Geometric derivation of quantum uncertainty

The solution to (3) through initial point φ0 is given by φτ = eφ0. Here e−iÂτ denotes the one-parameter group of unitary transformations generated by −iÂ, as described by Stone’s theorem. It follows that the integral curve through φ0 ∈ S2 will stay on the sphere. One concludes that, modulo the domain issues, the restriction of the vector field Aφ to the sphere S2 is a vector field on the sphere. Under the embedding, the inner product on the Hilbert space L2 gives rise to a Riemannian metric (i.e., pointdependent real-valued inner product) on the sphere S2 . For this one considers the realization L2R of the Hilbert space L2, i.e., the real vector space of pairs X = (Reψ, Imψ) with ψ in L2. If ξ, η are vector fields on S2 , one can define a Riemannian metric Gφ : TRφS2 × TRφS L2 −→ R on the sphere by