Space-Time Algebra for the Generalization of Gravitational Field
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Arxiv:2009.05574V4 [Hep-Th] 9 Nov 2020 Predict a New Massless Spin One Boson [The ‘Lorentz’ Boson] Which Should Be Looked for in Experiments
Trace dynamics and division algebras: towards quantum gravity and unification Tejinder P. Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] Accepted for publication in Zeitschrift fur Naturforschung A on October 4, 2020 v4. Submitted to arXiv.org [hep-th] on November 9, 2020 ABSTRACT We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang-Mills fields, and fermions. Dynamical variables are described by odd- grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space- time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electro-colour symmetry, as shown earlier by Furey. We arXiv:2009.05574v4 [hep-th] 9 Nov 2020 predict a new massless spin one boson [the `Lorentz' boson] which should be looked for in experiments. -
Notices of the American Mathematical Society
ISSN 0002-9920 of the American Mathematical Society February 2006 Volume 53, Number 2 Math Circles and Olympiads MSRI Asks: Is the U.S. Coming of Age? page 200 A System of Axioms of Set Theory for the Rationalists page 206 Durham Meeting page 299 San Francisco Meeting page 302 ICM Madrid 2006 (see page 213) > To mak• an antmat•d tub• plot Animated Tube Plot 1 Type an expression in one or :;)~~~G~~~t;~~i~~~~~~~~~~~~~:rtwo ' 2 Wrth the insertion point in the 3 Open the Plot Properties dialog the same variables Tl'le next animation shows • knot Plot 30 Animated + Tube Scientific Word ... version 5.5 Scientific Word"' offers the same features as Scientific WorkPlace, without the computer algebra system. Editors INTERNATIONAL Morris Weisfeld Editor-in-Chief Enrico Arbarello MATHEMATICS Joseph Bernstein Enrico Bombieri Richard E. Borcherds Alexei Borodin RESEARCH PAPERS Jean Bourgain Marc Burger James W. Cogdell http://www.hindawi.com/journals/imrp/ Tobias Colding Corrado De Concini IMRP provides very fast publication of lengthy research articles of high current interest in Percy Deift all areas of mathematics. All articles are fully refereed and are judged by their contribution Robbert Dijkgraaf to the advancement of the state of the science of mathematics. Issues are published as S. K. Donaldson frequently as necessary. Each issue will contain only one article. IMRP is expected to publish 400± pages in 2006. Yakov Eliashberg Edward Frenkel Articles of at least 50 pages are welcome and all articles are refereed and judged for Emmanuel Hebey correctness, interest, originality, depth, and applicability. Submissions are made by e-mail to Dennis Hejhal [email protected]. -
Introduction to Supersymmetry(1)
Introduction to Supersymmetry(1) J.N. Tavares Dep. Matem¶aticaPura, Faculdade de Ci^encias,U. Porto, 4000 Porto TQFT Club 1Esta ¶euma vers~aoprovis¶oria,incompleta, para uso exclusivo nas sess~oesde trabalho do TQFT club CONTENTS 1 Contents 1 Supersymmetry in Quantum Mechanics 2 1.1 The Supersymmetric Oscillator . 2 1.2 Witten Index . 4 1.3 A fundamental example: The Laplacian on forms . 7 1.4 Witten's proof of Morse Inequalities . 8 2 Supergeometry and Supersymmetry 13 2.1 Field Theory. A quick review . 13 2.2 SuperEuclidean Space . 17 2.3 Reality Conditions . 18 2.4 Supersmooth functions . 18 2.5 Supermanifolds . 21 2.6 Lie Superalgebras . 21 2.7 Super Lie groups . 26 2.8 Rigid Superspace . 27 2.9 Covariant Derivatives . 30 3 APPENDIX. Cli®ord Algebras and Spin Groups 31 3.1 Cli®ord Algebras . 31 Motivation. Cli®ord maps . 31 Cli®ord Algebras . 33 Involutions in V .................................. 35 Representations . 36 3.2 Pin and Spin groups . 43 3.3 Spin Representations . 47 3.4 U(2), spinors and almost complex structures . 49 3.5 Spinc(4)...................................... 50 Chiral Operator. Self Duality . 51 2 1 Supersymmetry in Quantum Mechanics 1.1 The Supersymmetric Oscillator i As we will see later the \hermitian supercharges" Q®, in the N extended SuperPoincar¶eLie Algebra obey the anticommutation relations: i j m ij fQ®;Q¯g = 2(γ C)®¯± Pm (1.1) m where ®; ¯ are \spinor" indices, i; j 2 f1; ¢ ¢ ¢ ;Ng \internal" indices and (γ C)®¯ a bilinear form in the spinor indices ®; ¯. When specialized to 0-space dimensions ((1+0)-spacetime), then since P0 = H, relations (1.1) take the form (with a little change in notations): fQi;Qjg = 2±ij H (1.2) with N \Hermitian charges" Qi; i = 1; ¢ ¢ ¢ ;N. -
Clifford Algebra and the Interpretation of Quantum
