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notation.lyx
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notation.lyx
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#LyX 2.2 created this file. For more info see http://www.lyx.org/
\lyxformat 508
\begin_document
\begin_header
\save_transient_properties true
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\textclass jcuthesis
\options sort&compress
\use_default_options true
\master thesis_main.lyx
\begin_modules
theorems-ams
theorems-ams-extended
theorems-sec
\end_modules
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\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine natbib
\cite_engine_type numerical
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
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\tracking_changes false
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\html_be_strict false
\end_header
\begin_body
\begin_layout Chapter*
Notation
\begin_inset CommandInset label
LatexCommand label
name "chap:Notation"
\end_inset
\end_layout
\begin_layout Standard
This thesis uses high-energy physics (HEP) conventions.
That is
\begin_inset Formula $\hbar=c=k_{B}=1$
\end_inset
and the metric is
\begin_inset Formula $\eta_{\mu\nu}=\mathrm{diag}\left(+1,-1,-1,-1\right)$
\end_inset
.
Loop counting factors of
\begin_inset Formula $\hbar$
\end_inset
will be kept explicit.
The repeated index summation convention is used.
Often the deWitt summation convention is used.
That is, for quantities with internal symmetry indices accompanied by spacetime
arguments, repeated indices imply integration over the corresponding spacetime
arguments.
So for example
\begin_inset Formula
\[
\phi_{a}\phi_{a}=\sum_{a=1}^{N}\int\mathrm{d}^{4}x\ \phi_{a}\left(x\right)\phi_{a}\left(x\right),
\]
\end_inset
and
\begin_inset Formula
\[
V_{abc}\phi_{a}\Delta_{bc}=\sum_{a,b,c=1}^{N}\int\mathrm{d}^{4}x\ \mathrm{d}^{4}y\ \mathrm{d}^{4}z\ V_{abc}\left(x,y,z\right)\phi_{a}\left(x\right)\Delta_{bc}\left(y,z\right).
\]
\end_inset
The final expression can always be reconstructed uniquely from the short
version of it.
Where the deWitt convention is not used the intended meaning will be obvious
from context.
Often functional convolutions are implied, so for example
\begin_inset Formula
\[
\mathrm{Tr}\left(\Delta_{0}^{-1}\Delta\right)=\Delta_{0ab}^{-1}\Delta_{ba}=\sum_{a,b=1}^{N}\int\mathrm{d}^{4}x\ \mathrm{d}^{4}y\ \Delta_{0ab}^{-1}\left(x,y\right)\Delta_{ba}\left(y,x\right),
\]
\end_inset
and
\begin_inset Formula
\[
\varphi K\varphi=\varphi_{a}K_{ab}\varphi_{b}=\sum_{a,b=1}^{N}\int\mathrm{d}^{4}x\ \mathrm{d}^{4}y\ \varphi_{a}\left(x\right)K_{ab}\left(x,y\right)\varphi_{b}\left(y\right),
\]
\end_inset
etc.
\end_layout
\begin_layout Standard
Time variables can be chosen to exist on different contours depending on
the application being considered.
Most manipulations are agnostic about the contour chosen.
Where relevant the contour in use is specified.
The choices are:
\end_layout
\begin_layout Itemize
\emph on
Zero temperature, Real time
\emph default
or
\emph on
Minkowski time
\emph default
contours (Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Minkowski-contour"
\end_inset
):
\begin_inset Formula $t$
\end_inset
runs from
\begin_inset Formula $-\infty+i\epsilon$
\end_inset
to
\begin_inset Formula $+\infty-i\epsilon$
\end_inset
where
\begin_inset Formula $\epsilon\to0^{+}$
\end_inset
is an infinitesimal convergence parameter,
\end_layout
\begin_layout Itemize
\emph on
Equilibrium, Imaginary time
\emph default
or
\emph on
Matsubara
\emph default
contours (Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Matsubarai-contour"
\end_inset
):
\begin_inset Formula $t$
\end_inset
runs from
\begin_inset Formula $0$
\end_inset
to
\begin_inset Formula $-i\beta$
\end_inset
with periodic boundary conditions,
\end_layout
\begin_layout Itemize
\emph on
Non-equilibrium, Closed time path
\emph default
or
\emph on
Schwinger-Keldysh
\emph default
contours (Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Schwinger-Keldysh-contour"
\end_inset
):
\begin_inset Formula $t$
\end_inset
runs on a multi-branch contour from
\begin_inset Formula $t_{0}$
\end_inset
up to
\begin_inset Formula $+\infty$
\end_inset
, then back down from
\begin_inset Formula $+\infty-i\epsilon$
\end_inset
to
\begin_inset Formula $t_{0}-i\epsilon$
\end_inset
then down the imaginary axis to
\begin_inset Formula $t_{0}-i\beta$
\end_inset
which is identified with
\begin_inset Formula $t_{0}$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
centering
\end_layout
\begin_layout Plain Layout
\backslash
begin{tikzpicture}
\end_layout
\begin_layout Plain Layout
\backslash
draw (0,-1) -- (0,1); % Axis
\end_layout
\begin_layout Plain Layout
\backslash
draw (-5.5,0) -- (5.5,0);
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
draw (4.8, 1) -- (4.8, 0.7) -- (5.1, 0.7);
\end_layout
\begin_layout Plain Layout
\backslash
node at (5, 0.9) { $t$ };
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
node at (-10pt, -5pt) { $0$ };
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
% Contour line
\end_layout
\begin_layout Plain Layout
\backslash
draw[thick,xshift=0pt,yshift=0pt,
\end_layout
\begin_layout Plain Layout
decoration={ markings, % This schema allows for fine-tuning the positions
of arrows
\end_layout
\begin_layout Plain Layout
mark=at position 0.25 with {
\backslash
arrow{latex}},
\end_layout
\begin_layout Plain Layout
mark=at position 0.75 with {
\backslash
arrow{latex}}},
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postaction={decorate}]
