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Jul 2nd, 2022
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1. PHYSICS BY GPT-J
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3. RESULT
4. "Three Fermion Generations from Octonions"
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6. by
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8. Arnaud Delorme
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14. Thank you to David Wess for reviewing this document.
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16. Thank you to all the students who took the time to send me email and questions. You all made this a very interesting and inspiring project.
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22. Introduction
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24. In 1998 I discovered that the octonion plane is related to the three dimensional Euclidean space and that one can construct a model of generation (the mass spectrum) of fermions using this idea. This was a "light bulb" moment for me.
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26. The model of generation uses the octonions to classify particles and the spinor representation of the octonions.
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28. I wrote an article for the American Journal of Physics, 1999:
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30. Arnaud Delorme, "Three Fermion Generations from Octonions"
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36. A Model of Generation
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38. Generations are the representations of a group. In the standard model of physics we have fermions which have generation 0, generation 1/2 and generation 3/2 (e.g. for quarks). In this paper, I construct a model of generation using the quaternion plane and the spinor representations of quaternions. The starting point of this model is an octonion plane.
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40. The Standard Model of Physics is based on the 3rd Euclidean Plane
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42. The vector representation of the 3rd Euclidean plane, or 3D Euclidean space, is the complex vector space.
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48. The Standard Model of physics uses a 4D Euclidean Plane
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50. There are many models of 3D Euclidean space, for example a three dimensional non-compact form of the split-complex numbers (the imaginary numbers ). In this model the dimension of is one less than the dimension of the real numbers. I will refer to this as the extended real numbers. In this extended real numbers, the sum of the complex numbers is also an extended real number. In this model, the complex numbers are self dual (like the vectors are self dual in 3D Euclidean space). So, there is no "norm". The sum of two complex numbers is the complex number, and the square of a complex number is the complex number.
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52. The complex numbers can be constructed from the real numbers using the Cauchy-Riemann equation. For example, the exponential of a complex number is a complex number (in the extended real numbers). The real and imaginary parts of the complex numbers can be multiplied together. In the complex numbers, the real part and the imaginary part are perpendicular to each other, and the real part is the "norm" of the complex number.
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54. The complex numbers also obey the orthogonality relations for the vectors of the 3D Euclidean space. For example, the two vectors,,, and are perpendicular to each other.
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56. The 3D Euclidean vector space has a spinor representation. A vector in the three dimensional Euclidean space is represented by a spinor (a two component complex number) on the three dimensional Euclidean plane. The magnitude of the vector is represented by the norm of the spinor. In a vector space, vectors are always perpendicular to each other. In the 3D Euclidean space, the vectors are self dual (because of the Cauchy-Riemann equation) and the perpendicular vectors are complex conjugate to each other. So, in the three dimensional Euclidean space, the real part of the spinor and the imaginary part of the spinor are perpendicular to each other. The real part of the spinor is perpendicular to the imaginary part of the spinor. The imaginary part of the spinor is a spinor (a self dual vector) in the three dimensional Euclidean space. In this 3D Euclidean space, the spinor has a norm equal to the imaginary part of the complex number.
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58. The four components of the vector (x, y, z, ) are represented by the four components of the spinor (r, i, s, t).
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60. The sum of the complex numbers are always a complex number. So, the sum of the two vectors, and, is also a complex number. The norm of the complex number is the norm of the sum of the two vectors.
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62. The norm of the vector in the 3D Euclidean space can be calculated using the Cauchy-Riemann equation. The norm of the vector is equal to the norm of the sum of the vectors and. The norm of the sum of the vectors is equal to the norm of the real part of the spinor.
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64. The sum of the vectors is always a complex number. The norm of the sum of the two vectors is equal to the norm of the complex number.
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66. So, the real part of the spinor is equal to the norm of the complex number.
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68. The complex conjugate of a complex number is also a complex number. The norm of the complex number is equal to the norm of the complex conjugate of the complex number.
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70. In the 3D Euclidean space, vectors are self dual. So, the sum of two vectors is a self dual vector. The complex conjugate of the complex number is also a complex number.
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72. So, the real part of the spinor is equal to the norm of the complex number.
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74. For a complex number, the norm is equal to the real part of the spinor.
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76. For a complex number, the norm is equal to the imaginary part of the spinor.
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78. In this 3D Euclidean space, the norm of the complex number is equal to the real part of the spinor and the imaginary part of the spinor.
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80. So, the norm of the spinor is equal to the real part of the complex number.
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82. So, the norm of the complex conjugate of the complex number is equal to the imaginary part of the spinor.
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84. In this 3D Euclidean space, the norm of the complex conjugate of the complex number is equal to the imaginary part of the complex number.
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86. The norm of the complex conjugate of the complex number is equal to the imaginary part of the spinor.
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88. The norm of the complex conjugate of the complex number is equal to the real part of the spinor.
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90. So, the norm of the complex conjugate of the complex number is equal to the real part of the spinor.
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92. The real part of the complex number is equal to the norm of the complex conjugate of the complex number.
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94. The real part of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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96. So, the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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98. The real part of the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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100. So, the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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102. The real part of the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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104. So, the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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106. The real part of the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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108. The real part of the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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110. The real part of the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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112. So, the norm of the complex conjugate of the complex number is equal to the real part of the norm of the complex conjugate of the complex number.
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114. The complex conjugate of the complex number is equal to the complex conjugate of the norm of the complex conjugate of the complex number.
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116. The complex conjugate of the complex number is equal to the complex conjugate of the complex conjugate of the norm of the complex conjugate of the complex number.
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118. So, the complex conjugate of the complex number is equal to the complex conjugate of the complex conjugate of the complex conjugate of the complex number.
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120. The complex conjugate of the complex number is equal to the complex conjugate of the complex conjugate of the complex conjugate of the complex number.
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122. So, the complex conjugate of the complex number is equal to the complex conjugate of the