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

\newcommand{\sqdiagram}[9]{Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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 $(a_{x},a_{y},a_{z})$, $(b_{x},b_{y},b_{z})$, $(c_{x},c_{y},c_{z})$, 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} (QF) 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 distinguished basis of a fusion algebra Failed to parse (unknown function "\A"): {\displaystyle \A} , that is, a finite-dimensional, unital, involutive algebra over $\mathbb {C}$ --the field of complex numbers. Moreover, the QF state assigns an element ${\phi }^{2}(f)$ --called an intertwiner-- of a Hilbert space $\mathbb {H} _{\phi }(f)={\mathbb {H} ^{\phi _{1}(ik)}}_{\phi _{1}(jk),\phi _{1}(ij)}$ to each face $f=(ijk)$ of the tetrahedron, such that $i\prec _{\phi }j\prec _{\phi }k.$

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.

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}

MapleForPlanetPhysics