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

Home , Spin network, Tullio Regge

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 reflection, 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 Amˆancio,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, AntônioTadeu, Roberta Bandeira, Juliana Oliveira Moraes, Claudionor Alburquerque, Elisane Sales, Mauricio Brandão,Carla Daiane, Itana Virginia, Renata Amorim, Thaiane, Nilton César,Jailma Costa, Rosival Barbosa, Bruno Eduardo, Jailton Barbosa, SérgioCarvalho, Justino Moysés, Angelo,ˆ 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 sharing 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. AntˆonioIatanilton, 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, AntˆonioSantos, 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 relationship 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 measure 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 development 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 estáinserida na Teoria Quântica de Momento Angular. Neste trabalho abordamos a teoria de acoplamento e reacoplamento do momento angular no estudo dos chamados s´ımbolos 3nj. As representa¸cõesgeométricas de árvore e de Yutsis foram exploradas, bem como suas propriedades matemáticas. Um outro ponto explorado foi a representa¸cãona tela da Curva Cáustica,que estádiretamente 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, análisee a resolu¸cãode problemas de redes de spin. Dentre as contribui¸cões,podemos citar as conexões da geometria projetiva com a Teoria Quântica de Momento Angular e os planos projetivos de Fanno, Desargues e Levi.

Palavras-chave: Rede de Spin, teoria quˆantica 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 symbols 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 triangles 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 ...... 87

5.2 Computational and Combinatorial Aspects ...... 88

5.2.1 Calculation of 6j Symbols ...... 88

5.2.2 The Triangle and Polygon Inequalities and the 4j-model of the 6j-symbol 89

5.2.3 Generation of Strings and Matrices of 6j Symbols ...... 92

5.3 Semiclassical and Geometrical Aspects ...... 93

5.3.1 Visualizations ...... 93

5.3.2 The volume operator and non-Euclidean extensions ...... 93

5.4 Couplings as Limits of Recouplings ...... 94

5.4.1 Regge symmetries and 3nj symbols and hydrogenic orbitals ...... 94

5.4.2 Momentum Space Wavefunctions and Sturmian Orbitals ...... 95

5.4.3 Connection to a Classical Mechanism and to Molecular Mechanics . . 97

Bibliography 98

6 Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages. 106

6.1 Introduction ...... 106

6.2 Background ...... 107

6.2.1 Quadrilaterals, quadrangles and tetrahedra ...... 107

ix 6.2.2 The screen ...... 108

6.3 Case studies and discussion...... 113

6.3.1 Correspondence between Regge symmetries and Grashof cases. . . . . 113

6.3.2 General case: illustrations...... 115

6.4 Concluding remarks ...... 125

6.4.1 A.1. Symmetries of 6j symbols and the Regge symmetries...... 126

6.4.2 A.2. The canonical order of Regge symbols and the Gashosf condition 128

Bibliography 129

7 Remarks 137 List of Figures

2.1 Triangular conditions for elements of 6j-symbols ...... 23

2.2 Tree scheme of 6j-symbol ...... 26

2.3 Yutsis diagram of the 6j-symbol ...... 27

2.4 Tetrahedron representation of the 6j-symbol ...... 27

3.1 Pentagonal scheme illustrating the Biedenharn-Elliot identity and representing connections among spin-nets, i.e. alternative recouplings of angular momenta. Solid arrows correspond to sums over the eliminated quantum number and are explicitly given by 6j coefficients. In the pentagon the sequence of the four original initially coupled angular momenta is maintained. Each coupling scheme (represented as a tree-like graph) can in turn be connected through a phase to an alternative graph, corresponding to a similar coupling scheme pertaining to a different sequence of angular momenta. This operation can also be thought of as a rotation around a pivot angular momentum (circled in the figure)...... 35

3.2 Hexagonal schemes obtained by extending the pentagonal scheme in Fig. 3.1. The corresponding sum rules obtained by following the edges of the hexagon produce the Racah sum rules...... 37

xi 3.3 Hexagons and pentagons leading to the truncated icosahedron, which constitues the abacus for recoupling coeﬃcients. On the left: half abacus, with explicit spin-nets at the vertices: on the right the whole abacus. Note that spin-nets at the antipodes (like those circled in red in the ﬁgure) are mirror images...... 38

3.4 The complete spin-network of the 120 coupling schemes, or spin-nets, classi- ﬁed according to two interconnected truncated icosahedrical abacus. Double dotted lines connect spin-nets onto the two abacus through a phase factor. A version in color,equivalent apart from minor details, is available in [54]. . . . 39

3.5 The highlighted dashed red path along the edges of the abacus corresponds to the deﬁnition of 9j symbols, eq. (3.5). The closed path obtained by following in sequence the edges along the dashed red path, the light green dotted path and the light blue dotted path is the graphical representation of the sum rule involving three 9j symbols...... 41

4.1 The 6j symbol and its four triads as a complete quadrilateral (left) and as complete quadrangles (center and right). In the ﬁrst graph, that represents

the 46 conﬁguration of incidence geometry, spins are associated to points and

triads to lines. In the center and right two graphs, that represent the 64 conﬁguration of incidence geometry, spins are associated to lines and triads to

points. In general, the Mn conﬁguration represent a graph with M lines and n points...... 56

4.2 Representation on the Fano plane of the network of seven spins. The illustration is provided by drawing correspondences between relations satisﬁed by the nj symbols and various collinearity properties of the appropriate diagrams; in the labeling scheme a triad is represented by three points on a line, and the three 6j symbols appear as complete quadrilaterals, related to the complete quadrangles by the point-line duality in the projective plane according to Fig. 4.1...... 57

xii 4.3 Representation of the Fano plane for network of seven spins, dual to that in Fig. 4.2. Here, the points are triads and the lines are angular momenta. The missing triad is represented as an empty triangle and indicates that no coincidence is enforced for the three lines x, y, z...... 61

4.4 Levi incidence diagram compacting the two networks of seven spins in Figs. 4.2 and 4.3. Triads are represented as triangles (the missing triad by an empty one); angular momenta are given as circles. The three corresponding quadrangles are given in the same order by which they appear in the Racah sum rule, Eq. (4.2)...... 63

4.5 Desargues conﬁguration for the networks of ten spins (a, b, c, d, e, f, p, q, r, x). The triads are represented as lines, and spins as dots, according to the traditional Fano-Racah assignment...... 66

4.6 (a) Dual version of the Desargues Theorem configuration of Fig.4.5, where role of lines and points as been interchanged using the principle of dualization valid in projective geometry, (b) Pentagram as a perfect figure, from which the Desargues configurations of figures can be labeled according to the text. 67

4.7 Desargues diagram of the network of ten spins corresponding to the 9j symbol of the second kind. The ten lines are labeled by spins of the Biedenharn- Elliott relationships Eq. (4.4), its ﬁve 6j are labeled by number from 1 to 5, represented by ﬁve quadrangles, two of them on the left-hand-side and three of them on the right-hand-side of the equation. The labels of 1 to 5 (from left to right) indicate the order of the 6j’s end side in Eq. (4.4), and are related with the order of the quadrangle as they appear there...... 68

xiii 4.8 Anti-Desargues diagram of the network of ten spins, corresponding to the 9j symbol of the first kind defined as a sum of three 6j-symbols. One can disentangle in the figure the symbols appearing in Eq. (4.6); the lines represent the triads, and the points are the angular momenta. The dashed line in this figure indicates that the triad (pqr) in general does not exist, namely that points p, q and r are not in general aligned, although they may be so accidentally and graphically simulated by...... 69

4.9 Conﬁguration of the anti-Desargues diagram Fig.(4.7), dual to that of Fig. (4.8), here considering points as triads, and lines as angular momenta. It is observed that in this case the missing triad is indicated as an empty triangle. The six vertices of the indicated as representation as an hexagon plus its three main diagonals corresponding to the 9j symbol can be recognized by points labeled by 9, while the other labels 1, 2 and 3¯ indicate the three quadrangles also disentangled and pictured in the ﬁgure, the numbers indicating the order by which they appear in Eq. (4.5)...... 71

4.10 Levi’s incidence diagram for the network of ten spins of Fig. (4.7) (left) and Fig.(4.9) (right). The 10 angular momenta are represented as empty circles and the ten triads are represented by full triangles. The empty triad (pqr¯) doesn’t exist in general but it may accidentally (see Fig. 4.9) and the dashed lines p, q andr ¯ do not meet unless forced to do so through a graphical approximation. 73

5.1 asym6j ...... 90

5.2 asym6j ...... 91

6.1 Comparison between a quadrangular coupling scheme of angular momentum, the 6j symbol (in the left), and a quadrangular scheme of bars, the four-bar linkage (in the right)...... 114

xiv 6.2 The screen of a quadrangle of sides a = 30, b = 45, c = 60, d = 55 (for the corresponding 6j symbol screen representation, see Fig. 1 in Ragni 2013). The diagram reports the four gates E, S, W, N and four further points SE, SW, NW, NE, the corresponding quadrangle, for these latter, is concave, biconcave, concave and convex, respectively...... 116

6.3 Illustration of the four-bar mechanism related to the caustic in Figure (6.2). For this system, the Grashof condition is veriﬁed. We report the four sides a, b, c and d, and the diagonals x and y. The side b is the “ground link” and a, the “input” or “output link”, acts as a “crank” when rotates clockwise (see the left column) or anticlockwise (see the right column). We report the conﬁgurations that the system assumes in correspondence of the gates N, S, W, E and the intermediate points NE, SE, NW, SW. For each of this point, there are a couple of mirror images originated by the clockwise and anticlockwise rotations...... 118

6.4 We report the Grashof conjugate of the object in Figure (6.3), which bars have lengths a0, b0, c0, d0 = 65, 50, 35, 40. In this case the Grashof condition is not verified. In this illustration, the configurations of the mechanism are made concrete by imposing a0 as fixed (the “ground link”) and following the behavior of b0 as input or output link...... 119

xv 6.5 a) Screen of a system characterized by a = 100; b = 110; c = 130; d = 140 (for the corresponding 6j symbol screen representation, see Fig. 5a in Ragni 2013). It corresponds to the case of symmetry v = 0; r = −10; u = −30. The North and West gates coincide. b) The screen of the system of sides a = 100; b = 140; c = 130; d = 110 (for the corresponding 6j symbol screen representation, see Fig5b in Ragni 2013) is reported. The symmetry parameters are u = 0; r = −10; v = 30 With respect to the system reported in a), there is an inversion in the shape of the curve, with the South and East gates coinciding. c) The screen is related to the system a = 100; b = 130; c = 140; d = 110 (for the corresponding 6j symbol screen representation, see Fig. 6a in Ragni 2013) and symmetry parameters r = 0; u = −10; v = 30. There is coalescence between the South and West gates...... 120

6.6 a) Screen of the system a = 100; b = 200; c = 100; d = 200 (for the corresponding 6j symbol screen representation, see Fig. 8a in Ragni 2013). The symmetry parameters are u = v = 0 and r = −100. The is coalescence between the gates North and West and the gates South and East. b) In this panel, we report the screen for the system a = 100; b = 110; c = 100; d = 110 (for the corresponding 6j symbol screen representation, see Fig. 8b in Ragni 2013). The symmetry parameters are u = v = 0 and r = −10...... 122

6.7 Screen for the system a = 100; b = 100; c = 1000; d = 1000 (for the corresponding 6j symbol screen representation, see Fig. 9a in Ragni 2013). In this case, the symmetry parameters are r = v = 0 and u = −900...... 124

6.8 Screen of the system a = b = c = d = 100 (for the corresponding 6j-symbol screen representation, see Fig. 9b in Ragni 2013), a quadrilateral with the four sides of identical length. The symmetry parameters are r = u = v = 0. . . . 124

xvi List of Tables

6.1 Classiﬁcation of input and output links bar movements ...... 128

xvii Chapter 1

Introduction

The quantum theory of angular momentum has been the scene of several investigations in the most diverse areas of the Physics. According to Edmonds (1957) [1], in his preface, a general theory of angular momentum algebra can be developed, from which it is possible to derive computational methods applicable to problems of atomic, molecular and nuclear spectroscopy, nuclear reactions and correlation angle of successive radiations from the nucleus.

On the other hand, Biedenharn and Louck (1981) [2] aﬃrm that the study of symmetries of physical systems are one of the main contemporary theoretical activities. These symmetries, which basically express the geometric structure of the physical system in question, must be clearly analyzed to understand the of dynamic system behavior. Analyzes of rotational symmetry, and the behavior of physical quantities under rotations, are common problems in the study of physical systems.

According to Varshalovich (1988) [3], the mathematical tools of this theory are widely used in atomic and molecular physics, in nuclear physics, and in elementary particle theory. This makes it possible to calculate atomic, molecular and nuclear structures, ground and excited state energies. Moreover, the tools are also very useful for calculating the probabilities of radioactive transition, cross section of various processes, such as elastic and inelastic scattering, diﬀerent decays and reactions (both chemical and nuclear), and to

1 Chapter 1. Introduction 2 study angular distribution and polarization of particles.

Today these tools are found in increasing use in solving practical on problems for quantum chemistry, kinetics, plasma physics, quantum optics, radiophysics and astrophysics.

The basic ideas of angular momentum theory were ﬁrst developed by M. Born, P. Dirac, W. Heisenberg and W. Pauli. However, the modern version of these tools was developed mainly in the works of E. Wigner, J. Racah, L. Biedenharn and others who applied the methods of group theory to problems in Quantum Mechanics [3].

The conception of angular momentum, initially deﬁned as the momentum of momentum was originated very early in classical mechanics as well seen in Kepler’s second Law, which contains this concept. In Wigner’s notes [2], for example, the general theorem of the momentum conservation is not mentioned. In fact, in the well-known work of Cajori [2] “History of Physics (1929 edition)” devotes exactly half a line to the conservation of angular momentum. We know that the concept of angular momentum is of great importance in quantum mechanics. For example the quantum of Planck action, h, has precisely the angular momentum dimension, and the Bohr quantization hypothesis speciﬁes the unit of h the (orbital) angular momentum for ~ = 2π . Therefore, the angular momentum theory and quantum physics are therefore clearly connected.

In this perspective, the general objective of this thesis is to contribute with the algebra of the coupling and recoupling of angular momentum theory with the introduction of the 3nj symbols as well as the spin-networks of the quantum angular momentum theory and relating the Regge symmetry with the four-bar problem and chirality.

That is, the present work seeks to collaborate with the development of the quantum angular momentum theory with respect to the coupling and re-coupling coeﬃcients of two or more angular moments, so-called 3nj symbols, developed by Clebsch-Gordan, Racah, Wigner, Regge and others. In addition, understanding and contributing to the associated procedures for their calculations, and the formulation of their semiclassical boundaries, which are basic to current areas of research such as quantum gravity, quantum information, node theory, and image reconstruction, that also present great importance in several areas of atomic, molecular and nuclear physics. Another point of interest in this work is the understanding and Chapter 1. Introduction 3 analysis of algebraic properties aiming to contribute to the literature in providing geometric interpretations of such symbols.

Theoretical molecular physics presents great interest in asymptotic expansions, which correspond to diﬀerent levels of semiclassical boundaries. However, such expansions are also useful for the motivation of some algorithms, for demanding computational problems of chemical reaction theory and molecular dynamics, where large angular moments are usually involved. The graphical and algebraic methods open the path to show combinatorial and geometric characteristics of the networks of seven and ten spins, associated respectively to symbols 6j and 9j. The graphic description of the properties of these structures of seven and ten spins is a valuable approach to discover and deepen not only the unexplored geometric and computational aspects of angular momentum algebra but also the combinatorial characteristics of general interest in a variety of applications such as quantum computation and quantum gravitation. Another interest in this work is to relate the theory of symbols 3nj to the Regge symmetry and to the 4-bar problem, so that the connection of such arguments provides information for the understanding of the processes of chirality that occurs in various molecules in nature, in particular by observing the spatial characteristics. Another interest is related to the geometric interpretations of the 3nj symbols and the projective geometry that provides an important result connected to the dualism between points and lines, and contributes to the relation of the projective planes of Fano and Desargues with the quantum angular momentum theory.

Thus, for this thesis we highlight specific objectives to follow: (i) analyze atomic and molecular processes with respect to the spatial aspects; (ii) to develop the Racah-Wigner algebra, introducing the spin networks, as well as the symbols 6j and 9j, allowing to discuss graphically its pentagonal and hexagonal relations; (iii) provide information about the graphical rules of a family of diagrams on the 7- and 10-spins network, which is called “abacus”, and is useful not only in spectroscopy and collision theory; (iv) develop and provide a combinatorial configuration for the family of quantum graphs that support an efficient “abacus”; (v) analyze and understand the network of 7-spins, as well as the network of 10-spins related to the symbols 6j and 9j, and their properties and applications in the various areas of sci- Chapter 1. Introduction 4 ence; (vi) understand and analyze the symbols 9j of the 1st and 2nd type, as well as their geometric interpretation; (vii) investigate the relation that allows us to write a symbol 9j as a sum over three symbols 6j;(viii) to understand the relation between the symbols 9j and the network of 10-spins with the Desargues geometry from a point of view of the projective Geometry; (ix) to investigate the Regge symmetry for the Racah-Wigner coefficients from a Quantum-Mechanical viewpoint using the four-bar problem; (x) investigate properties and relations of angular momentum algebra, using graphical methods.

The thesis is structured as follows. In Chapter 2, we review the standard recoupling theory for SU(2) angular momenta and it’s foundations. In Chapter 3 we discuss couplings and recouplings of four angular momenta: alternative 6j and 9j symbols and spin addition diagrams.This chapter was inspired by the paper “ Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams” published in the J. Mol. Model (2017) 23:147, with the respective authors: Robenilson F. Santos, Ana Carla P. Biten- court, Mirco Ragni, Frederico V. Prudente, Cecilia Coletti, Annalisa Marzuoli and Vincenzo Aquilanti. In Chapter 4 we establish the relationships between Projective Geometry and Angular Momenta. In this chapter we focus on the basic aspects of angular momentum theory and the spine network from the point of view of Projective Geometry. This chapter was inspired by the paper “Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations” published in the Journal of Molecular Spectroscopy, Volume 337, July 2017, pages 153-162, with the respective authors: Robenilson F. Santos, Manuela S. Arruda, Ana Carla P. Bitencourt, Mirco Ragni, Frederico V. Prudente, Cecilia Coletti, Annalisa Marzuoli and Vincenzo Aquilanti. In Chapter 5, we have introduced the origins of Tullio Regge Symmetry, as well as the combinatorics involved in Spin-Network. This chapter was inspired by the paper “Combinatorial and geometrical origins of Regge symmetries: their manifestations from spin-networks to classical mechanisms, and beyond” published in the ICCSA 2017, Part V, LNCS 10408, 2017, pages 314-327, with the respective authors: Vincenzo Aquilanti, Manuela S. Arruda, Cecilia Coletti, Robert Littlejohn and Robenilson F. Santos. In Chapter 6 the deﬁnition of Regge symmetry is given and then generalized to quadrilaterals and tetrahedral. In this chapter we have deduced an equation for the study of the Caustic curve as a function of the sides of a Chapter 1. Introduction 5 quadrilateral, as well as the diagonals of the quadrilateral. We also establish relationships between the four-bar motion, the Grashof condition and the Regge symmetry.This one was inspired by the paper “Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages” accepted for publication in the Rendiconti Lincei. Scienze Fisiche e Naturali, with the respective authors: Vincenzo Aquilanti, Ana Carla Peixoto Bitencourt, Concetta Cagli- oti, Robenilson Ferreira dos Santos, Andrea Lombardi, Federico Palazzetti, Mirco Ragni. In Chaper 7 we present the conclusions of this thesis, as well as the perspectives of work. Bibliography

[1] A. Edmonds, Angular Momentum in Quantum Mechanics, second edition Edi- tion,Princeton University Press, Princeton, New Jersey, 1960.

