Symmetry

This book presents the geometric structure of the Higgs field, the geometry that is beyond the standard model based on a modification of the structure that appears in Escher ́s Waterfall. The proposed structure is also a way to draw and visualize the since long time held idea that spacetime is quantized. It brings harmony between two colliding worlds in physics, general relativity and quantum mechanics. There is a geometric representation of the quarks, the leptons and the four types of gauge bosons, the weak isospin and the geometry and algebra of many interactions and decays. The values of the Weinberg angle, the Cabibbo angle and the parameters of the CKM matrix are also given by the structure, as well as the geometry of special relativity, the tensors of general relativity, the measurement problem, the uncertainty principle, the second law of thermodynamics, neutrino oscillation and the PMNS matrix. All the free parameters of the standard model find their precise values in the proposed structure. The fundamental constants -the gravitational constant, Boltzmann constant, Planck ́s constant and the vacuum permittivity- are related by the golden ratio. The second part is a description of the program that controls the perfectly symmetric distribution of the prime numbers among the natural numbers. The third part connects the natural numbers and the primes with the structure described in the first part. First part: Geometry beyond the standard model: Escher ́s Waterfall 13 I. The geometry of the elementary particles 13 1. Introduction [1-11] 13 2. Matrix representation 19 3. Antimatter 21 4. Weak isospin 23 5. Conservation of angular momentum 25 6. Symmetry [15, 16] 25 7. The gauge bosons 26 8. Examples of decays and interactions [6, 7, 8, 9, 10, 11, 17, 18, 19] 30 9. The Weinberg angle, θw 95 10. The Cabibbo angle, θc 96 11. The CKM matrix 97 11. The angles of the three generations 101 12. Mass of hadrons and leptons (in MeV) 102 II. Geometry of relativity and of some other questions in physics [23-36] 105 1. CPT. Duality of the structure 105 2. Intrinsic parity 109 3. The arrow of time 109 4. Linking basic units. Velocity 111 5. Frame time and proper time in different inertial frames 114 6. The relativity of simultaneity 115 7. Energy, momentum and mass [38-41] 116 8. The mass of the fermions 121 9. Dark matter and dark energy 124 10. The cosmological constant [42] 125 11. The flatness problem [43] 125 12. The wave function 127 13. Energy levels and the primes 128 14. Lorentz transformations [23] 129 15. Momenergy [23] 130 16. Parametric equations 131 17. Tidal forces 133 18. The stress-energy tensor 134 19. The Weyl tensor 142 20. Gravity and entropy gradients 144 21. The Ricci tensor 145 22. Neighbouring units: Gravity in action 146 23. What happens behind the scenes 148 24. The 8G factor. Spherical objects 150 25. Spherical object collapsing to form a black hole [46] 154 26. Geometric interpretation of the uncertainty principle and Planck ́s constant 156 27. Geometry of the measurement problem and Planck ́s constant 159 28. Temperature and Boltzmann constant 161 29. The infrared cut-off and the ultraviolet cut-off 162 30. The second law of thermodynamics [32, 47] 164 31. Neutrino oscillation and the PMNS matrix 165 32. The vacuum expectation value, the coupling constants, the mass of the top 169 33. The interactions and the golden ratio 175 34. The hidden transactions within the golden ratio 176 35. Range of the interactions 177 36. Table with the main values obtained from the structure 179 37. The naturalness problem 181 Second part: The symmetry of the prime numbers 187 1. The program of the primes 187 2. The twin primes 198 3. Chained symmetries for the even numbers 200 4. Program of the odd composite numbers 201 Third part: The prime numbers in the structure 205 1. The natural numbers and the energy levels. The primes 206 2. The Goldbach ́s conjecture and particle physics 208 3. The primes and the Riemann curvature tensor 213 Fourth part: The process of thought 217 1. Thought as the CP conjugate of reality 217 References 223