Wednesday, July 05, 2017

"A Man Who Knows Everything"

Gerald 't Hooft:
"Following his family's footsteps, he showed interest in science at an early age. When his primary school teacher asked him what he wanted to be when he grew up, he boldly declared, "a man who knows everything."
Some background: 't Hooft is a Dutch theoretical physicist. In graduate school at the University of Leiden, he worked under the also-now-famous Martinus Veltman to study a interesting theory that was emerging at the time, called the Yang-Mills theory of the strong interaction.

His last name is pronounced something close to"tooft."

(I sat in a talk Veltman once gave at Stony Brook, when I was a graduate student. I don't remember much of it, but the room was small and crowded and I remember he had a big kind-of jolly authoritative air about him. And that everyone very much respected him. One of the most prominent professors in our department, Peter van Nieuwenhuizen, co-discovered supergravity -- basically Yang-Mills + gravity -- and was also Veltman's student. I wrote about him here for Physics World.)

The strong interaction is also called the nuclear force -- it's what keeps all the positively charged protons in the nucleus from repulsively exploding outward -- it's why the atomic nucleus can exist. Those protons are all positively charged, of course -- so how do they all stay together in the nucleus? It's because there is a force even stronger than electromagnetism -- the strong force. Yang-Mills theory is the theory of the strong force, of interacting quarks and gluons.

This quantum field theory had the same problem as did the earlier quantum electrodynamics -- a quantum field theory of electrons and photons developed by Richard Feynman, Julian Schwinger and Tomonaga: calculations of real-world properties like how an electron careens off another electron gave results that were infinite.

Here's what separates physicists from mathematicians -- in the face of calculations that gave infinities and so were clearly(?) wrong when compared to the real world, the physicists plowed ahead anyway, looking more deeply into their equations and finding other infinities that cancelled out the first infinities. In essence they claimed

∞ - ∞ = some actual finite number that actually
agrees with experiments, like
maybe 1 over 137.036 times pi.

Talk about chutzpah! This looks absurd, but it works. Quantum electrodynamics was said to be "renormalizable." Freeman Dyson was the first to show this, when still a very young man. (He also showed that Feynman, Schwinger and Tomonaga's separate theories were equivalent [PDF].)

To my knowledge, quantum electrodynamics has never yet been proven, in a mathematically rigorous way, to give finite results like this. But some mathematicians are still trying. But the physicists didn't care (so much) for rigor, but for results.

And since then, physics students have been taught to "just shut up and calculate."

Anyway, 't Hooft, also as a very young man, showed that Yang Mills theories were also renormalizable -- infinities could be subtracted from infinities to give finite answers that also agreed with experiments.

't Hooft's calculations were more difficult than was Dyson's -- which was already tough enough -- because unlike electrodynamics, where the photons exchanged between electrons have no electric charge, the gluons of Yang-Mills theory that carry the force between quarks DO have a charge. It's just not an electric charge, but what's called "color charge."

As you know, electrons have a negative electric charge. Their antimatter partner, positrons, have a positive electric charge. And the photon, which carries the force between electrons and electrons, or electrons and protons, or electrons and positrons, etc. has no electric charge. That keeps things (relatively!) simple.

In the theory of the strong force, the particles that make up protons and neutrons, the quarks, also have a charge. It's not an electric charge, but was whimsically (and arbitrarily) given the name "color" charge. It comes in three values, not two: red, blue, and green. It's just a number, a property of quarks -- we're just not as "used to" color charge as we are electric charge. But if you think about it, we don't know what electric charge really is either -- we just know how particles and objects with it behave and interact, and we get used to not knowing more. (And maybe that's all there is to know about electric charge, anywayt?!)

So there are quarks with a red "charge," some with a blue "charge" and a green charge. And anti-quarks of the same color charges, or anti-red, anti-blue, etc.  These charges attract and repel in various ways -- there are eight gluons that take care of all that, emitted and absorbed by the quarks.

There's more, but this is our basic mental picture of it all.

The gluons also carry color -- actually, two colors, a color and an anti-color. This makes the Yang-Mills theory much more complicated, requiring "group theory," that you've probably heard mentioned at some point. The calculations are much more involved.

What 't Hooft showed, in the early 1970s, was that the calculations of Yang-Mills theory could also be made, despite all the infinities lurking everywhere, to give finite results. It was "renormalizable." That it, it was useful. It gave results, like 4.56309, that could be compared to the real world.

One of the ways 't Hooft and Veltman did this was simple but clever: instead of assuming the world had four dimensions -- time, length, width and height -- they assumed it has a little more of a dimension. Not five dimensions, just a little over four.

In they way mathematicians have used the letter "epsilon" to denote a very very small quantity -- this goes back to basic calculus -- 't Hooft and Veltman assumed space -- really spacetime, as we've known since Einstein -- had not four dimensions, but 4+epsilon dimensions.

It's just a mathematical trick. You assume that's what spacetime looks like, do your calculations with all the infinities, add and subtract them, and then in the end set episolon equal to zero. Simple, right?

Well, this technique, called dimensional regularization -- gives Yang-Mills answers that are finite, not infinite. Amazingly. It's a ludicrous trick, but it works.

For this, 't Hooft and Veltman received the 1999 Nobel Prize in Physics. I wonder if, after all that, 't Hooft thought, maybe for just a day or two, that he did indeed know everything.

There's a lot more to 't Hooft, which you can get a sense of by perusing his Web site.

1 comment:

William Connolley said...

There are precedents for the epsilon -> 0 stuff, so while it's a pretty neat trick and I'd never have thought of it, it's perhaps not totally surprising. The obvious example is differentiation, after all. http://mathworld.wolfram.com/AnalyticContinuation.html is also perhaps similar.