# 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 regular tetrahedron whose corners will have attached to them the 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 ${\displaystyle (a_{x},a_{y},a_{z})}$ , ${\displaystyle (b_{x},b_{y},b_{z})}$ , ${\displaystyle (c_{x},c_{y},c_{z})}$ , and ${\displaystyle (d_{x},d_{y},d_{z})}$ , that will be labeled, respectively, as ${\displaystyle 1,2,3,4}$ .

A \htmladdnormallink{quantum field {http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (QF) state} ${\displaystyle \phi }$  provides a total order denoted by ${\displaystyle \leq _{\phi }}$  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 ${\displaystyle e=(i,j)}$ , by an element ${\displaystyle \phi _{1}(e)}$  of ${\displaystyle B_{A}}$ , which is a distinguished basis of a fusion algebra $\displaystyle \A$ , that is, a finite-dimensional, unital, involutive algebra over ${\displaystyle \mathbb {C} }$  --the field of complex numbers. Moreover, the QF state assigns an element ${\displaystyle {\phi }^{2}(f)}$  --called an intertwiner-- of a Hilbert space ${\displaystyle \mathbb {H} _{\phi }(f)={\mathbb {H} ^{\phi _{1}(ik)}}_{\phi _{1}(jk),\phi _{1}(ij)}}$  to each face ${\displaystyle f=(ijk)}$  of the tetrahedron, such that ${\displaystyle i\prec _{\phi }j\prec _{\phi }k.}$

### Remarks

A \htmladdnormallink{topological {http://planetphysics.us/encyclopedia/CoIntersections.html} quantum field theory} (TQFT ) is described as a mathematical approach to quantum field theory that allows the computation of topological invariants of quantum state spaces (QSS), usually for cases of lower dimensions encountered in certain condensed phases or strongly correlated (quantum) superfluid states. TQFT has some of its origins in 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 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 concept of homological mirror (quantum) symmetry in relation to a mathematical conjecture in superstring theory.

[1]

## References

1. V. Kodyiyalam and V. S. Sunder. 2001. Topological Quantum Field Theories From Subfactors ., Chapman and Hall/CRC.

### Maple eps Graphics

\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle [{\it poincare},{\it generate_ic},{\it zoom},{\it hamilton_eqs}\\ \mbox{}]$ } \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}{ $\displaystyle H\, := \,1/2\,{{\it p1}}^{2}+1/2\,{{\it p2}}^{2}+1/24\,{e^{2\,{\it q2}+2\,\sqrt {3}{\it q1}}}\\ \mbox{}+1/24\,{e^{2\,{\it q2}-2\,\sqrt {3}{\it q1}}}+1/24\,{e^{-4\,{\it q2}}}-1/8$ } \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}{ $\displaystyle {\it _____________________________________________________}$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt H = .99005020}},\,\mbox {{\tt  Initial conditions:}},\,t=0,\,{\it p1}= 0.1,\,{\it p2}= 1.4,\,{\it q1}= 0.1,\,{\it q2}=0$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt Number of points found crossing the (p1,q1) plane: 127}}$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt Maximum H deviation : .5740000000e-5 \ \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ [itex]{\it _____________________________________________________}$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt Time consumed: 12 seconds`}}$ } \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}