[2] L. C. Biedenharn, J. D. Louck, The Racah-Wigner Algebra in Quantum Theory, no.Rota, GC. (Ed) in Encyclopedia of Mathematics and its Applications Vol 9, Addison-Wesley Publ. Co.: Reading MA, 1981.

[3] D. Varshalovich, A. Moskalev, V. Khersonskii, Quantum Theory of Angular Momentum, World Scientiﬁc, Singapore, 1988.

[4] R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Dover Publications, 2007.

[5] J.J. Sakurai, J. Napolitano. Modern Quantum Mechanics.2nd Edition. Bookman, 2013.

6 Chapter 2

Coupling and Recoupling Theory of 3nj Symbols

2.1 Foundations and Deﬁnitions

The purpose of this chapter is to provide the essential elements for the construction of the Coupling and Recoupling Angular Momentum Theory. In order to do this, we shall start from the classical point of view up to the principal properties of equations and symmetries from the theoretical point of view. For the section of Foundations and Deﬁnitions, the book “Angular Momentum: Understanding spatial aspects in chemistry and physics” from the author R.N Zare [1] was used. While in describing 3j-symbol we use Ref [1] and the books of Biedenharn and Louck [2], as well as the book “Modern Quantum Mechanics” by J.J. Sakurai [3] and “Angular Momentum in Quantum Mechanics” by Edmonds [4]; to deal with the 6j-symbol, Varshalovich’s book was used [5] as well as the previous references, and for the 9j-symbol we used the references previously mentioned and when dealing with spin network we used Nikiforov’s book, [6, 5] and Yutsis’s as well [7].

Initially it is important to note that the crucial rule of angular momentum has come to be understood as the quantity conserved under rotations in three-dimensional space, due

7 Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 8 to the isotropy medium. From a classical point of view the angular momentum is deﬁned as:

L = r × p, (2.1) where L is the angular momentum vector, r is the position vector and p is the linear momentum vector.

On the other hand, for the Quantum Mechanics formalism of the angular momentum it was represented as an operator, acting on a state vector of the physical system. According to Sakurai [3], in 1925 P.A.M. Dirac concluded that by replacing the classical Poisson brackets with quantum commutator, using the rule

[ , ]Quant [ , ]class → , (2.2) i~ it was possible to obtain various quantum relations (so that the classical Poisson brackets and the quantum mechanics commutator satisfy similar algebraic properties).

The quantum theory establishes that two observable can be simultaneously measured only if their operators commute; that is, considering Aˆ and Bˆ, as two operators, we must that: [A,ˆ Bˆ] = 0 (2.3)

The commutator [ , ] is established within the Lie algebra SU(2), which is a linear vector space V, on a body C, on which the SU(2) × SU(2) → SU(2) product is deﬁned, so that it satisﬁes Jacobi’s anti-symmetry, bilinearity and identity properties, as follows:

1. [X,X] = 0, ∀ X ∈ V (2.4)

2. [aX + bY, Z] = a[X,Z] + b[Y,Z] (2.5)

[Z, aX + bY ] = a[Z,X] + b[Z,Y ] ∀ a, b ∈ C and ∀ X,Y,Z ∈ V

3. [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0, ∀ X ∈ V (2.6)

For the position and the momentum vectors, we have the following commutation Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 9 relations (for simpliﬁcation we can use ~ = 1):

[ˆx, pˆx] = i; [ˆx, pˆy] = 0; [ˆx, pˆz] = 0 (2.7)

An important observation emerging from these relationships is the impossibility of simultaneously measuring with precision both the position and the linear momentum of a particle. In the case of Lˆ, the following commutation relations in Cartesian components are also established in the literature:

[Lˆx, Lˆy] = iLˆz;[Lˆy, Lˆz] = iLˆx;[Lˆz, Lˆx] = iLˆy (2.8)

Having established the above commutation relations, from this point on, we will adopt ˆj as the total quantum angular momentum operator of the system. By analogy with eq.(2.8), we have that the commutation relations of Cartesian components are given by:

[ˆjx, ˆjy] = iˆjz;[ˆjy, ˆjz] = iˆjx;[ˆjz, ˆjx] = iˆjy (2.9)

Once established the commutation relation of the total quantum angular momentum operator and its Cartesian components, it is possible to investigate its common eigenvalues and eigenfunctions. So we have:

ˆ2 2 2 2 j = jx + jy + jz . (2.10)

This is known in the literature as the Casimir operator and obeys the following commutation relations: 2 2 2 [ˆj , ˆjx] = 0; [ˆj , ˆjy] = 0; [ˆj , ˆjz] = 0 (2.11)

Observing eq.(2.11) and knowing that ˆj2 is hermitian, (and thus with real eigenvalues), we can construct the state |jmi that is simultaneously eigenfunction of ˆj2 and the Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 10 projection of ˆj onto the z-axis, that is:

2 ˆj |jmi = λj |jmi (2.12)

ˆjz |jmi = m |jmi (2.13)

We can multiply the two sides of eq. (2.12) and (2.13) by hjm| and obtain:

2 hjm| ˆj |jmi = hjm| λj |jmi (2.14)

hjm| ˆjz |jmi = hjm| m |jmi (2.15)

Knowing that λ is real, and the property of state vectors orthogonality, we obtain that:

2 hjm| ˆj |jmi = λj (2.16)

hjm| ˆjz |jmi = m (2.17)

ˆ2 ˆ2 ˆ2 ˆ2 Now we will apply the operator jx + jy = j − jz, to the |jmi state to get:

ˆ2 ˆ2 ˆ2 ˆ2 (jx + jy) |jmi = (j − jz) |jmi = (2.18)

2 = (λj − m ) |jmi (2.19)

2 2 From this equation we can conclude that (λj − m ) ≥ 0. This implies that m does not

2 exceed λj and that m has a minimum value mmin and a maximum value mmax for a given ˆj.

Next, we will analyze the behavior of the well known raising and lowering (ˆj±) operators:

ˆj± = ˆjx ± iˆjy (2.20) Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 11

These operators obey the following commutation relations:

2 [ˆj , ˆj±] = 0 (2.21)

[ˆjz, ˆj±] = ±ˆj± (2.22)

[ˆj+, ˆj−] = 2ˆjz (2.23)

With the use of eqs.(2.1) to (2.23), we can analyze the behavior of the function ˆj± |jmi. So we have:

2 2 (ˆj ˆj±) |jmi = (ˆj±ˆj ) |jmi = λjˆj± |jmi , (2.24) and we also obtain:

(ˆjzˆj±) |jmi = (ˆj±ˆjz ± ˆj±) |jmi (2.25)

= (m ± 1) |jmi (2.26)

2 Thus, we can also conclude that ˆj± |jmi it is eigenfunction of ˆj with eigenvalue λj and it is the eigenfunction of ˆjz with eigenvalue (m ± 1). It follows that ˆj± |jmi is proportional to the normalized eigenfunction |j(m ± 1)i, that is,

ˆj± |jmi = C± |j(m ± 1)i , (2.27)

where C± is a constant of proportionality.

Note that the so-called lowering and raising operators act on the state vector in order to change m by a ±1 unit. Since the values of m are limited between mmin and mmax, it follows that:

ˆj+ |jmmaxi = 0 (2.28)

ˆj− |jmmini = 0 (2.29) Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 12

We can now multiply the eqs. (2.28) and (2.29) by ˆj− and ˆj+, respectively, and use the identity

2 ˆj∓ˆj± = ˆj − ˆjz(ˆjz ± 1) (2.30)

So we have

ˆj−ˆj+ |jmmaxi = 0 (2.31)

λj |jmmaxi − mmax(mmax + 1) |jmmaxi = 0. (2.32)

Thus, we conclude that:

λj − mmax(mmax + 1) = 0 (2.33)

We can follow the same steps to get:

λj − mmin(mmin − 1) = 0 (2.34)

Using these two above equations to eliminate λj, we obtain the following relation with mmax and mmin.

mmax(mmax + 1) − mmin(mmin − 1) = 0 (2.35)

(mmax + mmin)(mmax − mmin + 1) = 0 (2.36)

Since mmax must be greater than or equal to mmin, we have the unique solution for this equation:

mmax + mmin = 0 (2.37)

mmax = −mmin (2.38) Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 13

Therefore, mmax = −mmin, hence we can conclude that:

mmax − mmin = 2j (2.39)

mmax + mmin = 0 (2.40)

Thus mmax = j and mmin = −j. Since 2j must be a integer number, then j is an integer or half integer number. This analysis allows us to establish 2j + 1 possible values of m, where m = −j, −j + 1, ..., j − 1, j for each value of j. Returning eqs.(2.33) and (2.34) and substituting mmax = j and mmin = −j, we obtain:

λj − j(j + 1) = 0 (2.41)

λj = j(j + 1) (2.42)

Re-examining eq.(2.27), we can now calculate the proportionality constant C±. In this case what follows is that:

ˆj± |jmi = C± |j(m ± 1)i , and its conjugate complex is given by:

hjm| ˆj∓ = hj(m ± 1)| C∓ (2.43)

2 hjm| ˆj∓ˆj± |jmi = |C±| hj(m ± 1) | j(m ± 1)i (2.44)

Therefore we have: 2 |C±| = hjm| ˆj∓ˆj± |jmi . (2.45)

2 We can now use the identity ˆj∓ˆj± = ˆj − ˆjz(ˆjz ± 1) in eq.(2.45) to ﬁnd: Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 14

2 2 |C±| = hjm| (ˆj − ˆjz(ˆjz ± 1)) |jmi (2.46)

= λj hjm | jmi − m(m ± 1) hjm | jmi (2.47)

2 |C±| = λj − m(m ± 1) (2.48)

We have, then, that the absolute value is given by:

1 C± = [λj − m(m ± 1)] 2 (2.49)

According to eq. (2.49) the absolute value of C± is determined, while the phase remains arbitrary. So we shall agree with the convention that C± is real and positive.

The quantum number j of the angular momentum can take integer or semi-integer 1 3 5 values such as: 0, , 1, , 2, , 3, ... Thus, with such tools in our hand, we shall now investigate 2 2 2 the physical signiﬁcance of C±, as well as of the sum of two or more angular momenta and of their properties.

2.2 The 3nj-symbols

The so-called 3nj-symbols were ﬁrst deﬁned algebraically by Racah and Wigner who independently arrived at same physical quantity. Such symbols were as related to elements of a matrix of unitary transformation between two distinct bases. In this context, we have that n corresponds to (N − 1) angular momentum. That is, for a 3j-symbol, we have the sum of two angular momenta and for n = 2, we have the sum of three angular momenta, which would correspond to a 6j-symbol. As a consequence we have the following relation: n = (N − 1); 3nj-symbol For n = 1, we have N = 2; 3j-symbol For n = 2, we have N = 3; 6j-symbol For n = 3, we have N = 4; 9j-symbol For n = 4, we have N = 5; 12j-symbol Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 15

2.2.1 The 3j-symbols

Let us now investigate what these symbols are, as well which are their properties. Consider a sum of two angular momentum vectors, given by:

ˆj = ˆja +ˆjb (2.50) where ˆj is the total angular momentum. According to Classical Mechanics, the sum of two angular momenta will be another angular momentum. On the other hand in Quantum Mechanics the sum of two angular momenta will be another angular momentum if it satisﬁes the rules of commutations given by eq.(2.9). So we have:

ˆjax, ˆjay = iˆjaz (2.51) h i ˆjbx, ˆjby = iˆjbz (2.52) h i and for any two pairs of diﬀerent operators we have that

ˆjax, ˆjby = 0 (2.53) h i

Thus the components of ˆja a and ˆjb satisfy the usual commutation relations for the angular momentum.

We can ﬁnd a complete basis set from operators that commute each other. For two ˆ2 ˆ2 ˆ ˆ angular momenta vectors, a complete basis set of commutating operators are: ja, jb , jaz, jbz .

Thus, a complete basis set can be formed by direct product of basis set ofn each ˆja and ˆjob operators, which can be given by:

|jama, jbmbi ≡ |jamai |jbmbi (2.54) Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 16

This state is simultaneously eigenstate of these operators.

We have that |jazma, jbmbi, which is called an uncoupled representation, and covers the space of dimension (2ja + 1) (2jb + 1).

On the other hand we can deﬁne another complete basis set from the following ˆ2 ˆ2 ˆ2 ˆ operators that commutes between them: ja, jb , j , jz . We can suppose that the state

|jajb; jmi ≡ |(jajb)jmi ≡ |jmi is simultaneouslyn eigenstateo of these operator set, so we have:

ˆjz |jmi = m |jmi (2.62)

The state |jmi, which is called a coupled representation, and covers the dimension space

2j + 1 for each value of j with |ja − jb| ≤ j ≤ ja + jb.

2 2 It is important to note that although ˆj , ˆjz = 0, we have ˆj , ˆjaz =6 0. This ˆ2 h i ˆ2 ˆ2 hˆ ˆ i amounts to say that we cannot add j to the set of operators ja, jb , jaz, jbz and in the ˆ ˆ n ˆ2 ˆ2 ˆ2 ˆo same way we cannot add jaz and even jbz to the set of operators ja, jb , j , jz . Thus we have two possible basis sets, which correspond to the two maximumn sets of mutuallyo com- patible observables. An interesting feature of these basis sets is the fact that they correspond to the same number of observables. Thus these two descriptions are equivalent and them representations are connected by a unitary transformation. So we have:

|jmi = C (jajbj; mambm) |jama, jbmbi (2.63) m ,m Xa b Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 17 and the inverse transformation is given by:

|jama, jbmbi = C (jajbj; mambm) |jmi . (2.64) j,m X where the elements of unitary transformation are called the coefficients of Clebsch-Gordan and are chosen by convention to be real. We can find these coefficients in the literature with the following notations:

C (jajbj; mambm) ≡ hjamajbmb|jmi ≡ hjm|jamajbmbi (2.65)

Additionally, they are also known as vector coupling coefficients, vector addition coefficients, and Wigner coefficients.

Consider the unit transformation that connects two bases of operators given by:

|jmi = |jmi (2.66)

|jama, jbmbi hjama, jbmb| = 1 (2.69) m ,m Xa b is the identity operator in the ket space of a given ˆja, ˆjb.To ensure the existence of the elements of this unit transformation matrix, which are the Clebsch-Gordan coeﬃcients, some properties must be endorsed, such as:

m = ma + mb (2.70)

|ja − jb| ≤ j ≤ |ja + jb| (2.71)

The eq.(2.71) is the triangular inequality and ma, mb and m are magnetic quantum Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 18 numbers. The dimensionality of the space generated by the two basis sets is equal as follows:

jmax

(2j + 1) = (2ja + 1) (2jb + 1) (2.72) j Xmin

So we have that, for a given ja, there are 2ja + 1 possible values of ma and then for a given jb, we have 2jb + 1 possible values. A unitary real matrix is orthogonal and therefore we have the orthogonality condition given by:

hj j ; m m |j j ; jmi hj j ; jm|j j ; m0 m0 i = δ δ (2.73) a b a b a b a b a b a b mama0 mbmb0 j m X X 0 0 As a special case of this, we can take ma = ma, mb = mb and we obtain:

2 | hjajb; mamb|jajb; jmi | = 1 j m X X

which follows from the orthonormality of |ja, jb; ma, mbi together with the fact that Clebsch-Gordan coeﬃcients are real. In the same way we have:

0 0 hjajb; mamb|jajb; jmi hjajb; mamb|jajb; j m i = δjj0 δmm0 (2.75) m m Xa Xb As a special case of this, we can take j0 = j, m0 = m and we obtain:

2 | hjajb; mamb|jajb; jmi | = 1 (2.76) m m Xa Xb which is the normalization condition for |jajb; jmi.

As already mentioned above, such coeﬃcients may appear in diﬀerent forms. Often Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 19 we can rewrite them as follows:

ja jb j 1 ja−jb−m − ≡ (−1) (2j + 1) 2 hjama, jbmb|j − mi (2.77)   ma mb m   1 ja jb j ja−jb+m hjama, jbmb|jmi ≡ (−1) (2j + 1) 2 (2.78)   ma mb −m   which is the deﬁnition of 3j Wigner’s symbol. The 3j-symbols can be rewritten in a number of ways, since they have symmetry properties that leave their value invariant. Within this context two main properties stand out: A permutation of a pair of columns as given below:

ja jb j jb j ja j ja jb = = , (2.79)       ma mb m mb m ma m ma mb       On the other hand an odd permutation between columns, which is equivalent to multiplying by a phase, as seen below:

ja jb j jb ja j = (−1)ja+jb+j (2.80)     ma mb m mb ma m     ja j jb = (−1)ja+jb+j (2.81)   ma m mb   j jb ja = (−1)ja+jb+j (2.82)   m mb ma   Finally, the symbol 3j has the following property:

ja jb j ja jb j = (−1)ja+jb+j (2.83)     ma mb m −ma −mb −m     Now we introduced the fundamental tools allowing us to continue towards summations of angular momenta, and establishing the main properties by adding three or more angular momenta. Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 20

2.2.2 The 6j-symbols

Consider three angular momenta vectors given by ˆja,ˆjb,ˆjc, that can be summed up, two by two, in diﬀerent ways as given below, generating a total ˆj.

ˆj = ˆja +ˆjb +ˆjc (2.84)

ˆjab = ˆja +ˆjb (2.85)

ˆjbc = ˆjb +ˆjc (2.86)

ˆj = ˆjab +ˆjc (2.87)

ˆj = ˆja +ˆjbc (2.88)

We have that the state given by |jmi does not completely characterize these sums in the sense that there are different ways of finding the total ˆj. We can then find the eigenfunction resulting from the sum of ˆja and ˆjb, which yields ˆjab and then sums ˆjab with ˆjc as it gives the total ˆj. It follows that:

|jabjmi = |jmi (2.89)

|jabjmi = hjabmab, jcmc|jmi |jabmab, jcmci (2.91) m b,m Xa c To ﬁnd the decoupled state on the right side of eq.(2.91), we can use the identity operator as follows:

|jabjmi = hjabmab, jcmc|jmi hjama, jbmb, jcmc|jabmab, jcmci ma,mb, mXab,mc × |jama, jbmb, jcmci (2.93)

Thus we observe that in order to go from a coupled state |jabjmi to an uncoupled state |jama, jbmb, jcmci, it was necessary to insert two Clebsch-Gordan coefficients. On the other hand, it is possible to find ˆj following a different way. That is, we can add ˆjb with ˆjc which gives ˆjbc, and then add ˆja to find the total ˆj. For this path we have the following basis:

As mentioned above, we can ﬁnd the following decoupled state:

|jajbcjmi = hjama, jbcmbc|jmi hjbmb, jcmc|jbcmbci |jama, jbmb, jcmci (2.96) mamb mXc,mbc

These states must lead to the similar result, since we only use alternative forms of coupling to ﬁnd the total ˆj. Then there must be a unitary transformation that connects these two coupling schemes with the same total angular momentum and projection m, and can be written as follows:

|jajbcjmi = hjabjcj|jajbcji |jabjcjmi (2.97) j Xab This unitary transformation given in terms of the Clebsch-Gordan coeﬃcients is the deﬁnition of Wigner’s 6j-symbol, which assumes the following form:

1 ja jb jab ja+jb+jc+j hjabjcj|jajbcji = (−1) [(2jab + 1)(2jbc + 1)] 2 (2.98)  j j j   c bc    Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 22

ˆ2 ˆ2 ˆ2 ˆ2 ˆ2 ˆ ˆ2 ˆ2 ˆ2 ˆ2 ˆ2 ˆ So, we have two complete basis sets given by: ja, jb , jab, jc , j , jz and by ja, jb , jc , jbc, j , jz .

