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Simplicial topological coding and homology of spin networks. (English) Zbl 1390.83233
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 11-20 (2017).
Summary: We study the commutation of the stabilizer generators embedded in the \( q\)-representation of higher dimensional simplicial complex. The specific geometric structure and topological characteristics of 1-simplex connectivity are generalized to higher dimensional structure of spin networks encoded in ordered complex via combinatorial optimization of a closed compact space. Obtained results of a consistent homology-chain basis are used to define connectivity and dynamical self organization of spin network system via continuous sequences of simplicial maps.
For the entire collection see [Zbl 1379.13001].
MSC:
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
83C27 Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory
83C47 Methods of quantum field theory in general relativity and gravitational theory
81T08 Constructive quantum field theory
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[1] 1. Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D. (ed.) Combinatorial Mathematics and its Applications, pp. 221-244. Academic Press, New York (1971) · Zbl 0216.43502
[2] 2. Rovelli, C., Smolin, L.: Loop space representation of quantum general relativity. Nucl. Phys. B 331 , 80-152 (1990)
[3] 3. Rovelli, C., Smolin, L.: Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442 , 593-619 (1995) · Zbl 0925.83013
[4] 4. Seth, M.A.: A spin network primer. Am. J. Phys. 67 (11), 972 (1999) · Zbl 1219.83060
[5] 5. Baez, J.C.: Spin networks in gauge theory. Adv. Math. 117 (2), 253 (1996) · Zbl 0843.58012
[6] 6. Baez, J.C.: Diffeomorphism-invariant spin network states. J. Funct. Anal. 158 , 253-266 (1998) · Zbl 0915.58019
[7] 7. Bartolo, C., Di Gambini, R., Griego, J., Pullin, J.: Consistent canonical quantization of general relativity in the space of Vassiliev invariants. Phys. Rev. Lett. 84 (11), 2314-2317 (2000) · Zbl 0959.83018
[8] 8. Grünbaum, B.: Convex Polytopes, 2nd edn. Springer, New York (2003) · Zbl 1024.52001
[9] 9. Ziegler, G.M.: Lectures on Polytopes. Springer, Berlin (1995) · Zbl 0823.52002
[10] 10. Whitehead, G.W.: Elements of Homotopy Theory. Springer, New York (1978) · Zbl 0406.55001
[11] 11. Gray, B.: Homotopy Theory. Pure and Appl. Math. 64, Academic Press, New York (1975) · Zbl 0322.55001
[12] 12. Griffiths, H.B.: The fundamental group of two spaces with a common point. Quart. J. Math. 5 , 175-190 (1954) · Zbl 0056.16301
[13] 13. Diestel, R.: Graph Theory, Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2005) · Zbl 1074.05001
[14] 14. Rosen, K.H.: Handbook of Discrete and Combinatorial Mathematics. CRC, Boca Raton (1999)
[15] 15. Seifert, H.: Konstruktion dreidimensionaler geschlossener Räume. Ber. Sächs. Akad. Wiss. 83 , 26-66 (1931) · JFM 57.0723.01
[16] 16. van Kampen, E.H.: On the connection between the fundamental group of some related spaces. Am. J. Math. 55 , 261-267 (1933) · JFM 59.0577.04
[17] 17. Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. thesis, California Institute of Technology (1997)
[18] 18. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, Cambridge Series on Information and the Natural Sciences, 1st edn. Cambridge University Press, Cambridge (2004) · Zbl 1049.81015
[19] 19. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001
[20] 20. Berec, V.: Phase space dynamics and control of the quantum particles associated to hypergraph states. EPJ Web Conf. 95 , 04007 (2015)
[21] 21. Berec, V.: Non-Abelian topological approach to non-locality of a hypergraph state. Entropy 17 (5), 3376-3399 (2015) · Zbl 1338.94033
[22] 22. Goebel, K., Kirk, W.A.: Topics in metric fixed point theory. In: Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge, New York (1990) · Zbl 0708.47031
[23] 23. Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54 , 1862 (1996)
[24] 24. Calderbank, A., Rains, E., Shor, P., Sloane, N.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78 , 405 (1997) · Zbl 1005.94541
[25] 25. Pemberton-Ross, P.J., Kay, A.: Perfect quantum routing in regular spin networks. Phys. Rev. Lett. 106 (2)
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