TURAEV-VIRO AND KAUFFMAN-LINS INVARIANTS FOR 3-MANIFOLDS COINCIDE

1992 ◽  
Vol 01 (02) ◽  
pp. 105-135 ◽  
Author(s):  
SERGEY PIUNIKHIN
Keyword(s):  
Quantum Group ◽  
Explicit Form ◽  
Network Theory ◽  
Spin Networks ◽  
Spin Network ◽  
6J Symbols

The presentation of link polynomials, arising from representations of quantum group SLq(2) by SLq(2)-spin networks is given. The explicit form of cabling formula for these polynomials is written. The connection between 6j-symbols in q-spin network theory and Rakah-Wigner q-6j-symbols is shown. The Kauffman's hypothesis about coincidence of his 3-manifold invariants and Turaev-Viro invariants for 3-manifolds is proved.

STATE SUM MODELS FOR TRIVALENT KNOTTED GRAPH INVARIANTS USING QUANTUM GROUP SLq(2)

1992 ◽  
Vol 01 (03) ◽  
pp. 253-278 ◽  
Author(s):  
SERGEY PIUNIKHIN
Keyword(s):  
Network Theory ◽  
Quantum Spin ◽  
Topological Invariants ◽  
Graph Invariants ◽  
Spin Network ◽  
Face Model ◽  
Polynomial Invariants ◽  
Root Of Unity ◽  
State Models ◽  
6J Symbols

Four approaches to construct polynomial invariants for trivalent knotted graphs in S3 are compared. The first approach is based on vertex model with R-matrices and Glebsh-Gordan coefficients, appearing in SLq(2)-representations theory, as Boltzman weights. The second approach is based on Kauffman's quantum spin network theory, the third one is based on Witten-Turaev area-coloring model (or face model) based on quantum 6j-symbols, where q is root of unity. The fourth approach is based on the same face (or area-coloring) model, but q is not root of unity. The coincidence (up to certain normalization) of topological invariants, arising from these four state models, is proved.

Fermions, q-deformed diffeomorphism and the evolution of spin networks

2015 ◽  
Vol 24 (10) ◽  
pp. 1550074 ◽  
Author(s):  
L. Mullick ◽  
P. Bandyopadhyay
Keyword(s):  
Quantum Gravity ◽  
Gauge Group ◽  
Degrees Of Freedom ◽  
Loop Quantum Gravity ◽  
Closed Loop ◽  
Direction Vector ◽  
Spin Networks ◽  
Spin Network ◽  
Diffeomorphism Invariance ◽  
Diffeomorphism Symmetry

We have considered here the emergence of diffeomorphism symmetry in quantum gravity in the framework of the quantization of a fermion. It is pointed out that a closed loop having the holonomy associated with the SU(2) gauge group is realized from the rotation of the direction vector associated with the quantization of a fermion depicting spin degrees of freedom which appear as SU(2) gauge bundle. During the formation of a loop, a noncyclic path with open ends can be mapped onto a closed loop when the holonomy involves q-deformed gauge group SUq(2). This gives rise to q-deformed diffeomorphism and helps to realize diffeomorphism invariance in quantum gravity through a sequence of q-deformed diffeomorphism in the limit q = 1. We can consider adiabatic iteration such that the quasispin associated with the quantum group SUq(2) gradually evolves as the time dependent deformation parameter q changes and in the limit q = 1, we achieve the standard spin. This essentially depicts the evolution of spin network as the loop is being formed and links fermionic degrees of freedom with loop quantum gravity.

scholarly journals q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

2007 ◽  
Vol 16 (03) ◽  
pp. 267-332 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SAMUEL J. LOMONACO
Keyword(s):  
Quantum Algorithms ◽  
Jones Polynomial ◽  
Braid Groups ◽  
Spin Networks ◽  
Spin Network ◽  
Topological Quantum ◽  
Jones Polynomials ◽  
Topological Quantum Computation ◽  
Bracket Polynomial ◽  
Colored Jones Polynomials

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

scholarly journals AN INTRODUCTION TO NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ON QUANTUM GROUPS

1993 ◽  
Vol 08 (10) ◽  
pp. 1667-1706 ◽  
Author(s):  
PAOLO ASCHIERI ◽  
LEONARDO CASTELLANI
Keyword(s):  
Differential Geometry ◽  
Quantum Group ◽  
Explicit Form ◽  
Quantum Groups ◽  
Differential Calculus ◽  
Gauge Theories ◽  
Classical Case ◽  
Contraction Operator ◽  
Lie Derivative ◽  
Tensor Fields

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case (q→1 limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan-Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group GL q(2) is given in detail. The softening of a quantum group is considered, and we introduce q curvatures satisfying q Bianchi identities, a basic ingredient for the construction of q gravity and q gauge theories.

