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  4. "Three Fermion Generations from Octonions"
  6. Abstract
  8. A study of three generations of fundamental fermions from the so-called "octonions",
  9. which is the largest non-associative algebra. The octonion algebra is an associative
  10. algebra with the multiplication rule o(x)o(y)=o(x+y) where x,y,x+y are non-negative
  11. integers. The octonion algebra, is based on the following identity: o(x)=cos(x)+sin(x)
  12. which is the definition of a 3-dimensional rotation in a plane, x=a+b+c. Octonion
  13. algebra was introduced by J. H. Conway and N. J. A. Sloane in 1963. This algebra
  14. has a structure similar to the quaternion algebra, but the octonions are associative
  15. algebras. The fundamental fermions of the Standard Model, quarks, leptons and
  16. the Higgs boson, are related to elements in this algebra. The minimal extension of
  17. the Standard Model which contains these fundamental fermions is based on the same
  18. algebra, but with higher dimensional transformations. From this point of view, the
  19. octonions are the universal description of all fundamental interactions. This paper
  20. studies the nature of fermions in the standard model and then explores the nature
  21. of the three generations of fermions using the octonions. This is the first time
  22. in which the octonions have been used to study these fundamental fermions. I am
  23. looking for inspiration and ideas from this huge algebra to make sense of all the
  24. information of the Standard Model.
  26. 1.
  27. Introduction
  29. 1.1. The Standard Model
  31. The Standard Model is a theory of the basic interactions of elementary particles
  32. and their symmetries, which explain the existing data at energies which are accessible
  33. to the accelerator experiments. The Standard Model is an excellent theory. However,
  34. one of its elements is the Higgs boson, which was introduced in 1964 by Peter Higgs.
  35. The Standard Model is constructed from the algebras and symmetries of quantum mechanics
  36. and general relativity. This means that the Higgs field plays a central role in
  37. the theory. There are three different fields that are fundamental in the Standard
  38. Model, quarks, leptons and the Higgs boson. Quarks and leptons are elementary and
  39. interact directly through the gauge interactions, which means that they only interact
  40. through the exchange of bosons, namely, gluons, photons, Z boson, and W boson. The
  41. electroweak interactions are described by the Higgs mechanism. The bosons are force
  42. carrying particles, which arise from a Higgs field. The Standard Model has the same
  43. symmetries as other theories of fundamental interactions, such as QED, QCD, or the
  44. Standard Model Extension (SME) [1,2,3].
  46. The standard model describes the weak, electromagnetic, and strong forces of nature.
  47. It is very complex and many different quantities are involved. Each fundamental
  48. interaction in the Standard Model is described by one or more gauge bosons. For
  49. example, the electric charge of a particle is described by the photon. The weak
  50. interaction is described by the W and Z gauge bosons, and the strong interaction
  51. is described by gluons. Each of these gauge bosons is associated with one of the
  52. fundamental interactions. For example, the photon is associated with the electromagnetic
  53. force, and the Z boson is associated with the weak force.
  55. 1.2. The three generations of fundamental fermions
  57. We have three different fundamental forces, electromagnetism, the weak force, and
  58. gravity. Each of these fundamental forces is associated with a different gauge boson,
  59. the photon, W and Z bosons, and the graviton, respectively. We have electromagnetic
  60. and weak forces that are transmitted by photons, and we have gravity that is transmitted
  61. by the graviton. The other important feature of the Standard Model is that it is
  62. based on three generations of fundamental fermions. The weak force can transmit
  63. electromagnetic and weak forces. Therefore, there should be two different fermions
  64. transmitted by the weak force, the u and d quarks. The third generation of fundamental
  65. fermions are the electrons and the muons, which are part of the third generation
  66. of fundamental fermions. They are leptons, which means that they do not transmit
  67. force, but interact directly through the exchange of bosons. The model predicts that
  68. there are three generations of fermions and that these fundamental fermions have
  69. quantized values of electric charge. The charge of a fermion is 1/3 of the charge
  70. of the electron.
  72. The model predicts that each generation of fundamental fermions is associated
  73. with an irreducible representation of SU(2) that is coupled with three generations
  74. of fundamental fermions to three generations of fundamental fermions, each fermion
  75. generation being associated with one such irreducible representation. This also
  76. means that there are three different representations of SU(2), which are a,b and
  77. c. A, b and c are not necessarily irreducible representations of SU(2), but are
  78. irreducible representations of SU(3) and correspond to the fundamental fermions of
  79. the Standard Model, quarks, leptons and the Higgs boson. The three fundamental fermions
  80. are: up, charm, and top quarks. Each of these three generations has a partner with
  81. the opposite sign of electric charge. This is why there are three generations of
  82. fundamental fermions. They correspond to the a, b and c irreducible representations
  83. of SU(3). For example, the electron corresponds to a representation of SU(3) and
  84. is associated with the first generation of fundamental fermions.
  86. 1.3. Octonions
  88. I will present an algebraic approach to the three generations of fundamental fermions
  89. using the non-associative algebra, the octonions. This will give a basis for a
  90. description of the Standard Model in the context of an algebraic theory, rather
  91. than a set of physical facts. I will first present the octonions and then study
  92. the nature of the three generations of fundamental fermions in the context of
  93. this huge algebra. The octonions are used in science for a wide range of problems,
  94. from quantum mechanics to particle physics and gravitational physics. The octonions
  95. are the algebra which give a common platform for these three generations of fundamental
  96. fermions. They are used in this theory for a unification of these fundamental interactions.
  97. They are part of the universal description of all fundamental interactions.
  99. There are many papers and books on octonions [4-9], but this paper will study
  100. octonions using a particular approach that is based on the study of the nature
  101. of three fundamental fermions. Octonions are the largest non-associative algebra.
  102. This algebra is a member of the exceptional series of algebras.
  104. 2.
  105. The Octonions
  107. 2.1. The non-associative algebra
  109. The octonions are the largest non-associative algebra. This algebra is also called
  110. the alternative algebra, as it is an alternative algebra. An alternative algebra
  111. is an algebra for which the associative identity does not hold. The associative
  112. identity is written A(A+A) = A+A(A) where A,A+A are elements of the algebra. An
  113. alternative algebra has the structure of the multiplication table and a basis of
  114. elements where the multiplication table satisfies the alternative identity. The
  115. alternative algebras are given by the following identity:
  117. o(x)=cos(x)+sin(x)
  119. where o(x) is the octonion number, x=a+b+c, a,b,c are non-negative integers,
  120. a≥0, b≥0, c≥0, a+b+c=0, and where A,A+A,A+A+A are arbitrary octonion elements
  121. which are written in the form:
  123. A = a1A1+a2A2+a3A3+a4A4,
  124. A+A = b1B1+b2B2+b3B3+b4B4,
  125. A+A+A = c1C1+c2C2+c3C3+c4C4
  127. where A1,A2,A3,A4,B1,B2,B3,B4,C1,C2,C3,C4 are octonion numbers. A1,A2,A3,A4,
  128. B1,B2,B3,B4,C1,
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