Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1233different metrics in the problem. Distances measured with respect to onemetric have been scaled to zero. However, the noncommutative theory ison a space with a different metric with respect to which all distances arenonzero. This guarantees that both on R n and on T n we end up with atheory with finite metric.6.8.1.1 Noncommutative Gauge TheoryFor R n with coordinates x i whose commutators are c−numbers, we write[x i , x j ] = iθ ij ,with real θ. Given such a Lie algebra, one seeks to deform the algebra offunctions on R n to a noncommutative, associative algebra A such thatf ∗ g = fg + 1 2 iθij ∂ i f∂ j g + O(θ 2 ),with the coefficient of each power of θ being a local differential expressionbilinear in f and g. The essentially unique solution of this problem (moduloredefinitions of f and g that are local order by order in θ) is given by theexplicit formulaf(x) ∗ g(x) = e i 2 θij∂∂ξ i∂∂ζ j f(x + ξ)g(x + ζ)| ξ=ζ=0 = fg + i 2 θij ∂ i f∂ j g + O(θ 2 ).This formula defines what is often called the Moyal bracket of functions;it has appeared in the physics literature in many contexts, includingapplications to old and new matrix theories [Witten (1986b);Seiberg and Witten (1999)]. We also consider the case of N × N matrix–valued functions f, g. In this case, we define the ∗ product to be the tensorproduct of matrix multiplication with the ∗ product of functions as justdefined. The extended ∗ product is still associative.The ∗ product is compatible with integration in the sense that for functionsf, g that vanish rapidly enough at infinity, so that one can integrateby parts in evaluating the following integrals, one has∫∫Tr f ∗ g = Tr g ∗ f,where Tr is the ordinary trace of the N ×N matrices, and ∫ is the ordinaryintegration of functions.