Conceptual Representation of Particles, Waves, and Heisenberg's Uncertainty Relation

This paper introduces a mathematical representation of the fundamental physical notions particles, waves, and Heisenberg's Uncertainty Relation in such a general way that neither Hilbert spaces nor the real or complex numbers are used. This new approach is based on the mathematical notion of a lattice as defined by Birkhoff (1940) who introduced lattices as a generalization of hierarchies in geometry, logic and algebra. In 1982 lattice theory has been connected by Wille (1982) with the philosophical construct of a concept using a mathematical definition of formal concepts and concept lattices.

Formal Concept Analysis (FCA), the mathematical theory of concept lattices, was then used by the author to introduce Temporal Concept Analysis which is based on Conceptual Time Systems where the notion of a state is introduced as a formal concept. The conceptual definition of life tracks of objects led to a generalization of the formal representation of objects in Conceptual Semantic Systems where distributed objects yield a clear mathematical representation of the idea of a wave packet together with a definition of particles and waves.

In this paper the author's previous definitions of particles and waves are extended, the notion of measurement is introduced and combined with the notion of a view and a (distributed) object to represent "how distributed" that object is represented by the measurement in the chosen view. That leads to a conceptual analogue of the notion of "simultaneously measurable" in Quantum Theory and to a conceptual analogue of Heisenberg's Uncertainty Relation.