Talk:PlanetPhysics/Mathematical Foundations of Quantum Field Theories

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: mathematical foundations of quantum field theories
%%% Primary Category Code: 00.
%%% Filename: MathematicalFoundationsOfQuantumFieldTheories.tex
%%% Version: 1
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% define commands here
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym}
\usepackage{xypic}
\usepackage[mathscr]{eucal}
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}
\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@
}}}
\numberwithin{equation}{section}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
\newcommand{\G}{\mathcal G}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathbb G}}
\newcommand{\dgrp}{{\mathbb D}}
\newcommand{\desp}{{\mathbb D^{\rm{es}}}}
\newcommand{\Geod}{{\rm Geod}}
\newcommand{\geod}{{\rm geod}}
\newcommand{\hgr}{{\mathbb H}}
\newcommand{\mgr}{{\mathbb M}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathbb G)}}
\newcommand{\obgp}{{\rm Ob(\mathbb G')}}
\newcommand{\obh}{{\rm Ob(\mathbb H)}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\ghomotop}{{\rho_2^{\square}}}
\newcommand{\gcalp}{{\mathbb G(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\glob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\oset}[1]{\overset {#1}{\ra}}
\newcommand{\osetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}

\begin{document}

 \subsection{Mathematical Foundations of Quantum Field Theories (QFT)}

\subsubsection{QED, QCD, Electroweak and Other Quantum Field Theories}

\begin{enumerate}
\item \textit{Quantum chromodynamics or \htmladdnormallink{QCD}{http://planetphysics.us/encyclopedia/LQG2.html}:} the advanced, standard mathematical and quantum physics treatment of strong \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} or \htmladdnormallink{nuclear interactions}{http://planetphysics.us/encyclopedia/HotFusion.html} such as those among \htmladdnormallink{quarks}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} and \htmladdnormallink{gluons}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html}, (or \htmladdnormallink{partons}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} and \htmladdnormallink{mesons}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html}), that have an intrinsic threefold, or eightfold \htmladdnormallink{quantum symmetry}{http://planetphysics.us/encyclopedia/HilbertBundle.html} described by the `\htmladdnormallink{quantum' group}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} {\em SU(3)} (which was first reported in 1964 by the US Nobel Laureate Murray Gell-Mann and others);
\item {\em \htmladdnormallink{quantum electrodynamics}{http://planetphysics.us/encyclopedia/QED.html} \htmladdnormallink{QED}{http://planetphysics.us/encyclopedia/LQG2.html}}: that involves {\em U(1)} symmetry, is the advanced, standard mathematical and quantum physics treatment of electromagnetic interactions through several approaches, the more advanced including the path-integral approach by Feynman, Dirac's \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} and QED equations, thus including either special or \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} formulations of electromagnetic phenomena;
\item Young--Mills theories;
\item Electroweak interactions: {\em SU(2)} Symmetry;
\item \htmladdnormallink{Algebraic Quantum Field Theories}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} (\htmladdnormallink{AQFT}{http://planetphysics.us/encyclopedia/SUSY2.html});
\item \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} (\htmladdnormallink{HQFT}{http://planetphysics.us/encyclopedia/QAT.html}) and \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{QFT's}{http://planetphysics.us/encyclopedia/HotFusion.html} (\htmladdnormallink{TQFT}{http://planetphysics.us/encyclopedia/SUSY2.html});
\item \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} (\htmladdnormallink{QG}{http://planetphysics.us/encyclopedia/SUSY2.html}) and related theories.
\end{enumerate}

\subsubsection{Extended Quantum Symmetries}

This obviates the need for `more fundamental' , or \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html}, such as those afforded by either several larger \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} such as $SU(3) \times SU(2) \times U(1)$ (and their \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}) in \htmladdnormallink{SUSY}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html}, or by spontaneously broken, multiple (`or localized') symmetries of a less restrictive kind present in `\htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}' as for example in \htmladdnormallink{weak Hopf algebra}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html} representations. More generally, such extended quantum symmetries can be realized as \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html}, {\em $G_{lc}$} {\em unitary} representations, and even more `powerful' structures to the higher dimensional (quantum) symmetries of \htmladdnormallink{quantum double groupoids}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html}, quantum \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}, \htmladdnormallink{quantum categories/}{http://planetphysics.us/encyclopedia/QuantumCategories.html} \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html} in \htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}, and/or quantum \htmladdnormallink{supersymmetry superalgebras}{http://planetphysics.us/encyclopedia/HamiltonianAlgebroid3.html} (or graded `\htmladdnormallink{Lie' algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html}, see- for example- the QFT ref. \cite{Weinberg2003} discussing \htmladdnormallink{superalgebras}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} in quantum gravity).

Thus, certain finite \htmladdnormallink{irreducible representations}{http://planetphysics.us/encyclopedia/PureState.html} correspond to `elementary' (quantum) \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} and \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} symmetry
representations have corresponding quantum obsevable \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}, such as the Casimir operators. A well-known case is that of Pauli \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} that are representations of the special unitary group $SU(2)$. \htmladdnormallink{Supersymmetry}{http://planetphysics.us/encyclopedia/Supersymmetry.html}, \htmladdnormallink{supergroups}{http://planetphysics.us/encyclopedia/Paragroups.html} and
superoperators further expand SUSY to quantum gravity and \htmladdnormallink{quantum statistical mechanics}{http://planetphysics.us/encyclopedia/QuantumStatisticalTheories.html}.

\begin{thebibliography}{9}
\bibitem{Weinberg2003}
S. Weinberg. 2003. Quantum Field Theories, vol. 1-3, Cambridge University Press: Cambridge, UK.
\end{thebibliography} 

\end{document}