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Spin-Network Calculus and Projective Geometry: Combinatorial and Incidence Aspects

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Spin-Network Calculus and Projective Geometry: Combinatorial and Incidence Aspects UNIVERSIDADE FEDERAL DA BAHIA INSTITUTO DE F´ISICA Programa de P´os-gradua¸c~aoem F´ısica Tese de Doutorado Spin-network calculus and projective geometry: Combinatorial and incidence aspects Robenilson Ferreira dos Santos 2018 UNIVERSIDADE FEDERAL DA BAHIA INSTITUTO DE F´ISICA PROGRAMA DE POS-GRADUAC¸´ AO~ Spin-network calculus and projective geometry: Combinatorial and incidence aspects Robenilson Ferreira dos Santos Orientador: Prof. Dr. Frederico Vascocellos Prudente Co-orientador: Prof. Dr. Vincenzo Aquilanti Tese apresentada ao Instituto de F´ısicada Universi- dade Federal da Bahia como parte dos requisitos para a obten¸c~aodo t´ıtulode Doutor em F´ısica. Salvador-BA, 2018 To My family and close friends. i ACKNOWLEDGEMENTS The act of thanking is part of my life from an early age. I believe that the human being achieves nothing without the contribution of a fellow being. On the road of life we encounter obstacles that seem impossible to overcome. They are moments of so much re- flection, crying and regrets. On the other hand, we find strength where there is nothing, and emotional and moral support with friends that cross our journey. I cannot measure the gratitude I have for the people who have somehow contributed to the accomplishment of this work. Facing such deep this reflection I would like to thank: God, for life, for all discernment, motivation and focus; To the Advisors, especially to Prof. Frederico Prudente and Prof. Vincenzo Aquilanti, who guided me on the road of knowledge, always seeking the best ways, advice and teaching. To the collaborators, especially Prof. Ana Carla, prof. Mirco Ragni, Prof. Federico Palazzetti, Profa. Cecilia Coletti, Profa. Manuela Arruda. Your contribution was extremely important at each stage of this research. Thank you. My Family, who did not let me give up; fundamental support to complete this work. In particular, for my mother, Eulina Ferreira, my father Benicio Am^ancio,my sisters Silmara Ferreira and Simone Ferreira also for my aunts Maria Edna, Roseli Ferreira and Marinalva. To my friends: Cecilia Castelo Branco, Marcos Dias, Erica Apr´ıgio,Augusto Gurgel, Geovana Dresch, Cristiano Quintino, Ant^onioTadeu, Roberta Bandeira, Juliana Oliveira Moraes, Claudionor Alburquerque, Elisane Sales, Mauricio Brand~ao,Carla Daiane, Itana Virginia, Renata Amorim, Thaiane, Nilton C´esar,Jailma Costa, Rosival Barbosa, Bruno Eduardo, Jailton Barbosa, S´ergioCarvalho, Justino Moys´es, Angelo,^ Emanoel Carlos, Clei- son, Tatiana Melo, Bebeto Melo, Juliana, Seu Carlos e Dona Lita. Especially for prof. Carlos Alberto (in memorian). Thank you, friends, for all the moments of sadness and joy, for shar- ing with me every minute of anguish and happiness. The word of each one of you was essential in my walk. I appreciate the professors and friends of the Institute of Physics of the Federal Uni- ii versity of Bahia for all teaching, Knowledge and discussion at the time of my formative years. In particular for the profs. Humberto, Micael, Moreira, Marcilio, Luiz, Ricardo and Aline. I am also grateful for my friends of Physics, in special for Beliato Santana Campos, Josenilton Nascimento, Andrea Sim~oes,Wanderson Silva, Helder Tanaka,Thiago Barbosa, Mariana Bez- erra,Tiago Silveira, Elymar Oliveira, Marivaldo, Lucas, Rafael, Leonardo, Eduardo. Thanks for all the support and encouragement. I would like to thank the Federal Institute of Alagoas, on behalf of Professor Ricardo de Alburquerque Aguiar, director of the IFAL-Campus Piranhas for having granted the with- drawal to carry out this research. My total respect and gratitude acknowledged here. I would also like to thank Prof. Ant^onioIatanilton, Head of Department of the IFAL-Campus Piran- has and Pedagogue Rendrikson Silva. Thanks for all encouragement and support entrusted to me. To my students in the Agroecology and Agroindustry classes of 2014 and 2015. In particular for Anderson Moraes, Ant^onioSantos, Tezequiel Douglas, ´Italo Pinheiro, Mariza Monteiro, Elvis, Pablo vin´ıcios,Valter. Thank you for being part of my story. You were fundamental to my professional growth and as a human being. Thank you. I have great pleasure in thanking you for all the experience you had in Italy. And on this journey I met people who were a gift from God. Therefore initially I would like to thank the research group of the chemistry department of the Perugia University for all the support in all the phases of my PhD and the acquired knowledge. I would also like to thank Prof. Simoneta Cavalli for being part of this team and contributing to this research. Thanks also to Manuela Arruda for the immense help in the whole process of installation to the new country and for all the scientific collaboration. To the Italian friends Alberto, Teresita