-------------------------------------------------------------- Why is quantum mechanics phrased in terms of linear operators? -------------------------------------------------------------- In an important sense, quantum mechnics is inherently linear. The tyoical textbook version of quantum mechanics is phrased in terms of linear operators. Even in more abstract settings, where observables are elements in some C^*-algebra. one still studies the linear representations of this algebra to extract information. Why is this so? The simple answer is: It is enough. The sophisticated answer is: One can rewrite every reversible nonlinear dynamical system as a reversible linear system in a much bigger space. This is a nontrivial generalization of the simple observation that one can represent any permutation of n objects as a linear operator in R^n. (Think of reversible motion on the list of objects as being a sequence of permutations....) Note that the space where the standard model lives in is truly very big. Many nonlinear systems are tractable by functional analytic techniques in a bigger linear space. So the rstriction to linear operators in quantum mechanics is not a real restriction. On the other hand, a linear problem may have a more tractable nonlinear representation; then this may be an important advantage. Thus, sometines, nonlinear representations are useful to study quantum mechanical problems. It depends a lot on what sort of questions one is trying to answer.