%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: quantum 6j-symbols and TQFT state
%%% Primary Category Code: 03.
%%% Filename: Quantum6jSymbolsAndTQFTStateOnTheTetrahedron.tex
%%% Version: 8
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\section{Topological Quantum Field (TQFT) State on the Tetrahedron}
Let us consider first a \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} tetrahedron whose corners will
have attached to them the \htmladdnormallink{TQFT}{http://planetphysics.us/encyclopedia/SUSY2.html} symbols representing a TQF state in terms of so-called `j-symbols' as further detailed next. The vertices of the tetrahedron are located at the points $(a_x, a_y, a_z)$, $(b_x, b_y, b_z)$, $(c_x, c_y, c_z)$, and $(d_x, d_y, d_z)$, that will be labeled, respectively, as $1,2,3,4$.
\begin{definition}
A \emph{\htmladdnormallink{quantum field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (\htmladdnormallink{QF}{http://planetphysics.us/encyclopedia/SUSY2.html}) state} $\phi$ provides a total order denoted by $ \leq_{\phi}$ on the
vertices of the tetrahedron, and thus also assigns a `direction' to each edge of the tetrahedron--from the
apparently `smaller' to the apparently `larger' vertices; a QF state also labels each edge $ e = (i,j)$,
by an element $\phi_1 (e)$ of $B_A$, which is a \emph{distinguished basis of a fusion algebra} $\A$, that is, a finite-dimensional, unital, involutive algebra over $\mathbb{C}$ --the \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} of complex numbers. Moreover, the QF state assigns an element ${\phi}^2 (f)$ --called an intertwiner-- of a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $$\H_{\phi}(f)= {\H^{\phi_1 (ik)}}_{\phi_1 (jk), \phi_1 (ij)}$$
to each face $f=(ijk)$ of the tetrahedron, such that $i\prec_{\phi} j \prec_{\phi}k .$
\end{definition}
\subsection{Remarks}
A \emph{\htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{quantum field theory}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}} ({\em TQFT}) is described as a mathematical approach to quantum field theory that allows the \htmladdnormallink{computation}{http://planetphysics.us/encyclopedia/LQG2.html} of \htmladdnormallink{topological invariants}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} of \htmladdnormallink{quantum state spaces}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} (\htmladdnormallink{QSS}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html}), usually for cases of lower dimensions encountered in certain condensed phases or strongly correlated (quantum) \htmladdnormallink{superfluid}{http://planetphysics.us/encyclopedia/QuantumStatisticalTheories.html} states. TQFT has some of its origins in \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} as well as Michael Atiyah's research; this was followed by Edward Witten, Maxim Kontsevich, Jones and Donaldson, who all have been awarded Fields Medals for \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} related to topological quantum field theory; furthermore, Edward Witten and Maxim Kontsevich shared in 2008 the Crafoord prize for TQFT related work. As an example, Maxim Kontsevich introduced the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of homological mirror (quantum) symmetry in \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} to a mathematical conjecture in \htmladdnormallink{superstring}{http://planetphysics.us/encyclopedia/10DBrane.html} theory.
\begin{thebibliography}{9}
\bibitem{VKVS2k1}
V. Kodyiyalam and V. S. Sunder. 2001.
\htmladdnormallink{{\em Topological Quantum Field Theories From Subfactors}}{http://planetmath.org/?op=getobj&from=books&id=174} ., Chapman and Hall/CRC.
\end{thebibliography}
\subsection{Maple eps Graphics}
\begin{maplelatex}\mapleinline{inert}{2d}{%[poincare, generate_ic, zoom, hamilton_eqs]}{
$[{\it poincare},{\it generate\_ic},{\it zoom},{\it hamilton\_eqs}\\
\mbox{}]$}
\end{maplelatex}\begin{maplelatex}\begin{Maple Heading 1}{\textbf{The Toda Hamiltonian}}\end{Maple Heading 1}
\end{maplelatex}
\begin{maplelatex}\begin{Maple Normal}{\textbf{Reference:}A.J. Lichtenberg and M.A. Lieberman, "Regular and Stochastic Motion", \textit{Applied Mathematical Sciences} 38 (New York: Springer Verlag, 1994).}\end{Maple Normal}
\textbf{H := 1/2*(p1\symbol{94}2 + p2\symbol{94}2) + 1/24*(exp(2*q2+2*sqrt(3)*q1) + exp(2*q2-2*sqrt(3)*q1) + exp(-4*q2))-1/8;}\end{maplelatex}
\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%H := p1^2/2+p2^2/2+1/24*exp(2*q2+2*3^(1/2)*q1)+1/24*exp(2*q2-2*3^(1/2)*q1)+1/24*exp(-4*q2)-1/8}{
$H\, := \,1/2\,{{\it p1}}^{2}+1/2\,{{\it p2}}^{2}+1/24\,{e^{2\,{\it q2}+2\,\sqrt {3}{\it q1}}}\\
\mbox{}+1/24\,{e^{2\,{\it q2}-2\,\sqrt {3}{\it q1}}}+1/24\,{e^{-4\,{\it q2}}}-1/8$}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\textbf{H, t=-150..150, \{[0,.1,1.4,.1,0]\}:}}{}
\end{mapleinput}
\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}{\textbf{poincare(%, stepsize=.05,iterations=5);}(12 sec.)}\end{Maple Normal}
}{}
\end{mapleinput}
\mapleresult
\begin{maplelatex}\mapleinline{inert}{2d}{%_____________________________________________________}{
\[{\it \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\]}
\end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{%`H = .99005020`, ` Initial conditions:`, t = 0, p1 = .1, p2 = 1.4, q1 = .1, q2 = 0}{
$\mbox {{\tt `H = .99005020`}},\,\mbox {{\tt ` Initial conditions:`}},\,t=0,\,{\it p1}= 0.1,\,{\it p2}= 1.4,\,{\it q1}= 0.1,\,{\it q2}=0$}
\end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{%`Number of points found crossing the (p1,q1) plane: 127`}{
$\mbox {{\tt `Number of points found crossing the (p1,q1) plane: 127`}}$}
\end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{%`Maximum H deviation : .5740000000e-5 %`}{
$\mbox {{\tt `Maximum H deviation : .5740000000e-5 \%`}}$}
\end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{%_____________________________________________________}{
\[{\it \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\]}
\end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{%`Time consumed: 12 seconds`}{
$\mbox {{\tt `Time consumed: 12 seconds`}}$}
\end{maplelatex}
\begin{maplelatex}\begin{Maple Normal}{Figure 1.a. shows a 2-D surface-of-section (2PS) over the q2=0 plane, with 127 intersection points lying on smooth curves. }\end{Maple Normal}
\end{maplelatex}\begin{mapleinput}
\mapleinline{active}{1d}{\begin{Maple Normal}{\textbf{poincare(%%,stepsize=.05,iterations=5,scene=[p2,q2]);}(13 sec.)}\end{Maple Normal}
}{}
\end{mapleinput}
\htmladdnormallink{MapleForPlanetPhysics}{http://planetphysics.org/encyclopedia/MapleForPlanetPhysics.html}
\end{document}