Their eigenstates are given, respectively, by n|jajbjabjcjmi and byo|jajbjcjbcnjmi, where jajbjc o ˆ2 are labels corresponding to the eigenvalues (ji + 1) of the ji operator. The coeﬃcients that make the transformation between two coupling schemes are also known as re-coupling coef- ﬁcients. The 6j-symbols must respect certain conditions for their existence, that are given below:

1. For the triangular condition must be respected, it follows:

|ja − jb| ≤ jab ≤ |ja + jb| (2.99)

|jb − jc| ≤ jbc ≤ |jb + jc| (2.100)

2. All momenta are nonnegative integers or semi-integers.

Wigner’s 6j-symbols express the transformation between two complete basis sets, them:

1 ja jb jab |j j j j ; jmi = [(2j + 1)(2j + 1)] 2 |j j j j ; jmi (2.101) a b ab c ab bc   a b c bc j jc j jbc Xbc     Alternatively, we can write these coefficients in terms of the so-called Racah coefficients, which was the first way introduced to characterize the re-coupling coefficients:

− 1 W (jajbjjc; jabjbc) ≡ [(2jab + 1)(2jbc + 1)] 2 hjabjcj|jajbcji (2.102)

Instead of Wigner’s 6j-symbol, Racah’s coefficients are often used, especially in spectroscopy [5]. These coefficients differ from the 6j-symbols by only one phase factor. Wigner defines a closed relation, for the 6j-symbol, which has high symmetry and is given below:

ja jb jab 1 ja+jb+jc+j − = (−1) W (jajbjjc; jabjbc) = [(2jab + 1)(2jbc + 1)] 2 hjabjcj|jajbcji  j j j   c bc  (2.103)   The 6j-symbols are widely used because they have properties of symmetries which Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 23

Figure 2.1: Triangular conditions for elements of 6j-symbols are now considered: Symmetry, orthogonality, expressions of 6j written as a function of the

3j-symbol and special case when one of ji is equal to zero.

1. The 6j-symbol is invariant for the exchange of two columns, and / or for the exchange of the upper and lower elements in each of two columns. We also consider the following property:

ja jb jab jb ja jab ja j jbc = = (2.104)  j j j   j j j   j j j   c bc   c bc   c b ab 

In order for 6j not to become zero, it is necessary to respect the triangular condition for

the four sets of three angular momenta, given as follows ∆(jajbjab); ∆(jajjbc); ∆(jcjjab). This condition can be illustrated as in ﬁg.(2.1):

2. Orthogonality

ja jb jab ja jb jab (2jab + 1)(2jbc + 1) = δj j (2.105)    0  bc bc0 j jc j jbc jc j j Xab    bc      3. Expression of 6j-symbol written as a function of 3j-symbol

ja jb jab  j j j0  = (−1)jc−mc+j−m+jbc−mbc  c bc    allm   X   ja jb jab ja j jbc jc jb jbc jc j jab ×         ma mb mab ma −m −mbc mc mb −mbc −mc m mab        (2.106) Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 24

4. Special case when an entry is zero:

ja jb 0 j j j − 1 a+ c+ bc 2 = (−1) [(2ja + 1)(2jc + 1)] δjajb δjcj (2.107)  j j j   c bc    It is fundamental to understand the mathematical properties of the 6j-symbol, since it works as a building block for theory. It is widely explored because it presents properties of interesting symmetries, and can be written in function of other expressions such as the

hypergeometric 4F3 functions. For other relations the reader is invited to see the books of Varshalovich et al [5] and Nikiforov et al [6].

2.2.3 The 9j-symbols

For the case of four angular momenta vectors, we will have several ways of combining

and recombining them to ﬁnd the total ˆj. Consider ˆja,ˆjb,ˆjc,ˆjd,ˆj, so that ˆjab = ˆja +ˆjb,

ˆjcd = ˆjc +ˆjd and ˆj = ˆjab +ˆjcd. There are several other ways to ﬁnd the same total ˆj. The passage between these various coupling schemes for the total angular momentum operator will be possible with the aid of the 9j-symbol. We can represent the eigenfunction by indicating the terms to be added two by two as follows:

|(jajb)jab(jcjd)jcdi = h(jajd)jad(jbjc)jbcjm|(jajb)jab(jcjd)i |(jajd)jad(jbjc)jbcjmi

jad jbc X X (2.108)

We can therefore establish the following equation as the deﬁnition of the 9j-symbol.

ja jb jab 1 2   h(jajb)jab(jcjd)jcdj|(jajd)jad(jbjc)jbcji = [(2jab+1)(2jcd+1)(2jad+1)(2jbc+1)] jc jd jcd     jad jbc j  (2.109)    The 9j-symbol has several important symmetry properties, that will be explored throughout this thesis. Here we will start from the most common properties, which are will reported below. Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 25

1. Symmetry:the 9j symbol is invariant by a row or column permutation, but must be

multiplied by the phase (−1)ja+jb+jc+jd+jab+jad+jbc+jcd+j. It is also invariant by an odd row or column permutation.

2. Orthogonality:

(2jab + 1)(2jcd + 1)(2jad + 1)(2jbc + 1) (2.110) j ,j abXcd ja jb jab ja jb jab     × jc jd jcd jc jd jcd = δjadj0 δjbcj0 (2.111)     ad bc    0 0  jad jbc j jad jbc j         3. Expression of 9j symbol can be written as a function of 6j-symbols

ja jb jab   2x jc jd jcd = (−1) (2x + 1) (2.112)     x jad jbc j X   ja jb jab jc jd jcd jad jbc j ×  j j x   j x j   x j j   cd   b bd   a c        4. Particular case that one entry is zero:

ja jb jab j j j j − 1   b+ ab+ c+ ad 2 jc jd jcd = (−1) [(2jab + 1)(2jad + 1)] (2.113)     jad jbc 0     ja jb jab × δjabjcd δjadjbc (2.114)  j j j   d c ad    Another relationships, amply exploited in the literature is the Pentagonal or Biedenharn- Elliott and will be explored in detail throughout this thesis (chapters). Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 26

Figure 2.2: Tree scheme of 6j-symbol

2.3 Spin network graphs

The 3nj-symbols may be represented diagrammatically, in order to make it possible to study the coupling and the re-coupling between the angular momenta. There are among these, we highlight the main ones that are: Diagram of trees [6]; Diagram of Yutsis [7] and the representation in tetrahedron form investigated by Ponzano and Regge [9]. The so-called tree representation is fundamental to understand the spin network that in the background is a sequence of all possible combinations and recombinations by adding (N + 1) angular momenta in diﬀerent ways. In Fig.(2.2) we present a basic unit of the tree-shaped diagram. In this particular case, the rods are angular momenta and the nodes are equivalent to the two-to-two couplings. We have that the 6j-symbol performs the transformation between the coupling schemes, or more exactly it performs a basis set transformation, which can be described as in the following equation:

j j j j +j +j +j 1 a b ab |(j j )j jmi = (−1) a b c [(2j + 1)(2j + 1)] 2 |j (j j )jmi (2.115) a b c ab bc   a b c j jc j jbc Xbc     The spin-network nomenclature was ﬁrst described by Roger Penronse [8] in 1971 to deal with problems of quantum gravitation. For the present case of Fig.(2.2) the two trees are related by a 6j-symbol which carries a coupled angular momentum scheme to another coupling scheme. For both coupling representations, the eigenstate is the same, and therefore we have: Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 27

Figure 2.3: Yutsis diagram of the 6j-symbol

ja+jb−jab |(jajb)jci = (−1) |(jbjc)jai (2.116)

The 6j-symbol may also be represented in the form of a Yutsis diagram. Figure 2.3 shows a quadrilateral and its ability to represent the coupled scheme. Since this diagram was ﬁrst amply exploited by Yutsis et al [7], it is named accordingly. Notice that the nodes are the coupling of three angular momenta, while each side represents a single angular momentum.

Figure 2.4: Tetrahedron representation of the 6j-symbol

Additionally, we also ﬁnd in the literature the representation of the 6j-symbol by means of a tetrahedron, as can be seen in Figure 2.4. This representation was studied by Ponzano and Regge [9], and describes a discretized three-dimensional geometry, given in terms of triangulation by simplexes (tetrahedral in three dimensions), such that the length of each edge corresponds to an irreducible representation of the Lie group SU(2). These tetrahedrons are described by 6j-symbols, whose asymptotic formula allows passage to the classical boundary, leading to an expression for the partition function representing a sum Chapter 2. Coupling and Recoupling Theory of 3nj Symbols 28 over a three-dimensional geometry.

In the following chapters we will investigate in detail the mathematical properties of these representations, their geometrical properties and their symmetries. Therefore with such tools in hand, we will deepen their discussion from the physical point of view. Bibliography

[1] R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Dover Publications, 2007.

[2] L. C. Biedenharn, J. D. Louck, The Racah-Wigner Algebra in Quantum Theory, no.Rota, GC. (Ed) in Encyclopedia of Mathematics and its Applications Vol 9, Addison-Wesley Publ. Co.: Reading MA, 1981.

[3] J.J. Sakurai, J. Napolitano. Modern Quantum Mechanics.2nd Edition. Bookman, 2013.

[4] A. Edmonds, Angular Momentum in Quantum Mechanics, second edition Edi- tion,Princeton University Press, Princeton, New Jersey, 1960.

[5] D. Varshalovich, A. Moskalev, V. Khersonskii, Quantum Theory of Angular Momentum, World Scientiﬁc, Singapore, 1988.

[6] A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag, Berlin, 1991.

[7] A. Yutsis, I. Levinson, V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum, Israel Program for Scientiﬁc Translation, 1962.

[8] R. Penrose, Angular momentum: an approach to combinatorial space-time, in Quan- tum Theory and Beyond, ed. T. Bastin, Cambridge University Press, Cambridge, pp. 151–180, 1971.

29 Bibliography 30

[9] G. Ponzano, T. Regge, Semiclassical limit of Racah coeﬃcients, Spectroscopic and Group Theoretical Methods in PhysicsF. Bloch et al (Eds.), NorthHolland, Amsterdam, pp. 1- 58. Chapter 3

Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams

The Wigner 9j symbols of the first kind also known as Fano X-coefficients serve to connect different addition schemes of four angular momenta, widely known examples being the LS and the jj couplings in atomic, molecular, and nuclear spectroscopies. Here, we also consider alternative sequences of binary couplings of four angular momenta, which are dealt through the 9j symbols of the second kind, and are explicitly given by the pentagonal (or Biedenharn–Elliott) identity. These coefficients are essential ingredients in the quantum- mechanical treatments of rotational and polarization phenomena in reaction dynamics and photoinduced processes. We also emphasize the combinatorial structure underlying the extended construction of a previously introduced truncated icosahedral “abacus”, and provide extensions useful for algebraical manipulations, semiclassical interpretations, and computational applications, including all the 120 addition schemes.

31 Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 32 3.1 Introduction

The general theory of quantum-mechanical angular momentum, developed originally to describe spectroscopic phenomena [1] in atomic, molecular, optical and nuclear physics, can be embedded in more formalized settings which emphasize in particular the underlying combinatorial aspects. This is one of the most notable applications of Lie group theory in physics, and is the basis for the description of the dynamics of anisotropic reactions. Wigner’s 3nj symbols [2, 3, 4, 5, 6], as well as the related problems of their calculations, general properties, asymptotic limits for large entries [7, 8, 9, 10], nowadays play a prominent role also in ﬁelds like quantum gravity [11, 12, 13, 14], quantum computing [15, 16, 17, 18, 19] and image reconstructions [20]. Recently, the apparatus and ingredients of this modern setup and of its extension to other Lie and quantum groups [21] are being studied in this and other laboratories [22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. We take advantage of recent results on 6j symbols [32] and on the associated 7-spin network, brieﬂy introduced and discussed by us in a previous report [33]. The important case of 9j symbols is considered here and details on the associated 10-spin networks will be presented elsewhere.

For the 6j symbols, which can be considered as particular cases of 9j symbols when one of the entries is equal to zero, explicit connections have been established with the mathematical theory of discrete orthogonal polynomials (the so-called Askey scheme), providing powerful tools based on asymptotic expansions, which correspond on the physical side to various levels of semi-classical limits [34, 35, 36, 37, 38, 39, 40]. These results are useful not only in theoretical molecular physics but also in motivating algorithms for the computationally demanding problems of molecular dynamics and chemical reaction theory, where large angular momenta are typically involved. Regarding quantum chemistry, applications of these techniques include selection and classiﬁcation of complete orthogonal basis sets in atomic and molecular problems, either in conﬁguration space (Sturmian orbitals) or in momentum space [41, 42, 43, 44, 45, 46, 47, 48, 49]. Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 33

3.2 Couplings and recouplings: Spin nets and 6j symbols

The full classification of the angular momentum algebra elements in terms of a scheme to be useful as an abacus for manipulations of the combinatorial features had an early motivation in Ref[41], where the presentation was mainly concerned with the trees of couplings of the (hyper)spherical harmonics arising from the separation of variables in Laplace-type equations. They are of importance for example as complete orthonormal sets for the kinetic energy operator in the quantum mechanics of few bodies. In quantum chemistry, the hydrogen atom basis sets in momentum space belong to this class. Indeed, such bases are essentially O(4) hyperspherical harmonics (mathematically related to hypergeometric polynomials), and their superposition coefficients which are Racah polynomials, i.e. a generalization of 6j coefficients, have much in common with the latter.

The introduction and a discussion of the abacus was given in [34], where it was developed in parallel for both hyperspherical harmonics and 6j coeﬃcients in terms of properties, allowing for a visual derivation of addition rules and relationships, otherwise often obscure, between angular momentum coupling and recoupling coeﬃcients.

Several applications have followed for the case of hyperspherical harmonics of any dimension [39, 42, 45, 46, 50], and their corresponding basis sets in direct space, i.e. Sturmian functions [47, 48, 51], of interest in quantum chemistry.

3.3 Orthogonality and the pentagonal relationship

In the present work we summarize some of relationships derived in ref. [34], dwelling on some less studied aspects and illustrating those features of the abacus, particularly relevant for the treatment of 9j symbols, to be treated later.

Let us consider an ordered sequence of four angular momenta, ja, jb, jc and jd, which couple in pairs to give j (for simplicity, a, b, c and d will instead be used in the ﬁgures). Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 34

As it will be stressed in the following, the overall symmetry of the coupling process makes any other choice, for instance jb, jc, jd and j coupling to give ja, equally legitimate. This is just one of a number of symmetry equivalences (either classical or of Regge type [4]) that hold at the algebraic level, recalling that alternative coupling schemes actually correspond to physically diﬀerent sets of commuting quantum observables.

The chosen angular momentum coupling scheme can be represented graphically by a tree-like or spin-net structure [5], such as those depicted in Figure 3.1, where successive couplings are taken into account to give the total angular momentum j.

Each coupling scheme (i.e. each tree in the ﬁgure) can undergo [34] two types of operations:

1. a recoupling operation (indicated by full arrows in Figure 3.1), connecting two alternative sequential couplings of angular momenta, which is essentially accomplished by a Racah coeﬃcient (6j symbol). The passage implies a sum over the eliminated angular momentum (explicitly indicated above the arrows in the ﬁgure).

2. a swap operation (indicated by dashed arrows in Figure 3.1), i.e. a reﬂection around an axis passing through a node, and corresponding to a simple phase factor, since it does not involve rearrangements in the coupling of angular momenta. This operation can also be seen as a rotation around a particular angular momentum, which plays the role of a pivot element (highlighted by a circle in Figure 3.1).

Going back and forth the solid arrows graphically represent the orthonormality relation:

ja jb jab ja jb jab (2jab + 1)(2jbc + 1) = δj ,j (3.1)    0  bc bc0 j jc j jbc jc j j Xab    bc  If we consider the same graphical sequence, including only recouplings and no swap operations, we obtain five possible different spin-nets which are linked according to the five- vertices scheme of Figure 3.1, leading to the well known pentagonal relationship in angular Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 35

Figure 3.1: Pentagonal scheme illustrating the Biedenharn-Elliot identity and representing connections among spin-nets, i.e. alternative recouplings of angular momenta. Solid arrows correspond to sums over the eliminated quantum number and are explicitly given by 6j coefficients. In the pentagon the sequence of the four original initially coupled angular momenta is maintained. Each coupling scheme (represented as a tree-like graph) can in turn be connected through a phase to an alternative graph, corresponding to a similar coupling scheme pertaining to a different sequence of angular momenta. This operation can also be thought of as a rotation around a pivot angular momentum (circled in the figure).. momentum theory, that is Biedenharn-Elliot identity:

ja jb jab jb jc jbc jd jbcd jbc (−)X (2j + 1) bc       j jc jabc jbc jd jbcd jcd ja jabc j Xbc       jab jcd j jab jcd j    = (3.2)  j j j   j j j   d abc c   bcd a b      Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 36

where X = ja + jb + jc + jd + jab + jbc + jcd + jabc + jbcd + j.

3.4 The hexagonal relationship and the truncated icosahedron abacus

Each of the spin-nets in the pentagon can be connected through a swap operation to another spin-net which only diﬀers for a phase factor, and corresponds to an alternative graphical sequence. The latter spin-net is in turn part of another pentagonal scheme. By developing this construction, we obtain a closed graph consisting of pentagons and hexagons (Figure 3.2). The relations which can be written down by performing the operations following the perimeter of the hexagons involve three 6j coeﬃcients and three phases and lead to the well known Racah sum rules, for instance:

ja jb jab ja jb jab ja jabc jbc (−)jab+jbc+jac (2j + 1) = (3.3) ab       j jc jabc jbc jabc jc jac jb jc jac Xab             The complete graph, obtained exhausting all ordered labelings, turns out to be isomorphic to that of a truncated icosahedron: thus our abacus is made of 60 vertices (the spin-nets) and of 12 pentagons and 20 hexagons (Figure 3.3).

In fact, this classiﬁcation does not cover all the possible spin-nets: those obtained by swapping two ”leaves” are to be found in an identical abacus gathering the other 60 possibilities (Figure 3.4). There are indeed 120 spin-nets, a number that can be obtained by considering that, when combining n+1 angular momenta (or ”leaves”), the resulting number of spin-nets is given by [36, 49]: (2n)! N = (3.4) n!

0 N (2n)! Note that only N = 2n = n!2n spin-nets, i.e. 15 in the present case, correspond to struc- turally diﬀerent coupling or recoupling schemes.

Other sum rules, besides those given above, can be obtained by performing close circuits along the abacus. Moreover, limits for large values of the angular momenta involved Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 37

Figure 3.2: Hexagonal schemes obtained by extending the pentagonal scheme in Fig. 3.1. The corresponding sum rules obtained by following the edges of the hexagon produce the Racah sum rules. can be important for discrete gravity models or for the semiclassical cases.

