Ivancevic_Applied-Diff-Geom

470 Applied Differential Geometry: A Modern IntroductionIt is by construction invariant with respect to deformed diffeomorphismsmeaning thatˆδξ Ŝ EH = 0.In this section we have presented the fundamentals of a noncommutativegeometry on the algebra  and defined an invariant Einstein–Hilbert action.There is however one important step missing which is subject of thefollowing section: We want to make contact of the noncommutative gravitytheory with Einstein’s gravity theory. This we achieve by introducing the⋆−product formalism.3.17.1.6 Star–Products and Expanded Einstein–Hilbert ActionTo express the noncommutative fields in terms of their commutative counterpartswe first observe that we can map the whole algebraic constructionof the previous sections to the algebra of commutative functions via thevector space isomorphism W introduced above. By equipping the algebraof commutative functions with a new product denoted by ⋆ be can renderW an algebra isomorphism. We define [Meyer (2005)]f ⋆ g : = W −1 (W (f)W (g)) = W −1 ( ˆfĝ), (3.291)and get (A, ⋆) ∼ = Â.The ⋆−product corresponding to the symmetric ordering prescription W isthen given explicitly by the Moyal product 17f ⋆ g = µ ◦ e i 2 θµν ∂ µ⊗∂ νf ⊗ g = fg + i 2 θµν (∂ µ f)(∂ ν g) + O(θ 2 ).It is a deformation of the commutative point–wise product to which itreduces in the limit θ → 0.In virtue of the isomorphism W we can map all noncommutative fieldsto commutative functions in AˆF ↦→ W −1 ( ˆF ) ≡ F.We then expand the image F in orders of the deformation parameter θF = F (0) + F (1) + F (2) + O(θ 3 ),17 This is an immediate consequence of (3.275).