Talk:PlanetPhysics/Categorical Quantum LM Algebraic Logic 2
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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: quantum field theories (QFT) %%% Primary Category Code: 03.65.Fd %%% Filename: CategoricalQuantumLMAlgebraicLogic2.tex %%% Version: 12 %%% 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}
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This topic links the general framework of \htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} to \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} symmetries
and other relevant mathematical \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} utilized to represent \htmladdnormallink{quantum fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} and their fundamental properties.
\subsection{Fundamental, mathematical concepts in quantum field theory }
\emph{Quantum field theory (QFT)} is the general framework for describing the physics of relativistic quantum \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}, such as, notably, accelerated elementary \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html}.
\htmladdnormallink{Quantum electrodynamics}{http://planetphysics.us/encyclopedia/QED.html} \emph{(QED)}, and QCD or quantum chromodynamics are only two distinct theories among several quantum field theories, as their fundamental \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} correspond, respectively, to very different-- $U(1)$ and $SU(3)$-- group symmetries. This obviates the need for `more fundamental' , or \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html}, such as those afforded by either larger groups such as $SU(3) \times SU(2) \times U(1)$ or spontaneously broken, special symmetries of a less restrictive kind present in `\htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/QuantumGroupoids.html}' as for example in \htmladdnormallink{weak Hopf algebra}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html} representations, or in \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html}, $G_{lc}$ unitary representations, and so on, 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},quantum \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperdiagramsAsHeterofunctors.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, their full development in a recent QFT textbook \cite{Weinberg2003} that lead to \htmladdnormallink{superalgebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html} in \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} or \htmladdnormallink{QCD}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html}.
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\bibitem{AS} E. M. Alfsen and F. W. Schultz: \emph{Geometry of State Spaces of Operator Algebras}, Birkh\"auser, Boston--Basel--Berlin (2003).
\bibitem{Y} D.N. Yetter., TQFT's from homotopy 2-types. \textit{J. Knot Theor}. \textbf{2}: 113--123(1993).
\bibitem{Weinberg2003} S. Weinberg.: \emph{The Quantum Theory of Fields}. Cambridge, New York and Madrid: Cambridge University Press, Vols. 1 to 3, (1995--2000).
\bibitem{Weinstein} A. Weinstein : Groupoids: unifying internal and external symmetry, \emph{Notices of the Amer. Math. Soc.} \textbf{43} (7): 744--752 (1996).
\bibitem{WB} J. Wess and J. Bagger: \emph{Supersymmetry and Supergravity}, Princeton University Press, (1983).
\bibitem{WJ1} J. Westman: Harmonic analysis on groupoids, \textit{Pacific J. Math.} \textbf{27}: 621-632. (1968).
\bibitem{WJ1} J. Westman: Groupoid theory in algebra, topology and analysis., \textit{University of California at Irvine} (1971).
\bibitem{Wickra} S. Wickramasekara and A. Bohm: Symmetry representations in the rigged Hilbert space formulation of quantum mechanics, \emph{J. Phys. A} \textbf{35}(3): 807-829 (2002).
\bibitem{Wightman1} Wightman, A. S., 1956, Quantum Field Theory in Terms of Vacuum Expectation Values, Physical Review, \textbf{101}: 860--866.
\bibitem{Wightman--Garding3} Wightman, A.S. and Garding, L., 1964, Fields as Operator--Valued Distributions in Relativistic Quantum Theory, Arkiv f\"ur Fysik, 28: 129--184.
\bibitem{Woronowicz1} S. L. Woronowicz : Twisted {\em SU(2)} group : An example of a non--commutative differential calculus, RIMS, Kyoto University \textbf{23} (1987), 613--665.
\end{thebibliography}
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