一般我們會使用希爾伯特空間中的向量代表量子力學中的物理狀態，但實際上非簡併的空間是映射希爾伯特空間。希爾伯特空間與映射希爾伯特空間都是凱勒流形。我們可以從薛丁格方程式與漢彌爾頓方程式看出凱勒流形在物理上的重要性。 凱勒積介紹了漢彌爾頓函數與量子物理中運算子之間的同構關係。對稱資料由在流形上某點的漢彌爾頓方程式及其導數組成，可以做為運算子的′值′的候選。;We usually consider a vector in Hilbert space to represent a physical state in quantum mechanics but the nondegenerate space is projective Hilbert space. Hilbert space and projective Hilbert space are both Kähler manifolds. We can show the importance of Kähler manifold to physics by the connection to Schrödinger equation and Hamilton’s equation. Kähler product introduce an isomorphism between Hamiltonian functions and operators in quantum physics. Symmetry data is a candidate of the “value” of the operator, which contains the Hamiltonian function and its derivatives at a point on the manifolds.