Leon, Raffaela, Ilenia, Domenico, Raffaele, Salvatore, Cristian, Adriana, Pedro, Gianmarco, Anderson. Thanks for the established rela- tionship of friendship. For my family in Italy: Alessandra Canaan, Monique Carbone Cintra, Gianni An- toniella, Thamara Brand~ao,Simone Mencarelli, Francesca Antoniella, Dino. I cannot mea- sure the gratitude I have for all of you. Thank you for sharing the most delicate moments iii and also for sharing the moments of joy. Thank you, thank you and thank you. To my friend Renato Freitas and your family (Viviane and Diana (daughter)), thank you for the conversations and discussions. We shared good times in Italy. These moments were fundamental for the resolution of bureaucratic problems or even for the good develop- ment of our researches. To Prof. Demetrius Filho, for sharing a little of his immense wisdom. A special being that taught a lot from his humility. I would also like to thank Profs. Federico Palazzetti and Andrea Lombardi, as well as Concetta Caglioti. I am immensely grateful for all support whether in research or in the installation process in Italy. I am also grateful for all the financial support given by Brazilian CAPES for the doctoral sandwich (PDSE88881.134388 / 2016-01) fellowship to the Perugia University and I also thank the financial and scientific support given by CNPQ, CAPES, IFAL,UFBA, UNIPG for this work. iv ABSTRACT The research developed in this doctoral thesis is inserted in the Quantum Theory of Angular Momentum. In this work we discuss the theory of coupling and re-coupling of angular momentum in the study of the so-called 3nj symbols. The geometric representations of tree and Yutsis were exploited as well as their mathematical properties. Another explored point was the on-screen representation of the Caustic Curve, which is directly related to the symbol 6j. The studies covered angular momentum re-coupling coefficients, encompassing the problem of the four bars, Regge symmetry, analysis and problem solving of spin networks. Among the contributions, we can mention the connections of the projective geometry with the Quantum Theory of Angular Momentum and the projective planes of Fanno, Desargues and Levi. Keywords: Spin Networks, Quantum angular momentum theory, projective geometry, Regge symmetry. v RESUMO A pesquisa desenvolvida nesta tese de doutorado est´ainserida na Teoria Qu^antica de Momento Angular. Neste trabalho abordamos a teoria de acoplamento e reacoplamento do momento angular no estudo dos chamados s´ımbolos 3nj. As representa¸c~oesgeom´etricas de ´arvore e de Yutsis foram exploradas, bem como suas propriedades matem´aticas. Um outro ponto explorado foi a representa¸c~aona tela da Curva C´austica,que est´adiretamente relacionada com o s´ımbolo 6j. Os estudos abrangeram os coeficientes de reacoplamento do momento angular, englobando o problema das quatro barras, simetria de Regge, an´alisee a resolu¸c~aode problemas de redes de spin. Dentre as contribui¸c~oes,podemos citar as conex~oes da geometria projetiva com a Teoria Qu^antica de Momento Angular e os planos projetivos de Fanno, Desargues e Levi. Palavras-chave: Rede de Spin, teoria qu^antica do momento angular, geometria projetiva, Simetria de Regge symmetry. vi Contents Contents vii List of Figures xi List of Tables xvii 1 Introduction 1 Bibliography 6 Bibliography . .6 2 Coupling and Recoupling Theory of 3nj Symbols 7 2.1 Foundations and Definitions . .7 2.2 The 3nj-symbols . 14 2.2.1 The 3j-symbols . 15 2.2.2 The 6j-symbols . 20 2.2.3 The 9j-symbols . 24 2.3 Spin network graphs . 26 Bibliography 29 Bibliography . 29 vii 3 Couplings and recouplings of four angular momenta: Alternative 9j sym- bols and spin addition diagrams 31 3.1 Introduction . 32 3.2 Couplings and recouplings: Spin nets and 6j symbols . 33 3.3 Orthogonality and the pentagonal relationship . 33 3.4 The hexagonal relationship and the truncated icosahedron abacus . 36 3.5 The 9j symbols and the 120 spin-net abacus . 38 3.6 9j symbol of the first kind . 40 3.7 9j symbol of the second kind . 42 3.8 Concluding Remarks . 44 Bibliography 45 4 Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configura- tions 51 4.1 Introduction . 52 4.2 Projective geometry and the spin network of seven spins . 53 4.2.1 Representation of angular momentum couplings and recouplings as tri- angles and quadrangles . 54 4.2.2 Incidence structure and projective features . 57 4.2.3 The quadrangle, its diagonal triangle and the network of seven spins . 60 4.3 The two networks of ten spins . 62 4.3.1 The 9j symbol of the second kind and Desargues theorem . 65 4.3.2 The 9j symbol of the first kind as an anti-Desargues configuration . 68 4.3.3 Incidence diagrams and the connections with the networks of ten spins 71 viii 4.4 Concluding remarks and perspectives . 72 Bibliography 76 5 Combinatorial and Geometrical Origins of Regge Symmetries: Their Man- ifestations from Spin-Networks to Classical Mechanisms, and Beyond 86 5.1 Introduction: Discovery and Developments .
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