This scheme can obviously be used to build more complex cycles, involving special matrix elements for further vectorial coupling, whose most relevant example is represented, as will be discussed in the following section, by 9j symbols, considered by Danos and Fano [52] to be at the basis of their graphical theory of angular momenta. Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 38

Figure 3.3: Hexagons and pentagons leading to the truncated icosahedron, which constitues the abacus for recoupling coeﬃcients. On the left: half abacus, with explicit spin-nets at the vertices: on the right the whole abacus. Note that spin-nets at the antipodes (like those circled in red in the ﬁgure) are mirror images.

3.5 The 9j symbols and the 120 spin-net abacus

In previous works [22, 34], mention was made about the use of the abacus for the deﬁnition of 9j symbols; in this section this combinatorial structure is now used in a systematic way to obtain their visualization and properties. Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 39 . n 2 / ! /n )! n = 15 is the cardinality 3 = (2 D n D ac ac 7 7 = 3 and the cardinality of the , , b d b d 6 6 = 3, n n ac ac c a 8 8 full spin networkof is labelled 120. trees with The notween distinction left number be- and righttorial is the double fac- Here For of the reduced spin network. , , b d b d b d 3 3 c a c a 8 2 , , b d b d 3 5 c a 8 8 ac , , b d 5 5 8 , b d 3 c a b d c a 8 8 , , , c a b d 5 5 c d b d ! Combinatorics c a . 8 8 b d , , b d ac 7 7 b d 5 5 c , , 2 a ! /n b a , 1 1 c d b ac d 5 )! c a 8 8 , , /n b d ac 7 7 +1 terminal b d n 5 5 , , c d b a b d b d 6 6 n 7 7 , , b d ac 8 8 ac ac 1 1 , , b a 3 3 + 1)! = (2 c d n n ( c d b d ac 6 5 C / , , b d b d b a 3 4 b d ac 7 7 7 7 )! , , , , b ac ac a 1 1 1 1 n + 1)! b n d = (2 c a b d b d 8 8 , , c a n c a 3 3 =( b d 8 8 , , C c a 3 3 n ˜ C b d c a b d 4 6 , , b d c a 7 1 b a c a The numberquadruple of factorial labeled trees is the Rooted binary trees with nodes. The numberis of the unlabeled Catalan trees number b d 8 8 , , ca db 8 7 c a 3 3 b , , d c d 1 2 b a c a b d 8 8 , , ca db 7 7 c a 3 3 , , b a c d 1 1 i ca db 7 7 , , 2 c d 6 6 / jm 1 ; b a db db 1 7 7 b a , , j ca db 7 7 ca ca 1 1 i , , ) c d +1) 6 6 ca db 8 8 3 , , c j d 5 5 23 jm 2 j j ); ( db db | (2 3 8 8 2 j , , ca ca / 5 5 23 2 1 j = j ( − i db 1 2 6 j j , ca | +1) 1 + jm 1 ; 12 j j 3 j Twist 1) ) (2 23 12 Rotation 2 j ca • j − j 4 + , • 1 db 3 7 j j =( ( 2 Expressions j j | + j ca ca i 2 7 7 j , , db db 6 6 + jm 3 1 1 j j j ); 3 1) j ca ca − 2 7 7 ( j , , db db ( 6 6 1 23 j j X | a a T b b c c 2 2 , , d d R 6 6 T R a a b b c c 2 2 , , d d R 6 6 a T b c 8 , d 7 Racah’s identity c d a 6 , b 5 c c d d a a 2 2 , , b b 6 6 R ca ac R 4 4 c c , , ca ac b d db 5 5 d d 4 4 R symbols T: phase , , a a 2 2 b d db 5 5 , , b b 6 6 j R R identity ca ac 2 2 , , b d db R: 6 6 6 c d ca ac 4 4 , , ca db 4 4 b d db 3 3 , , c b d a 5 5 ca ac Biedenharn-Elliot 8 4 ca db 2 2 , , The Spin-Network Graph b d , , db d 7 3 b a c 1 1 ca 2 db 4 , b , 1 a 3 c d ca db 4 4 c Top d , , b a 3 3 ca db 4 4 , , b a 3 3 d d c c Root Twists b d db b b 2 2 2 2 , , db Reduced spin network a a , , b d 1 1 ca ac 1 1 2 6 , ac , 1 ca 5 Group of and Root Twist d d b d db c c 4 4 b b 2 2 d , , , , Top Twist ca ac 3 3 a a c 1 1 b a b d db b 4 4 4 , , , a b d ac 4 4 d 3 ca ac 5 5 , , c c d 5 5 Trees b a b d db b 4 , 4 4 a , , 3 b d ac 2 2 ca ac 3 3 , , b a c d 6 6 Four moments b d ac 4 4 , , c d 5 5 b a b a d 2 , c 1 b a b d ac 4 4 , , c d b b 3 3 b d ac 6 8 a a , , c d Symmetric tree: Unsymmetric tree: 5 7 d d 2 2 , , c c 1 1 Top Twist Recoupling scheme b b a a d d 2 2 , , c c 1 1 Root Twist Planisphere Rotation Top Twist Internal Twist Legend Figure 3.4: The completenected spin-network truncated of the icosahedrical 120 abacus.factor. coupling A schemes, Double version or dotted in spin-nets, color,equivalent lines classiﬁed apart connect according from to spin-nets minor two onto details, intercon- is the available two in abacus [54]. through a phase Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 40

3.6 9j symbol of the ﬁrst kind

The deﬁnition of 9j symbols as a sum over three 6j symbols can be obtained by following the dashed path in Figure 3.5 along the abacus:

ja jb jab ja jb jab jc jd jcd ja jc jac   2jbcd jc jd jcd = (−1) (2jbcd+1)    j j j   j j j   j j j    jbcd  cd bcd   b bcd bd   bd bcd  jac jbd j X        (3.5)   which, apart from the use of the 6j symmetry properties over the last 6j symbol in the sum, is exactly the same deﬁnition of 9j symbols found in [4], p. 340, eq. (20).

Note that if we consider the important case of two quantum systems with, respectively, orbital angular momentum l1 and l2 and spin s1 and s2, respectively, coupled to give a state with total angular momentum j, according to either ls or jj couplings, the transformation between the two alternative schemes may involve the use of an additional phase

(−1)s1+s2−s to obtain a spin-net with the correct symmetry [22].

When dealing with 9j symbols, Yutsis diagrams [2] can be proﬁtably employed in order to graphically represent the genesis of the symbol. Following such an approach, a 6j symbol can be associated to a complete quadrangle (see [4, 2]) whose trivalent nodes correspond to triads of angular momentum variables according to Scheme I: Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 41

Figure 3.5: The highlighted dashed red path along the edges of the abacus corresponds to the deﬁnition of 9j symbols, eq. (3.5). The closed path obtained by following in sequence the edges along the dashed red path, the light green dotted path and the light blue dotted path is the graphical representation of the sum rule involving three 9j symbols.

Scheme I Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 42

In such a view, eq. (3.5) can be represented as Scheme II:

Scheme II where the hexagon plus its three diagonals is the Yutsis diagram associated to 9j symbols. The six triads can be associated to trivalent nodes and represent columns and rows of the array. Symmetries can also be easily visualized: the value of the coeﬃcient does not change, except for a phase, by permutations of columns and rows.

3.7 9j symbol of the second kind

The deﬁnition of the 9j symbols of the second kind (or type II [2]) corresponds to a diﬀerent sequence of couplings and turns out to be simply connected to the Biedenharn-Elliot relationship (eq. 3.2 and Fig. 3.1):

jabc ja jbc jb jc jbc jd jbcd jbc (−)X (2j + 1) bc       j jb jc jab jd jbcd jcd jj jabc j Xbc          ja  jb jbcd j j j j j j ab cd ab cd   = ≡ jab jcd j (3.6)  j j j   j j j   d abc c   bcd a b     jabc jc jd          an apparently trivial deﬁnition, since it only involves the product of two 6j’s and no sum. This conﬁguration can be graphically depicted as in Scheme III: where the last symbol is the so-called 9j symbol of the second kind in the rarely used notation of Yutsis [2]. Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 43

Scheme III

As in the case of 6j symbols, it is possible to derive from the abacus useful sum rules by following the edges corresponding to 9j symbols to complete a closed path, starting and ending at the same spin-net. Explicitly, as shown in Figure 3.5, the relationship allowing to write a 9j symbol as a sum over two 9j symbols can be obtained [4]:

ja jb jab ja jb jab

jcd     (−) (2jab + 1)(2jcd + 1) jc jd jcd jd jc jcd     jab,jcd     X jac jbd j jad jbc j     ja jd jad    2jb+jbc+jbd   = (−) jc jb jbc (3.7)     jac jbd j     which, according to Yutsis graphical method, can be depicted as in Scheme IV:

Scheme IV

Other addition relationships can be obtained by following selected close paths and by considering that the ”twin” abacus, connected to the ﬁrst by phase factors, can also be Chapter 3. Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams 44 exploited.

3.8 Concluding Remarks

Graphical methods can proﬁtably be used to investigate properties and relationships in angular momentum algebra. We showed that, when the coupling of four angular momenta is involved, an abacus graph can gather all the available combinations and can describe symmetry properties, relationships and sum rules, both for Wigner 6j and 9j symbols. The inclusion of these and other pictorial representations, extending the original Yutsis approach, enables to visualize additional properties of coupling and recoupling coeﬃcients.

The potentiality of graphical approches can be further exploited and the results obtained by different treatments can be related and analyzed under a different light. For instance, the hexagonal and pentagonal relations described in this work can be examined as follows: the hexagonal relationship, which represents Racah sum rule as an interconnected chain of three 6j coefficients, involves seven angular momenta and can thus be generalized introducing a 7-spin network [53] with remarkable projective geometry properties. Along the same line of thought, the pentagonal Biedenharn-Elliot identity deals with the coupling of ten angular momenta, leading to a Desargues configuration [3] admitting the interpretation of a 10-spin network. We have seen that the definition of the 9j symbol also involves ten angular momenta, so that an alternative 10-spin network can be introduced and will be illustrated elsewhere. The graphical description of properties of 7- and 10-spin networks constitutes a valuable approach to discover and deepen not only unexplored geometrical and computational aspects of angular momentum algebra, but also combinatorial features of general interest in a variety of applications.

For this reason, in future work, we will give a novel projective geometry presentation of the conﬁgurations of 9j symbols as 10-spin networks, remarkably connected to Desargues- type graphs and their variants. Bibliography

[1] R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Dover Publications, 2007.

[2] A. Yutsis, I. Levinson, V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum, Israel Program for Scientiﬁc Translation, 1962.

[3] L. C. Biedenharn, J. D. Louck, The Racah-Wigner Algebra in Quantum Theory, no.Rota, GC. (Ed) in Encyclopedia of Mathematics and its Applications Vol 9, Addison-Wesley Publ. Co.: Reading MA, 1981.

[4] D. Varshalovich, A. Moskalev, V. Khersonskii, Quantum Theory of Angular Momentum, World Scientiﬁc, Singapore, 1988.

[5] J. S. Carter, D. Flath, M. Sato, The Classical and Quantum 6j-symbols, Princeton University Press, Princeton, New Jersey, 1995.

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[7] G. Ponzano, T. Regge, Semiclassical limit of Racah coeﬃcients, Spectroscopic and Group Theoretical Methods in PhysicsF. Bloch et al (Eds.), NorthHolland, Amsterdam, pp. 1- 58.

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[54] https://sites.google.com/a/uefs.br/lece/download. Chapter 4

Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations

The basic ingredients of the quantum theory of orbital and spin angular momentum (vector coefficients, 3nj symbols) encounter continuing relevance in wide areas beyond the traditional ones (molecular, atomic and nuclear spectroscopies and dynamics). This work offers insight on the connection at the most elementary of levels with the diagrammatic approaches to projective geometry. In particular here we exhibit how the Fano, Desargues and related incidence configurations emerge in the Racah and in the Biedenharn-Elliott identities, corresponding respectively to the hexagonal and pentagonal relationships that provide the basis for the construction of 3nj symbols and of spin networks. It is shown that the treatment, although mostly confined to the quadrangulation of the real projective plane, permits however the introduction of networks involving seven and ten spins, and preludes to developments towards computational and asymptotic approaches for quantum and semi-

51 Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 52 classical applications to spectroscopy and dynamics.

4.1 Introduction

In this work we will be concerned with basic aspects of the theory of angular momentum and of spin networks from the point of view of projective geometry. Progress in quantum angular momentum theory, not only in the traditional areas of atomic, molecular and nuclear spectroscopies [1, 2, 3, 4, 5, 6, 7] and dynamics [8, 9, 10, 11], but also in areas such as quantum gravitation [12, 13], quantum computation [14, 15, 16, 17, 18, 19, 20] and image reconstruction [21] have been further considered in recent papers [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]

Recently [35, 36] we discussed the network of seven spins and the combinatorial diagrams corresponding to the angular momentum coupling and re-coupling schemes [37, 38, 39, 40].

The focus of the present work is the connections of the quantum theory of angular momentum with projective geometry, and specifically its foundations in the geometry of the projective plane [41, 42, 43, 44, 45, 46, 47]. The inspiration to the interpretation given here was first initiated by U. Fano and Racah [48] and also elaborated by Robinson [49], Biedenharn and Louck [41], Judd [45], and Labarthe [50, 51]. The strikingly symmetric nature of the network suggests mapping on the Fano plane, which was introduced by G. Fano in 1892 as the smallest nontrivial finite projective geometry with seven lines and seven points. Further aspects and related problems have been discussed previously [35, 36] and briefly resumed here. A main object of this work is the study of 9j symbols and the associated networks of ten spins. Their connection to projective geometry can be provided with the Desargues theorem, whose dual representation will be exploited here, as a promising advance in this line of research. Two recent articles ( [52, 53]) provide background information to establish the connections between the 9j coefficients and networks of ten spins and both the Biedenharn-Elliott identity and the Desarguesian geometry. A specific point to be highlighted is the introduction of the Levi graphs [54], that can be used for an interpretation of the Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 53 incidence features and a guide to spin network manipulations. We show that these graphs carry important information in the construction of spin networks, facilitating the geometric interpretation of the coupling and recoupling coefficients of the quantum theory of angular momentum.

As well known, the spin networks are of great interest for several areas of physical and quantum chemistry, and many open problems still require to be tackled by their study. Several have been indicated by a series of articles in relationship to the subject of discrete orthogonal polynomials [55, 56], and to molecular collision theories [57, 58, 59, 60, 61], mostly published by this research group. Details about the connection with projective geometry will be indicated, as well as their fundamental features which will be outlined in Section 4.2: there, an illustration will be presented with specific reference to the basic network of seven spins. Section 4.3 deals with the pentagonal relationship and the 9j symbols of the first and second kind and the associated 10-spin networks; it continues presenting their Desargues configuration and the Desargues-Levy incidence diagrams. In the final section we will summarize the key points of our investigation and discuss in detail some of the various open issues.

4.2 Projective geometry and the spin network of seven spins

We give an account here and provide an extension of the geometrical and combinatorial features involved in the recent developments [36] in the relationships between quantum theory of angular momentum and projective geometry. Initially, the attention is on the network of the seven spins. Papers by Rau [62, 63], dealing in particular with applications to quantum computation, exemplify previous interesting work on the involved geometrical and combinatorial features. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 54

4.2.1 Representation of angular momentum couplings and recouplings as triangles and quadrangles

In establishing the relationship between angular momentum theory and projective geometry, we chose to refer explicitly for the former to the well known collection of formulas [38], and for the latter to Coxeter’s treatise [43], one of the best known and authoritative textbooks. Speciﬁc points will be directly referred to as propositions and pages of those book.

Diagrammatical presentations of quantum angular-momentum coupling and recoupling coefficients have been an intense subject of investigations for several years, initially of interest in the traditional fields of spectroscopy and dynamics in atomic, molecular and nuclear physics, and in quantum chemistry. As outlined in the previous section, recent investigations are particularly related to quantum gravity (following initial work by Ponzano and Regge [64] and by Penrose [44]), to quantum computing, and to the expanding mathematical field of spin networks; they include q-extensions [65], knot coloring and several other associated areas of research.

The diagrammatic approaches naturally originated in atomic, molecular and nuclear spectroscopies from the observation that the process of coupling two angular momenta to give a resultant third one be represented:

(i) - either as the junction of three segments;

(ii) - or else by the three sides of a triangle;

(iii) - or, alternatively, by three points on a line in a projective plane (this was introduced in 1958 by U. Fano and G. Racah, who ﬁrst established a connection with projective geometry [48]).

ingredient of quantum angular momentum theory is the Racah recoupling coeﬃ- cient, essentially equivalent to the 6j symbol of Wigner that we will use in the following: as amply exploited for example in [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 66, 67, 68], graphical techniques allow manipulation of otherwise complicate formulations. The alternative presentations of 6j symbols are as follows, corresponding to (i) - (iii) above: Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 55

(i) - As pioneered in Edmonds’ 1957 textbook [3], recouplings are presented as graphs with no open ends. The elaboration by Yutsis and collaborators [37] permits explicit algebraic manipulations by graphical techniques (see also the comprehensive treatise by Varshalovich, and collaborators [38]). In the simplest formulation, 6j symbols are represented by six segments whose ends terminate at four junctions where the three segments branch. Viewing segments as parts of six lines in the projective plane, they are perceived as forming a complete quadrangle, consisting of four points and the six lines joining them. Their main properties, the hexagonal and pentagonal relationships, to be encountered next, are the basis for construction of the icosahedral “abacus” to deal with 9j symbols [34, 53] and a morphogenetic scheme [24] to deal with the higher 3nj symbols (and therefore the visualization of the often otherwise obscure angular momentum coupling and recoupling schemes). The content of such graphs is both combinatorial and topological. A detailed assessment of the relevant literature and of the state-of-the-art is given in Ref. [53].

(ii) - This approach has been instrumental to very insightful semiclassical asymptotic investigations initiated by Racah (1942) [2], Wigner (1959) [1], Ponzano and Regge (1968) [64], Schulten and Gordon [69, 70]. These and subsequent investigations re- vealed the insight gained by substantiating graphs with a metrical ﬂavor, interpreting the triads as triangular faces of polyhedra in a three-dimensional space, or possibly of higher dimensionality simplices (or polytopes): the important representation of a 6j symbol is generically as an irregular tetrahedron: see the quadrangles in Fig. 4.1, to be convincingly regarded as the planar projection of a three-dimensional tetrahedron. However, one has further to apply the Euler duality consisting in the exchange of roles between the four faces and the four points, in order to obtain correspondence between triads and points of incidence of three lines, and to maintain the representation of the six j’s of the symbol by the six lines of the graph.

(iii) - In their projective geometry approach U. Fano and Racah (1959) [48] considered the Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 56

pentagonal (or Biedenharn-Elliott) relationship and pointed out the Desarguesian nature of quantum angular momentum theory. Their construction (to be illustrated next in Fig. 4.5) was invariably exploited by all subsequent investigators of the relationship with projective geometry (Robinson [49], Biedenharn–Louck [41], Judd [71] and Labarthe [50, 51]). All these authors went also on attempting at establishing additionally a connection between the hexagonal (or Racah) identity with the Fano plane introduced in 1892 by Gino Fano (Ugo’s father). In 1995 and later years, Ugo Fano admitted that he was not aware of such a connection (personal communication to V. Aquilanti): see [53] and Fig. 4.2.

Figure 4.1: The 6j symbol and its four triads as a complete quadrilateral (left) and as complete quadrangles (center and right). In the first graph, that represents the 46 configuration of incidence geometry, spins are associated to points and triads to lines. In the center and right two graphs, that represent the 64 configuration of incidence geometry, spins are associated to lines and triads to points. In general, the Mn configuration represent a graph with M lines and n points.

