Profound Simplicity

Our best theories of the physical world appear complicated and difficult because they are profoundly simple, says Nobel Laureate and Academy Governor Frank Wilczek in his new book, The Lightness of Being.

Einstein is often quoted for his advice to "Make everything as simple as possible, but not simpler." After studying Einstein's general relativity, or his theory of fluctuations in statistical mechanics—two of his more intricate creations—you might wonder well whether he heeded his own advice. Certainly those theories are not "simple" in the usual sense of the word.

Modern physicists consider quantum chromodynamics (QCD) an almost ideally simple theory, yet we've seen how complicated it is to describe QCD in everyday words, and how challenging it is to work with (and not solve) that theory. Like Bohr's profound truth, profound simplicity contains an element of its opposite, profound complexity. This is a paradox, but its resolution is profoundly straightforward, as we'll now explore.

Perfection supporting complexity: Salieri, Joseph II, and Mozart

I learned what perfection means from the notoriously mediocre composer Antonio Salieri.¹ In one of my favorite scenes in one of my favorite movies, Amadeus, Salieri looks with wide-eyed astonishment at a manuscript of Mozart's, and says "Displace one note and there would be diminishment. Displace one phrase and the structure would fall."

In this, Salieri captured the essence of perfection. His two sentences define precisely what we mean by perfection in many contexts, including theoretical physics. You might say it's a perfect definition.

A theory begins to be perfect if any change makes it worse. That's Salieri's first sentence, translated from music to physics. And it's right on point. But the real genius comes with Salieri's second sentence. A theory becomes perfectly perfect if it's impossible to change it significantly, without ruining it entirely; that is, if changing the theory significantly reduces it to nonsense.

In the same movie, Emperor Joseph II offers Mozart some musical advice: "Your work is ingenious. It's quality work. And there are simply too many notes, that's all. Just cut a few and it will be perfect." The Emperor was put off by the surface complexity of Mozart's music. He didn't see that each note served a purpose—to make a promise or fulfill one; to complete a pattern or vary one.

Similarly, at first encounter people are sometimes put off by the superficial complexity of fundamental physics. Too many gluons!

But each of the eight color gluons is there for a purpose. Together, they fulfill complete symmetry among the color charges. Take one gluon away—or change its properties in any way—and the structure falls to the ground. Specifically, if you make such a change, then the theory formerly known as QCD begins to predict gibberish: some particles are produced with negative probabilities, and others with probability greater than one. Such a perfectly rigid theory, that doesn't allow consistent modification, is extremely vulnerable. If any of its predictions are wrong, there's nowhere to hide. No fudge factors or tweaks are available. On the other hand, a perfectly rigid theory, once it shows significant success, becomes very powerful indeed. Because if it's approximately right and can't be changed, then it must be exactly right!

Salieri's criteria explain why symmetry is such an appealing principle for theory-building. Systems with symmetry are well on the path to Salieri's perfection. The equations governing different objects and different situations must be strictly related, or the symmetry is diminished. With enough violations all pattern is lost, and the symmetry falls. Symmetry helps us make perfect theories.

So the crux of the matter is not the number of notes or the number of particles or equations. It is the perfection of the designs they embody. If removing any one would spoil the design, then the number is exactly what it should be. Mozart's answer to the Emperor was superb: "Which few did you have in mind, Majesty?"

Profound simplicity: Sherlock Holmes, Newton again, and young Maxwell

One sure way to avoid perfection is to add unnecessary complications. If there are unnecessary complications, they can be displaced without diminishment and removed without destruction. They also distract, as in this story of Sherlock Holmes and Dr. Watson:

Passing from the ridiculous to the sublime, you may recall that Sir Isaac Newton was not satisfied with his theory of gravity, which featured forces acting through empty space. But since that theory agreed with all existing observations and he could not discover any concrete improvement, Newton put his philosophical reservations aside and presented it unadorned. In the concluding General Scholium to his Principia, he made a classic declaration²:

The key phrase "I do not feign hypotheses" is "Hypothesis non fingo" in the original Latin. "Hypothesis non fingo" is the legend Ernst Mach put below the portrait of Newton in his influential Science of Mechanics. It is famous enough to have its own Wikipedia entry. It means, simply, that Newton refrained from loading his theory of gravity with speculation free of observable content.

(In his private papers, however, Newton worked obsessively to try to discover evidence for a medium filling space.)

Of course, the easiest way to avoid unnecessary complications is to say nothing at all. To avoid that pitfall, we need a dose of young Maxwell. According to an early biographer, as a small boy he was always asking "in the Gallowegian accent and idiom" "What's the go o' that?" and on receiving an unsatisfactory answer asking "but what's the particular go o' that?"

In other words, we must be ambitious. We must keep addressing new questions and strive for specific, quantitative answers. I'll shamelessly quote a paragraph from Chapter 4, since it's so relevant here: The phrase "scientific revolution" has been used for so many things that it has been devalued. The emergence of ambition to make precise mathematical world-models, and faith that one could succeed, was the decisive, inexhaustible Scientific Revolution.

