  We define operator manifolds as manifolds on which a spectral measure on a
Hilbert space is given as additional structure. The spectral measure
mathematically describes space as a quantum mechanical observable.
  We show that the vectors of the Hilbert space can be represented as functions
on the manifold. The arbitrariness of this representation is interpreted as
local gauge freedom. In this way, the physical gauge principle is linked with
quantum mechanical measurements of the position variable. We derive the
restriction for the local gauge group to be U(m), where m is the number of
components of the wave functions.
