Ivancevic_Applied-Diff-Geom

616 Applied Differential Geometry: A Modern Introductionand it is easy to see that ∇ ω is a connection on L ω whose curvature is ω/i.In terms of this connection, the definition of δ f becomesδ f = −i∇ ω X f+ f.The complex line bundle L ω = (L, π, M) together with its compatibleconnection and Hermitian structure is usually called the prequantum bundleof the symplectic manifold (M, ω).If (M, ω) is a quantizable manifold then the pair (H, δ) defines its prequantization.ExamplesEach exact symplectic manifold (M, ω = dθ) is quantizable, for the cohomologyclass defined by ω is zero. In particular, the cotangent bundle, withits canonical symplectic structure is always quantizable.Let (M, ω = dθ) be an exact symplectic manifold. Then it is quantizablewith the prequantum bundle given by [Puta (1993)]:L ω = (M × C, pr 1 , M);Γ(L ω ) ≃ C ∞ (M, C);∇ ω Xf = X(f) − i θ(X)f;((x, z 1 ), (x, z 2 )) x = ¯z 1 z 2 ; δ f = −i[X f − i θ(X f )] + f.Let (M, ω) = (T ∗ R, dp ∧ dq). It is quantizable with [Puta (1993)]:L ω = (R 2 × C, pr 1 , R 2 );Γ(L ω ) = C ∞ (R 2 , C);Therefore,∇ ω Xf = X(f) − i pdq(X)f; ((x, z 1) , (x, z 2 )) x= ¯z 1 z 2 ;[ ∂f ∂δ f = −i∂p ∂q − ∂f ]∂− p ∂f∂q ∂p ∂p + f.δ q = i ∂ ∂p + q, δ p = −i ∂ ∂q ,which differs from the classical result of the Schrödinger quantization:δ q = q, δ p = −i ∂ ∂q .Let H be a complex Hilbert space and U t : H → H a continuous one–parameter unitary group, i.e., a homomorphism t ↦→ U t from R to the group