How does non-linear behaviour arise from the inherently linear QM framework?

Question

Quantum mechanics is a linear theory, living in a Hilbert space with built-in linearity. It has even been argued that introducing non-linearity in the quantum theory would allow for superluminal signalling.

As far as I know there is also no experimental evidence showing that QM breaks down at a certain scale.

So why is it then that the world exhibits rich non-linear behaviour? Where does the non-linearity arise from mathematically?

EDIT: On examples of nonlinear behaviour:

• Properties of materials (electrical resistance, elasticity)
• Chaotic dynamics
• Complex systems

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user miha priimek

Comments

we can't know what examples of non-linearity you have in mind. Can you give some examples?

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user Sofia

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There are at least two points where it can enter: (1) A linear PDE can have a close relation to a non-linear ODE. Classical Hamilton-Jacobi theory allows you to formulate classical mechanics in the form of a linear PDE. (2) If you describe a subsystem by a density matrix, the evolution equation for the density matrix can have non-linear terms modeling the interaction of the subsystem with the environment. (I don't want to prevent anybody from writing a proper answer by this comment, even if it should use the same examples. I'm just too lazy to write a detailed answer.)

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user Thomas Klimpel

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as for the break-down of QM, see this answer physics.stackexchange.com/questions/159922/…. The scale where such break-down is expected to occur is given by Planck's length.

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user Phoenix87

Answer 1

Non-linearity arises when one takes the limit of quantum dynamics in some sense. Two standard examples are:

1) the semiclassical approximation (i.e. Born's correspondence principle) where in the limit of large quantum numbers ("$\hslash\to 0$") quantum linear dynamics becomes the classical (usually non-linear) one;

2) Mean field approximation (i.e. the limit of a very large number of particles), where the dynamics of each component of the system is modelled by an effective non-linear dynamics (i.e. Hartree or Gross-Pitaevskii equations as the mean field limit of many-bosons systems in condensed matter).

The subject of analyzing rigorously this classical or mean field limit is a very active subject in the domain of mathematical physics/analysis of PDEs.

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user yuggib