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First of all, I've read this other question Is the universe linear? If so, why? and I'm aiming at a different kind of answer.

Theories like General Relativity or QFT, which are believed to be quite fundamental, are strongly non-linear. However, in the end, both theories must be just low energy limits of an unified theory. So this question arises: Could this unified theory be linear? I'm looking for both a mathematical and a physical answer. In other words, I'd like to know if 1) it is possible to make a linear theory that has non-linear low energy aproximations and if 2) a non-linear universe would make any physical sense in view of the superposition principle.

Community
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P. C. Spaniel
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    Depends on what you mean by linear. QFT is still QM: the Schrödinger equation is linear and the superposition principle applies, but the field equations and interactions are nonlinear. – Javier Dec 19 '16 at 19:59

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To add to John Rennie's excellent answer I would like to note that the linearity of a theory depends of what you are asking of it, especially which observables you are interested in.

One example is the optical response of a 2-level atom (see e.g. https://en.wikipedia.org/wiki/Jaynes–Cummings_model). While being linear in the wavefunctions involved it is non-linear in the field strength of the incoming laser, i.e. would have an amplitude dependent effective refractive index.

(Note: my initial example was complete and utter non-sense. Thanks to Mark Mitchison for pointing that out.)

Wolpertinger
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  • The Lindblad equation is linear! What do you mean? – Mark Mitchison Dec 20 '16 at 13:24
  • @MarkMitchison true that, in the sense that the density matrix obeys the superposition principle. I guess I stopped the argument too early (cause there are non-linear effects from tracing out the environment), let me see if I can fix it. My bad. – Wolpertinger Dec 20 '16 at 21:01
  • One has to be very careful about what one means by "linear". In quantum mechanics, the state evolves linearly. This is always true, whether in a closed system or an open system interacting with an environment. However, observables may nonetheless obey highly non-linear equations of motion. Unfortunately, your answer seems to mix the two concepts, stating that QM is linear (always true for states) but then in interacting systems the evolution can be non-linear (never true for states, sometimes true for observables). Thus, the basic premise of your answer seems deeply flawed. – Mark Mitchison Dec 20 '16 at 21:34
  • For instance, take the given example: the quantum-optical Lindblad master equation describing spontaneous emission of a laser-driven two-level atom into the electromagnetic vacuum. This kind of evolution is as linear as it gets, since here even the observables evolve according to a strictly linear equation of motion, the celebrated Bloch equations. – Mark Mitchison Dec 20 '16 at 21:40
  • @MarkMitchison While everything you say is correct I don't think my answer is deeply flawed. In fact the example you just gave and described as "as linear as it gets" is a lot better than mine: a laser-driven two-level atom is non-linear in the driving field, i.e. the optical response would be non-linear. So I think my answer is fixable, but admittedly I carelessly didn't explain that I was thinking of optical response in the first place when mentioning the Lindblad equation. – Wolpertinger Dec 20 '16 at 23:34
  • That kind of non-linearity has nothing to do with the dissipative environment, it's actually just a result of the finite-dimensional Hilbert space of the two-level system. Yet the basic premise of your answer is that tracing out the environment induces some nonlinearity. Can you actually give an example where this is the case? You may be correct, but the answer as it is doesn't present any evidence in support of this claim. – Mark Mitchison Dec 20 '16 at 23:50
  • Indeed, the actual conclusion of the answer in its current form is just flat-out wrong, i.e. "trace out these degrees of freedom...this leads to a Master Equation which is non-linear". Therefore, I believe that the answer needs much more than just a superficial rewrite. Sorry if this seems harsh, it's not meant to be a personal attack in any way. – Mark Mitchison Dec 20 '16 at 23:59
  • @MarkMitchison no offence taken. And who cares if I was offended, you're right! Hope the new example is better. – Wolpertinger Dec 22 '16 at 07:42
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    +1, thanks, this is much improved. Merry Xmas :) – Mark Mitchison Dec 22 '16 at 11:05
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The obvious example is hydrodynamics.

The interactions in a fluid all originate from the interactions between atoms and molecules that are described by quantum mechnics, and QM is as far as we know linear. However the Navier-Stokes equations are (scarily) non-linear and produce all sorts of weird behaviour.

It's an interesting question whether general relativity is a fundamental theory or whether it's an emergent theory based on some more fundamental interactions. The simple answer is that we don't know as there is no evidence either way. The work on this idea is being led by Erik Verlinde.

John Rennie
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    Hi John. Did you mean counter example (to the universe being linear)? I suppose also we need to clarify what is meant by 'the universe'. Behavior at all scales? Or just the cosmic scale. I suppose also I would have posed the question "Is the nature of the universe fundamentally non-linear". I believe it is. Sometimes it may appear linear when one squints. – docscience Dec 19 '16 at 18:36
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    Isn't it fair to assume GM is an emergent theory? It operates on large scales after all. Why should anyone expect it to describe small scales accurately? (I'm thinking of the law of large numbers as an analogy here.) – user541686 Dec 19 '16 at 19:48
  • As anticipated by Javier's comment, this answer conflates two completely distinct types of linearity: linear evolution in phase space and linear evolution on Hilbert space, corresponding to different distance metrics. P. C. Spaniel is asking about the former, so "QM is as far as we know linear" is a red herring. The quantum theory of hydrodynamics is simultaneously linear on Hilbert space but nonlinear on phase space, and there's nothing inconsistent or mysterious about this. – Jess Riedel May 03 '17 at 00:13
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First things first, there is no proof that the universe is either. An outstanding question in philosophy is the ontological question of whether the universe is defined by mathematics, or if we created mathematics to understand the universe. Your question only makes sense in the former.

With sufficient feedback, you can create remarkable approximations of non-linear systems with linear systems. Likewise, linearity can be seen in many non-linear systems which operate on a manifold if one looks at small scale.

As for superposition, it only is useful for linear systems, or systems which have reasonable linear approximations. An excellent example appears in electrical engineering with "small signal" models. These are models of how a non-linear system (lots of transistors amplifying currents) can look linear within a small region around a chosen point. Many times that is more than sufficient for what we need.

Cort Ammon
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