Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 461ordered monomial [Meyer (2005)]W : A → Â, xµ ↦→ ˆx µ , x µ x ν ↦→ 1 2 (ˆxµˆx ν + ˆx ν ˆx µ ) · · · . (3.270)To study the dynamics of physical fields we need a differential calculuson the noncommutative algebra Â. Derivatives are maps on the deformedcoordinate space [Wess (2004)]ˆ∂ µ :  →  .This means that they have to be consistent with the commutation relationsof the coordinates. In the θ−constant case a consistent differential calculuscan be defined very easily by 12[ ˆ∂ µ , ˆx ν ] = δ ν µ( ˆ∂ µˆx ν ) = δ ν µ, [ ˆ∂ µ , ˆ∂ ν ] = 0. (3.271)It is the fully undeformed differential calculus. The above definitions yieldthe usual Leibniz-rule for the derivatives ˆ∂ µ( ˆ∂ µ ˆfĝ) = ( ˆ∂µ ˆf)ĝ + ˆf( ˆ∂µ ĝ). (3.272)This is a special feature of the fact that θ µν are constants. In the more complicatedexamples of noncommutative structures this undeformed Leibnizruleusually cannot be preserved but one has to consider deformed Leibnizrulesfor the derivatives [Wess and Zumino (1991)]. Note that (3.271) alsoimplies that( ˆ∂ µ ˆf) = ̂(∂µ f). (3.273)The Weyl ordering (3.270) can be formally implemented by the mapf ↦→ W (f) = 1 ∫d n k e ikµ ˆxµ ˜f(k)(2π) n 2where ˜f is the Fourier transform of f˜f(k) = 1 ∫d n x e −ikµxµ f(x).(2π) n 2This is due to the fact that the exponential is a fully symmetric function.Using the Baker–Campbell–Hausdorff formula one findse ikµ ˆxµ e ipν ˆxν = e i(kµ+pµ)ˆxµ − i 2 kµθµν p ν. (3.274)12 We use brackets to distinguish the action of a differential operator from the multiplicationin the algebra of differential operators.