scholarly journals SELF-ORGANIZED CRITICAL BEHAVIOR: THE EVOLUTION OF FROZEN SPIN NETWORKS MODEL IN QUANTUM GRAVITY

2008 ◽  
Vol 23 (24) ◽  
pp. 3891-3899 ◽  
Author(s):  
JIAN-ZHEN CHEN ◽  
JIAN-YANG ZHU
Keyword(s):  
Quantum Gravity ◽  
The Self ◽  
Spin Networks ◽  
Spin Network ◽  
Underlying Graph ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Propagation Rule ◽  
First Time ◽  
The Universe

In quantum gravity, we study the evolution of a two-dimensional planar open frozen spin network, in which the color (i.e. the twice spin of an edge) labeling edge changes but the underlying graph remains fixed. The mainly considered evolution rule, the random edge model, is depending on choosing an edge randomly and changing the color of it by an even integer. Since the change of color generally violate the gauge invariance conditions imposed on the system, detailed propagation rule is needed and it can be defined in many ways. Here, we provided one new propagation rule, in which the involved even integer is not a constant one as in previous works, but changeable with certain probability. In random edge model, we do find the evolution of the system under the propagation rule exhibits power-law behavior, which is suggestive of the self-organized criticality (SOC), and it is the first time to verify the SOC behavior in such evolution model for the frozen spin network. Furthermore, the increase of the average color of the spin network in time can show the nature of inflation for the universe.

scholarly journals Recurrence relation for the 6j-symbol of suq(2) as a symmetric eigenvalue problem

2015 ◽  
Vol 12 (10) ◽  
pp. 1550117 ◽  
Author(s):  
Igor Khavkine
Keyword(s):  
Eigenvalue Problem ◽  
Quantum Group ◽  
Recurrence Relation ◽  
Numerical Evaluation ◽  
Alternative Proof ◽  
Spin Network ◽  
Symmetric Eigenvalue Problem ◽  
Evaluation Algorithm

A well-known recurrence relation for the 6j-symbol of the quantum group su q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in Appendix A. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q = 1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley–Lieb recoupling theory to simplify intermediate calculations.

scholarly journals Q-Series and quantum spin networks

2020 ◽  
pp. 1-19
Author(s):  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
Jesse S. F. Levitt
Keyword(s):  
Natural Product ◽  
Jones Polynomial ◽  
Infinite Family ◽  
Product Structure ◽  
Quantum Spin ◽  
Spin Networks ◽  
Spin Network ◽  
Colored Jones Polynomial ◽  
The Family

The tail of a quantum spin network in the two-sphere is a [Formula: see text]-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by [Formula: see text]. We compute the [Formula: see text]-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure.

BLACK HOLE ENTROPY AND AREA QUANTUM ENTANGLEMENT ORIGINATED FROM SPIN NETWORKS

2010 ◽  
Vol 25 (07) ◽  
pp. 1339-1347
Author(s):  
D. SHAO ◽  
L. SHAO ◽  
C. G. SHAO ◽  
H. NODA
Keyword(s):  
Black Hole ◽  
Wilson Loop ◽  
Black Hole Entropy ◽  
Entangled States ◽  
Area Spectrum ◽  
Quantum Space ◽  
Spin Networks ◽  
Spin Network ◽  
Entropy Of Black Hole ◽  
Area Operator

By making use of the grasping action of the area operator as an antisymmetrizer of the grasped strands in spin network and the Penrose binor identity, an equidistant area spectrum [Formula: see text] is deduced. Utilizing the spectrum to calculate the quantized area of black hole horizon, we recalculate the entropy of black hole H(A) = (8πℏGN)-1kA ln 2. By taking advantage of the smallest area quantum "½" excited by the spectrum via the Wilson loop in edge of spin network to approach the possible origin of qubit, the existences of entanglement of the area quanta in quantum space, as well as the nonlocal property of the entangled states are demonstrated.

EFFECTS OF NOISE ON SPIN NETWORK CLONING

2006 ◽  
Vol 04 (03) ◽  
pp. 487-493
Author(s):  
GABRIELE DE CHIARA ◽  
ROSARIO FAZIO ◽  
SIMONE MONTANGERO ◽  
CHIARA MACCHIAVELLO ◽  
G. MASSIMO PALMA
Keyword(s):  
Harmonic Oscillators ◽  
Quantum Gates ◽  
Network Approach ◽  
Noisy Environment ◽  
Quantum Cloning ◽  
Spin Networks ◽  
Spin Network

We analyze the effects of noise on quantum cloning based on the spin network approach. A noisy environment interacting with the spin network is modeled both in a classical scenario, with a classical fluctuating field, and in a fully quantum scenario, in which the spins are coupled with a bath of harmonic oscillators. We compare the realization of cloning with spin networks and with traditional quantum gates in the presence of noise, and show that spin network cloning is more robust.

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