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quantum field state on the tetrahedron
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(Definition)
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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 $\A$ , 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 $$\H_{\phi}(f)= {\H^{\phi_1 (ik)}}_{\phi_1 (jk), \phi_1 (ij)}$$ to each face $f=(ijk)$ of the tetrahedron, such that $i\prec_{\phi} j \prec_{\phi}k .$
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'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.
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- V. Kodyiyalam and V. S. Sunder. 2001. Topological Quantum Field Theories From Subfactors ., Chapman and Hall/CRC.
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"quantum field state on the tetrahedron" is owned by bci1.
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Cross-references: theory, conjecture, relation, symmetry, Crafoord Prize, origins, quantum state spaces, topological invariants, quantum field theory, face, Hilbert space, complex numbers, involutive, unital, finite-dimensional, algebra, basis, labels, edge, total order, field, points, tetrahedron, vertices, terms, regular
There are 11 references to this entry.
This is version 34 of quantum field state on the tetrahedron, born on 2008-09-10, modified 2009-01-17.
Object id is 11016, canonical name is StateOnTheTetrahedron.
Accessed 1244 times total.
Classification:
| AMS MSC: | 57R56 (Manifolds and cell complexes :: Differential topology :: Topological quantum field theories) | | | 81T45 (Quantum theory :: Quantum field theory; related classical field theories :: Topological field theories) | | | 46H35 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: Topological algebras of operators) |
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Pending Errata and Addenda
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