A Non-causal Structure Mapping of Quantum State Space
Abstract
The concept of infinity long ago led to an understanding that proper subsets of an infinite set can be put in a one-to-one correspondence with the original set. This mechanism of ‘parts’ of a set ‘duplicating’ the original set took on deeper significance when in 1914 Hausdorff showed that the three-dimensional sphere S2 can be decomposed into a small finite number of pairwise disjoint subsets which may then be transformed by simple rotations to form two identical copies of S2. Further, the basic Hausdorff result may be extended to the case of the formation of any finite number of identical copies, and even to the case of countably infinite copies, if the restriction to finitely many initial pairwise disjoint pieces is relaxed to allow countably many such pieces.
We demonstrate the non-causal mechanics of state space multiplicity by the use of a free group of an independent pair of rotations acting on the state space of a qubit, S2, i.e. rotations from the group SO(3). We go on to speculate on the ontological implications of these rotational actions, namely that quantum state space is the ontological basis of a quantum system, and admits non-causal non-classical structure mappings which may provide a framework for superposition basis state co-existence, and even for a frequentist probability theory.
