0_researchnotes.tex

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% 0_researchnotes.tex                                                         %
% by: David A. Clarke                                                         %
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\documentclass[12pt]{book}


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% Normal preamble stuff.
\usepackage[title]{appendix}
\usepackage{mathtools,tensor,physics,slashed,textcomp}
\usepackage[version=4]{mhchem} % version is to suppress an annoying warning
\usepackage{microtype} % should help with hyphenation
\usepackage{xcolor,feynmf,subcaption} % stuff for feynman diagrams


% davitex and required packages
\usepackage{amsmath,amssymb,graphicx,multirow,tabularx}
\usepackage{davitex}


% I do not want to use this in general; I was just too lazy to make the
% replacements when porting code over.
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\newcommand{\st}{s_{\theta}}

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% If you click on citations or equation references, you will be take
% to the corresponding page. 
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\pdfminorversion=7
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% Put references at the end of each chapter, directly after the
% end of the chapter, call them "References" instead of
% "Bibliography", and use numbers with square brackets.
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\let\oldbibliography\bibliography
\renewcommand{\bibliography}[1]{{%
  \let\chapter\section%
  \oldbibliography{#1}}}
\renewcommand\bibname{References}
\setcitestyle{numbers,square}


% Make index and let it appear in table of contents.
\usepackage{makeidx}
\makeindex
\usepackage[totoc]{idxlayout}


% Prevent hyphenation.
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\overfullrule=2cm


% Theorems and stuff.
\usepackage{amsthm}
\usepackage[many]{tcolorbox}
\tcbuselibrary{theorems}
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% Some environments for some front matter.
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  {\centering\chapter*{}}
  {\clearpage}
\newenvironment{listofabbrev}
  {\chapter*{List of Abbreviations}}
  {\clearpage}


% Feynman rule references
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\newcommand{\F}[1]{\colorbox{light}{{F}{#1}}}


% Note that book class by default is formatted to be printed back-to-back.
\title{\bf Lattice Research Notes} 
\author{David Clarke}


\begin{document}
\frontmatter                            % roman page no., suppress chapter
\maketitle                              % print title page


\begin{frontstuff} % ----------------------------------------------- FRONT STUFF


\section*{Logic Symbols}
\begin{tabular}{ll}
$\exists$       & There exists \\
$\forall$       & For all \\
$\wedge$        & Logical and; wedge product \\
$\lor$          & Logical or \\ 
$\neg$          & Logical negation \\
$\in$           & Is a member of the set \\
$\subseteq$     & Is a subset of \\ 
$\subset$     & Is a proper subset of \\ 
\end{tabular}
\clearpage


\section*{Math Symbols}
\begin{tabular}{ll}
$\leq$          & Less than or equal to; is a subgroup of\\
$\approx$       & Approximate equality \\
$\cong$         & Isomorphic\\
$\equiv$        & Is defined as; modular congruency \\
$\sim$          & Equivalence relation \\
$\simeq$        & Asymptotic equality \\
$\propto$       & Is proportional to \\
$\mathbb{C}$    & Complex numbers \\
$\pr{X}$        & Probability of event $X$ \\
$\co_x$         & $\cos(x)$ \\
$\oplus$        & Direct sum; relativistic velocity addition  \\
$\otimes$       & Direct product \\ 
$\log$          & Natural logarithm \\
$\mathbb{N}$    & Natural numbers \\
$\mathcal{O}$   & Big O notation; topology on a set \\ 
$\mathbb{Q}$    & Rational numbers \\
$q$             & Gaussian or Student difference test; goodness-of-fit \\
$\mathbb{R}$    & Real numbers \\
$\s_x$          & $\sin(x)$ \\
$\SU(N)$        & Special unitary group of degree $N$ \\
$\U(N)$         & Unitary group of degree $N$ \\
O$(N)$          & Orthogonal group of dimension $N$ \\
$S_V$           & Symmetric group on the set $V$ \\
$\tauint$       & Integrated autocorrelation time \\
$\wedge$        & Logical and; wedge product \\
$\mathbb{Z}$    & Integers \\
$\mathbb{Z}_n$  & Integers modulo $n$ \\
$G$             & A group; Newton's gravitational constant\\
$g$             & A group element; $\SU(N_c)$ bare coupling\\
\end{tabular}
\clearpage


\section*{Physics Symbols}
\begin{tabular}{ll}
$a$             & Lattice spacing \\
$\beta$         & $\SU(N_c)$ coupling constant; beta function; critical exponent \\
$\chi_O$        & Susceptibility of observable $O$\\
$c$             & Speed of light\\
$G$             & A group; Newton's gravitational constant\\
$g$             & A group element; $\SU(N_c)$ bare coupling\\
$\hbar$         & Planck's constant \\
$k_B$           & Boltzmann's constant \\
$N_c$           & Number of colors \\
$N_f$           & Number of fermion flavors \\
$N_s$           & Lattice extension in a spatial direction \\
$N_\tau$        & Lattice extension in temperature/Euclidean time direction\\
$\sigma$        & String tension\\
$\sigma_i$      & A Pauli matrix\\
$T_c$           & Deconfining phase transition temperature\\
$u_0$           & Tadpole factor\\
\end{tabular}
\clearpage