In: J.S.R. Chisholm/A.K. Commons (Eds.), Cliord Algebras and their Applications in Mathematical Physics. Reidel, Dordrecht/Boston (1986), 321–346. CLIFFORD ALGEBRA AND THE INTERPRETATION OF QUANTUM MECHANICS David Hestenes ABSTRACT. The Dirac theory has a hidden geometric structure. This talk traces the concep- tual steps taken to uncover that structure and points out signicant implications for the interpre- tation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpreta- tion for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer con- nections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation. 1. INTRODUCTION The interpretation of quantum mechanics has been vigorously and inconclusively debated since the inception of the theory. My purpose today is to call your attention to some crucial features of quantum mechanics which have been overlooked in the debate. I claim that the Pauli and Dirac algebras have a geometric interpretation which has been implicit in quantum mechanics all along. My aim will be to make that geometric interpretation explicit and show that it has nontrivial implications for the physical interpretation of quantum mechanics. Before getting started, I would like to apologize for what may appear to be excessive self-reference in this talk. I have been pursuing the theme of this talk for 25 years, but the road has been a lonely one where I have not met anyone travelling very far in the same direction. -
A Clifford Dyadic Superfield from Bilateral Interactions of Geometric Multispin Dirac Theory
A CLIFFORD DYADIC SUPERFIELD FROM BILATERAL INTERACTIONS OF GEOMETRIC MULTISPIN DIRAC THEORY WILLIAM M. PEZZAGLIA JR. Department of Physia, Santa Clam University Santa Clam, CA 95053, U.S.A., [email protected] and ALFRED W. DIFFER Department of Phyaia, American River College Sacramento, CA 958i1, U.S.A. (Received: November 5, 1993) Abstract. Multivector quantum mechanics utilizes wavefunctions which a.re Clifford ag gregates (e.g. sum of scalar, vector, bivector). This is equivalent to multispinors con structed of Dirac matrices, with the representation independent form of the generators geometrically interpreted as the basis vectors of spacetime. Multiple generations of par ticles appear as left ideals of the algebra, coupled only by now-allowed right-side applied (dextral) operations. A generalized bilateral (two-sided operation) coupling is propoeed which includes the above mentioned dextrad field, and the spin-gauge interaction as partic ular cases. This leads to a new principle of poly-dimensional covariance, in which physical laws are invariant under the reshuffling of coordinate geometry. Such a multigeometric su perfield equation is proposed, whi~h is sourced by a bilateral current. In order to express the superfield in representation and coordinate free form, we introduce Eddington E-F double-frame numbers. Symmetric tensors can now be represented as 4D "dyads", which actually are elements of a global SD Clifford algebra.. As a restricted example, the dyadic field created by the Greider-Ross multivector current (of a Dirac electron) describes both electromagnetic and Morris-Greider gravitational interactions. Key words: spin-gauge, multivector, clifford, dyadic 1. Introduction Multi vector physics is a grand scheme in which we attempt to describe all ba sic physical structure and phenomena by a single geometrically interpretable Algebra. -
Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects
Advances in High Energy Physics Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects Guest Editors: Emil T. Akhmedov, Stephen Minter, Piero Nicolini, and Douglas Singleton Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects Advances in High Energy Physics Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects Guest Editors: Emil T. Akhmedov, Stephen Minter, Piero Nicolini, and Douglas Singleton Copyright © 2014 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board Botio Betev, Switzerland Ian Jack, UK Neil Spooner, UK Duncan L. Carlsmith, USA Filipe R. Joaquim, Portugal Luca Stanco, Italy Kingman Cheung, Taiwan Piero Nicolini, Germany EliasC.Vagenas,Kuwait Shi-Hai Dong, Mexico Seog H. Oh, USA Nikos Varelas, USA Edmond C. Dukes, USA Sandip Pakvasa, USA Kadayam S. Viswanathan, Canada Amir H. Fatollahi, Iran Anastasios Petkou, Greece Yau W. Wah, USA Frank Filthaut, The Netherlands Alexey A. Petrov, USA Moran Wang, China Joseph Formaggio, USA Frederik Scholtz, South Africa Gongnan Xie, China Chao-Qiang Geng, Taiwan George Siopsis, USA Hong-Jian He, China Terry Sloan, UK Contents Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects,EmilT.Akhmedov, -
Clifford Algebras, Spinors and Supersymmetry. Francesco Toppan
