Constructive Motives and Scattering M. D. Sheppeard Preface This elementary text is intended for anyone interested in combinatorial methods in modern particle physics. Advanced concepts are only mentioned when there is some chance at a simple explanation. There is a development of ideas through the book, but hopefully each chapter is also reasonably well contained. All diagrams and tables are embedded in the text along with the equations. At the heart of particle physics is the problem of emergence. As of 2013, nobody really understands what this is, but there is however general agreement that the answers involve the concept of motive. Throughout the book, our aim is to understand a little about motives, not from the standard mathematical point of view, but using a physicist's intuition. This can be done at an elementary level, because the underlying philosophy is a constructive one, meaning that theorems about motives should depend on their concrete construction. Motives are about both geometry and number theory, and hence about knots. Unfortunately, there are many relevant topics that cannot be covered. The essential physical ideas do not appear before chapter 6, but are an integral part of the methods discussed. If the reader really wants to skip the abstract nonsense on a ¯rst reading, they may do so. The whole book is typeset in LATEX, using mostly XY-pic for diagrams. It was written with no feedback, essentially no resources, and no doubt many errors remain. Thanks to wikipedia for an endless supply of free information. It cannot all be acknowledged. During the blogging years there were many conver- sations with keen theorists, notably Carl Brannen, Michael Rios, Louise Riofrio, Alejandro Rivero and Tony Smith. This work was made possible by the kindness of Kerie and Allan. It also owes a great debt to Graham Dungworth, whose ceaseless online enthusiasm and clari¯cation of the new cosmology has provided invaluable insights. °c Marni Dee Sheppeard 2013. Contents 1 Introduction 3 2 Numbers and Sets 8 2.1 The Word Monoid . 10 2.2 Continua and Quantum Numbers . 12 2.3 Union, Disjoint Union and Cohomology . 15 2.4 A Category of Relations . 17 3 Duality and the Fourier Transform 19 3.1 The Quantum Fourier Transform . 19 3.2 Unitary Bases and Decompositions . 21 3.3 Honeycombs and Hermitian Matrices . 24 4 The Ordinals and Discrete Duality 26 4.1 The d-Ordinals . 26 4.2 Categorical Strings . 28 4.3 Fourier Dualities and Topology . 30 5 Trees, Polytopes and Braids 32 5.1 Permutations and Planar Trees . 32 5.2 Solomon's Descent Algebra . 36 5.3 Associahedra, Permutohedra and Polygons . 38 5.4 Linear Orders and Forests . 42 5.5 Three Dimensional Traces . 46 5.6 Associated Braids and Knot Invariants . 48 6 Twistor Scattering Theory 53 6.1 Scattering Amplitudes . 55 6.2 N=8 Supergravity . 60 6.3 Grassmannians and Associahedra . 61 6.4 Symbology and Polylogarithms . 64 6.5 Decorated Polygons for Symbols . 66 7 The Ribbon Particle Spectrum 71 7.1 Braids and Ribbons . 72 7.2 The Single Generation Spectrum . 74 7.3 CPT and the Higgs Mechanism . 78 7.4 Electroweak Quantum Numbers . 82 7.5 The Burau Representation and Mirror Circulants . 83 7.6 Neutrino and Quark Mixing . 86 7.7 The Koide Rest Mass Phenomenology . 90 8 Knots and Ribbon Graphs 95 8.1 The Temperley-Lieb Algebra . 96 8.2 Bn and Khovanov Homology . 99 8.3 Chorded Braids . 101 8.4 Ribbons and Moduli Spaces . 101 9 Bootstrapping Adjoint Actions 105 9.1 Duality with S2 . 105 9.2 The Permutoassociahedron . 107 9.3 The Klein Quartic and S5 . 107 9.4 Nonassociative Braids . 108 10 Entanglement and Entropy 110 10.1 Entanglement with Trees and Jordan Algebra . 110 10.2 Categorical Entanglement . 114 10.3 Secondary Polytopes and Hyperdeterminants . 115 11 Non Local Cosmology 119 11.1 Modi¯ed Newtonian Dynamics . 121 11.2 AdS, dS and FRW Cosmologies . 123 11.3 Observational Notes . 127 11.4 Mirror Neutrinos and the CMB . 127 11.5 Discussion . 129 A Category Theory 131 A.1 Limits and Universality . 133 A.2 Monoidal, Braided and Tortile Categories . 135 A.3 Tricategories and Higher Dimensions . 138 A.4 The Crans-Gray Tensor Product . 140 B Braid Groups 141 C Elementary Algebra 144 C.1 Bialgebras and Hopf Algebras . 145 C.2 Shu²es and Lattice Paths . 147 C.3 Matrix Tensor Algebra and Distributivity . 147 D The Division Algebras 150 D.0.1 Particles from R ­ C ­ H ­ O . 151 D.0.2 Jordan Algebras . 