From Geometry to Topology

The ideal of classical field theory was geometry.

p, q

Action Principle (they form action, intersect, integrate, It is functional)

Symmetry (conservation)

Cannonical Commutation Relations (It is quantized)

Hamilton Equation (It is dual)

Fourier Transform (functional relation, continuous transformation)

Heisenberg Relations (Fourier nature of the manifold)

Transformation Theory of QM

Functional as a Bilinear Form