A new type of operator is required. Why? Because unitary transformations cannot move subspaces of a Hilbert space. They cannot move their own eigenvectors. Since their eigenfunctions span the whole Hilbert space will every subspace of dimension higher than one contain one or more eigenfunctions of every unitary transformation. Subspaces that represent objects must be moveable in order to implement the dynamics of those objects. Unitary transformations can only move one-dimensional subspaces. However, the single vector contained in a one-dimensional vector cannot carry on its own all the properties of an object. Further, a static unitary transform can only take a single step. A solution is to use a trail of infinitesimal unitary transforms. In that case the eigenvectors of subsequent members of the trail must differ. This is an odd construct. Due to the difference between the trail members interpreting the trail as a function of the trail progression parameter will cause problems.

A better solution is to introduce a new type of operator. It will be called a redefiner and as the regular symbol for it we take Ɽ. Where a unitary transform cannot move its own eigenvectors, but moves all other vectors, the redefiner moves its own “eigenvectors” and does not touch other vectors. Each closed subspace with dimension higher than one will contain eigenvectors of each redefiner. It is easy to interpret the redefiner as a function Ɽ_{t} of the progression parameter t. At every instance of t the set of eigenvectors contained in the current subspace span that subspace. At that same instance the corresponding element ΔU_{t} of the trail corresponds to Ɽ_{t}. The trail elements mimic the redefiner and take its eigenfunctions and its eigenvalues.

In the lattice isomorphism between the collection of closed subspaces of the Hilbert space and the collection of propositions in a quantum logic system the redefiner Ɽ plays the role of the redefiner of propositions. Where the subspace is redefined with respect to the eigenvectors of other operators, the proposition is redefined with respect to atomic propositions. These atomic propositions concern properties of the object treated by the enveloping proposition.

The unitary operators carry actions in their argument. These actions are also carried by the corresponding redefiner. In order to provide enough storage space for properties of these actions, it is sensible to tolerate high dimensional hyper complex numbers such as 2^{n}-ons as eigenvalues.

Both the trail concept and the higher dimensional eigenvalues open the possibility to adapt to the situation in which the local eigenvalues are topological elements of curved manifolds.

New type of operator

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