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%%% Primary Title: mathematical foundations of quantum field theories
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%%% Filename: MathematicalFoundationsOfQuantumFieldTheories.tex
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\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}
