Momentum as generator

Stone’s theorem applied to spatial translations gives:

U(a) = e^{−ip̂a/ℏ}

For a unique self-adjoint operator p̂, the generator of the group.

p̂ is the momentum operator.

The constant ℏ enters as the scale factor — the action-scale of the minimum record, derived above.

Position x̂ and momentum p̂ are observables — self-adjoint operators on the Hilbert space.

Position labels where on the manifold. Its spectrum is the set of spatial coordinates in the local chart.

Momentum generates displacement along the manifold.

They are linked by the structure of the manifold itself.