Beyond Quantum Mechanics

In this paper, an alternative formalism for modeling physics is proposed. The motivation for this approach arises from the tension between the countable (discrete) nature of empirical data and the uncountable sets (continuous functions) that form the foundations of modern physical theories. The foundation of this alternative formalism is the set of all base-2 sequences of length n. While this set is countable for finite n, it becomes uncountable in the limit that n goes to infinity, providing a viable pathway to correspondence with current theories. The mathematical construction necessary to model physics is developed by considering relationships among different base-2 sequences. Upon choosing a reference base-2 sequence, a relational system of numbers can be defined. Based on the properties of these relational numbers, the rules of angular momentum addition in quantum mechanics can be derived from first principles along with an alternative representation of the Clebsch-Gordan coefficients. These results can then be employed to model basic physics such as spin, as well as simple geometric elements such as directed edges. The simultaneous emergence of these modeling tools within this construction give hope that models for both matter and space-time may be supported by a single formalism.