## Tuesday, April 14, 2009 ... //

### Against the religion of chaos

One of the irrational misconceptions that I have considered to be dead but that have re-appeared in many recent interactions of mine is the idea that "nonlinear" and "chaotic" phenomena are cool while "linear" and "predictable" phenomena are unimaginative.

For many people, it is enough to say these adjectives if they want to feel smarter. Well, they also look smarter, but only in the eyes of other people whose mental abilities are equally limited. If you open the famous hoax paper by Alan Sokal from 1996,

Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity
that managed to be published by some pompous fools in the "Social Text" magazine, even though it is a long and manifestly fake sequence of ludicrous absurdities, you will see that the word "nonlinear" appears more than 20 times over there, and so does the word "chaos". They're combined in sentences showing the immense pride of the fictitious author, Alan Sokal, or real, dumb authors who are being cited by Sokal.

Steven Weinberg was amused and has reviewed Sokal's prank. Weinberg notices that the postmodernists use the words "quantum mechanics" and "chaos theory" to promote their ideas about the fragmentary nature of experience. Also, in this framework, "catastrophe theory" and "chaos theory" lead to social and economic liberation, Weinberg says. Yes, "catastrophe theory" is about the tipping points, the critical parameters in bifurcation theory.

Concerning the word "nonlinear", Weinberg has observed pretty much the same thing as I have written many times:
But for some postmodern intellectuals, "linear" has come to mean unimaginative and old-fashioned, while "nonlinear" is understood to be somehow perceptive and avant garde. In arguing for the cultural importance of the quantum theory of gravitation, Sokal refers to the gravitational field in this theory as "a noncommuting (and hence nonlinear) operator." Here "hence" is ridiculous; "noncommuting" does not imply "nonlinear," and in fact quantum mechanics deals with things that are both noncommuting and linear.
Well, the postmodernists - and unfortunately not only the "official ones" - like to use the words such as "chaos", "nonlinear", and "noncommutative" as synonyma. These are the words that they "endorse", regardless of the meaning, and such an endorsement makes them "cool" in their own eyes and in the eyes of their "peers".

Needless to say, this whole line of reasoning is irrational crap. These words are not equivalent and they are not "sexy" per se: only interesting, true, and non-trivial sentences combining these words and other words may be intriguing or stunning. And there's nothing generally "boring" about the words "linear" or "commuting", either.

Linear things are often cool

In fact, quantum mechanics, the most important insight of the 20th century science, implies that all observables (such as position, velocity, magnetic field, or spin) are linked to "operators", mathematical objects that are both noncommuting and linear. (There also exist objects that are nonlinear and commuting.) All the evolution in quantum mechanics is described by a linear equation (e.g. Schrödinger's equation in the corresponding picture).

This general proposition about Nature, much like the other basic postulates of quantum mechanics, seems to hold exactly and there are good reasons to expect that nothing will ever change about these basic rules of Nature. The inevitability of the quantum postulates has been partially demonstrated and all attempts to circumvent the principles proposed as of today have been ruled out, either theoretically or empirically.

Quantum mechanics is linear but there are many concrete situations - and approximate descriptions - in science that use linear dynamics in a very important way to obtain very profound results. It's not hard to see that virtually all classical dynamical laws seen in Nature and the society are nonlinear. The linear laws are of measure zero, as long as one adopts any continuous measure that covers both linear and nonlinear objects. But the linear laws and operators still exist and they may be very important. In many cases, they are more important than the nonlinear ones.

The "confusing", "unpredictable", and "chaotic" character of the observation seems to be the default state. This is the state of knowledge we have about any system that we haven't yet understood. There is nothing shocking about this state of ignorance - it's the standard situation. Interesting science only begins once we start to see patterns, regularities, and predictable features with nonzero information.

In this sense, I disagree that "chaos theory" belongs among the most important developments of the 20th century science. The methods of "linearization" have arguably been much more useful and led to more extensive and profound results than "chaos theory". In a related context, "perturbative" vs "nonperturbative", it could have been a tie even though the "perturbative" part of science has been much more accurately confirmed by experiments (QED...).

