Once upon a time there was a pendulum

Description

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 II. Geometry of relativity and of some other questions in physics [23-36] 103 1. CPT. Duality of the structure 103 2. Intrinsic parity 106 3. The arrow of time: 106 4. Linking basic units. Velocity 108 5. Frame time and proper time in different inertial frames 111 6. The relativity of simultaneity 112 7. Energy, momentum and mass [38-41] 113 8. The mass of the fermions 118 9. Dark matter and dark energy 121 10. The cosmological constant [42] 122 11. The flatness problem [43] 122 12. Energy levels and the primes 124 13. Lorentz transformations [23] 125 14. Momenergy [23] 126 15. Parametric equations 127 16. Tidal forces 129 17. The stress-energy tensor 130 18. The Weyl tensor 138 19. Gravity and entropy gradients 140 20. The Ricci tensor 141 21. Neighbouring units: Gravity in action 142 22. What happens behind the scenes 143 23. The 8G factor. Spherical objects 145 24. Spherical object collapsing to form a black hole [46] 149 25. Geometric interpretation of the uncertainty principle 151 28. Geometry of the measurement problem 155 29. The infrared cut-off and the ultraviolet cut-off 157 30. Temperature and the Boltzmann constant 159 31. The second law of thermodynamics [32, 47] 160 32. Neutrino oscillation and the PMNS matrix 161 33. The vacuum expectation value, the coupling constants, the mass of the top 165 34. The interactions and the golden ratio 171 35. The hidden transactions within the golden ratio 172 36. The gate 173 37. A possible geometric representation of the space axes for other universes 174 38. Range of the interactions 175 39. Table with the main values obtained from the structure 177 40. The naturalness problem 179 Second part: The symmetry of the prime numbers 183 1. The program of the primes 183 2. The twin primes 194 3. Chained symmetries for the even numbers 196 4. Program of the odd composite numbers 197 Third part: The prime numbers in the structure 201 1. The natural numbers and the energy levels. The primes 202 2. The Goldbach´s conjecture and particle physics 204 3. The primes and the Riemann curvature tensor 209