Complementarity, Quantum Variables, and the Relationships between Mathematics and Physics

Abstract

As part of his critique of Karl Popper’s argument against the Copenhagen interpretation of quantum mechanics in his “Karl Popper and the Copenhagen Interpretation” (Peres 2002), Asher Peres offers a lucid and elegant exposition both of quantum mechanics and of a certain interpretation of it. This interpretation is a version of the Copenhagen interpretation, which largely (albeit not completely) follows Bohr’s complementarity.20 In the course of this exposition, Peres makes the following, apparently strange, observation, used as my epigraph for this chapter: “in quantum theory, a particle also has a precise position and a precise momentum” (Peres 2002, p. 27). The observation may, however, not be as strange as it may seem, especially by virtue of its apparent conflict with the uncertainty relations. As I shall argue, Peres’s statement, on the contrary, reflects the precise mathematical and physical meaning of the uncertainty relations, and, I shall further argue, as such it also directs our attention toward a new type of relationships between mathematics and physics established in quantum mechanics, especially in the work of Heisenberg and Dirac. It is worth citing Peres’s elaboration where the proposition in question occurs more extensively. According to Peres: