When some members of a discussion group asked me to write a small course on Hilbert spaces, I decided to define that Hilbert space over the quaternions. Futher I wanted to indicate the relation between this Hilbert space and quantum logic on the one hand and the relation of the Hilbert space to quantum physics on the other hand.

When I looked for a thing that could push the representation of a physical item and the representation of the corresponding quantum logical proposition "This is the item" around in Hilbert space I took the idea to represent that manipulator by a trail of unitary-like transforms that are identified by a trail progession parameter t. I wanted to tolerate higher dimensional 2ⁿ-ons than quaternions as eigenvalues of operators. In this way it becomes possible that the eigenvalues of the trail elements belong to a curved manifold of which the elements behave locally as quaternions.

I encountered the waltz when I studied the effect of the manipulators on the results of observations. This effect stays hidden in complex Hilbert spaces. The trail element U_t and its local eigenvalue u_t affect the imaginary part of the observed value q_t = u_t^-1qu_t. Only the part that is perpendicular to the imaginary part of u_t is affected. A more detailed treatize is given at http://www.scitech.nl/English/Science/Exampleproposition.pdf. That article also reveals that the waltz is the source of the existence of a Minkowski metric in observed space. It clears the relation between time and space and shows that these things belong to different operators: the manipulator and the manipulated observable. The curved manifold of manipulator eigenvalues and the quaternion waltz both form the source of relativity.