# quantum field state on the tetrahedron

## 1 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 ‘6j-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 $1,2,3,4$.

###### Definition 1.1.

A quantum field (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 $\mathcal{A}$, 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  $\mathcal{H}_{\phi}(f)={\mathcal{H}^{\phi_{1}(ik)}}_{\phi_{1}(jk),\phi_{1}(ij)}$

to each face $f=(ijk)$ of the tetrahedron, such that $i\prec_{\phi}j\prec_{\phi}k.$

Notes: A topological 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 (http://planetmath.org/MichaelFrancisAtiyah)’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

• 1 V. Kodyiyalam and V. S. Sunder. 2001. http://planetmath.org/?op=getobj&from=books&id=174Topological Quantum Field Theories From Subfactors ., Chapman and Hall/CRC.
 Title quantum field state on the tetrahedron Canonical name QuantumFieldStateOnTheTetrahedron Date of creation 2013-03-22 18:22:30 Last modified on 2013-03-22 18:22:30 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 37 Author bci1 (20947) Entry type Definition Classification msc 46H35 Classification msc 81T45 Classification msc 57R56 Synonym state on the tetrahedron Related topic Tetrahedron Related topic QFT Related topic Grafix Related topic FoundationsOfQuantumFieldTheories Related topic MonoidalCategory Related topic CrafoordPrize Defines quantum field state on a tetrahedron Defines TQFT