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I dont know much about the subject, but the wave equations (classical and modern) as well as classical equations of motion all seem to be inherently linear differential equations. Presumably this also applies to string theory, which I know even less about.

What I know, however, is that nature (whatever that is) is basically non-linear, so is the linearity of the mentioned equations just a simplifying approximation to non-linear theories, and if so, what are they and can it in a few words be summarized what the simplifications are?

Qmechanic
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Jens
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    related: http://physics.stackexchange.com/q/95795 http://physics.stackexchange.com/q/141468 http://physics.stackexchange.com/q/296736 http://physics.stackexchange.com/q/1201 http://physics.stackexchange.com/q/33344 http://physics.stackexchange.com/q/299645 – AccidentalFourierTransform Jan 18 '17 at 12:51
  • Concerning the equations of evolution, usually non-linearity emerges only studying the effective behavior of a system under some suitable approximation (e.g. mean-field, semi-classical, ...) – yuggib Jan 18 '17 at 13:05
  • Do you mean you want a general outline of how we get solutions from non linear equations? As in Perturbation Theory –  Jan 18 '17 at 13:10
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    Countto10: No, perturbation theory is just a technique amongst others. Im only looking for imperfections in the physics. – Jens Jan 18 '17 at 14:04
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    Define "quantum based equation". – ACuriousMind Jan 18 '17 at 18:12

3 Answers3

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One of the most common nonlinear equations used in the study of quantum mechanics is the Gross-Pitaevskii equation, which is a special type of nonlinear Schrodinger equation and emerges naturally from the many-body Schrodinger equation under the Hartree-Fock approximation. Also, the classical equations of motion for any interacting quantum field theory are nonlinear, which is why most of these theories can only be approximately solved perturbatively. And the classical equations of motion for all nonabelian gauge theories are nonlinear, even those like Yang-Mills theory that consist only of a kinetic term, due to the presence of terms cubic and quartic in fields when you expand out the kinetic term (which physically correspond to three- and four-gluon scattering vertices in the context of QCD). This is why there are no trivially solvable nonabelian gauge theories.

tparker
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  • Nitpick: In two dimensions, gauge theories have no local propagating degrees of freedom and are trivial apart from global phenomena related to the structure of the spacetime. – ACuriousMind Jan 18 '17 at 20:33
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There are plenty of non-linear components to quantum theory, it's just when you are starting off is not useful to look at the non-linear parts. Here is a list of equations that are common to quantum theory and you will see second and higher order differential equations are present. Sorry if I miss understood what you were asking and this is of no help to you.

Michael H.
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  • https://en.wikipedia.org/wiki/List_of_equations_in_quantum_mechanics – Michael H. Jan 18 '17 at 12:51
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    Second and higher order partial derivatives do not institute non-linearity in it self. As an example of a possible non-linear equation you could imagine adding a pertubation proportional to E^3 in one of the Maxwell equations to explain some thermal diffraction effects in solids or you could add a force proportional to x^3 in the elongation x of a spring extended beyond the practical linear limit. – Jens Jan 18 '17 at 13:24
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The answer is no. This becomes especially evident in many-body problems, where the modulus squared of wave functions appears in the in the potential V(r) that appears in the Schrödinger equation, say.

Ben S
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