PlanetPhysics/Quantum 6J Symbols and TQFT State on the Tetrahedron

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Topological Quantum Field (TQFT) State on the Tetrahedron

Let us consider first a [[../CoIntersections/|regular]] tetrahedron whose corners will have attached to them the [[../SUSY2/|TQFT]] 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 $1,2,3,4$.

A \htmladdnormallink{quantum field {http://planetphysics.us/encyclopedia/CosmologicalConstant.html} ([[../SUSY2/|QF]]) state} $\varphi$ provides a total order denoted by ${\le }_{\varphi }$ 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 ${\varphi }_1(e)$ of $B_A$, which is a distinguished basis of a fusion algebra Failed to parse (unknown function "\A"): {\displaystyle \A} , that is, a finite-dimensional, unital, involutive algebra over $ℂ$ --the [[../CosmologicalConstant2/|field]] of complex numbers. Moreover, the QF state assigns an element ${\varphi }^2(f)$ --called an intertwiner-- of a [[../NormInducedByInnerProduct/|Hilbert space]] ${ℍ}_{\varphi }(f)={{ℍ}^{{\varphi }_1(ik)}}_{{\varphi }_1(jk),{\varphi }_1(ij)}$ to each face $f=(ijk)$ of the tetrahedron, such that $i{\prec }_{\varphi }j{\prec }_{\varphi }k.$

Remarks

A \htmladdnormallink{topological {http://planetphysics.us/encyclopedia/CoIntersections.html} [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theory]]} (TQFT ) is described as a mathematical approach to quantum field theory that allows the [[../LQG2/|computation]] of [[../ModuleAlgebraic/|topological invariants]] of [[../QuantumSpinNetworkFunctor2/|quantum state spaces]] ([[../QuantumSpinNetworkFunctor2/|QSS]]), usually for cases of lower dimensions encountered in certain condensed phases or strongly correlated (quantum) [[../QuantumStatisticalTheories/|superfluid]] states. TQFT has some of its origins in [[../PhysicalMathematics2/|theoretical physics]] 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 [[../Work/|work]] 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 [[../PreciseIdea/|concept]] of homological mirror (quantum) symmetry in [[../Bijective/|relation]] to a mathematical conjecture in [[../10DBrane/|superstring]] theory.

References

Maple eps Graphics

\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle [{\it poincare},{\it generate_ic},{\it zoom},{\it hamilton_eqs}\\ \mbox{}]} } \end{maplelatex}\begin{maplelatex}\begin{Maple Heading 1}{The Toda Hamiltonian }\end{Maple Heading 1} \end{maplelatex} \begin{maplelatex}\begin{Maple Normal}{Reference: A.J. Lichtenberg and M.A. Lieberman, "Regular and Stochastic Motion", Applied Mathematical Sciences 38 (New York: Springer Verlag, 1994).}\end{Maple Normal} 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}{ Failed to parse (syntax error): {\displaystyle 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}{H, t=-150..150, \{[0,.1,1.4,.1,0]\ :}}{} \end{mapleinput} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}{poincare( }{} \end{mapleinput}

\mapleresult \begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle {\it _____________________________________________________}} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \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}{ Failed to parse (syntax error): {\displaystyle \mbox {{\tt `Number of points found crossing the (p1,q1) plane: 127`}}} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \mbox {{\tt `Maximum H deviation : .5740000000e-5 \ \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ <math>{\it _____________________________________________________}} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \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}{poincare( }{} \end{mapleinput}

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