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The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors 被引量:1
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作者 Andrew Walcott Beckwith 《Journal of High Energy Physics, Gravitation and Cosmology》 2017年第4期727-753,共27页
The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “... The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial Octonionic geometry embedded in a larger, nonlinear “pilot model” (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a “big crunch” is picked as the embedding structure, so as to avoid the “instants” of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards “Schrodinger equation” states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to;making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry. 展开更多
关键词 octonionic Geometry CYCLIC CONFORMAL COSMOLOGY (Penrose) Modified HUP
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Pre-Octonionic to Octonionic Gravity and Could Kinetic Energy Be Larger than Potential Energy before Inflation? 被引量:1
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作者 Andrew Walcott Beckwith 《Journal of High Energy Physics, Gravitation and Cosmology》 2017年第4期693-707,共15页
We look at early universe space-time which is characterized by a transition from Pre-Planckian to Planckian space-time. In doing so we also invoke the geometry of Octonionic non-commutative structure and when it break... We look at early universe space-time which is characterized by a transition from Pre-Planckian to Planckian space-time. In doing so we also invoke the geometry of Octonionic non-commutative structure and when it breaks down. Doing so is also equivalent to a speculation given earlier by the author as to the kinetic energy of Pre-Planckian space-time being significantly larger than the Potential energy, which is the opposite of what happens after the onset of Inflation, with the assumption as to how this is justified given in a (Pre- Planckian) Hubble Parameter set as of Equation (16), and we close with a comparison of this proposal with string cosmology, as represented in the 2nd reference in this paper. 展开更多
关键词 octonionic Geometry Cyclic CONFORMAL COSMOLOGY (Penrose) Modified HUP
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Projection of 5 Dimensions into Four Is When the Octonionic Structure Kicks in as an Emergent Gravity Phenomenon
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作者 Andrew Walcott Beckwith 《Journal of High Energy Physics, Gravitation and Cosmology》 2018年第4期731-742,共12页
The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “... The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial octonionic geometry embedded in a larger, nonlinear (semi-classical) embedding structure. We detail some of what the quantum HUP is, in terms of deterministic 5-dimensional geometry and show that the projection of 5 dimensions into four is when the octonionic structure kicks in as an emergent gravity phenomenon. The example of such is to consider what would happen if there was an aftermath to a presumed initial causal discontinuous structure, after math being the generation of millions of Planck mass black holes, which would in themselves generate emergent gravity. 展开更多
关键词 octonionic Geometry Modified HUP EMBEDDING of Quantum HUP in DETERMINISTIC STRUCTURE
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AN IMPROVEMENT OF THE OCTONIONIC TAYLOR TYPE THEOREM 被引量:3
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作者 廖建全 李兴民 《Acta Mathematica Scientia》 SCIE CSCD 2011年第2期561-566,共6页
We prove that the octonionic polynomials Vl 1k……lk. are independent of the associative orders k. This improves the oetonionic Taylor type theorem.
关键词 OCTONIONS associative order Taylor type theorem
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Dirac and Maxwell Systems in Split Octonions
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作者 Merab Gogberashvili Alexandre Gurchumelia 《Journal of Applied Mathematics and Physics》 2023年第7期1977-1995,共19页
The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear inva... The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions. 展开更多
关键词 Split Octonions TRIALITY SO(4 4) Dirac and Maxwell Equations
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Investigating Initial Conditions of the WdW Equation in Flat Space in a Transition from the Pre-Planckian Physics Era to the Electroweak Regime of Space-Time
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作者 Andrew W. Beckwith 《Journal of Modern Physics》 2012年第9期1285-1288,共4页
This document is due to reviewing an article by Maydanyuk and Olkhovsky, of a Nova Science conpendium as of “The big bang, theory assumptions and Problems”, as of 2012, which uses the Wheeler De Witt equation as an ... This document is due to reviewing an article by Maydanyuk and Olkhovsky, of a Nova Science conpendium as of “The big bang, theory assumptions and Problems”, as of 2012, which uses the Wheeler De Witt equation as an evolution equation assuming a closed universe. Having the value of k, not as the closed universe, but nearly zero of a nearly flat universe, which leads to serious problems of interpretation of what initial conditions are. These problems of interpretations of initial conditions tie in with difficulties in using QM as an initial driver of inflation. And argue in favor of using a different procedure as far as forming a wave function of the universe initially. The author wishes to thank Abhay Ashtekar for his well thought out criticism but asserts that limitations in space-time geometry largely due to when is formed from semi classical reasoning, i.e. Maxwell’s equation involving a close boundary value regime between Octonionic geometry and flat space non Octonionic geometry is a datum which Abhay Ashekhar may wish to consider in his quantum bounce model and in loop quantum gravity in the future. 展开更多
关键词 Wheeler De Witt EQUATION Planck’s Constant WAVEFUNCTION of the UNIVERSE octonionic Geometry Quantum Mechanics
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Is Nature Fundamentally Continuous or Discrete, and How Can These Two Different But Very Useful Conceptions Be Fully Reconciled?
