geometry from Fedosov's deformation quantization

Castro, Carlos

doi:10.1016/s0393-0440(99)00044-3

Cited by 24 publications

(35 citation statements)

References 40 publications

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“…Higgs matter fields in the adjoint representation were introduced also with the typical quartic potential terms which generated an infinte tower of massive spin 2 fields (massive higher spin fields in the case of w ∞ gauge theories) after an sponteaneous symmetry breaking. These gauge theories based on the infinite-dim Virasoro and w ∞ , w ∞ ∞ algebras are essential ingredients to understand further what is W ∞ Geometry [62], [72]. The importance of non-critical W ∞ strings within the context of Vasilev's higher spin theories [63] and ( super ) membranes in D = 11, 27 dimensions, Moyal deformations of gravity, the higher-dim Quantum Hall Effect, Chern-Simons Branes ( High-dimensional version of Knots ) and Topological Chern-Simons Matrix models was analyzed in [71] .…”

Section: Gravity As Gauge Theories Of Diffs and Holograpymentioning

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: Gravity As Gauge Theories Of Diffs and Holograpymentioning

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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“…It is interesting that the Moyal product was recently found to be relevant to the M theory in its M(atrix) formulation [32,33,34,35]. A parallelism between HS gauge theories and the Fedosov quantization was recently emphasized in [36]. Another interesting parallelism is due to the analysis of N = 2 critical open superstring in [37] where it was shown that physical degrees of freedom of the model describe self-dual fields of arbitrary high spin and can naturally be described in terms of a hyperspace with spinor commuting coordinates.…”

Section: Introductionmentioning

confidence: 99%

Higher-spin gauge interactions for massive matter fields in 3D AdS space-time

Prokushkin¹,

Vasiliev²

1999

Nuclear Physics B

326

1,034

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A remarkable feature of the models with interactions exhibiting higher-spin (HS) gauge symmetries in d > 2 is that their most symmetric vacua require (anti)-de Sitter (AdS) geometry rather than the flat one. In striking parallelism to what might be expected of M theory HS gauge theories describe infinite towers of fields of all spins and possess naturally space-time SUSY and Chan-Paton type inner symmetries. In this paper, we analyze at the level of the equations of motion the simplest non-trivial HS model which describes HS gauge interactions (on the top of the usual supergravitational and (Chern-Simons) Yang-Mills interactions) of massive spin-0 and spin-1/2 matter fields in d = 2 + 1 AdS space-time. The parameter of mass of the matter fields is identified with the vev of a certain auxiliary field in the model. The matter fields are shown to be arranged into d3 N = 2 massive hypermultiplets in certain representations of U (n) × U (m) Yang-Mills gauge groups. Discrete symmetries of the full system are studied, and the related N = 1 supersymmetric truncations with O(n) and Sp(n) Yang-Mills symmetries are constructed. The simplicity of the model allows us to elucidate some general properties of the HS models. In particular, a new result, which can have interesting implications to the higher-dimensional models, is that our model is shown to admit an "integrating" flow that proves existence of a non-local Bäcklund-Nicolai-type mapping to the free system.

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“…There were proposed different sophisticate constructions with formal and partial solutions for quantum gravity and field interactions theories. We cite here the BRST quantization methods for non-Abelian and open gauge algebras [1,2,3,4], deformation quantization [5,6,7,8], quantization of general Lagrange structures and, in general, BRST quantization without Lagrangians and Hamiltonians [9,10], W -geometry and Moyal deformations of gravity via strings and branes [11,12,13] and quantum loops and spin networks [14,15,16].…”

Section: Introductionmentioning

confidence: 99%

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

Anastasiei

Vacaru

2009

Journal of Mathematical Physics

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We provide a method of converting Lagrange and Finsler spaces and their Legendre transforms to Hamilton and Cartan spaces into almost Kähler structures on tangent and cotangent bundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on effective phase spaces. This allows us to define the corresponding Fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on Lagrange-and Hamilton-Fedosov manifolds.

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geometry from Fedosov's deformation quantization

Cited by 24 publications

References 40 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

Higher-spin gauge interactions for massive matter fields in 3D AdS space-time

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

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