IV Escola do CBPF – Rio de Janeiro, 15-26 de julho de 2002 Algebraic Structures and the Search for the Theory Of Everything: Clifford algebras, spinors and supersymmetry. Francesco Toppan CCP - CBPF, Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil abstract These lectures notes are intended to cover a small part of the material discussed in the course “Estruturas algebricas na busca da Teoria do Todo”. The Clifford Algebras, necessary to introduce the Dirac’s equation for free spinors in any arbitrary signature space-time, are fully classified and explicitly constructed with the help of simple, but powerful, algorithms which are here presented. The notion of supersymmetry is introduced and discussed in the context of Clifford algebras. 1 Introduction The basic motivations of the course “Estruturas algebricas na busca da Teoria do Todo”consisted in familiarizing graduate students with some of the algebra- ic structures which are currently investigated by theoretical physicists in the attempt of finding a consistent and unified quantum theory of the four known interactions. Both from aesthetic and practical considerations, the classification of mathematical and algebraic structures is a preliminary and necessary require- ment. Indeed, a very ambitious, but conceivable hope for a unified theory, is that no free parameter (or, less ambitiously, just few) has to be fixed, as an external input, due to phenomenological requirement. Rather, all possible pa- rameters should be predicted by the stringent consistency requirements put on such a theory. An example of this can be immediately given. It concerns the dimensionality of the space-time. -
What Is Gravity and How Is Embedded in Mater Particles Giving Them Mass (Not the Higgs Field)? Stefan Mehedinteanu
What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)? Stefan Mehedinteanu To cite this version: Stefan Mehedinteanu. What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?. 2017. hal-01316929v6 HAL Id: hal-01316929 https://hal.archives-ouvertes.fr/hal-01316929v6 Preprint submitted on 23 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License What Is Gravity (Graviton) and How Is Embedded in Mater Particles Giving Them Mass (Not the Higgs Field)? Stefan Mehedinteanu1 1 (retired) Senior Researcher CITON-Romania; E-Mail: [email protected]; [email protected] “When Einstein was asked bout the discovery of new particles, the answer it was: firstly to clarify what is with the electron!” Abstract It will be shown how the Micro-black-holes particles produced at the horizon entry into a number of N 106162 by quantum fluctuation as virtual micro-black holes pairs like e+ e- creation, stay at the base of: the origin and evolution of Universe, the Black Holes (BH) nuclei of galaxies, of the free photons creation of near mass-less as by radiation decay that condensate later at Confinement in to the structure of gauge bosons (gluons) . -
Non-Abelian Gravitoelectromagnetism and Applications at Finite Temperature
Hindawi Advances in High Energy Physics Volume 2020, Article ID 5193692, 8 pages https://doi.org/10.1155/2020/5193692 Research Article Non-Abelian Gravitoelectromagnetism and Applications at Finite Temperature A. F. Santos ,1 J. Ramos,2 and Faqir C. Khanna3 1Instituto de Física, Universidade Federal de Mato Grosso, 78060-900, Cuiabá, Mato Grosso, Brazil 2Faculty of Science, Burman University, Lacombe, Alberta, Canada T4L 2E5 3Department of Physics and Astronomy, University of Victoria, BC, Canada V8P 5C2 Correspondence should be addressed to A. F. Santos; alesandroferreira@fisica.ufmt.br Received 22 January 2020; Revised 9 March 2020; Accepted 18 March 2020; Published 3 April 2020 Academic Editor: Michele Arzano Copyright © 2020 A. F. Santos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. Studies about a formal analogy between the gravitational and the electromagnetic fields lead to the notion of Gravitoelectromagnetism (GEM) to describe gravitation. In fact, the GEM equations correspond to the weak-field approximation of the gravitation field. Here, a non-abelian extension of the GEM theory is considered. Using the Thermo Field Dynamics (TFD) formalism to introduce temperature effects, some interesting physical phenomena are investigated. The non-abelian GEM Stefan- Boltzmann law and the Casimir effect at zero and finite temperatures for this non-abelian field are calculated. 1. Introduction between equations for the Newton and Coulomb laws and the interest has increased with the discovery of the The Standard Model (SM) is a non-abelian gauge theory with Lense-Thirring effect, where a rotating mass generates a symmetry group Uð1Þ × SUð2Þ × SUð3Þ. -
Clifford Algebras, Multipartite Systems and Gauge Theory Gravity
