 Closer to home, while the classical world is recognised as having been all the time a dynamical pattern emerging from quantum mechanics, it turns out to be less classical than we might have expected. One interesting example is the description of classically chaotic systems. A straightforward application of the techniques allowing one to derive Newtonian trajectories at the level of components has been employed by Zurek and Paz (1994) to derive chaotic trajectories in quantum mechanics. The problem with the quantum description of chaotic behaviour is that prima facie it should be impossible. Chaos is characterised roughly as extreme sensitivity in the behaviour of a system on its initial conditions, in the sense that the distance between the trajectories arising from different initial conditions increases exponentially in time (see the entry on  chaos).  Since the Schrödinger evolution is unitary, it preserves all scalar products and all distances between quantum state vectors. Thus, it would seem, close initial conditions lead to trajectories that are uniformly close throughout all of time, and no chaotic behaviour is possible (‘problem of quantum chaos’). The crucial point that enables Zurek and Paz’s analysis is that the relevant trajectories defined by decoherence are at the level of components of the state of the system. Unitarity is preserved because the vectors in the environment, to which these different components are coupled, are and remain orthogonal: how the components themselves more specifically evolve is immaterial. Explicit modelling yields a picture of quantum chaos in which different trajectories branch (a feature absent from classical chaos, which is deterministic) and then indeed diverge exponentially. (As with the crossing of trajectories in de Broglie–Bohm theory in  Section 3.2,  one has behaviour at the level of components that is qualitatively different from the behaviour derived for wave functions of an isolated system.) The qualitative picture is the same as we mentioned above in  Section 1.3,  of classical trajectories that are kicked slightly off course (trajectories with slight kinks). In the case of classically chaotic systems, however, this has a dramatic effect. This means that systems like the weather turn out to be ‘branching’ all the time due to decoherence interactions, so that what we usually think of as classical unpredictability is in fact quantum indeterminism! (For an excellent discussion, see Wallace 2012a, Chapters 3 and 10.) 