Analysis and Geometry on Complex Homogeneous Domains

Faraut, Jacques; Kaneyuki, Soji; Korányi, Adam; Lu, Qianyun; Roos, Guy; Birkenhake, Christina; Lange, Herbert

doi:10.1007/978-1-4612-1366-6

Cited by 53 publications

(57 citation statements)

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“…We may recall that the conventional von Neuman Hyperfinite II1 factor is roughly an infinitedimensional version of the spinor representation of Complex Clifford algebras, which have periodicity 2 and so are like an infinite limit of what Baez calls "... the fermionic Fock space over C ( 2n) ..." and then generalize it to the case of Real Clifford Algebras with periodicity 8 so that one gets is an infinite limit of a tensor product of a lot of copies of 256-dim Cl(8), 7 Each Cl(8) would describe physics locally in the neighborhood of a given spacetime point, as described in the physics model 8 All the Cl(8) factors in the generalized Hyperfinite II1 factor (roughly an infinite tensor product) would be linked together to form (at the next higher energy level above our quaternionic 4-dim physical spacetime plus 4-dim CP 2 internal symmetry space) a higher-energy real/octonionic 8-dim spacetime as described in When one takes quantum superpositions in the many-worlds quantum theory, quantum loops/graphs of higher and higher order appear, whose description involves the prime numbers 10 and which may be closely related to the p-adic geometry. For additional references pertaining the topics discussed in this section see [49] , [50] , [51] , [52] , [53], [54] , [55] , [56], [57] , [? ].…”

Section: The Standard Model and Gravity From 8d Clifford Structuresmentioning

confidence: 99%

The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification

Castro

2005

Found Phys

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We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper and lower length scales (infrared/ultraviolet cutoff ). The invariance symmetry leads naturally to the real Clifford algebra Cl(2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. We proceed with an extensive review of Smith's 8D model based on the Clifford algebra Cl(1, 7) that reproduces at low energies the physics of the Standard Model and Gravity; including the derivation of all the coupling constants, particle masses, mixing angles, ....with high precision. Further results by Smith are discussed pertaining the interplay among Clifford, Jordan, Division and Exceptional Lie algebras within the hierarchy of dimensions D = 26, 27, 28 related to bosonic string, M, F theory. Two Geometric actions are presented like the Clifford-Space extension of Maxwell's Electrodynamics, Brandt's action related the 8D spacetime tangent-bundle involving coordinates and velocities ( Finsler geometries ) followed by a discussion ( based on results by Cho et al ) why Einstein's gravity in m + n dimensions is equivalent to an m-dim Yang-Mills-like theory of diffemorphisms of an internal n-dim space which admits a holographic reduction to lower dimensions in the case of AdS m ×S n and dS m ×H n backgrounds. Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit.

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Section: The Standard Model and Gravity From 8d Clifford Structuresmentioning

confidence: 99%

The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification

Castro

2005

Found Phys

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“…Shilov boundaries of homogeneous (symmetric spaces) complex domains, G/K [70][71][72] are not the same as the ordinary topological boundaries (except in some special cases), because the action of the isotropy group, K, of the origin is not necessarily transitive on the ordinary topological boundary. Shilov boundaries are the minimal subspaces of the ordinary topological boundaries, which implement the Maldacena-'t Hooft-Susskind holographic principle [73] in the sense that the holomorphic data in the interior (bulk) of the domain is fully determined by the holomorphic data on the Shilov boundary.…”

Section: Evaluation Of the Coupling Constantsmentioning

confidence: 97%

A Clifford algebra-based grand unification program of gravity and the Standard Model: a review study

Castro

2014

Can. J. Phys.

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A Clifford Cl(5, C) unified gauge field theory formulation of conformal gravity and U(4) × U(4) × U(4) Yang-Mills in 4D, is reviewed along with its implications for the Pati-Salam (PS) group SU(4) × SU(2) L × SU(2) R , and trinification grand unified theory models of three fermion generations based on the group SU(3) C × SU(3) L × SU(3) R . We proceed with a brief review of a unification program of 4D gravity and SU(3) × SU(2) × U(1) Yang-Mills emerging from 8D pure quaternionic gravity. A realization of E 8 in terms of the Cl(16) = Cl(8) R Cl(8) generators follows, as a preamble to F. Smith's E 8 and Cl(16) = Cl(8) R Cl(8) unification model in 8D. The study of chiral fermions and instanton backgrounds in CP 2 and CP 3 related to the problem of obtaining three fermion generations is thoroughly studied. We continue with the evaluation of the coupling constants and particle masses based on the geometry of bounded complex homogeneous domains and geometric probability theory. An analysis of neutrino masses, Cabbibo-Kobayashi-Maskawa quark-mixing matrix parameters, and neutrino-mixing matrix parameters follows. We finalize with some concluding remarks about other proposals for the unification of gravity and the Standard Model, like string, M, and F theories and noncommutative and nonassociative geometry. Résumé : Nous passons en revue une formulation en théorie du champ unifié de Clifford Cl(5, C) de la gravité conforme et de Yang-Mills U(4) × U(4) × U(4) en 4D, avec ses implications pour le groupe de Pati-Salam SU(4) × SU(2) L × SU(2) R et les modèles GUT de trinification de trois générations de fermions basés sur le groupe SU(3) C × SU(3) L × SU(3) R . Nous commençons avec une brève revue du programme d'unification de la gravité 4D et Yang-Mills SU(3) × SU(2) × U(1) qui émerge de la gravité quaternionique pure en 8D. Nous poursuivons avec une réalisation de E 8 en termes des générateurs de Cl(16) = Cl(8) R Cl(8), comme introduction au modèle d'unification de F. Smith E 8 et Cl(16) = Cl(8) R Cl(8) en 8D. Nous étudions en profondeur les fermions chiraux et les fonds d'instantons dans CP 2 et CP 3 reliés à la difficulté d'obtenir trois générations de fermions. Nous continuons avec l'évaluation des constantes de couplage et des masses des particules, sur la base de la géométrie des domaines homogènes complexes bornés et de la théorie de probabilité géométrique. Suit une étude des masses des neutrinos, des paramètres de la matrice de mélange de Cabbibo-Kobayashi-Maskawa et des paramètres de la matrice d'oscillation de neutrinos. Nous concluons avec des remarques sur les propositions faites pour unifier la gravité et le modèle standard, comme la théorie des cordes, la théorie M et F, et la géométrie non associative et non commutative. [Traduit par la Rédaction]

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“…The notation and terminology used here is largely the same as in [10,14,26,29,30,32], to which we refer the reader for detailed accounts, including the complete axiomatic settings of the theory of Jordan-triple structures. In what follows, A denotes a JB * -triple or a JBW * -triple with triple product…”

Section: Jb(w) * -Triples Inner Derivations and Gridsmentioning

confidence: 99%

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

Hügli

2011

Integr. Equ. Oper. Theory

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It is shown that if P is a weak * -continuous projection on a JBW * -triple A with predual A * , such that the range P A of P is an atomic subtriple with finite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of P A, then P is contractive if and only if it commutes with all inner derivations of P A. This provides characterizations of 1-complemented elements in a large class of subspaces of A * in terms of commutation relations.Mathematics Subject Classification (2010). Primary 17C65; Secondary 47D27.

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Analysis and Geometry on Complex Homogeneous Domains

Cited by 53 publications

References 0 publications

The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification

The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification

A Clifford algebra-based grand unification program of gravity and the Standard Model: a review study

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

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