Formulation of Quantum Theory Using Computable and Non-Computable Real Numbers

It is shown that in two-state quantum theory, a generic quantum state can be described by a non-computable real number. In terms of this, the criterion for measurement outcome is simply and deterministically defined. This demonstration is based on a construction of the Riemann sphere whose points represent, not complex numbers, but divergent sequences with bivalent elements. Complex structure arises from self-similar properties of a set of operators which generate these sequences. In general, a rotation of (the coordinates of) the sphere maps a computable real to a non-computable real. This is interpreted physically as a mapping of a physically-measurable state to a counterfactual state. Implications for non-locality, null measurements, many worlds and so on, are discussed. The possible role of the Euler equation as the counterpart of the Schrodinger equation for real-number quantum state evolution is also outlined.