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作者 Andrew Walcott Beckwith 《Journal of High Energy Physics, Gravitation and Cosmology》 2018年第4期796-813,共18页
Our contention is that reality is actually analog, but at a critical limit, when the Octonian gravity condition kicks in, for a time it is made to appear discrete. This is due to an initial phase transition just at th... Our contention is that reality is actually analog, but at a critical limit, when the Octonian gravity condition kicks in, for a time it is made to appear discrete. This is due to an initial phase transition just at the start of the big bang. Our second consideration is that symmetry breaking models, i.e. the Higgs boson, are in themselves not appropriate or necessary for the formation of particles with mass just before Octonionic gravity which could arise in pre-Planckian physics models without a potential. Finally, the necessity of potentials for pre-Octonionic gravity physics can be circumvented via judicious use of Scherrer k essence physics. 展开更多
关键词 CONTINUOUS and DISCRETE SPACE-TIME octonionic Geometry ANALOG Physics
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Looking at Graviton Properties, as Either Classical or QM, in Nature, via Alicki-Van Ryn Experimental Realization
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作者 Andrew Beckwith 《Open Journal of Microphysics》 2012年第4期49-52,共4页
Recently, the author read the Alicki-Van Ryn test as to behavior of photons in a test of violations of classicality. The same thing is propoosed via use of a spin two graviton, using typical spin 2 matrices. While the... Recently, the author read the Alicki-Van Ryn test as to behavior of photons in a test of violations of classicality. The same thing is propoosed via use of a spin two graviton, using typical spin 2 matrices. While the technology currently does not exist to perform such an analysis yet, the same sort of thought experiment is proposed in a way to allow for a first principle test of the either classical or quantum foundations of gravity. The reason for the present manuscript topic is due to a specific argument presented in a prior document as to how h is formed from semiclassical reasoning. We referred to a procedure as to how to use Maxwell’s equations involving a closed boundary regime, in the boundary re- gime between Octonionic Geometry and quantum flat space. Conceivably, a similar argument could be made forgravi- tons, pending further investigations. Also the anlysis of if gravitons are constructed by a similar semiclassical argument is pending if gravitons as by the Alicki-Van Ryn test result in semiclassical and matrix observable eigenvalue behavior. This paper also indirectly raises the question of if Baysian statistics would be the optimal way to differentiate between and matrix observable eigenvalue behavior for reasons brought up in the conclusion. 展开更多
关键词 Planck’s CONSTANT octonionic Geometry Quantum MECHANICS Alicki-Van Ryn Test
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TAYLOR SERIES AND ORTHOGONALITY OF THE OCTONION ANALYTIC FUNCTIONS 被引量:12
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作者 李兴民 彭立中 《Acta Mathematica Scientia》 SCIE CSCD 2001年第3期323-330,共8页
The Taylor series of the Octonion analytic function is given. And the orthogonal formula for the Octonion analytic functions is also obtained.
关键词 OCTONION O-analytic function Taylor series ORTHOGONALITY
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THE ALL-ASSOCIATIVITY OF OCTONIONS AND ITS APPLICATIONS 被引量:2
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作者 Jianquan Liao Jinxun Wang Xingmin Li 《Analysis in Theory and Applications》 2010年第4期326-338,共13页
Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are ... Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained. 展开更多
关键词 OCTONIONS ASSOCIATIVITY PERMUTATION DETERMINANT
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Inhomogeneous Cauchy-Riemann equation in Octonion space
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作者 GONG Ding-dong 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2015年第4期447-452,共6页
The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the ... The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed. 展开更多
关键词 OCTONION integral representation of differentiable functions inhoraogeneous Cauchy-RiemannEquation Cousin problem.