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(-5.5,4pt) -- (5.5,-4pt);
\end_layout
\begin_layout Plain Layout
\backslash
end{tikzpicture}
\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:Minkowski-contour"
\end_inset
The Minkowski contour in the complex time plane.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
centering
\end_layout
\begin_layout Plain Layout
\backslash
begin{tikzpicture}
\end_layout
\begin_layout Plain Layout
\backslash
draw (0,-5.5) -- (0,1); % Axis
\end_layout
\begin_layout Plain Layout
\backslash
draw (-.5,0) -- (1,0);
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
draw (.8, 1) -- (.8, 0.7) -- (1.1, 0.7);
\end_layout
\begin_layout Plain Layout
\backslash
node at (1, 0.9) { $t$ };
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
node at (-10pt, -5pt) { $0$ };
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
draw (-4pt,-5) -- (4pt,-5);
\end_layout
\begin_layout Plain Layout
\backslash
node at (-15pt, -5) { $- i
\backslash
beta$ };
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
draw[thick,xshift=2pt,yshift=0pt,
\end_layout
\begin_layout Plain Layout
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(0,0) -- (0,-5);
\end_layout
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\backslash
end{tikzpicture}
\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:Matsubarai-contour"
\end_inset
The Matsubara contour in the complex time plane.
\begin_inset Formula $0$
\end_inset
and
\begin_inset Formula $-i\beta$
\end_inset
are identified.
\end_layout
\end_inset
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wide false
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status open
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\backslash
draw (0,-5.5) -- (0,1); % Axis
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\backslash
draw (-1,0) -- (5.5,0);
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
draw (4.8, 1) -- (4.8, 0.7) -- (5.1, 0.7);
\end_layout
\begin_layout Plain Layout
\backslash
node at (5, 0.9) { $t$ };
\end_layout
\begin_layout Plain Layout
\end_layout
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\backslash
node at (-10pt, -5pt) { $0$ };
\end_layout
\begin_layout Plain Layout
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\backslash
draw (-4pt,-5.25) -- (4pt,-5.25);
\end_layout
\begin_layout Plain Layout
\backslash
node at (-15pt, -5.25) { $- i
\backslash
beta$ };
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
% Contour line
\end_layout
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\backslash
draw[thick,xshift=0pt,yshift=-2pt,
\end_layout
\begin_layout Plain Layout
decoration={ markings, % This schema allows for fine-tuning the positions
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\begin_layout Plain Layout
mark=at position 0.2 with {
\backslash
arrow{latex}},
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\begin_layout Plain Layout
mark=at position 0.8 with {
\backslash
arrow{latex}}},
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postaction={decorate}]
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(0,0) -- (5.5,0) arc (90:-90:2pt) -- (0.075,-5pt);
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\end_layout
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\backslash
draw[thick,xshift=2pt,yshift=-7pt,
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decoration={ markings, % This schema allows for fine-tuning the positions
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\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Schwinger-Keldysh-contour"
\end_inset
The Schwinger-Keldysh contour in the complex time plane.
The initial time is taken as
\begin_inset Formula $t_{0}=0$
\end_inset
.
\begin_inset Formula $0$
\end_inset
and
\begin_inset Formula $-i\beta$
\end_inset
are identified.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage clearpage
\end_inset
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\begin_layout Standard
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<lyxtabular version="3" rows="63" columns="2">
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:Symbols"
\end_inset
Commonly used symbols and their meanings
\end_layout
\end_inset
\end_layout
\end_inset
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<cell multicolumn="2" alignment="left" valignment="top" usebox="none">
\begin_inset Text
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\begin_inset Text
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Symbol
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Meaning
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\begin_inset Text
\begin_layout Plain Layout
QFT
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Quantum Field Theory
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
nPI
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n-Particle Irreducible (
\begin_inset Formula $n=1,2,3,\cdots$
\end_inset
)
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
nPIEA
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
nPI Effective Action
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
SI-nPIEA
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Symmetry Improved nPIEA
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
SSI-nPIEA
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Soft Symmetry Improved nPIEA
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
CTP
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Closed Time Path
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
WI
\end_layout