The (Dimensionless) Constants of Nature

As noted yesterday, someone going by “who” (who may or may not be a doctor) took me to task in the comments to the dorky poll for talking about fundamental constants that have units, preferring dimensionless ratios instead:

I would be really interested to hear what your readers come up with if the popularity constest was about DIMENSIONLESS physical constants—-which are significant RATIOS built into the universe. Like the ratio of mass of proton to mass of electron, or the ratio of Planck mass to proton mass.

these are things you can’t simply make become = 1 and in effect “go away” merely by adjusting the system of units. and so they are enduring mysteries.

At first glance, this might seem just like the sort of cranky objection made by a theorist who has been mocked once too often for setting Planck’s constant and the speed of light both equal to one. But there is an important practical point here, especially when it comes to the discussion of possible changes in the fundamental constants, as mentioned recently by Nathan at ICAP, and discussed in more detail by Rob Knop. There is a very real sense in which these dimensionless ratios are the most important fundamental constants, and there’s even a whole book taking that view (from which I stole the post title).

I’m going to talk about the meaning and measurement of one of these dimensionless constants, starting from the practical question of how you measure changes in fundamental quantities (because I find it easier to understand things when there’s a concrete starting point). This ended up being really long, so it will be split over two posts, beginning after the cut:

The question of whether the fundamental-constants-with-units– things like the speed of light or Planck’s constant– are really constant is a fascinating one, and a topic that has received a fair amount of experimental and observational attention over the years. There’s no reason why these things couldn’t change, after all– we tend to think of them as fixed and unchanging values, just because it’s easier to deal with a universe whose fundamental properties aren’t changing over time, but there’s no reason why the universe should be arranged in such a way as to make things easy for physicists. There would be all sorts of interesting consequences if the speed of light were different now than ten billion years ago, and that sort of thing is enough to get at least some scientists to attempt the fiendishly difficult measuremetns you need to do to detect these effects. The change, if there is a change, is expected to be very, very small, so in order to have any hope of measuring it, you need to look at these values over a very long time.

The big problem, of course, is that there’s no direct way to measure the speed of light now and compare it to the speed of light ten billion years ago. The universe doesn’t come with one of those fixed arrays of clocks and meter sticks that you see in textbooks on special relativity, after all, so we have no way of doing a direct measurement of the speed of light. We don’t have a way to measure the time taken for light to move a known distance ten billion years ago, and compare that to the time light takes to cover the same distance today. We could, in principle at least, measure the time light takes to travel a fixed distance now, and again a year from now, and get a rate of change from that, but the precision that would require is ridiculously unreasonable.

What we can do is measure a ratio between the speed of light (if that’s the quantitity we care about) and something else. If we can measure how big the speed of light was relative to some other fundamental quantity ten billion years ago, and compare that to the same ratio today, that lets us put a limit on the change in the speed of light (assuming the other quantity remained constant).

It turns out that there’s a very natural way to do this, through the dimensionless ratio known as the fine structure constant. This is the ratio of the electron charge squared to Planck’s constant times the speed of light (with a fudge factor or two in there depending on what system of units you’re using). Whatever units you’re using, it has the same value, roughly 1/137, and is usually represented by the Greek letter alpha, because “alpha” is easier to type than “fine structure constant.”

There are a couple of different ways to understand this as a physical concept– historically, it shows up as the ratio of the speed of an electron in the Bohr model of hydrogen to the speed of light (which is useful in that it shows that you can treat the electron’s motion non-relativistically, to a very good approximation). In more abstract terms, you can look at it as a quantification of the strength of the electronmagnetic interaction between two particles– another easy way to arrive at the fine structure constant is to take the ratio of the energy of two electrons separated by some distance to the energy of a single photon having a wavelength equal to the distance between the electrons (which is the key limit in the quantum electrodynamics picture where the interaction is described in terms of charged particles passing photons back and forth).

It’s also a convenient way to characterize the strength of the electromagnetic interaction– alpha shows up as the “coupling constant” in QED, the thing that determines how strong the interaction between particles is. You can also construct similar coupling constants for the other forces, usually in terms of some ratio between a constant and hc, and you’ll sometimes see those thrown around as a means of comparing the strengths of different forces.

However you understand it, the important thing is that this constant shows up all over the place, causing effects that allow it to be measured over long stretches of time. Relatively easily, anyway. But that’s the topic for the next post…