Aren't all physical relations non-linear?

Question

It is well-known that Hooke's Law is only approximately true and thus that linear relation is merely an idealization not strictly corresponding to the reality. Wouldn't it be necessary/appropriate that all linear relations decribing physical phenomena be reformulated to contain non-linear terms for use in cases where higher accuracy is desired? In particular, what would oppose Maxwell's Equations eventually being modified to have non-linear terms? (Or put in other words, why these equations "must" be linear?)

Answer 1

Real-world situations are often messy and they usually lead to non-linear relationships that may only be approximated – i.e. "linearized" – by linear relationships such as Hooke's Law and many others. But it's not true that "everything" in physics is non-linear. It's not true that every linear relationship is just an approximation.

In particular, the action of any observable (as well as any evolution operator or any other physically meaningful transformation) on the wave function is always exactly linear. This linearity is a universal postulate of quantum mechanics, applies to all objects in the Universe and beyond, and there are no empirically known violations of the linearity postulate. In fact, it seems mathematically impossible to construct a local, mathematically consistent, nonlinear deformation of quantum mechanics.

While the linearity in the previous paragraph was one dependent on the addition of wave functions and not observables (wave functions are not observables), there are also exact linear relationships between observables. For example, $E=mc^2$ is completely exact, too – and one may say that it is a linear relationship between the total energy and the total mass (not the rest mass) of a physical system. Physically, we say that $E$ and $m$ is conceptually the same thing so the relationship becomes trivial but it's still true that for others, it's an exactly linear relationship between observables.

There are also lots of linear relationships whose accuracy is immensely good or becomes arbitrarily good in certain important limits. For example, $pV=nRT$ isn't exact for real gases. But it's valid for ideal gases and it's important to know that real gases are close to ideal gases in a majority of situations we care about – and especially in some situations we may describe by extra conditions (very low densities etc.). Physicists often say that $pV=nRT$ means that the relationship between pressure or volume on one side and temperature on the other side is exactly linear (if the third quantity is kept fixed) for ideal gases. Ideal gases don't exist in the real world, strictly speaking, but it's still important for physicists to understand them and they're "almost enough" to predict a huge portion of questions about real-world gases, too.

So one must be very careful and discuss the situations separately. Sometimes the nonlinearities are large, sometimes they're small, sometimes their being zero represents important approximations that must be taken seriously, and sometimes the linearity is totally exact and linked to the consistency of any physical theory.

In particular, Maxwell's equations for pure electromagnetic fields belong to the group where the nonlinearities exist but they are extremely small and only appear if one goes beyond a certain approximation. The linearity of Maxwell's equations boils down to the fact that the Maxwell Lagrangian density $-F^{\mu\nu}F_{\mu\nu}/4$ is quadratic in the derivatives of the electromagnetic potential $A_{\mu}$. The Lagrangian has to be gauge-invariant and the other, nonlinear terms, already have a mass dimension higher than 4 (the units are ${\rm mass}^\Delta$ where $\Delta\gt 4$). If that's so, these extra interactions are non-renormalizable. But yes, these non-linear corrections do appear in the effective action.

If one considers the effect of an "electron box" (Feynman diagram), i.e. two initial photons's splitting into electron-positron pairs, one electron and one positron (from opposite photons) reannihilate immediately and the other pair annihilates after both the electron and positron emit a photon (thus getting the final two photons), one learns that two photons (and therefore two electromagnetic waves) have a nonzero probability that they will interact with each other. This mutual interaction is equivalent to nonlinearity of the equations and/or to higher-order (quartic) terms in the action such as $c(F_{\mu\nu})^4$. However, the probability is really tiny, especially for low-energy photons such as the visible light.

Nonlinear modifications of Maxwell's equations have been proposed in the past, especially the so-called Dirac-Born-Infeld action. It turned out that this action naturally describes the electromagnetic field propagating in the world volume of "branes" in string/M-theory. This effective action is important in string theory. However, one must realize that the appearance of this "historical" non-linear action is a sort of coincidence and the original motivation saying why such nonlinearities "had to be" a part of Nature around us is known to be wrong (they thought it was needed to regulate the divergent electron's self-energy – for a few decades, we haven't worried about it because the electron mass comes out finite after renormalization; renormalization ultimately summarizes the effects of shorter-distance "neglected" new physical effects such as new particles and nonlinearities that they produce but we don't have to discuss the form of these nonlinearities of the short-distance theory – it's a key point of renormalization that the long-distance physics is independent of these short-distance details).

Answer 2

I think that all physics is linear (or some other order) approximation to laws of nature. What I mean is that all formulas that are linear (or some other order) like gravity F = M*m*G/(R^2) in fact should be non linear functions like F = F(M,m,R). The only reason why it appears not so is that we perceive world and do experiments in very limited range of orders of magnitude of variables. Probably if you increase R much beyond or below usual observation you will see that gravity is not proportional to R^(-2). If gravity works differently on the scale R similar to size of universe then you won't detect that at all.

"In particular, the action of any observable (as well as any evolution operator or any other physically meaningful transformation) on the wave function is always exactly linear."

That's true in the theory that was based on limited experiments. Experiments performed for different scales or energies will contradict the theory and open doors for some new theory.

Answer 3

Let me note that (under some conditions) a nonlinear theory can be precisely (rather than approximately) described by a linear theory using, e.g., Carleman linearization (embedding). See, e.g., http://engineering.ucsb.edu/~bamieh/pubs/doylefest_seq.pdf , p.38). The price you have to pay - the linear theory "lives" in an infinite-dimensional space.