The very observation that complex enough systems are generically chaotic - i.e. that they sensitively depend on the initial conditions - seems to be completely obvious. I don't believe that Isaac Newton would have been surprised by such things. In his era, people simply focused on simple enough systems that they had a chance to solve or approximately solve, and that was a very wise decision.

But if one considers systems with many degrees of freedom - e.g. a real gas composed out of many atoms that Ludwig Boltzmann had to consider in his atomic picture of the world - it is completely clear that the microscopic details will be effectively unpredictable. The result of each collision depends on the detailed initial positions and velocities (imagine that you shoot a planet against the Sun and study where it deflects) and the large number of collisions "thermalizes" the initial information much more rapidly: the entropy keeps on increasing, after all.

All these statements can be seen to hold within minutes, one doesn't have to be a PhD to see why. And at a general level, that's pretty much the end of the story. I don't understand in what sense this "discipline" may be equally important as e.g. atomic theory (based on quantum mechanics) or the relativistic description of black holes. It is surely not equally far-reaching or extensive.

And what's much worse, the existence of "chaos" in various theoretical models (think of the atmospheric or climate models) is often incorrectly assumed to be sufficient for these models to be right. Well, the (non-)existence of chaos carries at most one bit of information which is surely not enough to verify a hypothesis. Much more specific, quantitative, "non-chaotic" tests are needed to do so.

Is the world chaotic?

Well, the world is exactly described by quantum mechanics whose state vector is governed by a linear law. Because the Hamiltonian is Hermitian and/or the evolution is unitary, it follows that the distances between the initial states will not be modified by the evolution at all. That's what unitarity means: a unitary operator is just a complex "rotation" that doesn't change the distances in the (Hilbert) space. Unitarity of our quantum world makes the whole world completely non-chaotic in the classical sense. The chaotic behavior only emerges in a classical description - if you pick some particular classical observables and the "distance between configurations" based on these observables.

That doesn't mean that there is nothing deserving the word "chaos" in a quantum theory: "quantum chaos" has become a whole subdiscipline of quantum physics. It studies the complicated spectra and eigenstates of various Hamiltonians whose existence is linked to the appearance of chaotic (initial-condition-sensitive and otherwise unpredictable) behavior in the classical limit.

But yes, quantum mechanics implies that some conclusions about the "chaotic behavior" are mere artifacts of the classical approximation. Moreover, every system with a finite phase space - or every quantum system with a finite-dimensional Hilbert space - exhibits the Poincaré recurrences. They imply that the "asymptotic behavior in the far future" used to define some concepts in chaos theory (those where one assumes that the number of degrees of freedom is infinite) cannot be taken literally.

But again, what I find most unscientific in this ideological framework is the idea that the foggy word "chaos" should ever be the end of the story. Many things in Nature look incomprehensible at the beginning. But the more we study them, the more we learn. Various outcomes may look "random" but there is not just one kind of "randomness". Every set of random data in Nature follows some statistical distribution. In every context, this statistical distribution may be, in principle, found - both by experimental and theoretical methods.

The overall distribution for one quantity may be a big victory relatively to the complete ignorance but it doesn't have to be the last word, either. One can try to measure and/or calculate more complicated correlations, too. Even in the absence of a detailed knowledge about the microstates and their evolution, one can ask new well-defined questions (e.g. statistical questions) about this evolution. With a sufficient effort, one can answer them, too.

One can assign the word "chaos" with specific meanings but any meaning of this sort removes the fog from the word and makes it relatively mundane or even uninteresting. For example, whole classes of theories can be instantly proved to increase/decrease the size of the initial perturbations. Many theories and situations may be exactly solved while many more don't have a compact solution - even though this statement is, once again, not terribly interesting because it depends on the inevitably non-canonical definition of a "compact solution". Which functions are allowed to appear in a compact solution? There is clearly no universal answer to this question.

