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Why is quantum mechanics phrased in terms of linear operators?
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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.