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__Sign Up__- PHYSICS BY GPT-J
- RESULT
- "Three Fermion Generations from Octonions"
- Abstract
- A study of three generations of fundamental fermions from the so-called "octonions",
- which is the largest non-associative algebra. The octonion algebra is an associative
- algebra with the multiplication rule o(x)o(y)=o(x+y) where x,y,x+y are non-negative
- integers. The octonion algebra, is based on the following identity: o(x)=cos(x)+sin(x)
- which is the definition of a 3-dimensional rotation in a plane, x=a+b+c. Octonion
- algebra was introduced by J. H. Conway and N. J. A. Sloane in 1963. This algebra
- has a structure similar to the quaternion algebra, but the octonions are associative
- algebras. The fundamental fermions of the Standard Model, quarks, leptons and
- the Higgs boson, are related to elements in this algebra. The minimal extension of
- the Standard Model which contains these fundamental fermions is based on the same
- algebra, but with higher dimensional transformations. From this point of view, the
- octonions are the universal description of all fundamental interactions. This paper
- studies the nature of fermions in the standard model and then explores the nature
- of the three generations of fermions using the octonions. This is the first time
- in which the octonions have been used to study these fundamental fermions. I am
- looking for inspiration and ideas from this huge algebra to make sense of all the
- information of the Standard Model.
- 1.
- Introduction
- 1.1. The Standard Model
- The Standard Model is a theory of the basic interactions of elementary particles
- and their symmetries, which explain the existing data at energies which are accessible
- to the accelerator experiments. The Standard Model is an excellent theory. However,
- one of its elements is the Higgs boson, which was introduced in 1964 by Peter Higgs.
- The Standard Model is constructed from the algebras and symmetries of quantum mechanics
- and general relativity. This means that the Higgs field plays a central role in
- the theory. There are three different fields that are fundamental in the Standard
- Model, quarks, leptons and the Higgs boson. Quarks and leptons are elementary and
- interact directly through the gauge interactions, which means that they only interact
- through the exchange of bosons, namely, gluons, photons, Z boson, and W boson. The
- electroweak interactions are described by the Higgs mechanism. The bosons are force
- carrying particles, which arise from a Higgs field. The Standard Model has the same
- symmetries as other theories of fundamental interactions, such as QED, QCD, or the
- Standard Model Extension (SME) [1,2,3].
- The standard model describes the weak, electromagnetic, and strong forces of nature.
- It is very complex and many different quantities are involved. Each fundamental
- interaction in the Standard Model is described by one or more gauge bosons. For
- example, the electric charge of a particle is described by the photon. The weak
- interaction is described by the W and Z gauge bosons, and the strong interaction
- is described by gluons. Each of these gauge bosons is associated with one of the
- fundamental interactions. For example, the photon is associated with the electromagnetic
- force, and the Z boson is associated with the weak force.
- 1.2. The three generations of fundamental fermions
- We have three different fundamental forces, electromagnetism, the weak force, and
- gravity. Each of these fundamental forces is associated with a different gauge boson,
- the photon, W and Z bosons, and the graviton, respectively. We have electromagnetic
- and weak forces that are transmitted by photons, and we have gravity that is transmitted
- by the graviton. The other important feature of the Standard Model is that it is
- based on three generations of fundamental fermions. The weak force can transmit
- electromagnetic and weak forces. Therefore, there should be two different fermions
- transmitted by the weak force, the u and d quarks. The third generation of fundamental
- fermions are the electrons and the muons, which are part of the third generation
- of fundamental fermions. They are leptons, which means that they do not transmit
- force, but interact directly through the exchange of bosons. The model predicts that
- there are three generations of fermions and that these fundamental fermions have
- quantized values of electric charge. The charge of a fermion is 1/3 of the charge
- of the electron.
- The model predicts that each generation of fundamental fermions is associated
- with an irreducible representation of SU(2) that is coupled with three generations
- of fundamental fermions to three generations of fundamental fermions, each fermion
- generation being associated with one such irreducible representation. This also
- means that there are three different representations of SU(2), which are a,b and
- c. A, b and c are not necessarily irreducible representations of SU(2), but are
- irreducible representations of SU(3) and correspond to the fundamental fermions of
- the Standard Model, quarks, leptons and the Higgs boson. The three fundamental fermions
- are: up, charm, and top quarks. Each of these three generations has a partner with
- the opposite sign of electric charge. This is why there are three generations of
- fundamental fermions. They correspond to the a, b and c irreducible representations
- of SU(3). For example, the electron corresponds to a representation of SU(3) and
- is associated with the first generation of fundamental fermions.
- 1.3. Octonions
- I will present an algebraic approach to the three generations of fundamental fermions
- using the non-associative algebra, the octonions. This will give a basis for a
- description of the Standard Model in the context of an algebraic theory, rather
- than a set of physical facts. I will first present the octonions and then study
- the nature of the three generations of fundamental fermions in the context of
- this huge algebra. The octonions are used in science for a wide range of problems,
- from quantum mechanics to particle physics and gravitational physics. The octonions
- are the algebra which give a common platform for these three generations of fundamental
- fermions. They are used in this theory for a unification of these fundamental interactions.
- They are part of the universal description of all fundamental interactions.
- There are many papers and books on octonions [4-9], but this paper will study
- octonions using a particular approach that is based on the study of the nature
- of three fundamental fermions. Octonions are the largest non-associative algebra.
- This algebra is a member of the exceptional series of algebras.
- 2.
- The Octonions
- 2.1. The non-associative algebra
- The octonions are the largest non-associative algebra. This algebra is also called
- the alternative algebra, as it is an alternative algebra. An alternative algebra
- is an algebra for which the associative identity does not hold. The associative
- identity is written A(A+A) = A+A(A) where A,A+A are elements of the algebra. An
- alternative algebra has the structure of the multiplication table and a basis of
- elements where the multiplication table satisfies the alternative identity. The
- alternative algebras are given by the following identity:
- o(x)=cos(x)+sin(x)
- where o(x) is the octonion number, x=a+b+c, a,b,c are non-negative integers,
- a≥0, b≥0, c≥0, a+b+c=0, and where A,A+A,A+A+A are arbitrary octonion elements
- which are written in the form:
- A = a1A1+a2A2+a3A3+a4A4,
- A+A = b1B1+b2B2+b3B3+b4B4,
- A+A+A = c1C1+c2C2+c3C3+c4C4
- where A1,A2,A3,A4,B1,B2,B3,B4,C1,C2,C3,C4 are octonion numbers. A1,A2,A3,A4,
- B1,B2,B3,B4,C1,

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