If "chaos theory" is supposed to be long-lived - to be more than just a simple problem to find the sign of a Lyapunov exponent - it must dynamically change the meaning of the chaos as we are learning additional things. That's why there exists a complementarity - the discipline is uninteresting whenever it is sharply defined and it is vague whenever it has a chance to become interesting.

Many fragmented pieces of knowledge are being incorporated to "chaos theory". For example, complicated systems often exhibit a self-similar, fractal behavior. Again, this is not hard to understand if the effective laws describing the situation are recursive (like in the case of the Mandelbrot set) or scale-invariant (like in conformal field theories). In the latter case, we often care about the scaling exponents.

For example, the primary fields in the Ising model have dimensions 0, 1/2, and 1/16. The third one looks pretty complicated - and be sure that it describes a feature of some self-similar pictures of situations involving many small magnets, among other things. Is this result, 1/16, supposed to be a part of chaos? Well, the simplest derivation I am familiar with doesn't look chaotic in any way. You can get it as the difference between the ground state energies in the periodic and antiperiodic sectors of a fermion.

Because 1+2+3+.... = -1/12 while 0.5+1.5+2.5+... = +1/24, their difference is 3/24=1/8 for a complex fermion (without 1/2 for the ground state energies) i.e. 1/16 for a real fermion. If someone suffers from the zeta-regularization-phobia, the argument can be easily modified to avoid any divergent sums while the result is guaranteed to be equal: study various exponents and factors in the modular functions. Is there something "chaotic" about these derivations?

I don't see it. I don't understand how such derivations could be separated from conventional theoretical physics - in this case, from conformal field theory - and how could they deserve a natural umbrella of a new discipline. The state-operator correspondence (annulus vs cylinder map) has been used and it is very deep and "non-chaotic". And it sounds very counterproductive if someone would like to suppress all these deep methods in his attempt to preserve the "chaotic image" of a system that can actually be understood by proper, predictable, calculable science.

I could give you many more examples of this phenomenon. For example, "attractor points" belong to "chaos theory". But it is clear that the clearer, more transparent, more conceptually organized, and more brilliant derivation of the attractor points and their properties we find (think about the derivations of the attractor mechanism affecting black holes in supergravity, issues that were recently very active in string theory), the less chaotic the whole set of phenomena looks.

What makes certain insights ready to be included into a new discipline called "chaos theory" is precisely the fogginess of their existing description - and this is surely not a feature that should justify the creation of a new discipline. Let me apologize to the readers-believers in advance and explain why I have used the word "religion" in the title. The reason is that this "worshipping of chaos" has the same internal logic as religion. As Feynman said:
God was invented to explain mystery. God is always invented to explain those things that you do not understand. Now, when you finally discover how something works, you get some laws which you're taking away from God; you don't need him anymore. But you need him for the other mysteries. So therefore you leave him to create the universe because we haven't figured that out yet; you need him for understanding those things which you don't believe the laws will explain, such as consciousness, or why you only live to a certain length of time -- life and death -- stuff like that. God is always associated with those things that you do not understand. Therefore I don't think that the laws can be considered to be like God because they have been figured out.
By the way, I am not the first person who invented the religion of chaos. It has been around for some time and it is also called discordianism. There's no consensus whether this religion is real or just a parody. ;-) By the way, one of the important practices of this religion is Operation Mindfuck which is often manifested as civil disobedience, activism, trolling, and other things. It may explain a lot!

Summary

In real science, it is impossible to impress rational people just by repeating some words often. It would be great if the postmodernists ultimately understood this principle. Meanwhile, it seems that the opposite dynamics is taking place - that many people are effectively becoming postmodernists. Sokal's hoax is so similar to the contemporary writing by so many people - including those who claim to like physics - that I am skeptical about the dynamics of the society. The evolution seems to lack the proper selection mechanisms and it is becoming pretty, ehm, chaotic.

And that's the memo.