An additional representation, as incidence diagrams combining both the dual ones, is an extension of novelty introduced here. Its discussion needs the geometrical detour sketched in [Aquilanti and Marzuoli [52] and Santos et al. [53]]. For popular introductions to concepts of projective and ﬁnite geometries, see [46] and [47].

Our task will therefore be to demonstrate that by dualization of the construction Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 57

Figure 4.2: Representation on the Fano plane of the network of seven spins. The illustration is provided by drawing correspondences between relations satisﬁed by the nj symbols and various collinearity properties of the appropriate diagrams; in the labeling scheme a triad is represented by three points on a line, and the three 6j symbols appear as complete quadrilaterals, related to the complete quadrangles by the point-line duality in the projective plane according to Fig. 4.1. of these authors one gets also a direct connection with the established graphical methods of Yutsis [37] and Varshalovich [38].

4.2.2 Incidence structure and projective features

Discussions of geometrical aspects of angular momentum algebra and spin networks using any of the above alternatives should benefit of advantages of sharing several viewpoints: beneficial is also their ample interchangeability, due to appropriately individuated and exploited duality properties. However, we will next show the remarkable simplicity of the relationships with the foundational geometrical aspects of the projective and incidence diagrammatic approach emphasized here, and will document its consistency with Refs. [52, 53]. A purpose of this investigation has been therefore to establish how extensive is the conceptual overlap of two apparently separate subjects belonging to physical and mathematical areas Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 58 and how promising is its study.

In the following, extensive reference will be made to Coxeter’s book [43], Projective Geometry, Second Edition (the initial discussion is on p.15). In the axiomatic construction of projective geometry, the ﬁrst two axioms are written using the unifying concept of incidence when the only primitive objects of the theory, “points” and “lines”, meet:

Axiom I - (Coxeter [43] 2.13, p. 15) Any two distinct points are incident with just one line.

Axiom II - (Coxeter [43] 3.11, p. 24) Any two lines are incident with at least one point.

Any asymmetry between points and lines is effectively eliminated by their duality valid in projective geometry, [Coxeter, p. 25], but not in affine or Euclidean geometries. De- noting configurations consisting of points and lines by the number of points and the numbers of lines as subscripts, a triangle – three points and three lines – is 33 and therefore self-dual: however, since this configuration is shown to be too simple for a geometry with no angles nor lengths, in order to proceed one assumes the existence of the ”complete quadrangle”:

Axiom III - (Coxeter [43] 3.12, p. 24) There exist four points of which no three are collinear.

Note that the previous investigations [49, 41] used diﬀerent axioms, not on line with mainstream projective geometry, nor with he present discussion: indeed, this axiom provides strikingly a foundational motivation to our approach, in turn establishing a bridge to the more common graphical techniques of quantum angular momentum theory.

This is the simplest conﬁguration to build a plane and assign coordinates,as emphasized by M. Hall [72] in 1943, the four points being denoted e.g. as (00), (01), (10), Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 59

(11), using only the simplest of fields, F2. A notation for the configuration of the complete quadrangle of Yutsis diagrams is 46, and that of its dual, the complete quadrilateral, is 64. Fig. 4.1 establishes the connection between the 6j symbol and the associated graphs, one view for the complete quadrilateral and two for the complete quadrangle. Quadrangles and quadrilaterals are different from an affine viewpoint. Adding a line and three points to both cases gives two projectively dual 77 Fano configurations.

A consistent labelling of the complete quadrilateral and of the complete quadrangle in terms of the corresponding labelling of the 6j symbols is given in Figs. 4.2 and 4.3 respectively, employing small case letters in italics for SU(2) angular momentum variables and faithful for exhibiting the 24 permutational symmetries of the 6j symbols.

In the notation that we use an apparently slight lexicographical asymmetry still persists, originating from the fact that a basic facet of 6j symbols is their being elements of orthogonal matrices where indices x and y mark either rows or columns. Their ranges are determined since the entries must obey triangular relationships, indicated for example {a b c} and meaning |a−b| ≤ c ≤ a+b (and cyclically), see ([30, 66, 67, 92]). Besides combinatorial aspects, crucial is the role of Regge symmetries, see [26, 27, 74]. Orthogonality and normalization have thus to be endorsed as a property of 6j symbols (this is the ﬁrst of the three deﬁning properties of the 6j symbols, according to [48], and denoted A, B and C below):

Property A of 6j symbols (Orthonormality[See [38], Eq. (7) p. 463])

a b x c d x δ (2x + 1) = yy0 {a d y}{b c y} (4.1)    0  2y + 1 x c d y a b y X     where, as indicated above, the symbol{...} means that the three entries obey the triangular relationship. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 60

4.2.3 The quadrangle, its diagonal triangle and the network of seven spins

The second deﬁning property of 6j symbols, according to Fano and Racah [48], Robinson [49], Biedenharn and Louck [41], is known as the Racah sum rule (1942) [2], the equation below being labeled according to our conventions:

Property B of 6j symbols (Additivity: Racah sum rule)

a b x c d x c a y (−1)z+y+x(2x + 1) = (4.2)       x c d z b a y d b z X             which, using Eq. (4.1), Eq. (4.2) becomes ([38], Eq. (23) p. 467)

a b x a c y a d z0 (−1)x+y+z(2x + 1)(2y + 1)       xy c d z d b x b c y X       δzz    = 0 {a d z}{b c z} (4.3) 2z + 1

The hexagonal illustration of the formulas, which involve seven angular momenta, is for example in [34, 53].

Eq. (4.3) shows an apparent symmetry of the seven angular momenta, spoiled by the fact that only six triads are seen to be involved. Attempts at their representation on the Fano plane are given in Figs. 4.2 and 4.3. Appropriate then becomes to establish the connection to the projective geometric construction. Let’s introduce the following axiom, where reference is made to the so-called “diagonal triangle of a quadrangle” and known as G. Fanos’s axiom: Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 61

Figure 4.3: Representation of the Fano plane for network of seven spins, dual to that in Fig. 4.2. Here, the points are triads and the lines are angular momenta. The missing triad is represented as an empty triangle and indicates that no coincidence is enforced for the three lines x, y, z.

Axiom IV - (Coxeter [43] 2.17, p. 15) The three diagonal points of a quadrangle are never collinear.

Fig. 4.2 and 4.3 illustrate in two equivalent ways the important association of a unique diagonal triangle to any quadrangle. The content of both is compacted in the Levi incidence diagram in Fig. 4.4 [43] (formally this is recognized as a Heawood diagram known in graph theory since 1890). The new three diagonal points are added at the incidence of the three couples of opposite lines, but Fano axiom forbids their collinearity: in the angular momentum graphical assignment, we admit the new triads acz and bdz. The z segment joins them, but doesn’t correspond to any xyz triad! The resulting configuration contains the graphical properties of Racah’s sum rule, since the addition of z results in the appearance of the two quadrangles corresponding to the two 6j’s in the right-hand side of Property B. Such a configuration is often referred to as anti-Fano (at times also quasi-Fano): this apparently Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 62 contradictory terminology is due to the fact that, admitting the extension of the z segment as a full line including also a new unphysical triad xyz, would generate a special configuration

77 violating G.Fano’s axiom. This one is indeed the configuration that appearis in finite geometries initiated by him, who studied the effect of such violation. These finite geometries had a wealth of developments and important applications in expanding fields. Not long ago Roger Penrose noted, in his influential book “The Road to Reality” [44]: “Although there is a considerable elegance to these geometric and algebraic structures, there seems to be little obvious contact with the workings of the physical world”. However, this work conveys the message that this may not be so, and, as far as for computational science and block design are concerned, see [36, 62, 63] and below.

4.3 The two networks of ten spins

The next important step consists in introducing the celebrated Desargues conﬁgu- ration:

Axiom V - (Coxeter [43] 2.32, p. 19) If two triangles are perspective from a point they are perspective from a line.

This axiom corresponds to the construction of the 1010 configuration of general va- lidity and basic to planar projective geometry:it needs be introduced due to the well known impossibility of proving the Desargues theorem in the plane without assuming the existence of at least a point off the plane itself. Actually the theorem in space is a very simple one to be proven and is at the foundation of all geometries including affine and Euclidean ones. Its central role in the discussion from the spin network viewpoint emerges in related recent papers [52, 53]. The striking relationship can now be established between this configuration and the third and last basic property characterizing the 6j symbol. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 63

Figure 4.4: Levi incidence diagram compacting the two networks of seven spins in Figs. 4.2 and 4.3. Triads are represented as triangles (the missing triad by an empty one); angular momenta are given as circles. The three corresponding quadrangles are given in the same order by which they appear in the Racah sum rule, Eq. (4.2).

Property C of 6j symbols (Associativity):

This third (and last deﬁning, [48]) property of the 6j symbol is the Biedenharn- Elliott identity, relabeled according to our convention. It provides the associative relationship Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 64 schematically visualized as pentagons in [34].

φ a b x c d x e f x x (−1) (2x + 1)  c d p   e f q   b a r  P       p q r  p q r      = (4.4)  f b c   e a d          where φ = a + b + c + d + e + f + p + q + r + x. [See [38], Eq. (18), p. 466]

Using orthonormality, Eq. (4.1), one obtains from eq. (4.4)

a b c a e f c f p b e q x+y xy (−1) (2x + 1)(2y + 1)  d x y   g x y   g d x   g d y  P         δpq a b c        = (−1)Φ {dgp} (4.5) 2p + 1  p f e      where Φ = −a + b + c − d + e + f − g − p. [See [38], Eq. (35), p. 470]

As also documented elsewhere [52, 53], this is the property that, from a projective viewpoint, makes Desarguesian the projective geometry underlying angular momentum theory [48, 49, 71]. One can see there the 1010 conﬁguration of ten lines and ten points, consistent with the observation that in Eqs. (4.4) and (4.5) ten angular momenta and ten triads appear. A uniﬁed vision of e.g. a network of ten spins involving also relationship with the 9j symbol is discussed in [52, 53].

In the following subsections the progress in understanding the connections between angular momentum theory and projective geometry is further expanded, in particular giving a unifying diagrammatic view at the two 9j symbols previously discussed [24, 53, 37] and ﬁnally establishing the transformation between them. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 65

4.3.1 The 9j symbol of the second kind and Desargues theorem

In the definition of the 9j symbol of the second kind, recently discussed in Ref. [53] (see also references therein), the involved sum can be carried out thanks to the Biedenharn- Elliott or pentagonal identity, Eq.(4.4) and leads to the product of two 6j symbols. A total of ten angular momenta are involved in the spin network: as we have seen previously, the striking result that was found by Fano and Racah [48] is a projective geometry feature underlying the configuration identity and the celebrated Desargues’ configuration, Fig. 4.5.

The ”self dual” configuration involves ten points and ten lines: each point occurs at the incidence of three lines, and each line contains three points. As is well known the duality among points and lines lies at the foundation of projective geometry. The construction is guaranteed by Desargues proposition on two triangles perspective on a point being also perspective on a line (Coxeter, par. 3.2 and [47]). Fano and Racah [48] have shown that it is possible, in order to establish the connection with the theorem, to assign as points the ten spins appearing in the Biedenharn-Elliott relationship and as lines the triads. All subsequent studies [49, 41, 51, 50] used the same assignment, as in Fig. 4.5. However, we stress that this assignment is at variance with the convention universally used in application of angular momentum theory, that of Yutsis: as emphasized in the previous section, we found it interesting to provide a new labeling exploiting the point-line duality, in order to preserve the more insightful assignment as lines for spins and points for triads. The details of the construction are not given here [52], it is sketched with the aid of Fig. 4.6 and the result is shown in Fig. 4.7. See [52], where we also transcribe the Biedenharn- Elliott (or pentagonal) identity, Eq. (4.4), using a much simpler labeling, justified by the progressively more and more evident underlying symmetry emerging as we go on in this work. The renaming of the entries in the five 6j appearing in Fig. 4.7 is connected with the respective quadrangles of the Desargues configuration, as will now be illustrated in detail.

Let’s vertices of a pentagon be labeled with numbers, e.g. from 1 to 5 (Fig. 4.6), in order to exhibit the origin of the Desargues construction in Fig. 4.7, that remarkably contains a proper combination of the five quadrangles as they appear in the pentagonal relationship. Fig. 4.6a represents the dual labeling scheme for the Desargues Configuration. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 66

Figure 4.5: Desargues conﬁguration for the networks of ten spins (a, b, c, d, e, f, p, q, r, x). The triads are represented as lines, and spins as dots, according to the traditional Fano-Racah assignment.

It consists of ten points and ten lines, and we represent a point, formed by the intersection of 3 lines, by a circle. The ﬁve quadrangles, which are obtained from the complete pentagon, are represented in Fig. 4.6b. Each quadrangle is obtained from the complete pentagon by eliminating one of its points. In this way, if we eliminate, for example, the point 1 in Fig. 4.6b, we form the quadrangle whose vertices in Fig. 4.6a are represented by points 12, 13, 14 and 15. The numbers inside the circles, in Fig. 4.6a, always represent the eliminated vertex of the complete pentagon and one of the vertices that are not eliminated. By convention we always write the smaller number ﬁrst. If we eliminate, for example, point 3 in the complete pentagon, the quadrangle (1,2,4,5) in Fig. 4.6a has the following vertices: 13, 23, 34, 35.

Interestingly, the Desargues conﬁguration so obtained is encountered also in the description of the isomerization in pentacoordinated molecules. See for example [75], where an alternative graphical representation is given.

In the Desargues conﬁguration, a line is numbered by a set of three dots (or three circles), for example 23, 25 and 35. In the dual representation, we can designate a line with a single label, which consists of the numbering of the dual point to this line written in brackets Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 67

Figure 4.6: (a) Dual version of the Desargues Theorem configuration of Fig.4.5, where role of lines and points as been interchanged using the principle of dualization valid in projective geometry, (b) Pentagram as a perfect figure, from which the Desargues configurations of figures can be labeled according to the text. and in the figure also inside a quadrangle. The line numbered by points, 23, 25 and 35, for example, is labeled as [14]. Simply stated, the dual line of a given point will be one whose 3-point set, which represents the line in the Desargues configuration, does not contain the numbers labeling the point dual to the line. The line dual to point 12, for example, will be the line 34, 35, 45 (the set of points that does not contain the numbers 1 and 2) and in dual notation this line will be labeled as [12]. The overall result is an insightful perception of the Desargues configuration where the disentangling into the five quadrangles entering the Biedenharn-Elliott identity is clearly exhibited. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 68

Figure 4.7: Desargues diagram of the network of ten spins corresponding to the 9j symbol of the second kind. The ten lines are labeled by spins of the Biedenharn-Elliott relationships Eq. (4.4), its ﬁve 6j are labeled by number from 1 to 5, represented by ﬁve quadrangles, two of them on the left-hand-side and three of them on the right-hand-side of the equation. The labels of 1 to 5 (from left to right) indicate the order of the 6j’s end side in Eq. (4.4), and are related with the order of the quadrangle as they appear there.

4.3.2 The 9j symbol of the ﬁrst kind as an anti-Desargues conﬁg- uration

Turning now to the 9j symbol of the ﬁrst kind, again at variance with some previous projective interpretations [45, 48], we display our result in Fig. 4.8, without giving uninformative details of its construction, which requires an accurate arrangement and labeling. Important background information can be retrieved with reference to [53].

In the same notation as in [24, 38, 45] the 9j symbol can be written as a sum over three 6j symbols, as indicated in Eq. (4.6). Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 69

Figure 4.8: Anti-Desargues diagram of the network of ten spins, corresponding to the 9j symbol of the first kind defined as a sum of three 6j-symbols. One can disentangle in the figure the symbols appearing in Eq. (4.6); the lines represent the triads, and the points are the angular momenta. The dashed line in this figure indicates that the triad (pqr) in general does not exist, namely that points p, q and r are not in general aligned, although they may be so accidentally and graphically simulated by.

a f r¯ a b x c d x e f x (−1)2x(2x + 1) =  d q e  (4.6)         x c d p e f q a b r¯   X        p c b            There are alternative ways to use orthonormality to obtain interesting formulas showing how networks can be compacted in cycles and give then an identity matrix as sum. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 70

Ref.[38] lists explicitly both

x y z x y z x a d y b e   xyz (2x + 1)(2y + 1)(2z + 1) d e f      0   0    f c g b g c d f g P  a b c       

b d δbb δdd         = (−1)2( + ) 0 0 {abc}{def}{bdg} (4.7) (2b + 1)(2d + 1)

and

a b x a b x c d y e f g (−1)2z(2x + 1)(2y + 1)(2z + 1)  c d y  xyz           f z d z e a x y z P  e0 f 0 g       

δee δff         = 0 0 {ace}{bdf}{egf}  (4.8) (2e + 1)(2f + 1)

For more information about symmetry properties and sum rules for the 6j and 9j see Refs [52, 53].

The Yutsis representation of the 9j symbol by the nine segments of a hexagon plus its three main diagonals. The resulting diagram is classified [24] as the well known complete bipartite graph K3,3. In Fig. 4.9, the triad structure of the 9j symbol in eq. (4.6)- (4.8) can be read following the six points labeled as 9, while 1, 2 and 3¯ permit to recognize the three 6j’s occurring there as Yutsis quadrangles. These labels also identify them according to the order of their appearance in Eq. (4.6), namely 1 represents the first 6j, 2 the second 6j, 3¯ the third 6j (barred to distinguish it with respect to its counterpart indicated by r in the previous section and appearing in different triads); finally 9 can be checked, labels the triads occurring in the 9j symbols, and to correspond to its Yutsis graphical representation. The profound difference from the previous case cannot go unnoticed. Here also we have ten spins incident in triads, but accurate enumeration of the latter appearing in the three 6j’s and in the 9j symbol, and in the configuration, encounters only nine triads. Accordingly, we have indicated the missing triad as an empty triangle, because there is no Desargues-like construction to guarantee that for generic lines there is their crossing in a point. This is one of Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 71

the ten 103 conﬁgurations known since long time [46] and still currently investigated [76, 77]. The spectroscopically important symmetries of the 9j symbols [5] can be easily classiﬁed within the framework of our approach.

Figure 4.9: Conﬁguration of the anti-Desargues diagram Fig.(4.7), dual to that of Fig. (4.8), here considering points as triads, and lines as angular momenta. It is observed that in this case the missing triad is indicated as an empty triangle. The six vertices of the indicated as representation as an hexagon plus its three main diagonals corresponding to the 9j symbol can be recognized by points labeled by 9, while the other labels 1, 2 and 3¯ indicate the three quadrangles also disentangled and pictured in the ﬁgure, the numbers indicating the order by which they appear in Eq. (4.5).

4.3.3 Incidence diagrams and the connections with the networks of ten spins

The further step needed to complete our effort in interpreting the spin network configurations by incidence geometry graphs is shown in Figs. 4.8, 4.9 and 4.10. We show the result for the two cases of the Figs. 4.8 and 4.9 compacted in Fig.4.10 in the spirit of the construction of the incidence diagrams named after Levi by Coxeter [54]. In short, the far from trivial lengthy procedure consists in finding a Hamiltonian cycle in the graphs and Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 72

“coloring” diﬀerently points and lines. The signiﬁcance in terms of the relationships among

103 conﬁgurations is visually evident comparing Fig.4.10, with that in [77], p.244, where however labels are not given. Of course in Fig. 4.10 our labels continue to be the same as in Figs. 4.8 and 4.9, namely triangles for the triads, and spins as open circles; numbers have also the same meaning as before. The careful comparison of the networks for the two 9j symbols is thus made possible. Additionally evident is the relationship between them, essentially embedded in that between the quadrangles labeled as 3 and 3,¯ that can be transcribed in terms of 6 j symbols. The relationship in Fig. 4.10 follows then by noting that one can give the connection between 3 and 3¯ utilizing the Racah formula, Eq. (4.2), properly relabeled. Finally one obtains:

a b x a b x a e r (−1)x+r+¯r(2x + 1) = (4.9)       xyz f e r e f r¯ b f r¯ X       that is the interesting formula providing the connection  between the 9j symbols of the ﬁrst and of the second kind. This is on illustration of a key step of the 3nj construction and can be of help for extensions to more and more complex structures involving webs of spins.

