Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 459is defined as:f ∗ g = fg +∞∑ n C n (f, g),n=1where each C n is a bi–differential operator of order n with the followingproperties:1. f ∗ g = fg + O() – deformation of the pointwise product.2. f ∗ g − g ∗ f = i{f, g} + O( 2 ) – deformation in the direction of thePoisson bracket.3. f ∗ 1 = 1 ∗ f = f – the 1 of the un–deformed algebra is the 1 in thenew algebra.4. f ∗ g = g ∗ f – the complex conjugate is an anti–linear anti–automorphism.Let the Poisson structure on a symplectic manifold M be defined byω = ω ij ∂ i ∧ ∂ j .If this structure is constant, that is, if ω ij do not depend on the localcoordinates on M 11 , then the Moyal product of two functions f, g ∈ M canbe defined asf ∗ g = fg + ω ij (∂ i f)(∂ j g) + 22 ωij ω km (∂ i ∂ k f)(∂ j ∂ m g) + . . .where is the (reduced) Planck constant.For example, in Weyl deformation quantization [Weyl (1927)], the symplecticphase–space of classical mechanics is deformed into a noncommutativephase–space generated by the position and momentum operators, usingthe Moyal product.3.17.1.2 Noncommutative Space–Time ManifoldsIn physical field theories one usually considers differential space–time manifolds.Now, in the noncommutative realm, the notion of a point is no longerwell–defined and we have to give up the concept of differentiable manifolds.However, the space of functions on a manifold forms an algebra. A generalizationof this algebra can be considered in the noncommutative case. Wetake the algebra freely generated by the noncommutative coordinates {ˆx i },which respects commutation relations of the type [Madore (1995)][ˆx µ , ˆx ν ] = C µν (ˆx) ≠ 0. (3.266)11 Such a form can always be found at least locally by Darboux’s Theorem