Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 615(2) δ λf = λδ f , for each f ∈ C ∞ (T ∗ R n , R) and λ ∈ R;(3) δ 1R n = Id H ; and(4) [δ f , δ g ] = (δ f ◦ δ g − δ g ◦ δ f ) = iδ {f,g}ω , for each f, g ∈ C ∞ (T ∗ R n , R);The pair (H, δ), whereH = L 2 (R n , C); δ : f ∈ C ∞ (T ∗ R n , R) ↦→ δ f : H → H;δ f = −iX f − θ(X f ) + f; θ = p i dq i ,gives a prequantization of (T ∗ R n , dp i ∧ dq i ), or equivalently, the answer tothe Dirac problem is affirmative [Puta (1993)].Now, let (M = T ∗ Q, ω) be the cotangent bundle of an arbitrary manifoldQ with its canonical symplectic ( structure ω = dθ. The prequantizationof M is given by the pair L 2 (M, C),δ θ) , where for each f ∈ C ∞ (M, R),the operator δ θ f : L 2 (M, C) →L 2 (M, C) is given byδ θ f = −iX f − θ(X f ) + f.Here, symplectic potential θ is not uniquely determined by the conditionω = dθ; for instance θ ′ = θ +du has the same property for any real functionu on M. On the other hand, in the general case of an arbitrary symplecticmanifold (M, ω) (not necessarily the cotangent bundle) we can find onlylocally a 1–form θ such that ω = dθ.In general, a symplectic manifold (M, ω = dθ) is quantizable (i.e., we candefine the Hilbert representation space H and the prequantum operator δ fin a globally consistent way) if ω defines an integral cohomology class. Now,by the construction Theorem of a fiber bundle, we can see that this conditionon ω is also sufficient to guarantee the existence of a complex line bundleL ω = (L, π, M) over M, which has exp(i u ji /) as gauge transformationsassociated to an open cover U = {U i |i ∈ I} of M such that θ i is a symplecticpotential defined on U i (i.e., dθ i = ω and θ i = θ i + d u ji on U i ∩ U j ).In particular, for exact symplectic structures ω (as in the case of cotangentbundles with their canonical symplectic structures) an integral cohomologycondition is automatically satisfied, since then we have only one setU i = M and do not need any gauge transformations.Now, for each vector–field X ∈ M there exists an operator ∇ ω X on thespace of sections Γ(L ω ) of L ω ,∇ ω X : Γ(L ω ) → Γ(L ω ), given by ∇ ω Xf = X(f) − i θ(X)f,