Quantum Fractals. Geometric modeling of quantum jumps with conformal maps

Abstract

Positive matrices in SL(2,C) have a double physical interpretation; they can be either considered as "fuzzy projections" of a spin 1/2 quantum system, or as Lorentz boosts. In the present paper, concentrating on this second interpretation, we follow the clues given by Pertti Lounesto and, using the classical Clifford algebraic methods, interpret them as conformal maps of the "heavenly sphere" S^2. The fuzziness parameter of the first interpretation becomes the "boost velocity" in the second one. We discuss simple iterative function systems of such maps, and show that they lead to self--similar fractal patterns on S^2. The final section of this paper is devoted to an informal discussion of the relations between these concepts and the problems in the foundations of quantum theory, where the interplay between different kinds of algebras and maps may enable us to describe not only the continuous evolution of wave functions, but also quantum jumps and "events" that accompany these jumps. Paper dedicated to the memory of Pertti Lounesto.