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Operator Guide, 144+ Pages, Pdf, Latex /Memoir Format Operator Guide to the Standard Model Operator Guide to the Standard Model Snuarks and Anions, for the Masses Carl Brannen Liquafaction Corporation Redmond, Washington, USA January 20, 2008 ( Draft ) This edition published on the World Wide Web at www.brannenworks.com/dmaa.pdf January 20, 2008 Copyright c 2008 by Carl A. Brannen. All rights reserved. For Mom and Dad Contents Contents vii List of Figures x List of Tables xi Preface xiii Introduction xvii The Thief's Lover . xvii 0 Fundamentals 1 0.1 Particle Waves . 1 0.2 Wave Function Density Matrices . 3 0.3 Consistent Histories Interpretation . 4 1 Finite Density Operators 5 1.1 Traditional Density Operators . 5 1.2 An Alternative Foundation for QM . 6 1.3 Eigenvectors and Eigenmatrices . 7 1.4 Bras and Kets . 11 1.5 Naughty Spinor Behavior . 14 1.6 Linear Superposition . 15 2 Geometry 17 2.1 Complex Numbers . 18 2.2 Expectation Values . 21 2.3 Amplitudes and Feynman Diagram . 24 2.4 Products of Density Operators . 29 3 Primitive Idempotents 35 3.1 The Pauli Algebra Idempotents . 35 3.2 Commuting Roots of Unity . 37 3.3 3 × 3 Matrices . 40 4 Representations 47 vii viii Contents 4.1 Clifford Algebras . 47 4.2 The Dirac Algebra . 52 4.3 Matrix Representations . 57 4.4 Dirac's Gamma Matrices . 61 4.5 Traditional Gamma Representations . 65 5 Algebra Tricks 71 5.1 Exponentials and Transformations . 71 5.2 Continuous Symmetries . 76 5.3 Velocity . 77 5.4 Helicity and Proper Time . 79 6 Measurement 81 6.1 The Stern-Gerlach Experiment . 81 6.2 Filters and Beams . 84 6.3 Statistical Beam Mixtures . 85 6.4 Schwinger's Measurement Algebra . 87 6.5 Clifford Algebra and the SMA . 90 6.6 Generalized Stern-Gerlach Plots . 95 7 Force 101 7.1 Potential Energy, Some Guesses . 101 7.2 Geometric Potential Energy . 105 7.3 Snuarks as Bound States . 108 7.4 Binding Snuarks Together . 111 7.5 The Feynman Checkerboard . 112 7.6 Adding Mass to the Massless . 114 7.7 A Composite Checkerboard . 116 7.8 Bound State Primitive Idempotents . 121 8 The Zoo 127 8.1 The Mass Interaction . 127 8.2 Snuark Antiparticles . 129 8.3 Quarks . 131 8.4 Spinor and Operator Symmetry . 136 8.5 Weak Hypercharge and Isospin . 141 8.6 Antiparticles and the Arrow of Time . 143 9 Mass 145 9.1 Statistical Mixtures . 145 9.2 The Koide Relation . 149 10 Cosmic Haze 151 11 Conclusion 153 Bibliography 155 Contents ix Index 157 List of Figures 0.1 The Two Slit Experiment: Quantum particles impinge on a barrier with two slits. Particles that pass the barrier form an image on the screen that exhibits interference. 2 2.1 Two adjoining spherical triangles on the unit sphere. Suvw is addi- tive, that is, SABCD = SABC + SABD, and therefore the complex phase of a series of projection operators can be determined by the spherical area they encompass. 32 3.1 Drawing of the lattice of a complete set of commuting idempotents of the Pauli algebra, see text, Eq. (3.17). 39 3.2 Drawing of the lattice of a complete set of commuting idempotents of the algebra of 3 × 3 matrices, see text, Eq. (3.21). The primitive idempotents are ρ−−+, ρ−+−, and ρ+−−. 41 6.1 The Stern-Gerlach experiment: Atoms are heated to a gas in an oven. Escaping atoms are formed into a beam by a small hole in a plate. A magnet influences the beam, which then forms a figure on a screen. 82 6.2 Two Stern-Gerlach experiments, one oriented in the +z direction, the other oriented in the +x direction. 83 6.3 Stern-Gerlach experiment that splits a beam of particles according to two commuting quantum numbers, ι1 and ι2. 96 6.4 Stern-Gerlach experiment with the three non independent opera- tors, ι1, ι2, and ι2ι1. Spots are labeled with their eigenvalues, ±1, in the order (ι1; ι2; ι2ι1). The cube is drawn only for help in visual- ization. 97 6.5 Stern-Gerlach plot with the three operators (ι1ι2; ι2ι3; −ι2). The labels for each corner are the ι1, ι2 and ι3 quantum numbers in order (ι3; ι2; ι1). The antiparticles have ι3 = −1 and are drawn as hollow circles. 98 6.6 Stern-Gerlach plot of 12 snuarks and their antiparticles. See Eq. (6.33) for the quantum numbers. The antiparticles have ι3 = −1 and are drawn as hollow circles. 99 x 7.1 Feynman diagrams that contribute to a left handed primitive idem- potent (snuark) becoming right handed. 119 7.2 The mass interaction as an exchange of three \gauge bosons". 124 8.1 Quantum number plot of the eight snuarks with zero ixyzstd and ixyztd . The primitive idempotents are marked with small solid circles and form a cube aligned with the axes. 132 8.2 Quantum number plot of the eight leptons and sixteen quarks in snuark form relative to the primitive idempotents. Primitive idem- potents are small filled circles, leptons are large hollow circles and quarks (×3) are large filled circles. 134 8.3 Weak hypercharge (t0) and weak isospin (t3) quantum numbers for the first generation particles and antiparticles. 141 8.4 Weak hypercharge, t0, and weak isospin, t3, quantum numbers plot- ted for the first generation standard model quantum states. Leptons are hollow circles and quarks (×3) are filled circles. Electric charge, Q, and neutral charge, Q0 also shown. 143 List of Tables 4.1 Signs of squares of Clifford algebra MVs with n+ positive vectors and n− negative vectors. 51 xi Preface The chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction|a direction obvious from an unfashionable view of field theory{who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unfashionable point of view; one that he may have to invent for himself. Richard Feynman, Stockholm, Sweden, December 11, 1965 s I write this in October 2006 particle physics is in trouble. Two books A stand out on the physics best sellers list: Lee Smolin's The Trouble With Physics, and Peter Woit's Not Even Wrong. These books show that mankind's centuries long effort to understand the nature of the world has come to a quarter century long pause. The methods that worked between 1925 and 1980 have not succeeded in pushing back the frontier since then. This book defines an alternative path, a path that may once again allow nature's mysteries to unfold to us. Lee Smolin writes, \When the ancients declared the circle the most perfect shape, they meant that it was the most symmetric: Each point on the orbit is the same as any other. The principles that are the hardest to give up are those that appeal to our need for symmetry and elevate an observed symmetry to a necessity. Modern physics is based on a collection of symmetries, which are believed to enshrine the most basic principles." In this book we will reject symmetries as the most basic principle and instead look to geometry, more specifically, the geometric algebra[1] of David Hestenes. But instead of applying geometric algebra (a type of Clifford algebra) to spinors, we will be applying it to density operator states. The geometric algebra is elegant and attractive and several authors have applied it to the internal symmetries of the elementary particles. [2] This book has the advantage over previous efforts in that it derives the relationship between the quarks and leptons, the structure of the generations, and provides exact formulas for the lepton masses. On the other hand, this book suffers from the disadvantage of requiring a hidden dimension and that the geometric xiii xiv PREFACE algebra be complex. We will attempt to justify these extensions; in short, they are required because the usual spacetime algebra is insufficiently complicated to support the observed standard model particles. This book is intended as a textbook for graduate students and working physicists who wish to understand the density operator foundation for quantum mechanics. The density operator formalism is presented as an alternative to the usual Hilbert space, or state vector, formalism. In the usual quantum mechanics textbooks, density operators (or density matrices) are derived from spinors. We reverse this, and derive spinors from the density operators. Thus density operators are at least equal to spinors as candidate foundations for quantum mechanics. But we intend on showing more; that the density operator formalism is vastly superior. In the state vector formalism, one obtains a density operator by multiplying a ket by a bra: ρ = jAihAj. Thus a function that is linear in spinors becomes bilinear in density operators. And a function that is linear in density operators becomes non linear when translated into spinor language. This means that some problems that are simple in one of these formalisms will become nonlinear problems, difficult to solve, in the other. To take advantage of both these sorts of problem solving, we must have tools to move back and forth between density operator and state vector form. While most quantum textbooks provide no method of obtaining spinors from density operators, we will, and we will show how to use these methods. The standard model of the elementary particles has been very good at predicting the results of particle experiments but it has such a large number of arbitrary constants that it has long been expected that it would be eventually replaced with a deeper theory.
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