The Lucky Linearity Of Quantum Mechanics

Classically, the universe behaves in a weakly chaotic manner. All it`s clockwork-like cyclicality is mostly illusory. The apparent simple periodicity observed in the universe belies a complex struggle between bifurcations of plausible orbits in the short term against the pull of long-term stability. That we have such stability at all in our rotating solar system is profoundly due to the fact that space is 3 dimensional.

The multiplicity of words to describe rotation: periodicity, cyclicality and orbits are a tribute to some simple governing inverse square law, but the precession of the orbits allude to this uneasy state of equilibrium.

That chaos is just a small perturbation away is due to the non-linearity of the (gravitational field) equations describing the system`s trajectories: a system of interacting particles given a little kick (perturbation) by the addition of another small body, will behave in a manner that will embody more than the sum of these two parts. Its behaviour will not be deterministic.

The simplicity of Linearity is present in the addition of a static electric charge to a set of other static charges. The (apparently) natural superposition principle gives the resultant Coulomb force as being equal to just the linear sum of all the Coulomb forces of the charges.

Happily for those seeking to reduce our patterned but ultimately turbulent world to a clockwork mechanics of primitive basic building blocks these simple rules of linearity preside over the realms of the very small. The mechanics of the quantum, be they point, string or brane-like form is linear in nature. That is, the possible states of a physical object at its most basic level form a linear space - think matrix based rotations - that are eloquently described by Sophius Lie`s (continuos) Group theory.

The supreme Reductionists express the fundamental symmetries of their universe through the theory of linear representations of Lie algebras.