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Positive matrices in $$SL(2, {\mathbb{C}})$$ 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. [ABSTRACT FROM AUTHOR]

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Möbius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO( n + 1) invariant measure on S n under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra $$ \mathcal{C}(n+1) $$ as generalized Lorentz “spin–boosts”, and their action as Möbius transformation on n-sphere, and a decomposition of any element of Spin+(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO( n + 1) invariant Riemannian metric with respect to the associated Möbius transformation. [ABSTRACT FROM AUTHOR]