4.4 Concluding remarks and perspectives

In retrospect the pentagonal relationships were introduced [8, 9] to classify the alternative coupling cases for rotating open shell systems and involved not only the 6j symbols under focus here, but also the Wigner d-matrices and 3j symbols: as has been documented later [11], they can be dealt within the semiclassical limit and therefore the present theory can be applied, working out the details of interest in view of perspective applications. As- pects regarding the 9j’s are specifically relevant to extend the attention initially restricted to relatively simple Hund- type coupling cases for light diatomic molecules to the more con- gested high resolution spectra such as those involving hyperfine coupling under extreme laser- cooling and astrophysical environments (see e.g. [78, 79]. In these and other situations ex- Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations 73

Figure 4.10: Levi’s incidence diagram for the network of ten spins of Fig. (4.7) (left) and Fig.(4.9) (right). The 10 angular momenta are represented as empty circles and the ten triads are represented by full triangles. The empty triad (pqr¯) doesn’t exist in general but it may accidentally (see Fig. 4.9) and the dashed lines p, q andr ¯ do not meet unless forced to do so through a graphical approximation. pressions involving long strings or even large matrices of 9j (and possibly higher) symbols are frequently needed and algorithms serving for their manipulations and eﬃcient computation are being developed exploiting the work presented here. A note accounts for applications to the torsional modes around the O-O and S-S bonds of classes of peroxides and persulﬁdes [80], as exemplary of the insight that can be currently provided in molecular structure and dynamics.

Emphasis of this work has been on molecular spectroscopy and dynamics. The often obscure combinational machinery of recoupling in angular momentum algebra is conveniently compacted in the form of the abacus, proposed some time ago [34] and in a recent paper [53], and in this one illustrated from the viewpoint of projective geometry and incidence graphs.

Regarding the construction of the geometry of the projective plane presented in Sec. Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 74

4.2, we have not considered a further axiom on projectivity, Axiom VI (2.18 in Coxeter, pp. 31 and 93). This axiom appears as of no relevance here, and is actually superﬂuous in many geometries. Another remark is in order and concerns the ﬁnal observation in Judd’s 1984 [45] paper that no example is known in angular momentum theory about role of Pappus theorem, which is arguably related to commutativity [71]. Judd’s conjecture is that its non-occurrence is due to non commutativity in quantum mechanics and as far as we know it has not been proven nor confuted yet.

An important question demanding further investigation concerns account of symmetry: note that the seven objects displayed in the complete Fano plane, admit a total of 168 permutational symmetries. The 6j symbols have 144 (24 permutational plus the others due to the intriguing Regge ones). Relationships are pointed out in [49, 41, 50, 51]: however detailed connections need to be established, to remove the unphysical restriction of a ﬁnite geometry.

These techniques are also of relevance in view of the mathematical relationship between angular momentum algebra and sets of orbital of Sturmians and hydrogenoid type, of interest in the quantum theory of electronic structure [81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91]. Updated information on 9j properties and the generalized abacus are provided in Ref. [53] and references therein. Progress in the asymptotic limits regarding both aspects of semiclassical disentangling and of extensive systematic investigations [73, 92] of various limits as extensions of Ponzano-Regge theory is continuing [93, 94, 95, 96, 97, 98]. The emphasis has been placed on those semiclassical versions of the construction, which turn out to play a crucial role in many diﬀerent contexts, ranging from molecular physics and quantum computing and discrete quantum gravity models. Not tackled here is the associated to metric these graphs, when labelled by spins (i.e. by SU(2) and similar “colorings”). For in their calculations see e.g. [26, 27, 28, 29] and references herein: it is expected further progress exploiting relationships suggested by the present work.

The approach to 9j symbols [99] has been extended [22, 23, 92], while for higher symbols [24] we hope viewpoints indicated here may be insightful.

The Fano conﬁguration 77 and the Fano plane where the simplest of geometries can Chapter 4. Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence conﬁgurations 75

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Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond

Tullio Regge discovered new symmetries in 1958, hidden in formulas for calculations of the coupling and recoupling coeﬃcients of quantum angular momentum theory, as developed principally by Wigner and Racah: the only known (limited) application appeared computational. Ten years later, in a paper with Ponzano, Regge provided a semiclassical interpretation showing relevance to the basic geometry of quadrilaterals and tetrahedra, and opening also a promising road to quantum gravity, still currently being explored.

New facets are here indicated, continuing a sequence of papers in this Lecture Notes series and elsewhere. We emphasize how an integrated combinatorial and geometrical interpretation is emerging, and also examples from the quantum mechanics of atoms and molecules are brieﬂy documented. Attention is dedicated to the recently pointed

86 Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 87

out connection between the quantum mechanics of spin recouplings and the Grashof analysis of four-bar linkages, with perspective implications at the molecular level.

5.1 Introduction: Discovery and Developments

On September 23 and October 9, 1958 Tullio Regge (1931-2014) submitted two very brief letters [1, 2], where he reported what are now known as Regge symmetries for the Clebsch-Gordan’s coupling coeﬃcients and Racah’s recoupling coeﬃcients, or respectively the 3j and 6j symbols of Wigner. Ref. [2], one of the shortest paper ever, consists of only 13 lines of text and of three equations, so that it can be cited practically in full:

We have shown in a previous letter [1] that the true symmetry of Clebsch-Gordan coeﬃcients is much higher that is was before believed. A similar result has been now obtained for Racah’s coeﬃcients. Al- though no direct connection has been established between these wider symmetries it seems very probable that it will be found in the future. We shall merely state here the results which can be checked very easily with the help of the well known Racah’s formula.

(here, the Racah’s formula for 6j symbols [3] is given. We will discuss it in Sec.5.2). The only citation is to paper [1], which is only slightly longer than [2] and has no citations: it acknowledges discussions and encouragement from Racah. Paper [2] continues:

From the usual tetrahedral symmetry group of the [6j] we know already that: a b c b a c a e f c e d = = = = etc. d e f e d f d b c f b a Our results can be put into the following form:

a b c a (b + e + c − f)/(2) (b + c + f − e)(2) = d e f d (b + e + f − c)/(2) (c + e + f − b)/(2) = etc.

Six forms of the new symmetries are given, Sec.5.2. The paper concludes: Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 88

Only the ﬁrst of these symmetries is essentially new, the others can be obtained from it and [the permutational symmetries, Sec.5.2]. We see therefore that there are 144 identical Racah’s coeﬃcients. These new symmetries should reduce by a factor 6 the space required for the tabulation of [the 6j]. It should be pointed out that this wider 144-group is isomorphic to the direct product of the permutation groups of 3 and 4 objects.

Papers [1] and [2] are reprinted in [4], a collection of articles relevant to quantum angular momentum theory compiled in 1965. At p.12, they cite [1] as a “surprise.” In [4], papers [1] and [2] are followed by [5], where Bargmann cites them as “Regge’s intriguing discovery of unsuspected symmetries of 3j and 6j symbols.” Although (see above) Regge had the intuition that symmetries in [1] and [2] were connected, early attention was focused on that for 3j [1], apparently deceivingly simpler. For example, ten years later, in [6] we read “intriguing symmetry of the Clebsch-Gordan coefficients discovered in 1958 by Regge”, that “came as quite a surprise”, and “rather bizarre”. In the famous 1968 paper, Regge with Ponzano [7] briefly noted that properties of 3j’s can be obtained by a semiclassical limiting procedure from those of the 6j’s giving the latter the crucial, basic building role that now they occupy in quantum mechanics. We present these notes as an ordered series of remarks, exhibiting how the recent results fit into a scheme not necessarily coincident with the chronological succession of events.

5.2 Computational and Combinatorial Aspects

5.2.1 Calculation of 6j Symbols

In [2] indicated was application of new symmetries to compact tables of 6j’s: this is actually a very limited usefulness, now that their calculations and compilations can be carried out routinely. The explicit formulas given by Racah as ﬁnite summations [3] can be written in various way, and the same for 3j formulas, found by Wigner [8]. A later discovery was the connections of these expressions with generalized hypergeomet-

ric series (4F3 and 2F1 of argument one, for 6j’s and 3j’s respectively) and are listed Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 89

in Varshalovich [9], pp. 293- 295. As described at length elsewhere [...], they are also related to important classes of orthogonal polynomials of discrete variables, described and classiﬁed in books [10, 11]. Refs [7, 12, 13, 14, 15, 16, 17, 18, 19, 20], report the relationships with spherical and hyperspherical harmonics, and with the orthonormal basis sets as the basic ingredients of quantum mechanical applications to atomic, molecular and nuclear sciences. To be noted is that most of formulas for the calculations as series as listed in the literature are incomplete, because sometimes the summation variable has to be chosen according to rules dictated by Regge symmetries: the precise rules are given on pp.246 and 259 in [11].

5.2.2 The Triangle and Polygon Inequalities and the 4j-model of the 6j-symbol

We make a remark on a generalization of the triangle inequalities before presenting the 4j-model of the 6j-symbol. Here we summarize ﬁrst the developments in [21], in order

to extend it to relevance for the Regge Symmetry. If (`1, `2, `3) are three nonnegative

lengths, the usual triangle inequalities are |`i−`j| ≤ `k ≤ `i+`j, where (i, j, k) = (1, 2, 3)

and cyclic permutations. We generalize these as follows. Let {`i, i = 1, . . . , n} be a set

of lengths, ì ≥ 0, i = 1, . . . , n. Then this set is said to satisfy the “polygon inequality” if n 1 max{` } ≤ ` . (5.1) i 2 i i=1 X This is equivalent to the triangle inequalities when n = 3. In general, it represents the necessary and sufficient condition that line segments of given, nonnegative lengths can be fitted together to form a polygon with n sides (in RN , N > 0). A different way of writing the 6j-symbol as a scalar product begins with Fig. 5.1. With respect to the 6j symbols encountered in Sec.5.1, we introduce a useful relabeling in Fig. 5.1 and will refer it as the “asymmetric” labeling, which is more appropriate for the 4j-model [21].

The quantum numbers j12 and j23 of the intermediate angular momenta range in integer Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 90

Figure 5.1: Asymmetric labeling of the 6j-symbol, according to the 4j model [21]. Upper: the plane Yutsis diagram versus the Wigner- Regge tridimensional tetrahedron . Note the point-face duality, i.e. triads are represented as points on the left and triangles on the right. Lower: Alternative coupling schemes (or spin nets) as labeled trees represented on the left as a bra and a ket, the bracket forming the 6j symbol as the simplest of spin network.

steps between the bounds

j12,min ≤ j12 ≤ j12,max, j23,min ≤ j23 ≤ j23,max, (5.2)

where the maximum and minimum values are given in terms of the four ﬁxed jr, r = 1,..., 4 by

j12,min = max(|j1 − j2|, |j3 − j4|), j12,max = min(j1 + j2, j3 + j4),

j23,min = max(|j2 − j3|, |j1 − j4|), j23,max = min(j2 + j3, j1 + j4). (5.3)

Then the dimension of D is given by

D= j12,max − j12,min + 1 = j23,max − j23,min + 1.(5.4) Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 91

An expression for D can be given that is symmetrical in (j1, j2, j3, j4). Using the fact that |x| = max(x, −x), the diﬀerence between j12,max and j12,min is now computed since j12,min can be written:

j12,min = max{j1 − j2, j2 − j1, j3 − j4, j4 − j3} (5.5)

Then by Eq.(5.5) D is the diﬀerence between the minimum of one set of two numbers and the maximum of another set of four numbers, Fig.(5.2).

Figure 5.2: The ﬁgure illustrates two sets of numbers as points on a line, in which the maximum of the a-set is less than the minimum of the b-set. In a case like this, the diﬀerence between the minimum of the b-set and the maximum of the a-set is the minimum distance between all NaNb pairs of points that can be formed from the a-set and the b-set, that is, all pairs of the form (ai, bj).

That is, when the minimum of the b-set in Fig.(5.2) is greater than the maximum of the a-set, then

{bi, i = 1, ..., Nb} − max{aj, j = 1, ..., Na} = {bi − aj, i = 1, ..., Nb, j = 1, ..., Na} (5.6)

When the minimum of the b-set is less than the maximum of the a-set (that is, when the two sets overlap), then the right hand side of Eq. (5.4) can still be computed, but it is negative.

In particular, when j12,max is greater than or equal to j12,min, the diﬀerence j12,max − j12,min is given by the minimum value of 2 x 4 = 8 numbers, Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 92

j1 + j2 − (j1 − j2) = 2j2, j1 + j2 − (j2 − j1) = 2j1, j1 + j2 − (j3 − j4) = j1 + j2 − j3 + j4 = 2(s − j3), j1 + j2 − (j4 − j3) = j1 + j2 + j3 − j4 = 2(s − j4), j3 + j4 − (j1 − j2) = −j1 + j2 + j3 + j4 = 2(s − j1), j3 + j4 − (j2 − j1) = j1 − j2 + j3 + j4 = 2(s − j2), j3 + j4 − (j3 − j4) = 2j4, j3 + j4 − (j4 − j3) = 2j3. (5.7)

But if the ji, i = 1, ...4 are all ≥ 0 and if they satisfy the polygon inequality (5.1), then these eight numbers are all ≥ 0, and so j12,max ≥ j12,min

Therefore D becomes the shortest distance between one set of four numbers, {j1 − j2, j2 − j1, j3 − j4, j4 − j3}, and another set of two numbers, {j1 + j2, j3 + j4}. But this is the minimum of the distance between all eight possible pairs taken from the two sets Eq. (5.7). Thus

D = 2 min(j1, j2, j3, j4, s − j1, s − j2, s − j3, s − j4) + 1, (5.8) where the list in Eq.(5.7) is rearranged and given in terms of the semiperimeter s,

1 s = (j + j + j + j ). (5.9) 2 1 2 3 4

More precisely, if D computed by eq. (5.8) is ≤ 0, then subspace is trivial (D = 0); otherwise it is crucial for computational purposes.

5.2.3 Generation of Strings and Matrices of 6j Symbols

The formula (5.8) not only bears an interesting relationship to the Regge symmetry of the 6j-symbol (Varshalovich 1981, Eq. (9.4.2.4)), but serves for the best known procedure to obtain whole strings and matrices of the symbols. It is essentially based on a set of 3- Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 93 term recursion relationships, ﬁrst suggested by Neville [22], and independently rederived and computationally demostrated by Schulten and Gordon [14, 28].

The derivation in the previous subsection demonstrates that, as in the case of the direct summation formulas (Sec. 5.1). one has to carefully choose which of the two equivalent symbols connected by Regge symmetry has to be chosen for actual calculations. This has been demonstrated and implemented in our papers [43, 24, 25, 26, 38]

5.3 Semiclassical and Geometrical Aspects

5.3.1 Visualizations

As explained in [24, 25, 26], the use of Regge symmetry, and especially the deﬁnition of ranges of the j12 and j23 variables, permitted to deﬁne uniquely the choice of the symbol between the two options. This had been overlooked in previous work, that independently derived [22] or put on a rigorous basis [14, 28] the semiclassical treatment developed by Ponzano and Regge [7]. Associating a 6j symbol to a Euclidean tetrahedron Fig. (5.1) the extraordi- nary results of [7] are that the symmetry discovered in the quantum mechanical formulas by Regge also holds for properties, in particular volume, products for face areas, dihedral angles, of the most basic tridimensional geometric object under study since the ancient Greeks and had gone unnoticed for long. For the exploit of Regge symmetry to consistently represent images of computed 6j and 3j symbols, of use for discussion of properties and for image reconstructions [61], see for example the “screen” representation in Refs. [15, 25].

5.3.2 The volume operator and non-Euclidean extensions

The Ponzano-Regge paper played a signiﬁcant role also in the foam-model and loop- gravity approaches to the still open problem of developing a quantum mechanical foundation to general relativity. In our current work [32, 58] and in preparation, a basic ingredient in the theory, the volume operator, and the representation of its eigenfuntions by discrete polynomial sets, has been recently discussed, and the part played by the Regge symmetries are Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 94 emerging. Under study is their role in the “regularization”, i.e. the removal of divergences in the theory [58] of interest in general for the extension to non-Euclidean elliptic and hyperbolic geometries, see the exploratory study presented in a companion paper in this Lecture Notes Series [59]

5.4 Couplings as Limits of Recouplings

5.4.1 Regge symmetries and 3nj symbols and hydrogenic orbitals

The preceding results can be extended to 3j symbols by a limiting procedure illustrated in detail elsewhere [12, 13, 15] The Regge symmetries for 3js play a crucial role in the understanding of the properties of orbitals in quantum chemistry.

For instance, Schrödinger equation for the hydrogen atom in configuration space is separable in different coordinate sets [70, 76, 74, 66, 67]. Most widely employed are polar coordinates (r, θ, φ) for which the separation of variables is always possible in the case of a central symmetric field. The corresponding basis set can be indicated by using Dirac notation and the quantum numbers labelling the wavefunctions as |nlmi.

An important alternative set, also leading to analytic close solutions of Schr¨odinger equation and to the separation of variables, is the parabolic set. This set is particularly advantageous when there is a privileged direction in space, as for atoms in electric or magnetic ﬁelds. The notation |nµmi can be used for the parabolic wavefunctions, where µ is one among various choices to indicate the parabolic quantum number [67, 68, 70, 76, 74].

The connection between the two basis sets is recognized as alternative parametriza- tions of the hypersphere S3 embedded in a four-dimensional space (Fock’s projection [85]). It is given by a coupling (Clebsch-Gordan) coeﬃcient [69, 70]:

hnµm|nlmi =

1 (n−1)+ m − µ 1 1 µ m µ m (−) 2 2 2 · h (n − 1), (n − 1); − + , + |lmi (5.10) 2 2 2 2 2 2 Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 95

An interesting representation of these matrix elements was proposed by us making use of the Regge symmetry for Clebsch-Gordan coeﬃcients [76]:

1 1 1 1 h (n − 1), (n − 1); (m − µ), (m + µ)|lmi = 2 2 2 2 1 1 µ µ h (n + m − 1), (n − m − 1); − , |l0i (5.11) 2 2 2 2

Such formulation has the additional advantage of providing a physical meaning for the µ quantum number. A Clebsch-Gordan coeﬃcients with zero projection of the orbital angular momentum suggests that µ can be interpreted as a helicity quantum number. Indeed, the choice of the reference frame for which the component of the orbital angular momentum vector L is zero is equivalent to change the quantization axis on the plane of the orbit, where the Runge-Lenz vector lies. This interpretation becomes much clearer when analyzed from the momentum space point of view, where the symmetry group for the three-dimensional hydrogen atom manifestly appears to be O(4) [69, 70, 71, 75, 67].

This operation finds its analogues in various contexts: the passage from symmetric and asymmetric coordinates in the hyperspherical treatment of the three body problem [83, 84], the space fixed-body fixed transformation in molecular collisions, and the transformation between the Hund’s cases (e) → (c) in molecular spectroscopy and atomic scattering [81, 82].

5.4.2 Momentum Space Wavefunctions and Sturmian Orbitals

The momentum space perspective allows connecting hydrogenic wave-functions, or specifically Coulomb Sturmian orbitals, with four-dimensional hyperspherical harmonics, as obtained by Fock projection from the Fourier transform of Schrödingerequation in configuration space [85].

Accordingly, the transformation coeﬃcients between alternative Coulomb Sturmi- ans can be calculated as superposition integrals between sets of hyperspherical harmonics pertaining to alternative parameterizations of the four-dimensional sphere S3 [69, 71, 75]. This approach has been extended to spaces of any dimension [70, 73, 67]. Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 96

In the d-dimensional space the superposition between the two harmonics is essentially a (d − 1)−dimensional integral over four Jacobi polynomials [80, 79] and, apart from a phase factor, can be identiﬁed with a Racah orthonormalized coeﬃcient, the discrete analogue of Jacobi polynomials.

In turn, Racah orthonormalized coeﬃcients can also be written as a generalized

1 6j coeﬃcient [72] whose entries can be integers, half integers or proportional to 4 depending on the dimensions of the subspaces involved in the construction of the d−dimensional hypersphere.

When such spaces are even in dimension the coefficient reduces to an ordinary 6j symbol with integer and/or half integer entries only. In all cases. an alternative expression for the superposition coefficient can be obtained making use of Regge symmetry for 6j coefficients by taking care that the selection rules for hyperspherical harmonics that still hold valid [67]. In this case in each entry -i.e. for every (hyper-)angular momentum- it appears only a specific quantum number ti and the corresponding numbers di counting the dimensions of the subgroup reduction chain:

1/2 t+t +t t +d +d +d 12 23− 2 1 2 3 d1 + d2 d2 + d3 T = (−) 2 (t + − 1)(t + − 1) × t12,t23 12 2 23 2 t1 d1 t2 d2 t12 d1+d2 − 2 − 4 2 + 4 2 + 4 − 1 ×   (5.12)      t3 d3 t d1+d2+d3 t23 d2+d3  − 2 − 4 2 + 4 − 1 2 + 4 − 1     For spaces of low dimensions, hyperspherical harmonics (and the expression of the connecting 6j coeﬃcient) may simplify.

For the d-dimensional hydrogen atom, the overlaps between polar and parabolic wavefunctions are generalized 6j symbols, which simplify either to a 3j coeﬃcient, as seen above in eq. (5.10), or to a generalized 3j coeﬃcient [86], depending on the even or odd value of d. These functions are relevant as basis sets in quantum chemistry. Chapter 5. Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond 97

5.4.3 Connection to a Classical Mechanism and to Molecular Me- chanics

Regge symmetries enter also in the discussion of the eigenfunctions of volume operator. See Ref.[32], where a geometrical picture is presented,as see well as relationships with diverse topics, such as hybridization theory in quantum chemistry [77] and quaternion calculus [32, 78]. In [32] it was noted that the transformation of a quadrangle involved in the Regge symmetry of a tetrahedron showed analogies to the parametrization of the four-bar mechanism, that accompanied the evolution of mechanics from Watt to robotics: it was a concern for example of Cayley and Grashof, and continues to be currently investigated (see [45, 46] and references therein). With the analysis developed here in Sec. 5.2, we are now able to establish the connection between the choice of the convention for the 6j according to Regge symmetry, and the Grashof condition written considering four bars of length k, l, m, M where, m is the shortest bar, M is the longest and k, l are the intermediate ones:

m + M ≤ k + l (5.13)

This is the condition for the bar of length m being able to act as a crank, namely to perform a 360◦ turn: for this mechanism, the dimension D in Eq. (5.8) can be identified omitting +1 since we are dealing with min(m, s − M), with a classical measure of differences in lengths. The identification of m with a in our convention requires m ≤ s − m = (m − M + k + l)/2, rather than on counting number of points in discrete interval. Beyond this application, recently implications for the systematics of the molecular mechanics of chirality changing processes there were pointed out the and are subject of a companion note [60]. Bibliography

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Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.

6.1 Introduction

The four-bar linkage (see for example [1] and [2]) was introduced during the In- dustrial Revolution and is nowadays applied to a wide area that spans from engineering to biomechanics, microbiology and chemistry. Generally, the links can be assembled in three dimensions, although the most common configurations concern planar displacements; this is the case upon which we focus in this work, starting however from a spatial perspective. The ratios between the lengths of the bars determine different paths of the joints that permit to classify the performances of the linkage. One of the four links is usually fixed, the ground link, and is directly connected to an input link and an output link. The remaining link, op-

106 Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.107 posite to the ground link, is called floating link. Assuming that the ground link is horizontal, the input and output links can make four kinds of rotations: the crank, the bar rotate by the full 360 degree range; the rocker, the rotation is allowed within a limited range that does not include 0 and 180; the 0-rocker, similar to the rocker, but the rotation excludes 0; the -rocker, again similar to the rocker and the previous case, but excludes 180. Quadrilateral linkages are conveniently classified according to the ratios of lengths of the bars: they are said to accomplish the condition of Grashof, if the shortest link can rotate fully with respect to the neighboring links, if the sum of the longest and the shortest bars is less than or equal to the sum of the remaining two. Four-bar linkages are also classified into eight cases, related to the Grashof condition, according to the positive or negative sign of appropriate combinations of the length of the ground, floating, input and output links. In this work, we reconsider this classification exploiting similarity with features encountered in case studies of the coupling of angular momentum ([3]): to shed further light on the four-bar mechanism problem, we establish connections between the conditions of Grashof and the symmetries of Regge of the 6j symbols, through the screen representation ([4], [5], [6]). The screen originates from the geometrical interpretation of the 6j symbols in the asymptotic limit, through the Ponzano-Regge theory ([7]), the application of the properties of tetrahedra and quadrangles, and the symmetry properties of the 6j symbols ([8],[9], [10]). It was initially developed for the representation of the Regge symmetries of 6j symbols, the building blocks of spin networks ([11],[12],[13], [14],[15], [16]), and then applied to compact the available data of peroxides ([17], [18], [19], [20]) and persulphides ([21]), by displaying the distances ([22]) related to chirality changes ([23], [4], [70]).

6.2 Background

6.2.1 Quadrilaterals, quadrangles and tetrahedra

A quadrilateral is a polygon, that in planar Euclidean geometry is defined as a geometric shape with four sides, the sum of the four inner angles being 360. The opposite vertices of quadrilaterals are joined by diagonals. Their classification relies on the displacement of Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.108 the diagonals: convex (two inner diagonals); biconcave (two outer diagonals); concave (one inner and one outer diagonal). In projective and affine geometries, a complete quadrilateral is formed by four points and six lines, being diagonals considered as further sides. A complete quadrangle includes four incident lines in six points. In projective geometry, quadrangles and quadrilaterals are the same object, since duality in RP2 permits exchange of the role of points and lines. In space, objects of non-zero volume are seen as tetrahedral, being formed by four vertices and six edges ([24], [25]).In the four-bar mechanism, the four links (see Sec. 6.1) and the diagonals connecting the opposite joins can thus be seen as the edges of a tetrahedron, where the planar displacement represents those of a tetrahedron of zero volume. The length of the four bars and of the two diagonals represent a 6-distance system that shows analogies with the 6j symbols of Wigner, Racah coefficients. Wigner and Racah gave a geometrical interpretation of these elements, attributing to the six entries of the 6j, as the edges of a general irregular tetrahedron. ([7], [26], [27], [28], [29], [30], [31], [32]).

6.2.2 The screen

A 4j model ([33], [35]) permits to represent the allowed range of a tetrahedron through a two-dimensional plot of a couple of discrete variables for the 6j symbol ([34],

[36], [37])), which entries are designates as j1, j2, j3, j4, j12, j23 (see also [38], [39], [40]). An analogous 4d model ([22]) is introduced here to represent the six distances and the geometrical properties of the associated tetrahedron described in Section 6.2.1. According to the “canonical” ordering ([6]), we denote the edges as follows:a is the shortest one, c is opposite to a and d is the longest one between the remaining two edges. These four edges are ﬁxed by construction and in the quadrangle (the tetrahedron projected on the plane) they are its sides. The remaining two edges, namely x and y, are variables and for the planar case they are the diagonals of the quadrilateral. The resulting “screen” by on the plane, deﬁned the Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.109

allowed ranges xmin, xmax, ymin, ymax: is a square.

xmin = b − a

xmax = b + a

ymin = d − a

ymax = d + a (6.1)

Sometimes it is expedient to confine the screen into a square, which sides range between 0 and 1, one has to reduce the distances corresponding to x and y, respectively, by b−a and d−a and normalized by b+a and d+a. In the screen, we define two significant curves called caustic and ridge. Thecaustic delimit the existence of a quadrilateral and surrounds an area corresponding to a tetrahedron (the caustic corresponds to a tetrahedron of zero volume). This line touches four points of the axes of the screen, called gates, indicated by the four cardinal points north N, south (S), east (E) and west (W). The north gate identifies the configuration for which the value of x corresponds to y = 1, while S is the counterpart for y=0. Analogously, E and W are the values of y corresponding to x=0 and x=1, respectively. The ridges are curves that mark configurations of the associate tetrahedron when two specific pairs of triangular faces are orthogonal. The equation of the caustic has been derived by the Heron formula that permits to calculate the area of a triangle as a function of the sides: the quadrangle can be seen as two triangles with a common side: abx and cdx, whose areas are x x Fa,b and Fc,d.

1 [(a + b − x)(−a + b + x)(a − b + x)(a + b − x)] 2 F x = (6.2) a,b 16 1 [((a + b) − x)((−a + b) + x)((a − b) + x)((a + b) − x)] 2 F x = (6.3) a,b 16

Eqs. (6.2 and 6.3) are rearranged to give: Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.110

1 [(a + b)2 − x2][x2 − (b − a)2] 2 F x = (6.4) a,b 16 1 [(a + b)2 − x2][x2 − (b − a)2] 2 F x = (6.5) c,d 16

Thus, the area of the quadrangle is

1 1 [(a + b)2 − x2][x2 − (b − a)2] 2 [(a + b)2 − x2][x2 − (b − a)2] 2 F x + F x = + (6.6) a,b c,d 16 16

The area of the quadrangle, K, can also be calculated by the Bretschneider generalized formula ([67]): 1 [4x2y2 − (b2 + d2 − a2 − c2)2] 2 K = (6.7) 16 Thus x x K = Fa,b + Fc,d (6.8)

By substituting equation 6.8 with equations 6.3 and 6.4 one obtains:

1 1 1 [4x2y2−(b2+d2−a2−c2)2] 2 [(a+b)2−x2][x2−(b−a)2] 2 [(c+d)2−x2][x2−(d−c)2] 2 16 = 16 + 16 (6.9) n o n o n o In order to eliminate the square root of both sides, we square the two terms of Eq. 6.8:

2 x x 2 x 2 x 2 x x K = (Fa,b + Fc,d) = (Fa,b) + (Fc,d) + 2Fa,bFc,d (6.10)

In such a way, the root terms are eliminated and obtained by a diagonal analytical expression as a side function. By replacing both sides of Eq. (6.10) by Eqs. (6.4), (6.5) and (6.7), one obtains the following expression: Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.111

4 [(b2 − a2) + (d2 − c2)]2 y2 = + (F x )2(F 2 )2 + 2F x F x x2 16 a,b c,d a,b c,d to

(6.12) [(b2 − a2) + (d2 − c2)]2 4[(F x )2 + (F x )2] 8F x F x y2 = + a,b c,d + a,b c,d (6.13) x2 x2 x2 [(b2 − a2) + (d2 − c2)]2 4[(F x )2 + (F x )2] y2 = + a,b c,d (6.14) x2 x2

Equation 6.14 is named Caustic and relates the diagonals of a quadrilateral with only the respective sides and respective two-sided areas that share the same side. An alternative way consists in obtaining the same expression only as a function of its sides, in this case we substitute Eqs. (6.4) and (6.5) in Eq. (6.11):

(4x2y2 − (b2 + d2 − a2 − c2)2) 16 [(a + b)2 − x2][x2 − (b − a)2] [(c + d)2 − x2][x2 − (d − c)2] = + 16 16 2 2 2 2 2 2 2 2 1 [[[(a + b) − x ][x − (b − a) ][(c + d) − x ][x − (d − c) ]] 2 ] ±2 (6.15) 16 obtaining

1 y2 = (b2 + d2 − a2 − c2)2 + [(a + b)2 − x2][x2 − (b − a)2] + [(c + d)2 − x2][x2 − (d − c)2] 4x2 2 2 2 2 2 2 2 2 1 ±2[[(a + b) − x ][x − (b − a) ][(c + d) − x ][x − (d − c) ]] 2 (6.16)

This latter equation relates the diagonals and the sides of the quadrilateral. It corresponds to a tetrahedron when its volume is zero. For Ponzano-Regge [7] the volume of Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.112 a tetrahedron is given by: 2 V = F x F x sin(θ) (6.17) 3x a,b c,d From Eqs. (6.3) and (6.4), we can formulate Eq.6.17 as a function of the edges of the tetrahedron:

1 1 2 [(a + b)2 − x2][x2 − (b − a)2] 2 ([(c + d)2 − x2][x2 − (d − c)2]) 2 V = sinθ 3x 16 16 (6.18)

π The ridges. The maximum value of the volume of a tetrahedron is given for θ = 2 . From Eq. (6.18), one obtains the equations of the ridges that correspond to the maximum volume for the tetrahedron obtained by folding the planar quadrilaterals either on the x or on then y diagonal:

1 y2 = (b2 + d2 − a2 − c2)2 + [(a + b)2 − x2][x2 − (b − a)2] R 4x2 +[(c + d)2 − x2][x2 − (d − c)2] (6.19) 1 x2 = (b2 + d2 − a2 − c2)2 + [(d + a)2 − y2][y2 − (d − a)2] R 4y2 +[(c + b)2 − y2][y2 − (c − b)2] (6.20)

The gates. Finally, the four gates are given by the following equations: For x = b–a, corresponding to the West, one has

1 y2 = [(b − a2) + (d2 − c2)]2 + [(c + d)2 − (b − a)2][(b − a)2 − (d − c)2] (6.21) W 4(b − a)2

while for x = b + a, corresponding to the East:

1 y2 = [(b2 − a2) + (d2 − c2)]2 + [(c + d)2 − (b + a)2][(b + a)2 − (d − c)2]. (6.22) E (b + a)2 Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.113

For y = d − a, corresponding to the South:

1 x2 = [(b2 − a2) + (d2 − c2)]2 + [(b + c)2 − (d − a)2][(d − a)2 − (c − b)2] (6.23) S 4(d − a)2

and for y = d + a, corresponding to the North:

1 x2 = [(b2 − a2) + (d2 − c2)]2 + [(b + c)2 − (d + a)2][(d + a)2 − (c − b)2]. (6.24) N (d + a)2

6.3 Case studies and discussion.

6.3.1 Correspondence between Regge symmetries and Grashof cases.

A four-bar linkage displays different configurations, according to the relative lengths of the bars. As mentioned in the Introduction, the bars are classified as ground, input, output and floating links ([41]). Systems are classified according to appropriate combinations of the length of the four bars, that also provide information regarding the Grashof condition:

T0 = g + a + h + b (6.25)

T1 = g − a + h − b (6.26)

T2 = g − a − h + b (6.27)

T3 = −g − a + h + b (6.28)

where g is the ground link, a is the input link, b is the output link and h is the ﬂoating link. Equation (6.25) has to be added to build up an orthogonal transformation upon a, g, b, h. In Figure 1, we show the comparison between a quadrangle originated by the coupling scheme of angular moments and a similar quadrangle representing a four-bar linkage. In the ﬁrst case, the sides and diagonals of the quadrangle are represented by the angular moments, while in the second case they are represented by the length of the four Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.114 bars and the two distances between the opposite angles of the quadrangle.

Figure 6.1: Comparison between a quadrangular coupling scheme of angular momentum, the 6j symbol (in the left), and a quadrangular scheme of bars, the four-bar linkage (in the right).

The Regge symmetry relations (Section A.1) can be formulated by a set of variables s, u, v and r introduced to describe the topology of the corresponding screen:

a b x a0 b0 x s − a s − b x c − r d + r x = = = =  c d y   c0 d0 y   s − c s − d y   a − r b + r y               b − u a − u x  d − v c + v x = (6.29)  d + u c + u y   b + v a − v y          where the variables are deﬁned as follows: s is the semiperimeter:

1 s = (a + b + c + d) (6.30) 2 u is the diﬀerence between lines:

1 u = [(a + b) − (c + d)] (6.31) 2 Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.115 r is the diﬀerence between columns:

1 r = [(a + c) − (b + d)] (6.32) 2 v is the diﬀerence between diagonals:

1 v = [(a + d) − (b + c)] (6.33) 2

Comparison between Eqs. (6.25) – (6.28) and (6.30) – (6.33) gives the following relations:

T0 = 2s,

T1 = 2r;

T2 = 2u,

T3 = 2v. (6.34)

Table I shows the sign (+, −, 0) of the Regge variables (r, u and v) for each of the 27 four- bar linkage types, classiﬁed according to the input and output links. Systems I–I0,II − II0,III–III0 and IV –IV 0 are couples of Regge conjugates, for all of them the value of r, u and v is diﬀerent from zero. Systems labelled from 1 to 19 are instead characterized by at least one of the Regge variables equal to 0.

6.3.2 General case: illustrations.

In this Section, we discuss the screen (Figure 6.2) for a system characterized by the sides: a = 30, b = 45, c = 60, d = 55. It belongs to the case IV 0, being u = −20, r = −5 and v = −10. The range of the diagonals is 15 ≤ x ≤ 75 and 25 ≤ y ≤ 85

a b x 30 45 x a0 b0 x 65 50 x = = = (6.35)  c d y   60 55 y   c0 d0 y   35 40 y                  Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.116

Figure 6.2: The screen of a quadrangle of sides a = 30, b = 45, c = 60, d = 55 (for the corresponding 6j symbol screen representation, see Fig. 1 in Ragni 2013). The diagram reports the four gates E, S, W, N and four further points SE, SW, NW, NE, the corresponding quadrangle, for these latter, is concave, biconcave, concave and convex, respectively.

Since the caustic does not present symmetry features, the case is indicate as general. Figures (6.3 and 6.4) report the configurations of the four-bars linkage in correspondence of the four gates N, W, E, S and in the intermediate points NW, NE, SW, SE. Figure (6.3) shows the four-bar linkage, for which the Grashof condition is verified. Diagonals x and y vary according to the caustic reported in Fig.(6.2). The side b is considered the ground link, while the side a, the input or output link, works as a crank, being allowed to rotate of 360° clockwise (left column) or anticlockwise (right column). The first configuration we report (left column) consists of the sides a and b aligned and the x diagonal is at its maximum length; it Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.117 corresponds to the E gate. The evolution of the system leads to concave quadrilaterals SE, then in S it assumes the lowest value of the y diagonal, with a lying over d. The passage to the gate SW leads to a biconcave quadrilateral and subsequently to W, where x reaches the minimum length, with a folded over b. A further rotation to NW, concave quadrilateral, and N yields the highest length of y, with a and d aligned, and finally a convex quadrilateral in correspondence of NE. After a 360° rotation of the side a in correspondence of E, we obtain a configuration that is specular to the initial one (compare the E configurations of the first and second column). By further rotating the a side, one obtains configurations specular to those reported in the first column (see the second column), until reaching the E configuration in the first column after a second 360° rotation. In Figure (6.4), we illustrate the four-bar linkage for the Regge conjugate of the object shown in Figure (6.3). The Regge symmetry produces an object for which the Grashof condition is not accomplished. In this case, the bar b0 (left column) moves towards the ground link a0. As seen previously for the case of Fig. (6.3), in correspondence of the gates two bars are aligned, forming a triangle: the diagonals (x for (E) and y for N) reach the maximum length corresponding to c0 + d0 and c0 + b0, respectively. At W and (S) the diagonals (x for W and for at S) assume the minimum value, given by a0 − b0 and a0 − d0, respectively. In correspondence of the intermediate points SE, SW, NW and NE, the configurations are those of a concave, biconcave, concave, and convex quadrilateral, respectively. Being a non-Grashof case, b’ does not work as a crank, but inverts the direction of rotation. From a topological point of view, the sequence of configurations for both the Grashof and the non-Grashof case cover the single surface of a Moebius band.