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The Analyticity for the Product of Analytic Functions on Octonions and Its Applications
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作者 Jianquan Liao Jinxun Wang 《Advances in Pure Mathematics》 2017年第12期692-705,共14页
Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove?fλ that is a left Oc-analytic function for any const... Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove?fλ that is a left Oc-analytic function for any constants λ∈Oc if and only if is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with Cauchy-Kowalewski product are also considered. 展开更多
关键词 OCTONIONS Oc-Analytic Functions Stein-Weiss Conjugate Harmonic System Cauchy-Kowalewski PRODUCT
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Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE
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作者 Sergey Victor Ludkovsky 《Advances in Pure Mathematics》 2012年第2期63-103,共41页
Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. A... Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered. 展开更多
关键词 Laplace Transform Quaternion Skew Field OCTONION ALGEBRA Cayley-Dickson ALGEBRA Partial Differential Equation NON-COMMUTATIVE Integration
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Powers of Octonions
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作者 W. E. Ahmed 《Applied Mathematics》 2021年第2期75-84,共10页
As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to... As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to derive suitable formulas from these methods. In this paper, we present a novel way to compute the powers of an octonion number using formulas derived from the binomial expansion. 展开更多
关键词 OCTONION Matrix Algebra
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The Cyclic Universes Model Based on the Split Division Algebras
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作者 Ding-Yu Chung 《Journal of Modern Physics》 2018年第13期2257-2273,共17页
The proposed cyclic universes model based on the split division algebras accounts for the inflation, the Big Bang, gravity, dark energy, dark matter, the standard model, and the masses of all elementary particles. The... The proposed cyclic universes model based on the split division algebras accounts for the inflation, the Big Bang, gravity, dark energy, dark matter, the standard model, and the masses of all elementary particles. The split algebras (complex quaternion and complex octonion) as the Furey model generate the fixed spacetime dimension number for the observable universe with the fixed 4-dimensional spacetime (4D) standard model particles and the oscillating spacetime dimension number for the oscillating universes (hidden or dark energy) with the oscillation between 11D and 11D through 10D and between 10D and 10D through 4D. 11D has the lowest rest mass, the highest speed of light, and the highest vacuum energy, while 4D has the highest rest mass, the lowest (observed) speed of light, and zero vacuum energy. In the cyclic universes model, the universes start with the positive-energy and the negative-energy 11D membrane-antimembrane dual universes from the zero-energy inter-universal void, and are followed by the transformation of the 11D membrane-antimembrane dual universes into the 10D string-antistring dual universes and the external dual gravities as in the Randall-Sundrum model, resulting in the four equal and separate universes consisting of the positive-energy 10D universe, the positive-energy external gravity, the negative-energy 10D universe, and the negative-energy external gravity. Under the fixed spacetime dimension number, the positive-energy 10D universe is transformed into 4D standard model particles through the inflation and the Big Bang. Dark matter is the right-handed neutrino, exactly five times of baryonic matter in total mass in the universe. Under the oscillating spacetime dimension number, the other three universes oscillate between 10D and 10D through 4D, resulting in the hidden universes when D > 4 and dark energy (the maximum dark energy = 3/4 = 75%) when D = 4. Eventually, all four universes return to the 10D universes. 展开更多
关键词 CYCLIC UNIVERSES MODEL Division Algebras Furey COMPLEX Quaternion COMPLEX OCTONION DARK Energy DARK Matter Standard MODEL Gravity
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The Paley-Wiener theorem in the non-commutative and non-associative octonions 被引量:3
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作者 LI XingMin PENG LiZhong QIAN Tao 《Science China Mathematics》 SCIE 2009年第1期129-141,共13页
The Paley-Wiener theorem in the non-commutative and non-associative octonion analytic function space is proved.
关键词 octonionic exponential function octonionic Taylor expansion Fourier transform Paley-Weiner theorem 42B35 30G35 17A35
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Octonion Analysis of Several Variables 被引量:2
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作者 Haiyan Wang Guangbin Ren 《Communications in Mathematics and Statistics》 SCIE 2014年第2期163-185,共23页
The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional o... The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional octonion analysis is initiated in this article,which extends the theory of several complex variables,such as the Bochner–Martinelli formula,the theory of non-homogeneous Cauchy–Riemann equations,and the Hartogs principle,to the non-commutative and non-associative realm. 展开更多
关键词 Several octonionic variables Bochner–Martinelli formula Hartogs theorem Non-homogenous Cauchy–Riemann equations
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Three-line Theorem on the Octonions 被引量:1
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作者 XingMinLI LiZhongPENG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2004年第3期483-490,共8页
The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sj?lin from the associative Clifford algebra to non-associative octonion algebra.
关键词 OCTONION O-analytic functions 3-line theorem
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Split Octonion Reformulation for Electromagnetic Chiral Media of Massive Dyons
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作者 B.C.Chanyal 《Communications in Theoretical Physics》 SCIE CAS CSCD 2017年第12期701-710,共10页
In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2... In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2 × 2 Zorn's vector matrix realization of split-octonion and its dual Euclidean spaces, we represent the unified structure of split octonionic electric and magnetic induction vectors for chiral media. As such, in present paper, we describe the chiral parameter and pairing constants in terms of split octonionic matrix representation of Drude-Born-Fedorov constitutive relations. We have expressed a split octonionic electromagnetic field vector for chiral media, which exhibits the unified field structure of electric and magnetic chiral fields of dyons. The beauty of split octonionic representation of Zorn vector matrix realization is that, the every scalar and vector components have its own meaning in the generalized chiral electromagnetism of dyons. Correspondingly, we obtained the alternative form of generalized Proca–Maxwell's equations of massive dyons in chiral media. Furthermore, the continuity equations, Poynting theorem and wave propagation for generalized electromagnetic fields of chiral media of massive dyons are established by split octonionic form of Zorn vector matrix algebra. 展开更多
关键词 OCTONIONS Zorn vector matrix chiral media constitutive relations Poynting theorem
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Associative Cones and Integrable Systems
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作者 Chuu-Lian TERNG 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2006年第2期153-168,共16页
Abstract We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S^6. It is known that a cone over a surface M in S^6 ... Abstract We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S^6. It is known that a cone over a surface M in S^6 is an associative submanifold of R^7 if and only if M is almost complex in S^6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S^6 are the equation for primitive maps associated to the 6-symmetric space G2/T^2, and use this to explain some of the known results. Moreover, the equation for S^1-symmetric almost complex curves in S^6 is the periodic Toda lattice, and a discussion of periodic solutions is given. 展开更多
关键词 OCTONIONS Associative cone Almost complex curve Primitive map
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