Mathematical Phyiscs Clifford Algebras, Multipartite Systems and Gauge Theory Gravity Marco A. S. Trindadea Eric Pintob Sergio Floquetc aColegiado de Física Departamento de Ciências Exatas e da Terra Universidade do Estado da Bahia, Salvador, BA, Brazil bInstituto de Física Universidade Federal da Bahia Salvador, BA, Brazil cColegiado de Engenharia Civil Universidade Federal do Vale do São Francisco Juazeiro, BA, Brazil. E-mail: [email protected], [email protected], [email protected] Abstract: In this paper we present a multipartite formulation of gauge theory gravity based on the formalism of space-time algebra for gravitation developed by Lasenby and Do- ran (Lasenby, A. N., Doran, C. J. L, and Gull, S.F.: Gravity, gauge theories and geometric algebra. Phil. Trans. R. Soc. Lond. A, 582, 356:487 (1998)). We associate the gauge fields with description of fermionic and bosonic states using the generalized graded tensor product. Einstein’s equations are deduced from the graded projections and an algebraic Hopf-like structure naturally emerges from formalism. A connection with the theory of the quantum information is performed through the minimal left ideals and entangled qubits are derived. In addition applications to black holes physics and standard model are outlined. arXiv:1807.09587v2 [physics.gen-ph] 16 Nov 2018 1Corresponding author. Contents 1 Introduction1 2 General formulation and Einstein field equations3 3 Qubits 9 4 Black holes background 10 5 Standard model 11 6 Conclusions 15 7 Acknowledgements 16 1 Introduction This work is dedicated to the memory of professor Waldyr Alves Rodrigues Jr., whose contributions in the field of mathematical physics were of great prominence, especially in the study of clifford algebras and their applications to physics [1]. -
Pure Spinors to Associative Triples to Zero-Divisors
Pure Spinors to Associative Triples to Zero-Divisors Frank Dodd (Tony) Smith, Jr. - 2012 Abstract: Both Clifford Algebras and Cayley-Dickson Algebras can be used to construct Physics Models. Clifford and Cayley-Dickson Algebras have in common Real Numbers, Complex Numbers, and Quaternions, but in higher dimensions Clifford and Cayley-Dickson diverge. This paper is an attempt to explore the relationship between higher-dimensional Clifford and Cayley-Dickson Algebras by comparing Projective Pure Spinors of Clifford Algebras with Zero-Divisor structures of Cayley-Dickson Algebras. 1 Pure Spinors to Associative Triples to Zero-Divisors Frank Dodd (Tony) Smith, Jr. - 2012 Robert de Marrais and Guillermo Moreno, pioneers in studying Zero-Divisors, unfortunately have passed (2011 and 2006). Pure Spinors to Associative Triples ..... page 2 Associative Triples to Loops ............... page 3 Loops to Zero-Divisors ....................... page 6 New Phenomena at C32 = T32 ........... page 9 Zero-Divisor Annihilator Geometry .... page 17 Pure Spinors to Associative Triples Pure spinors are those spinors that can be represented by simple exterior wedge products of vectors. For Cl(2n) they can be described in terms of bivectors Spin(2n) and Spin(n) based on the twistor space Spin(2n)/U(n) = Spin(n) x Spin(n). Since (1/2)((1/2)(2n)(2n-1)-n^2) = (1/2)(2n^2-n-n^2) = = (1/2)(n(n-1)) Penrose and Rindler (Spinors and Spacetime v.2) describe Cl(2n) projective pure half-spinors as Spin(n) so that the Cl(2n) full space of pure half-spinors has dimension dim(Spin(n)) -
Theory of Trigintaduonion Emanation and Origins of Α and Π
Theory of Trigintaduonion Emanation and Origins of and Stephen Winters-Hilt Meta Logos Systems Aurora, Colorado Abstract In this paper a specific form of maximal information propagation is explored, from which the origins of , , and fractal reality are revealed. Introduction The new unification approach described here gives a precise derivation for the mysterious physics constant (the fine-structure constant) from the mathematical physics formalism providing maximal information propagation, with being the maximal perturbation amount. Furthermore, the new unification provides that the structure of the space of initial ‘propagation’ (with initial propagation being referred to as ‘emanation’) has a precise derivation, with a unit-norm perturbative limit that leads to an iterative-map-like computed (a limit that is precisely related to the Feigenbaum bifurcation constant and thus fractal). The computed can also, by a maximal information propagation argument, provide a derivation for the mathematical constant . The ideal constructs of planar geometry, and related such via complex analysis, give methods for calculation of to incredibly high precision (trillions of digits), thereby providing an indirect derivation of to similar precision. Propagation in 10 dimensions (chiral, fermionic) and 26 dimensions (bosonic) is indicated [1-3], in agreement with string theory. Furthermore a preliminary result showing a relation between the Feigenbaum bifurcation constant and , consistent with the hypercomplex algebras indicated in the Emanator Theory, suggest an individual object trajectory with 36=10+26 degrees of freedom (indicative of heterotic strings). The overall (any/all chirality) propagation degrees of freedom, 78, are also in agreement with the number of generators in string gauge symmetries [4].