Symmetric cases According to references ([41], [42]) there are twenty-seven possible cases that describe the motion of a four-bar linkage, and among them, for nineteen cases the Grashof condition is accomplished. Speciﬁcally, the nineteen cases are those for which at least one of the three variables r, u, v are zero. Among them, for twelve cases only one of the variables (r, u, v) is zero, for six cases two of the variables (r, u, v) are equal to zero, and there is only one case, where all the three variables are zero. In this section, we discuss the screens of some of the cases 1 – 19, introduced in Section (Correspondence between Regge symmetries and Grashof cases). These case, characterized by having at least one of the variables r, u, v equal to 0, are called symmetric, because of symmetry features presented by Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.118

Figure 6.3: Illustration of the four-bar mechanism related to the caustic in Figure (6.2). For this system, the Grashof condition is veriﬁed. We report the four sides a, b, c and d, and the diagonals x and y. The side b is the “ground link” and a, the “input” or “output link”, acts as a “crank” when rotates clockwise (see the left column) or anticlockwise (see the right column). We report the conﬁgurations that the system assumes in correspondence of the gates N, S, W, E and the intermediate points NE, SE, NW, SW. For each of this point, there are a couple of mirror images originated by the clockwise and anticlockwise rotations. Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.119

Figure 6.4: We report the Grashof conjugate of the object in Figure (6.3), which bars have lengths a0, b0, c0, d0 = 65, 50, 35, 40. In this case the Grashof condition is not verified. In this illustration, the configurations of the mechanism are made concrete by imposing a0 as fixed (the “ground link”) and following the behavior of b0 as input or output link. the related caustic. It is important to note that for the symmetric cases, the system presents a folded configuration during the rotation of its bars, corresponding to four-bar linkages with

T1,T2 or T3 = 0 ([41]). For the symmetric cases a single revolution along the caustic lead to Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.120 the initial object. In Figure (6.5-a), we report the screen for a system characterized by the following sides: a = 100, b = 110, c = 130 and d = 140; the linkage presents v = 0. According to the 6j notation, it is written as follows:

a b x 100 110 x = (6.36)  c d y   130 140 y         

Figure 6.5: a) Screen of a system characterized by a = 100; b = 110; c = 130; d = 140 (for the corresponding 6j symbol screen representation, see Fig. 5a in Ragni 2013). It corresponds to the case of symmetry v = 0; r = −10; u = −30. The North and West gates coincide. b) The screen of the system of sides a = 100; b = 140; c = 130; d = 110 (for the corresponding 6j symbol screen representation, see Fig5b in Ragni 2013) is reported. The symmetry parameters are u = 0; r = −10; v = 30 With respect to the system reported in a), there is an inversion in the shape of the curve, with the South and East gates coinciding. c) The screen is related to the system a = 100; b = 130; c = 140; d = 110 (for the corresponding 6j symbol screen representation, see Fig. 6a in Ragni 2013) and symmetry parameters r = 0; u = −10; v = 30. There is coalescence between the South and West gates.

The linkage is classiﬁed as case 8 (see Table I). The diagonals vary within the ranges 10 ≤ x ≤ 210 and 40 ≤ y ≤ 240. The N and W gates coincides and are located in the upper left corner. This is a particular case of Grashof, for which the summation of the longest and shortest bars are equal to the summation of the intermediate ones, being all the bars of diﬀerent length. In correspondence of the coalescence the bars are folded and lie on the same line, with b and a overlapped and b and d displaced in sequence. The passage to the gate E, a triangle with a and b aligned, occurs through a convex quadrilateral, and the passage to Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.121 the gate S, with b and d overlapped, through a concave quadrilateral. Finally, the passage between S and the coalescence NW involves a biconcave quadrilateral. In Figure (6.5-b), we show the screen for a = 100, b = 140, c = 130 and d = 110, with u = 0:

a b x 100 140 x = (6.37)  c d y   130 110 y          There is a coalescence of the caustic at SE (Figure 6.5-b). It corresponds to the case 3 (see Table I), with input and output links of crank type. The ranges of the diagonals are 40 ≤ x ≤ 240 and 10 ≤ y ≤ 210. The conﬁguration of the four bars is similar to that of the previous case, but the coalescence at SE indicates the folded system involves a and b aligned, and b and d overlapped. In Figure (6.5-c), the screen for a = 100, b = 130, c = 140 and d = 110, with r = 0 is reported:

a b x 100 130 x = (6.38)  c d y   140 110 y          This caustic presents coalescence at SW. It corresponds to the case 4 (see Table I), where both input and output links are of crank type. The ranges of the diagonals are 30 ≤ x ≤ 230 and 10 ≤ y ≤ 210. The coalescence at SW characterizes a completely folded system, where the four bars are overlapped. The quadrilateral involved are a convex, at NE, and the concave at NW and SE. The gate N is represented by a triangle, with the bars a and d aligned, while the gate E is given by a triangle with the bars a and b aligned. In Figure (6.6) (left panel), we plot the screen for a = 100, b = 200, c = 100 and d = 200, corresponding to the case 16 (Table I)

a b x 100 200 x = (6.39)  c d y   100 200 y          Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.122

Figure 6.6: a) Screen of the system a = 100; b = 200; c = 100; d = 200 (for the corresponding 6j symbol screen representation, see Fig. 8a in Ragni 2013). The symmetry parameters are u = v = 0 and r = −100. The is coalescence between the gates North and West and the gates South and East. b) In this panel, we report the screen for the system a = 100; b = 110; c = 100; d = 110 (for the corresponding 6j symbol screen representation, see Fig. 8b in Ragni 2013). The symmetry parameters are u = v = 0 and r = −10.

This case presents a higher Regge symmetry, with u = v = 0, it implies that the folding of the system occurs in two diﬀerent conﬁgurations. It presents the Piero line with a double coalescence at NW and SE gates. The range of the diagonals are 100 ≤ x ≤ 300 and 100 ≤ y ≤ 300. A linkage with these characteristics presents the four bars aligned in correspondence of the two coalescences. More precisely, a and b lie on the same line and are overlapped to c and d at SE, while a and d lie on the same line and are overlapped with b and c at NW. A convex quadrilateral characterizes NE, while a biconcave quadrilateral characterizes SW. In the right panel, we report the case for a = 100, b = 110, c = 100 and d = 110, corresponding to the case 16 (Table I).

a b x 100 110 x = (6.40)  c d y   100 110 y          As seen in the previous figure, u = v = 0. It presents Piero’s line and coalescences are at NW and SE gates. The ranges of the diagonals are 10 ≤ x ≤ 210 and 10 ≤ y ≤ 210. We note that r qualitatively indicates a change in shape of the caustic and the ridges. The Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.123 linkage is substantially similar to that reported for the previous case, although the bars a and d and the bars b and c are overlapped between SW and SE, as well as the bars c and d and the bars a and b are overlapped between NW and SW, giving a configuration similar to the hands of the clock. In Figure (6.7), the case for a = 100, b = 100, c = 1000 and d = 1000 is illustrated and corresponds to the case 17 (Table I), with r = v = 0 (the system folds in two different configurations)

a b x 100 110 x = (6.41)  c d y   1000 1000 y          The ranges of x and y are 0 ≤ x ≤ 200 and 900 ≤ y ≤ 1100. The caustic presents a coalescence at NW and SW. At NW linkage presents the bars a and d aligned and overlapped to the bars b and c, while at SW the system is completely folded, with the four bars superimposed. At E, a and b are aligned, forming a triangle, at NE the conﬁguration is that of a convex quadrilateral, while SE is characterized by a concave quadrilateral.

In ﬁgure (6.8) reports the case for a = 100, b = 100, c = 100 and d = 100, corresponding to the case 19

a b x 100 100 x = (6.42)  c d y   100 100 y          It represents the case with the highest symmetry, with r = u = v = 0. Under these conditions, the system folds in three diﬀerent conﬁgurations and the input and output links rotate completely. The ranges of the diagonals are 0 ≤ x ≤ 200 and 0 ≤ y ≤ 200. The screen presents the Piero’s line and coalescence at NW and SE. The coalescence point at NW is characterized by a and d aligned and overlapped with b and c, while at SE a and b are aligned and overlapped with c and d. The intermediate point at NE is represented by a convex quadrilateral. Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.124

Figure 6.7: Screen for the system a = 100; b = 100; c = 1000; d = 1000 (for the corresponding 6j symbol screen representation, see Fig. 9a in Ragni 2013). In this case, the symmetry parameters are r = v = 0 and u = −900.

Figure 6.8: Screen of the system a = b = c = d = 100 (for the corresponding 6j-symbol screen representation, see Fig. 9b in Ragni 2013), a quadrilateral with the four sides of identical length. The symmetry parameters are r = u = v = 0. Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.125

6.4 Concluding remarks

In this work, we have related the tools developed for the theory of angular momentum in quantum mechanics and the four-bar linkage. Specifically, we have investigated relations between the Regge symmetries and the classification of the four-bars mechanism, especially for what concern the Grashof condition. The screen representation has been adopted for the study of some significant case of four-bar linkage, that have been classified in symmetric and general cases. The symmetric cases are those for which at least one of the Regge variables are equal to 0. We have observed that systems characterized by one of the Regge variables equal to 0, present a folded configuration, during the rotation of the bars. Systems characterized by two Regge variables equal to 0, fold in correspondence of two configurations. Finally, systems with all the three Regge variables equal to 0 present folding correspondent to three different configurations. The screens present coalescence of a couple or both couple of gates, for all the linkages the Grashof condition is verified and systems are invariant under Regge symmetry operations. Regarding the general case, it is characterized by having all the Regge variables different from 0, the corresponding screen presents separated gates and the system is not invariant under Regge symmetry operation. For this latter aspect, we have observed that the Regge conjugate has an opposite Grashof behavior, with respect to the starting system. The variation of the diagonals lengths, as represented by the caustic, produces an object which is specular with respect to the initial one, thus it is necessary to do two whole circuits to obtain the initial configuration. Future developments will concern the extension of the present work to the most recent application of the four-bar linkage. Some important examples concern the evolutionary trend in the biomechanics of many species ([43], [44], [45]), often applied to the robotics and engineering, like the kinematics and anatomy of the human knee ([46], [47], [48]) and the morphologic characteristics of elephants that allow them to sustain the huge amount of body weight ([49]). Other significative examples are found in spearing appendages of mantis shrimps ([50], [51]) and in the operculus of fishes, a protective covering for the gills ([52]) and in the prey-processing mechanism of salmon-type fishes. Vertebrates skull in general show a four-bar linkage mechanism, (see the example of birds, fishes and lizards) this model permits to study the relation between the functionality Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.126 and the structure of these systems and their evolution ([53]). The feeding system of seahorses, composed by a quick elevation of the head followed by a rotation, is also modelled by a four-bar mechanism ([54]). A similar mechanism is also found in the kinematic of insects’ wings along the translational and rotational degrees of freedom ([55]) and bacteria ([56]). It is noteworthy also the application of four-bars mechanism to engineering based on the emulation of animals’ motions, e. g. amphibian robots that mimic the skill of the basiliscus lizard to run on the water surface and walk on the ground ([57]). For possible involvements in other areas of chemistry see also ([58], [59], [60], [61]).

APPENDIX

6.4.1 A.1. Symmetries of 6j symbols and the Regge symmetries.

Similarly to the Wigner-Racah-Regge approach for the 6j symbol, we arrange the six distances as follows,

a b x (6.43)  c d y      where the third column corresponds to the diagonals. According to the invariance under permutation of the three columns, other choices of diagonals are possible:

a b x a x b = = ... (6.44)  c d y   c y d          in this way, there are 3! = 6 identical symbols. The choice of the diagonal is useful to monitor speciﬁc properties of the system ([62]). The ﬁrst row of the 6j symbol represents a triangular face of the tetrahedron into a vertex, called triad, while the second row corresponds to the convergence of three edges of the tetrahedron. The tetrahedron is composed by four Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.127 triads: a, b, x, c, b, y, a, d, y, c, d, x. The invariance of the 6j symbol under the interchange of upper and lower arguments of any two columns, generated by the combined invariance with respect to permutations of the four triads and the triangles, gives for example the following relations:

a b x c d x = = ... (6.45)  c d y   a b y          (four ways). In total, there are 24 four symmetries. Each symbol reported above has in addition six replicas that Regge discovered and presented in two famous letters (Regge 1958, 1959). Here, we report in full the thirteen lines of the text that Tullio Regge wrote in his second paper: “we have shown in a previous letter that the true symmetry of Clebsch-Gordan coeﬃcients is much higher than before believed. A similar result has been now obtained for Racah’s coeﬃcients. Although no direct connection has been established between these wider symmetries it seems very probable that it will be found in the future. We shall merely state here the results which can be checked very easily with the help of the well known Racah’s formula.” Continues with the symmetry relations: “From the usual tetrahedral symmetry group of the 6j we know already that:

a b c b a c a e f c e d = = = (6.46)  d e f   e d f   d b c   f b a                  Our results can be put into the following form:

a b c a (b+e+c−f) (b+c+f−e) = 2 2 = etc”. (6.47)  d e f   b (b+e+f−c) (c+e+f−b)     2 2      The paper concludes: “Only the ﬁrst of these symmetries is essentially new, the others can be obtained from it and the permutational symmetries. We see therefore that Chapter 6. Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages.128 there are 144 identical Racah’s coeﬃcients. These new symmetries should reduce by a factor 6 the space required for the tabulation of the 6j. It should be pointed out that this wider 144-group is isomorphic to the direct product of the permutation groups of three and four objects.” Letters ([63]) and ([64]), and the reprinted ([65]) are a collection of articles relevant to quantum angular momentum theory, cited by Bargmann 1962 as “Regge’s intriguing discovery of unexpected symmetry of 3j and 6j symbols”.

6.4.2 A.2. The canonical order of Regge symbols and the Gashosf condition

The Grashof condition states that “in a four-bar linkage, if the sum of the length of the shortest, s, and the longest, l, bar is lower than the sum of the two intermediate ones, p and q, the shortest bar can make a full rotation with respect the adjacent bars”:

s + l ≤ p + q (6.48)

A consequence of this condition is that only quadrilateral that accomplish the above mentioned relation are deﬁned within the range of a screen.

Table 6.1: Classiﬁcation of input and output links bar movements r u v Input link Output link r u v Input link Output link r u v Input link Output link - - + crank crank I + 0 + crank 0-rocker 2 - 0 - 0-rocker 0-rocker 11 + + - 0-rocker π rocker I’ - 0 + crank crank 3 0 - - 0-rocker crank 12 − + - - rocker crank II 0 - + crank crank 4 0 0 + crank crank 13 - + + π rocker π rocker II’ + + 0 crank π rocker 5 0 + 0 crank π rocker 14 − − − − - + - rocker rocker III - + 0 π rocker π rocker 6 + 0 0 crank crank 15 − − + - + π rocker 0-rocker III’ + - 0 π rocker crank 7 - 0 0 crank crank 16 − − + + + crank rocker IV - - 0 crank crank 8 0 - 0 crank crank 17 - - - 0-rocker 0-rocker IV’ 0 + - 0-rocker π rocker 9 0 0 - 0-rocker crank 18 − 0 + + crank π rocker 1 + 0 - 0-rocker crank 10 0 0 0 crank crank 19 − (Roman numerals refer to Regge’s symmetric in the case where r, u, v are diﬀerent from zero, while the Arabic numbers refer to the other 19 symmetry∗ cases where r, u, v = 0). Bibliography

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Remarks

The present research developed around the Angular Momentum Theory and Spin Network was aimed at relating and deepening important aspects of the Racah-Wigner algebra, in addition to important subjects such as Regge’s Symmetry, Projective Geometry, Symbols 3nj and its geometrical representations, as well as the four bar movements, initially studied by Grashof.

Therefore, it is useful to point out that we start from the understanding of coupling and re-coupling of angular momentum theory, in particular for the geometric representations in tree diagrams and diagrams of Yutsis. On the other hand, the mathematical aspects were also taken into account for the construction of these ones. In chapter 3 we study the spin network combinatorial. In addition, another fundamental aspect was the construction of the “Abacus”, a quantum graph, which combines the possible coupling and re-coupling combinations of angular momentum and can describe symmetry properties, relations and sum rules for the symbols 6j and 9j. The pentagonal relationship, or also known as Biedenharn- Elliott identity, and the hexagonal relation or sum of Racah, were discussed and related to the 7 and 10 spin-network. Thus, it is necessary to reaﬃrm that the description of these structures constitute a valid model for discovering and deepening unexplored geometric and computational aspects of angular momentum algebra, but also combinatorial characteristics of general interest in a range of applications in the various areas of Physics.

137 Chapter 7. Remarks 138

One of the aspects that also marked this work was the connection of the Projective Geometry with the Angular Moment Theory, initially proposed in the works of Judd and Robson. In this line, we explore and propose the 7 and 10 spin-network in the Fano plan, Desargues and also in the Levi incidence diagram. In order to understand such connections, we start with the dual representation analysis of lines and points in the projective planes and in the complete quadrilateral (6 lines and 4 points) and the complete quadrilateral (4 lines and 6 points).

The literature analysis allows to identify the great contributions given by the symmetry of Regge, as well as its proposal of the tetrahedron model associated with the symbol 6j. In this context, in chapter 5 we start from the analysis of the origins of the Regge symmetry, as well as the combinatorial and aspects related to the conditions of existence of the symbols 3nj. We also sought to relate this symmetry to the classical four-bar motion in order to understand the processes of chirality in molecules, so we also deduced the Caustic curve and the Ridge according to the sides of a quadrilateral and its diagonals (x and y).

The advances provided with this thesis are conﬁgured as a solid base that allows the opening of new paths for the quantum study of complex systems. The systematization in the larger framework of angular momentum re-coupling coeﬃcients, which encompasses the four-bar problem, will allow the confrontation, analysis and problem solving of n − spin networks with n > 10.

As an example, the chirality of molecules with more chiral centers and, in particular, enantiomers and diastereomers, can be considered. These structures, because they have geometries of the speciﬁc angular moments, are strictly connected, by means of bijectors functions, to mechanical systems like the one of the 4 bars. The study of these molecules represents an interesting aspect for the scientiﬁc development of the area.

Therefore, the advance in the quanta mechanical description, and the discoveries of symmetries of the mathematical objects used in the calculation of the observables implies in the simpliﬁcation of the mathematical representation and the obtaining of the results of interest, thus inserting the continuation of the present study in the current researches, quantum computing and quantum gravitation.