  Recently, weak measurements have attracted a lot of interest as an
experimental method for the investigation of non-classical correlations between
observables that cannot be measured jointly. Here, I explain how the complex
valued statistics observed in weak measurements relate to the operator algebra
of the conventional Hilbert space formalism and show that the algebra of
operators originates from more fundamental relations between the physical
properties of a quantum system that can be expressed in terms of complex
conditional probabilities. In particular, commutation relations can be
identified with fundamental imaginary correlations that characterize the
relations between physical properties in terms of their transformation
dynamics. Non-commutativity thus originates from a definition of relations
between physical properties that replaces the assumption of joint reality with
a complex-valued probability reflecting the dynamical response of the system to
external forces, e.g. in measurement interactions.