\section*{Abbreviations}
\begin{tabular}{ll}
BC      &       Boundary condition \\
CW      &       Clockwise \\
CCW     &       Counterclockwise\\
CDF     &       Cumulative distribution function \\
CLT     &       Central limit theorem\\
CW      &       Clockwise\\
EM      &       Electromagnetic\\
FLOP    &       Floating point operation\\
HB      &       Heat bath\\
HISQ    &       Highly improved staggered quarks\\
HMC     &       Hybrid Monte Carlo\\
HRG     &       Hadron resonance gas\\
HVP     &       Hadronic vacuum polarization\\
LFT     &       Lattice field theory\\
LGT     &       Lattice gauge theory\\
LQCD    &       Lattice QCD\\
LHS     &       Left hand side\\
LLN     &       Law of large numbers\\
MCMC    &       Markov chain Monte Carlo\\
MCOR    &       Monte Carlo plus over-relaxation \\
MPI     &       Message passing interface \\
OOP     &       Object-oriented programming \\
OR      &       Over-relaxation \\
PDF     &       Probability distribution function \\
QCD     &       Quantum chromodynamics \\
QFT     &       Quantum field theory \\
QM      &       Quantum mechanics \\
RG      &       Renormalization group \\
RHS     &       Right hand side \\
RNG     &       Random number generator \\
SM      &       Standard Model \\
SSB     &       Spontaneous symmetry breaking\\
SVD     &       Singular value decomposition\\
UV      &       Ultraviolet \\
VP      &       Vacuum polarization \\
WLOG    &       Without loss of generality
\end{tabular}
\clearpage


\section*{Constants in Various Unit Systems}
\index{units!SI}\index{units!Gaussian}
\begin{tabularx}{\linewidth}{LlLll}
 & {\it SI} && {\it Gaussian}&\\ 
\vspace{2mm}\\
$\hbar$  & $1.05457\times 10^{-34}$ & [J\,s]
         & & \\
$k_B$    & $1.38065\times 10^{-23}$ & [J\,K$^{-1}$]
         & & \\
$c$      & $2.99792\times 10^8$ & [m\,s$^{-1}$]
         & & \\
$e$      & $1.60218\times 10^{-19}$ & [C] 
         & $4.80321\times 10^{-10}$ & [esu]\\
$G$      & $6.674\times 10^{-11}$ & [m$^3$kg$^{-1}$s$^{-2}$]
         & & \\
$m_e$    & $9.10938\times 10^{-31}$ & [kg]
         & & \\
$m_p$    & $1.67262\times 10^{-27}$ & [kg]
         & & \\
\end{tabularx}
\clearpage


\end{frontstuff} % ----------------------------------------------------- PREFACE


\chapter{Acknowledgements}

I would like to start off by thanking some people with whom discussions
have influenced the material in these notes in one way or another.
These people are
D. Bollweg,
P. Boyle,
H. Dick,
L. Dini,
A. X. El-Khadra,
G. Gagliardi,
J. Goswami,
O. Kaczmarek,
F. Karsch,
A. Kronfeld,
S. Lahert,
A. Lahiri,
L. Mendoza,
Q. Mabanta,
L. Mazur,
E. Neil,
L. Reina,
H. Roch,
H. Sandmeyer,
M. Sarkar,
A. Sciarra,
C. Schmidt,
P. Scior,
M. Semaan,
S. Sharma,
S. Singh,
and
A. Yunesi.
Hopefully I haven't missed anyone.

Next, I would like to give thanks to G. Endr\"odi whose lattice seminar
has helped me learn many aspects of lattice QCD in an accessible way while
providing a forum for judgement-free and open discussion.

Finally I want to especially thank B. Berg, who took my desire to be a
physicist seriously and has continually been a patient advocate for me,
and C. DeTar, who besides being extremely knowledgable and enthusiastic about
science, is perhaps the most empathetic physicist I've ever worked with.


\chapter{Preface}
This ``book" is a collection of research notes that I made while researching in
physics. The first chapters are dedicated to mainly mathematics, and the last
chapters are dedicated to mainly physics, with an aim toward lattice field
theory. 

Throughout the book, particularly in the math sections, I have set aside
Propositions, Lemmas, and Theorems, which usually contain proofs. 
In a somewhat arbitrary and
subjective way, I have mostly used the proof environment as a way to encapsulate
a fact that I deem particularly important along with its argument.
In some cases, these proofs are fully rigorous, while in other cases, they are 
rather a proof sketch,
containing the most salient points of an argument. 

Attached at the end are also some appendices. 
\apref{ap:spec_math} includes some special topics in mathematics 
that, while sometimes useful for a physicist, do not really fit in one of 
the mathematics chapters at the beginning.
Later appendices are for topics in physics that I found interesting but are
not particularly relevant to my research.
I may eventually want
to develop some sections of these appendices into full chapters.


\tableofcontents                        % print table of contents
\mainmatter                             % arabic page no., include chapter


% These are the chapters of the book.
\include{ch_algebra}
\include{ch_complex_analysis}
%\include{ch_differential_geometry}
\include{ch_probability_and_statistics}
\include{ch_computer}
\include{ch_statistical_physics}
\include{ch_RG}
\include{ch_QFT}
\include{ch_LGT_prelim}
\include{ch_LFT_fermions}
\include{ch_LFT_improved}
\include{ch_LFT_MCMC}
\include{ch_LFT_realPhysics}


% Here are the appendices.
\begin{appendices}
\include{ap_spec_math}
\include{ap_spec_neutrino}
\include{ap_spec_higgs}
\include{ap_spec_SM}
%\include{ap_spec_cosmology}
\include{ap_spec_bell}
\end{appendices}


\printindex


\end{document}