Hard Calculations

[snousle]

Something very interesting has happened in the world of physics, but it isn't getting a lot of press. After thirty years, physicists have finally been able to account for mass in terms of the most fundamental constituents of matter. Most of the mass in ordinary matter is composed of hadrons (protons and neutrons), and these, in turn, are composed of quarks and gluons. The combined mass of the quarks is much less than the mass of the hadrons, and that extra mass comes from the energy associated with their interaction. Accounting for that interaction is the tricky part.



The standard model, along with the basic equations of quantum chromodynamics, have been known since the 70s. It is wildly counterintuitive, but the fundamentals are not that complicated. The basic theory is reasonably terse when you lay it all out. And for the most part - especially now - it is believed that QCD is extremely, extremely accurate, even if it does embody some basic contradictions (which might possibly be resolved by the LHC if they ever get it to work.)

The thing that is so strange is the absurd difficulty of putting that theory to use. Here we have a system of just three quarks and some gluons flying around, yet it took thirty years to convincingly account for the most basic properties of the most basic entities they form. The Science article describes the calculations in general, and they employ math that I only touched on as an undergrad; four more years of grad school is required if you really want to get into it. And that was for just an approximate solution - it happens to be a very accurate approximation, but it still involves a whole lot of "cheating" to get the equations into that approximately-correct form which can actually be solved.

It was a real tour de force, but it does make me wonder. What does it mean to have discovered what appear to be the most fundamental principles of matter, yet still have to work so very very hard to answer the simplest questions?

I think that most students of physics are traumatized by this. You start with a bit of a fallacy - that atoms are like billiard balls bouncing around - and it leads you to believe that everything in the universe is similarly causal and analytically transparent. Later, you learn that it's a little more complicated, and you have to do some somewhat peculiar things to understand the hydrogen atom. Then they start to talk about helium atoms, and - wham - it's like being run over by a truck, because all of a sudden you are faced with the realization that it isn't at all like billard balls, and that the "space" in which you perform the calculations is not just bigger in size, but bigger in kind than the space you thought you lived in. Like the way a cube is always "bigger" than a square, because it occupies a different kind of space - but in truth, it's even worse than that.

The development of the standard model is sort of like that joke where you order a "complete model airplane kit" out of the back of a comic book, but what you receive is nothing more than a block of wood and a knife. Yeah, it's all you need, but making the airplane is so fantastically harder than you expected that you almost wish you hadn't bothered.

It reminds me that our universe is a very, very big place that is full of unexpected things. There is a certain conceit, particularly among those with only a casual interest in science but also affecting professional scientists, that our brains give us god-like powers of understanding, and that given the right facts, our power of reason will take us anywhere we care to go.

For all its success and utility, science offers only the weakest of assaults on the subtleties of the universe, and can hardly make a dent in the most elementary problems. It's kind of humbling. I don't subscribe to the idea that physics tells us much about metaphysical issues, but I do think that it's sort of pointless to speculate about big questions when the very smallest ones are so intractable.

Comments

[sig-info.livejournal.com]

I was going to say that pure mathematicians experienced a similar crisis in the 1970s, when a proof of the four-color conjecture was announced. Though the conjecture is simple to state, the proof, most of which was done by a computer, was too long for a human to understand. This caused quite a bit of debate in the mathematical ranks -- was it really a proof at all, if no person could understand the entire thing?

But then I remembered there have been many mathematical crises throughout the 19th and 20th centuries—Cantor's diagonal argument and classes of infinite sets, pathological curves, and the Banach-Tarski paradox, to name a few. So maybe the physicists should ask their mathematician colleagues for some advice on weathering a radical paradigm shift.

This reminds me that it's time to reread Morris Kline's Mathematics: The Loss of Certainty, a book I commend to you without reservation.

[Then start thinking about the infinite set of theorems that cannot be proven in this finite physical universe because there's not enough aggregate processing power. And then think about how much basic math we cannot do because our universe is finite (we can't even add numbers that are sufficiently large!). Suddenly, math starts taking on a whole new aspect, one of kids playing in a very small sandbox.]

[snousle.livejournal.com]

Very good points. The difference, though, is that the four color conjecture is not so concrete a question; it's about proving the absence of something (a map that cannot be colored), which you might imagine is more "inherently" hard, while the hadron spectrum is about computing some particular and fairly concrete numbers. That's why I find it an especially good example of this difficulty!