Universal Star Products

Ammar, Mourad; Chloup, Véronique; Gutt, Simone

doi:10.1007/s11005-008-0240-0

Cited by 12 publications

(35 citation statements)

References 8 publications

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“…is a formal power series in y which begins with δ λ µ and whose coefficients are smooth in x. By these properties it immediately follows [26,81]…”

Section: Fedosov Manifolds and Global Deformation Quantizationmentioning

confidence: 89%

“…29 Furthermore, Un(ξ, Π φ , · · · , Π φ ) = 0 for n ≥ 2 if ξ is a linear vector field [81]. Thus X in eq.…”

Section: Fedosov Manifolds and Global Deformation Quantizationmentioning

confidence: 99%

“…In the context of Poisson geometry (M, Π), generally speaking, one cannot find a "Poisson connection" ∇ P such that ∇ P Π = 0 since parallel transport preserves the rank of the Poisson tensor, so the Poisson manifold must be regular in order for such a connection to exist. But it turns out [25,26,27,81] that it is enough to consider only a torsion free linear connection to formulate a star product on any Poisson manifold. Thus one can work only with an affine torsion free connection and construct the formal exponential map (5.67) for the torsion free connection.…”

Section: Generalization To Poisson Manifoldsmentioning

confidence: 99%

“…is a covariant symmetric tensor of rank n and smoothly depends on x ∈ M . It turns out[26,81] that f φ is a particular example of a section of the jet bundle E → M (where E is the bundle F (M ) × GL(d,R) R[[y 1 , · · · , y d ]] associated to the frame bundle F (M ) on M ) with the fiber R[[y 1 , · · · , y d ]] (i.e., formal power series in y with real coefficients) and transition functions induced from the transition functions of T M . In general any section of E is of the form σ(x; y) = 2 ···λn (x)y λ 1 y λ 2 · · · y λn (···λn define covariant tensors on M .…”

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confidence: 98%

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Quantization of emergent gravity

Yang

2015

Int. J. Mod. Phys. A

122

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Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as spacetime admits a symplectic structure, in other words, a microscopic spacetime becomes noncommutative (NC). If gravity emerges from U(1) gauge theory on NC spacetime, this picture of emergent gravity suggests a completely new quantization scheme where quantum gravity is defined by quantizing spacetime itself, leading to a dynamical NC spacetime. Therefore the quantization of emergent gravity is radically different from the conventional approach trying to quantize a phase space of metric fields. This approach for quantum gravity allows a background independent formulation where spacetime as well as matter fields is equally emergent from a universal vacuum of quantum gravity.

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“…is a formal power series in y which begins with δ λ µ and whose coefficients are smooth in x. By these properties it immediately follows [26,81]…”

Section: Fedosov Manifolds and Global Deformation Quantizationmentioning

confidence: 89%

“…29 Furthermore, Un(ξ, Π φ , · · · , Π φ ) = 0 for n ≥ 2 if ξ is a linear vector field [81]. Thus X in eq.…”

Section: Fedosov Manifolds and Global Deformation Quantizationmentioning

confidence: 99%

Section: Generalization To Poisson Manifoldsmentioning

confidence: 99%

mentioning

confidence: 98%

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Quantization of emergent gravity

Yang

2015

Int. J. Mod. Phys. A

122

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“…It results [2] from the explicit expression of the form A and the operator δ −1 that the star product constructed in this way is universal.…”

Section: Universal Star Productmentioning

confidence: 92%

Deformation Quantisation and Connections

Gutt

2009

European Women in Mathematics

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After a brief introduction to the concept of formal Deformation Quantisation, we shall focus on constructions of star products, enhancing links between star products and linear connections. We first consider the symplectic context: we recall how any natural star product on a symplectic manifold determines a unique symplectic connection and we recall Fedosov's construction which yields a star product, given a symplectic connection. In the more general context, we consider universal star products, which are defined by bidifferential operators expressed by universal formulas for any choice of a linear torsionfree connection and of a Poisson structure. We recall how formality implies the existence (and classification) of star products on a Poisson manifold. We present Kontsevich formality on R d and we recall how globalisation of this result proves the existence of a universal star product. Finally, we present some special constructions of star products on some symplectic manifolds endowed with special connections.

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