Non-linear Operators: Physical Reasons Explained

[QuantumCosmo]

Hi,
I was wondering: What is the physical reason for only choosing linear operators to represent observables?

 

[dextercioby]

I don't know if there's a certain physical reason for which quantum mechanics is linear, but there's certainly a mathematical one: the models built from a linear QM explain (up to gravity) just about everything we know in the microscopical realm.

OTOH, the mathematics of linear operators is a lot simpler and far more studied than the one of nonlinear operators.

 

[QuantumCosmo]

Hm,
yeah, it's certainly true that the mathematics is simpler and a lot of things we use - for example the spectral theorem - require operators to be linear. But I can't seem to find a physical reason for why that should be so.
The only thing that comes to mind is the requirement of real eigenvalues. For that you need a self adjoint (and therefore linear, since we are dealing with operators on Hilbert spaces, not star algebras or the like) operator. Since we measure those quantities, the should be real. But I'm not sure if this qualifies as a "physical reason"...

 

[A. Neumaier]

Hm,
yeah, it's certainly true that the mathematics is simpler and a lot of things we use - for example the spectral theorem - require operators to be linear. But I can't seem to find a physical reason for why that should be so.
The only thing that comes to mind is the requirement of real eigenvalues. For that you need a self adjoint (and therefore linear, since we are dealing with operators on Hilbert spaces, not star algebras or the like) operator. Since we measure those quantities, the should be real. But I'm not sure if this qualifies as a "physical reason"...

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...)

By the way, the space where the standard model lives in is truly very big.

 

[QuantumCosmo]

Ok, so it is really a practical decision rather than a physical necessity?
Thank you! :)

 

[A. Neumaier]

Ok, so it is really a practical decision rather than a physical necessity?

Many nonlinear systems are tractable by functional analytic techniques in a bigger linear space. On the other hand, a linear problem may have a more tractable nonlinear representation; then this may be an important advantage. It depends a lot on what sort of questions one is trying to answer.