Ivancevic_Applied-Diff-Geom

458 Applied Differential Geometry: A Modern Introductionis satisfied. For case I, there is an unbounded sub–domain of the spacecoordinates (q 1 , q 2 , q 3 ) for which the weak energy condition holds. For caseII, it is easily seen that the restrictions on the energy–momentum tensorare inconsistent, so that the weak energy condition never holds.3.17 Applied Unorthodox Geometries3.17.1 Noncommutative GeometryIn this subsection we give review of noncommutative geometry and its maingravitational applications. In the last section of the book, we will give itsapplications to string theory.3.17.1.1 Moyal Product and Noncommutative AlgebraNoncommutative geometry is concerned with the possible spatial interpretationsof algebraic structures for which the commutative law fails; that is,for which xy does not always equal yx. The challenge of the theory is toget around the lack of commutative multiplication, which is a requirementof previous geometric theories of such structures (see [Connes (1994)]).Recall that an ordinary differentiable manifold can be characterized bythe commutative algebra of smooth functions defined on it, and the spaceof smooth sections of its tangent bundle, cotangent bundle and other fiberbundles. All these spaces are modules over the commutative algebra ofsmooth functions. The concepts of exterior derivative, Lie derivative andcovariant derivative are also important elements in understanding derivationsover this algebra. In the noncommutative case, the algebras in questionare noncommutative. To handle differential forms, one must work withthe graded exterior algebra bundle of all p−forms under the wedge productand look at its algebra of smooth sections. A ‘differential’ is taken tobe an anti–derivation (or, something more general) on this algebra, whichincreases the grading by 1 and is quadratically nilpotent.Historically first noncommutative product was the Moyal product[Moyal (1949)], 10 that is an associative, noncommutative ‘star’–product.For any two functions f, g on a Poisson manifold M, the Moyal product ∗10 The Moyal product is also sometimes called Weyl–Moyal product.