the scheme.
 The immediate practical consequence is that any modification of the 
\begin_inset Formula $n$
\end_inset

PIEA which does not derive from a consistent modified variational principle
 is very likely to be inconsistent.
 So, for instance, the consistency of recent attempts to improve the symmetry
 properties of 2PIEAs 
\begin_inset CommandInset citation
LatexCommand citep
key "Pilaftsis2013"

\end_inset

 is guaranteed by the existence of a suitable constrained variational principle,
 however, ad hoc attempts to modify the equations of motion to satisfy symmetrie
s will fail.
\end_layout

\begin_layout Standard
From the 
\begin_inset Formula $n$
\end_inset

PIEA one obtains a set of coupled integro-differential equations of motion
 for the 
\begin_inset Formula $1$
\end_inset

- through 
\begin_inset Formula $n$
\end_inset

-point correlation functions which are similar in spirit to the Schwinger-Dyson
 or BBGKY equations.
 However, the 
\begin_inset Formula $n$
\end_inset

PI equations of motion are automatically closed and require no further approxima
tion than a truncation of the diagram series to be used in practice.
 Further, time evolution of the equations of motion is well behaved compared
 to other schemes.
 The reason for this is that the non-linear nature of the equations of motion
 eliminates the secularities present in perturbation theory (see, e.g.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Berges2004,Berges2015"

\end_inset

).
 This gives 
\begin_inset Formula $n$
\end_inset

PIEAs an appealing degree of mechanisation and suitability especially for
 non-equilibrium time evolution problems.
 There are two types of outstanding difficulties however, both due to the
 non-linear nature of the equations of motion.
 The first is the renormalisation procedure, which has not been discussed
 in detail here.
 A renormalisation procedure based on a Bogoliubov-Parasiuk-Hepp-Zimmerman
 (BPHZ) type of analysis will be discussed in Chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:chap4"

\end_inset

 where it is applied to the symmetry improved 3PIEA.
 The renormalisation of 
\begin_inset Formula $n$
\end_inset

PIEA remains an open problem in general, however.
 The second problem is violation of symmetries by truncations of 
\begin_inset Formula $n$
\end_inset

PIEA.
 Discussion of this problem and attempts at its resolution form the bulk
 of the rest of this thesis.
\end_layout

\end_body
\end_document