153 1 Introduction The task of emergent geometry is to recover the rich mathematical structures underlying quantum ¯eld theory and general relativity. This means no less than unveiling classical geometry itself, in an axiomatic setting capable of transcending the limitations of set theory. Much progress has been made in recent decades, and the foundations of category theory are now essential to any serious endeavour in theoretical physics. This basic text covers a range of combinatorial and categorical techniques lying behind the modern approach. These begin with the discrete permuta- tion and braid groups, but the overall aim is to understand a constructive continuum, wherein the complex numbers and other division algebras ap- pear in the motives of a universal cohomology. A motive is a gadget much beloved by mathematicians, although nobody really understands what it is. To a physicist, cohomology is an algorithm for cutting spaces down to their essential physical content. In a quantum universe, we would also like to do the reverse: start with measurement, and build the geometry de¯ned by its logic. This means that concrete diagrams are interpreted not as classical spaces, but rather as symbols representing measurement questions. In ¯eld theory, Feynman diagrams are replaced by diagrams for twistor spaces. We now know for certain that a motivic formulation of particle physics exists. Modern twistor methods for scattering amplitudes use motivic meth- ods, as do studies of renormalisation algebras. Amplitudes for n particles are computed on a space whose dimension appears to increase with n, sug- gesting the increasing complexity of abstract information rather than an external reality of a ¯xed number of dimensions. In order to explain clearly the choice of topics here, it is necessary to take a ¯rm point of view on the physics. Our position on the Lagrangian is the following. The local theory is exactly the Standard Model, with Majorana neutrinos and no proper neutrino oscillations. The only observable local states come from this SM Lagrangian, and they are enumerated by a special set of ribbon diagrams. In the non local theory: the neutrinos, the Higgs boson, and in fact all neutral particles, may exhibit novel features. There is a natural spectrum of mirror fermions, but no additional bosons, suggesting that mirror fermions are merely a non local aspect of baryonic matter. The algebraic structure of the local SM Lagrangian is outlined in [1] using adjoint actions for the division algebras. In this scheme, the unbroken SU(2) £ U(1) symmetry comes essentially from the algebra H ­ C. The octonions are responsible for color SU(3) symmetry. Here we focus mostly on the complex numbers and the quaternions, but we insist these ¯elds occur only in a way that respects the underlying structure of all the division algebras. With emergence, we can start with the broken symmetries, which from a measurement perspective are more fundamental. Right handed states need 3 not be singlet states, since the states are not de¯ned with respect to the classical gauge symmetry, itself an emergent structure. Including right handed (mirror) neutrinos, we can speak about standard Dirac neutrinos. At ¯rst sight there is no see-saw mechanism, since the right handed neutrinos do not set a large mass scale. On the other hand, a dual mass scale may de¯ne an e®ective see-saw. The mirror neutrino scale in the non local theory is identical to that of the left handed triplet, ¯xing the temperature of the CMB at T = 2:725 K. These mirror neutrinos are only observed in their manifestation as CMB photons. They come from a land of supersymmetric information, which dissolves the distinction between fermions and bosons. Every electroweak boson may be viewed as a Fourier § § dual to a fundamental lepton state: W from e , γ from ºL, and Z from a composite of three right handed lepton states. Although we speak about the possibility of mirror dark matter, it is un- clear whether a mirror Lagrangian serves any useful purpose. It is natural to consider the right handed mirror neutrino as the only additional particle, and to disallow localisation for all other mirror states. However, since the dark matter problem is thoroughly addressed by mirror matter proposals, we consider it in the ¯nal chapter. Motivated by the mirror neutrino CMB photon, one might view all physical electroweak bosons as transformed mir- ror states, since the Fourier supersymmetry transform may be applied to the mirror sector. More radically, perhaps the mirror fermions stand for known particles: the protons and neutrons. In these two cases, there is no dark matter, and general relativity must be abandoned on large scales. This is quite plausible, since the non local ribbon diagrams display a preon as- pect to any particle, and one imagines zooming in and out of a complicated network of bunched, knotted strands. The particle states are speci¯ed by the most basic ribbon diagrams in three dimensions. As quantum numbers, spin and rest mass must emerge algebraically in a natural way from such diagrams, along with the Poincare group symmetries.