There is creative friction between the conflicting demands of economizing on assumptions while providing particular answers to many questions. Profound simplicity is stingy on the input side, generous on the output side.

Compression, decompression, and (in)tractability

Data compression is a central problem in communication and information technology. I think it gives us a fresh and important perspective on the meaning and importance of simplicity in science.

When we transmit information, we want to take best advantage of the available bandwidth. So we boil down the message, removing redundant or inessential information. Acronyms like MP3 and JPEG are familiar to users of iPods and digital cameras; MP3 is an audio compression format, JPEG is an image compression format. Of course, the receiver at the other end has to take the boiled-down data and unpack it to reproduce the full intended message. When we want to store information, similar problems arise. We want to keep the records compact, but ready to unfold.

From a larger perspective, many of the challenges humans face in making sense of the world are data compression problems. Information about the external world floods our sensory organs. We must fit it into the available bandwidth of our brains. We experience far too much to keep accurate memory of it all—so-called "photographic memory" is rare and limited, at best. We construct working models and rules of thumb that allow us to use small representations of the world, adequate to function in it. "There's a tiger coming!" compresses gigabytes of optical information, plus perhaps megabytes of audio from the tiger's roar, and maybe even—this means trouble—a few kilobytes of her odor and the wind she stirs up, into a tiny message. (For experts: 23 bytes in ASCII.) A lot of information has been suppressed—but we can unfold some very useful consequences from the little that's there.

Constructing profoundly simple theories of physics is an Olympian³ game in data compression. The goal is to find the shortest possible message—ideally, a single equation—that when unpacked produces a detailed, accurate model of the physical world. Like all Olympian games, this one has rules. Two of the most important are:

• Style points are deducted for vagueness.
• Theories that produce wrong predictions are disqualified.

Once you understand the nature of this game, some of its strange features become less mysterious. In particular: For the ultimate in data compression, we must expect tricky and hard-to-read codes. Consider, for example, "Take this sentence in English." Eliminating the vowels, we make it shorter:

Tk ths sntnc n nglsh.
...............................

This is harder to read, but there's no real ambiguity about what sentence it represents. According to the rules of the game, it's a step in the right direction. We might go further, eliminating the spaces

Tkthssntncnnglsh.
...........................

That starts to get more questionable. It could be mistaken for

Took those easy not nice nine ogles, he.
...........................................................

Of course, English is so quirky that this kind of code loses heavily on style points, for vagueness. It's hard to be sure exactly what counts as a legitimate sentence. In the game of profound simplicity, we must do our decompression using precisely defined mathematical procedures. But as this simple example suggests, we must expect that short codes will be less transparent than the original message, and that decoding them will require cleverness and work.

After centuries of development, the shortest codes could become quite opaque. It could take years of training to learn how to use them, and hard work to read any particular message. And now you understand why modern physics looks the way it does!

Actually, it could be a lot worse. The general problem of finding the optimal way of compressing an arbitrary collection of data is known to be unsolvable. The reason is closely related to Gödel's famous Incompleteness Theorem, and (especially) to Turing's demonstration that the problem of deciding whether a program will send a computer into an infinite loop is unsolvable. In fact, looking for the ultimate in data compression runs you straight into Turing's problem: you can't be sure whether your latest wonderful trick for constructing short codes will send the decoder into an infinite loop.

But Nature's data set seems far from arbitrary. We've been able to make very short codes that describe large parts of reality fully and accurately. More than this: In the past, as we've made our codes shorter and more abstract, we've discovered that unfolding the new codes gives new messages, which turn out to correspond to new aspects of reality.

When Newton encoded Kepler's three laws of planetary motion into his law of universal gravity, explanations of the tides, the precession of the equinoxes, and many other tilts and wobbles tumbled out. In 1846, after almost two centuries of triumph upon triumph for Newton's gravity, small discrepancies were showing up in the orbit of Uranus. Urbain Le Verrier found that he could account for these discrepancies by assuming the existence of a new planet. When observers turned their telescopes where he suggested they look, there was Neptune! (Today's dark matter problem is an uncanny echo, as we'll see.)

Ever more compressed, profoundly simple equations to start, ever more complex calculations to unfold them, ever richer output that the world turns out to match. This, I think, is a down-to-earth interpretation of what Einstein meant in saying "Subtle is the Lord, but malicious He is not." In striving for further unification, we're betting that our luck will continue.

¹ Salieri's mediocrity is a question debated by serious music critics. Regardless, he's notorious for being mediocre.

² Warning: May induce déjà vu. I quoted this before, in Chapter 7.

³ It's not in the Olympics, of course, so it's not an Olympic event. But as a challenge worthy of the Greek gods and goddesses, it's Olympian.

From the book The Lightness of Being: Mass, Ether, and the Unification of Forces, by Frank Wilczek. Reprinted by arrangement with Basic Books (www.basicbooks.com), a member of the Perseus Books Group. Copyright 2008.