The redefiner Ɽ_t can be mimicked by a trail of infinitesimal unitary transforms. Each subsequent trail element has eigenvectors that differ from those of its predecessor. These eigenvectors are also eigenvectors of Ɽ_t. For a single vector, which is not an eigenvector of Ɽ_t, the action of Ɽ_t can be represented by the integrated activity of this trail on that vector. This can be interpreted as the activity of a genuine unitary transform U_t. When a redefiner Ɽ_t is applied to the eigenvector |q> of an operator Q with eigenvalue q, then the eigenvector is transferred into another vector |U_t q>. The expectation value for |QU_t q> is no longer q, but

<q U_t|QU_t q> = <q|U_t^†QU_t q> ≈ u_t^*qu_t.

where u_t is the expectation value <q|U_t q> of U_t for vector |q>. With real and complex numbers this just equals q, but for higher 2ⁿ-ons q can be affected by this number transformation. This number transformation is the number waltz. It causes a partial precession of the imaginary part of q and does not affect the real part of q.  The effect depends on the argument of u. When the action of U_t proceeds further in a stationary way, then it causes a full precession of q. That is why the transformation u_t^*qu_t gets the name number waltz.

All observations that go together with a redefinition are affected by the described number waltz.

However, in some cases the number waltz has no effect. This occurs when the action of u_t is parallel to the imaginary part of q.

Relativity

The trail that is taken by the actions of the redefiner can be studied with the toolkit of differential geometry. The study of inertia shows that geodesics traveled with uniform speed are not affected by inertia. Therefore we take the steps such that the action curve features unit speed and becomes a geodesic.

The tantrix e₀ of that curve delivers the main part of the action S. The eigenvalue u can be specified as a function of the trail progression parameter t. The Taylor expansion gives:

u_{t + Δt} = u_t·(1 + Δs_t)

Δs_t = e_t0·Δt  + e_t1·(Δt)²·χ _t1/2  + e_t2·(Δt)³·χ _t1 ·χ _t2/6 + order(Δt)³

u_t = ∏(1 + Δs_τ), τ = 0…t

Δu_t ≈ u_t·Δs_t

Thus all trail elements u_t are close to unity. The expansion also holds when u_t is a higher dimensional 2ⁿ-on. Since 2ⁿ-ons for n>1 don’t commute, the ordering of the products is important. The tantrix is the direct cause of the waltz of the observations. The result of the action step is:

q_t = u_t^-1·q·u_t ≈ (1 – Δs_rt)·u_t0·q_t0·u_t0·(1 + Δs_rt)

The real part of q is not affected. So we only consider the imaginary part.

q_t ≈ (1 – Δs_t)·q_t0·(1 +Δs_t)

Δq_t ≈ q_t0·Δs_t – Δs_t·q_t0 – Δs_t·q_t0·Δs_rt ≈ 2·q_t0×Δs_t   (1)

Thus the space step Δq_t is perpendicular to both q_t0 and Δs_t0. Let us define the local spacetime step Δσ_t as the action step Δs_t.

Δσ_t  := Δs_t  ≈ e_t0·Δt

Due to the waltz the space step occurs perpendicular to the spacetime step. The rectangular triangle can be closed with a step that we will relate to the coordinate time interval Δτ_t.

Δτ_t := Δq_t/c + e_ort·Δt

In this setting the factor 2·q_t0 is ignored. e_t0 is a unit size vector. The formula that relates the three steps corresponds to a time-like convention:

|Δt|² = |Δτ|² – |Δq|²/c²

The steps define a Minkowski metric and as a consequence it raises special relativity. The curvature of the manipulator trail causes an additional curvature of the trail of the observed position. The corresponding manifold is not Riemannian, but pseudo-Riemannian. More precisely, it is Lorentzian. The action in the argument of the current manipulator can be related to the gravitation field. The curvature in action space brings general relativity into the picture.

Coordinate time has only local significance. It only exists in combination with an observation of position in Q space. The combination of coordinate time and Q space no longer corresponds to quaternions. These specimen are better treated with Clifford algebras. In this way the quaternion waltz introduces other algebras than 2ⁿ-ons.

In the above procedure we ignored the relation (1).

We only looked at the dynamic steps.

We introduced a new progression concept: the coordinate time.

With these measures we introduced a maximum speed of information transfer c.

Division

The number waltz causes a separation between two very different worlds. The manipulators play their dynamic game in Hilbert space with progression parameter t, which comes close to proper time. They are not affected by a maximum speed c. The manipulated subjects live in the observed space, which has a Minkowski/Lorentzian metric and a quite different progression parameter τ, which we call coordinate time.