Author:
(1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University. Table of Links Abstract and Introduction Space-time quanta and Becken Universal bound Shape of space-time quanta Symmetry of space-time quanta Space-time quanta and Spectral mass gap Phenomenological implications Conclusion, Acknowledgments, and References III. SHAPE OF SPACE-TIME QUANTA A natural question arises, what is the geometric shape of the space-time quanta? To answer this question, we need first to apply the energy-momentum relation that is satisfied by Snyder’s algebra [38]: where m is the mass of the physical object. Substituting Eq. (3) and Eq. (4) into Eq. (8), we get: This means that the mass of the space-time quanta is completely determined by its unique length and it is strictly a “positive real value”. It is worth mentioning that generalized forms of Snyder algebra with de Sitter background imply modification of dispersion relation [39] that may introduce corrections to the mass of space-time quanta obtained in Eq.(9). The flatness of quantum Lorentzian space-time suggests that the space-time quanta be described by 4-polytope geometry [40]. We use elementary particle physics as a guide to building the quanta of space-time. The standard model of particle physics has 25 fundamental particles that include 12 fermions (quarks and leptons), 4 gauge bosons that carry electromagnetic force and weak nuclear force, 8 gluons that carry strong nuclear force, and 1 Higgs scalar field. The quanta of space-time must carry a signature of information from all these fundamental particles that constitute the fundamental structure of nature. The space-time quanta must be “self-dual” as well to preserve its uniqueness. Therefore, we look for a highly symmetric 4-dimensional geometric object that is self-dual and identified by only unique length which could represent the fundamental particles of nature on its vertices. This could be a uniform 4-polytope which is a 4-dimensional object with flat sides/faces and is vertex-transitive symmetric which means an isometric map of any vertex onto any other. A regular 4-polytope has the highest degree of symmetry as its faces/cells are regular polytopes and transitive on the symmetries of the polytope. More symmetries are found in regular 4-polytopes that define the convex region as a subset that intersects every line into a single line segment. There are two self-dual convex regular polytopes that are 5-cell, which has 5 vertices, and 24-cell which has 24 vertices. We choose the 24-cell because it has enough vertices to assign with elementary particles. A representation in a plane of a regular 24-cell is given in Fig. (1): The 24-cell geometric properties can be summarized as follows: • Its boundary in 3-dimensions forms 24 octahedral cells with six meeting at each vertex, and three at each edge • It has 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube • self-dual [41] • identified by one length where edge length equals the distance between the center and vertex (radius) The 24-cell exists in 4- dimensional Euclidean geometry, but Minkowski space-time is 4- dimensional “pseudo-Euclidean”. This can be simply resolved if time is represented as an imaginary spatial dimension which is well-established in quantum field theory [42] and in Euclidean quantum gravity [43]. To put it another way, the space-time quanta is represented by a 24-cell with considering time as an imaginary spatial dimension. The covariance principle requires that the space-time quanta should represent the elementary particles of the standard model [44]. In the next section, we show how to do this representation. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University. Author: Author: (1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University. Table of Links Abstract and Introduction Abstract and Introduction Space-time quanta and Becken Universal bound Space-time quanta and Becken Universal bound Shape of space-time quanta Shape of space-time quanta Symmetry of space-time quanta Symmetry of space-time quanta Space-time quanta and Spectral mass gap Space-time quanta and Spectral mass gap Phenomenological implications Phenomenological implications Conclusion, Acknowledgments, and References Conclusion, Acknowledgments, and References III. SHAPE OF SPACE-TIME QUANTA A natural question arises, what is the geometric shape of the space-time quanta? To answer this question, we need first to apply the energy-momentum relation that is satisfied by Snyder’s algebra [38]: where m is the mass of the physical object. Substituting Eq. (3) and Eq. (4) into Eq. (8), we get: This means that the mass of the space-time quanta is completely determined by its unique length and it is strictly a “positive real value”. It is worth mentioning that generalized forms of Snyder algebra with de Sitter background imply modification of dispersion relation [39] that may introduce corrections to the mass of space-time quanta obtained in Eq.(9). The flatness of quantum Lorentzian space-time suggests that the space-time quanta be described by 4-polytope geometry [40]. We use elementary particle physics as a guide to building the quanta of space-time. The standard model of particle physics has 25 fundamental particles that include 12 fermions (quarks and leptons), 4 gauge bosons that carry electromagnetic force and weak nuclear force, 8 gluons that carry strong nuclear force, and 1 Higgs scalar field. The quanta of space-time must carry a signature of information from all these fundamental particles that constitute the fundamental structure of nature. The space-time quanta must be “self-dual” as well to preserve its uniqueness. Therefore, we look for a highly symmetric 4-dimensional geometric object that is self-dual and identified by only unique length which could represent the fundamental particles of nature on its vertices. This could be a uniform 4-polytope which is a 4-dimensional object with flat sides/faces and is vertex-transitive symmetric which means an isometric map of any vertex onto any other. A regular 4-polytope has the highest degree of symmetry as its faces/cells are regular polytopes and transitive on the symmetries of the polytope. More symmetries are found in regular 4-polytopes that define the convex region as a subset that intersects every line into a single line segment. There are two self-dual convex regular polytopes that are 5-cell, which has 5 vertices, and 24-cell which has 24 vertices. We choose the 24-cell because it has enough vertices to assign with elementary particles. A representation in a plane of a regular 24-cell is given in Fig. (1): The 24-cell geometric properties can be summarized as follows: • Its boundary in 3-dimensions forms 24 octahedral cells with six meeting at each vertex, and three at each edge • It has 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube • self-dual [41] • identified by one length where edge length equals the distance between the center and vertex (radius) The 24-cell exists in 4- dimensional Euclidean geometry, but Minkowski space-time is 4- dimensional “pseudo-Euclidean”. This can be simply resolved if time is represented as an imaginary spatial dimension which is well-established in quantum field theory [42] and in Euclidean quantum gravity [43]. To put it another way, the space-time quanta is represented by a 24-cell with considering time as an imaginary spatial dimension. The covariance principle requires that the space-time quanta should represent the elementary particles of the standard model [44]. In the next section, we show how to do this representation. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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What Is the Shape of Space-Time Quanta?

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