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

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
• we can't know what examples of non-linearity you have in mind. Can you give some examples? – Sofia Jan 18 '15 at 17:21
• 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.) – Thomas Klimpel Jan 18 '15 at 17:41
• as for the break-down of QM, see this answer physics.stackexchange.com/q/159922. The scale where such break-down is expected to occur is given by Planck's length. – Phoenix87 Jan 18 '15 at 19:56
• Sorry, you're conflating totally different things, @miha. Schrödinger's equation for the wave function is linear - and all actions on the wave function or density matrix is given by linear operators. This requirement ultimately results from the fact that the wave function is made of probability amplitudes and probailities also add linearly under "or" - the linearity follows from basic probability calculus. On the other hand, Heisenberg equations of motion for the operator as as nonlinear as dynamical equations in classical physics always were. – Luboš Motl Jun 1 '16 at 14:13
• Possible duplicates: physics.stackexchange.com/q/1201/2451 and links therein. – Qmechanic Aug 17 '18 at 8:12

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 is just completely wrong. Nonlinearity never arises from linear equations. The point is that the linear equation and the nonlinear equations are completely different kinds of equations. – Luboš Motl Jun 1 '16 at 14:11
• I deleted some inappropriate comments; feel free to consider the discussion while remaining civil. – David Z Jun 1 '16 at 18:01
• @LubošMotl It is not completely wrong. It is something commonly observed on physical systems. The mean field non-linear equations such as Hartree or Gross-Pitaevskii are used and observed to describe the evolution of the one-particle wavefunction in Bose-Einstein condensates, – yuggib Jun 1 '16 at 18:55
• and has been proved that they can be obtained from the full linear $N$-body dynamics of bosons with pair interaction e.g. considering reduced density matrices here and here (to cite two different approaches). Also the nonlinear dynamics of classical probability distributions can be recovered in a similar fashion considering the limit $\hbar\to 0$ of linearly evolved quantum states. This also has been rigorously proved. – yuggib Jun 1 '16 at 18:55

From experimentation with and intuition of the quantum equations, I would suggest the following as at least a partial resolution:

1. The first is to clarify the status of the linearity of quantum mechanics and the nonlinearity of classical mechanics: that is, what is linear in quantum and not linear in classical? The trick here is in recognizing that the Schrodinger equation, which is the linear equation in quantum mechanics, does not describe directly the motion of a point in physical space, but rather the motion of a point in Hilbert space, an often infinite-dimensional vector (linear) space. Whereas in classical mechanics the governing equations (e.g. Newton's Laws of Motion) govern directly the motion of a point in physical space. The correspondence arises in that the elements of Hilbert space can be represented as complex-valued functions $\psi(x)$ subject to suitable constraints, while this $x$ represents the usual position we know of from classical mechanics. This function $\psi(x)$ represents a form of ambiguous information about the position $x$, namely, it's a "norm-2, complex" probability distribution: you can say a particle's position is more clearly defined if the distribution is tightly packed and peaked, and less clearly defined if the distribution is spread out over a wide range of $x$-values. When imagined to evolve over time, we may write $[\psi(t)](x)$ instead, using the "currying" method from computer programming: this is a function that returns a function for each parameter $t$, which is then evaluated at $x$, to maintain the intuition.
2. With that down, approximate "classical motion", in the case of a single particle, corresponds to the case where a tightly-peaked peak in the $\psi(t)$ functions for each time $t$ moves while remaining tightly peaked. The location of this peak is the "most likely position".
3. The nonlinearity or exponential divergence in classical chaos reflects a divergence between positions in position space after a small perturbation, i.e. between positions $x(t)$ and $x^{*}(t)$ where $x^{*}(0) = x(0) \pm \epsilon$ for a small $\epsilon > 0$. Namely, at least for a time, $|x^{*}(t) - x(t)|$ grows up at least exponentially from $\epsilon$. This would correspond in the "classical quantum" of point 2 above to the exponentially growing separation between peaks of two separate $\psi(0)$ and $\psi^{*}(0)$ initial wave functions whose peak centers are separated by a small $\epsilon$.
4. But, given point 1, the relevant divergence when it comes to the fact that the quantum equations are linear is not divergence in $x$-space, which is really just an index we use to describe an element of Hilbert space, but the divergence within Hilbert space itself! And for that, you cannot measure it using the distance between peaks alone, but have to measure using the proper notion of distance in Hilbert space, indeed having such a notion is partially what constitutes the idea of Hilbert space itself. If $\psi$ and $\phi$ are two elements (Hilbert vectors), represented as functions in the manner above, then the distance in Hilbert space is $$d(\psi, \phi) = -\int_{-\infty}^{\infty} |\psi(x) - \phi(x)|^2\ dx$$ (This is just taking the norm of the vectorial difference $\psi - \phi$.) Now suppose that $\psi(x)$ and $\phi(x)$ are two equally tight, but widely-spaced, peaks. It is not hard to see from the subtraction that the function integrated is just a single function with two widely spaced in $x$, peaks. The integral, however, doesn't care about the distance between them in the $x$-coordinate, only the area under them, and in fact this is essentially constant regardless of their separation as it's just the sum of areas under each peak individually! Thus even if the two peaks are separating exponentially in classical ($x$)-space, the two Hilbert space elements to which they correspond, and which are what the quantum equations govern, are not separating at all! Thus there is no inconsistency, on this basis, between linear behavior of the Hilbert dynamics and highly sensitive, nonlinear behavior of the peak location.
5. A final attribute that may be important is the fact that the states of bound quantum systems (i.e. analogous to the bound classical ones that are used in the study of chaos) are subject to recurrence - that is, the $\psi(t)$ evolution will always eventually repeat itself after a suitably long interval, if not exactly, than it will come arbitrarily close. The reason for this is that any such bound state $\psi$ can be expressed as a sum of bound energy eigenstates whose dynamics under the Schrodinger equation are periodic, and the period is inversely proportional to the energy of each state. For at least an always-bound system (e.g. the quantum harmonic oscillator or better the quantum double pendulum) the bound states will form a (generalized, allowing for infinite linear combinations) basis set for Hilbert space, meaning that any given Hilbert element can be expressed as an (infinite) sum of these, effectively a "wave function of energy", "$\psi(E)$". The lowest energy will have the longest period, periods decreasing with increasing energy. In order for the sum to converge, the higher-energy terms must, of course, have lower amplitudes, and that means after a while we can effectively neglect them to yield a very high-accuracy approximation of whatever initial state we started the system in, perhaps up to the limits of any contaminating error, and thus we can regard it for theoretical analysis as even being in such a state. This state will eventually repeat itself, after a period equal to the least common multiple of all the composite periods, even if the position of the initial wave peak is not on a classical periodic orbit. How is this consistent with the classical behavior? Well, what happens is that it will, for a time, execute a chaotic-like motion, then the peak will slowly spread out and eventually will fill up the angular space of the double pendulum, interfering with itself and becoming a non-classical mist state, where the pendulum is now in a strange, flickering quantum fuzz of orientations that has no counterpart in classical chaotic theory, and this can persist for an egregiously long time until finally the internal components find each other again and it comes out to recollect back into a single packet at the original position, where it repeats itself again. By doing something WEEEEEEEERD after being left for a long time (intuitive rule: if you find something "should be" nice and intuitive by a reasonably good theory yet according to a better theory it will do something contradictory to the predictions of that theory like repeat itself or be nonchaotic, then it is almost sure it will have to WEEEERD its way there), it sneaks around the limitations of the classical chaos and meets the demands of the quantum theory.

I'd rather answer your question with my question. Classical electrodynamics with Maxwell equations is a linear theory. Yet, there are plenty effects there where we see non-linearity. Non-linearity comes from interaction with matter when it's properties start to depend on electric or magnetic fields. Yet, the equations themselves are linear.

Others have already mentioned that the point here is that you have linearity in equation for particular variable (e.g., wave function in QM). When you try to calculate other variables (measurable ones) in an interacting system, it's not a surprise you get a non-linear result. Take for example two-level atom interacting with electromagnetic wave. If you solve the equations precisely (without assumptions about weak EM field or frequencies being far off resonance, etc.), the result is that electromagnetic properties of matter are always non-linear because state of atom has changed after interaction with light and now it interacts differently (some electrons went to higher energy state). But both Schroedinger and Maxwell equations here are linear.

Quantum mechanics has two parts: well understood linear evolution, and rather mysterious measurement process. The measurement is not linear. One can find nice discussion of quantum measurement in Roger Penrose book "The Emperors's New Mind", where it is tentatively related to consciousness of observer.

The case of a quantised version of a classical chaotic system has been discussed in many papers. If the system evolves in complete isolation then its expectation values don't closely match those of classical chaotic systems because of interference. But real systems on the scale where we observe chaotic behaviour don't evolve in isolation, they interact with the environment. That interaction introduces additional terms to the system's equation of motion that make the quantum and classical expectation values match one another on the relevant scales of length and time "Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time" by Zurek:

https://arxiv.org/abs/quant-ph/9803042

and many similar papers. It should be noted that this is not the same as taking the limit $\hbar\to 0$. Nor is it the same as simply letting wavepackets diverge, which would give results that do not match classical choatic behaviour at all. For example, in classical chaotic systems, the relevant behaviour is that trajectories that are very close together can exhibit very different behaviour over time. But in wavepacket spreading the whole idea of trajectories is a bad approximation because of interference terms. The trajectory approximation only works because of decoherence.