Physics Notes: The Numerology of Physical Constants

When you learn physics, you learn many constants.

The one that almost everyone knows is the speed of light, c = 300,000,000 m/s. Almost as well known is the gravitational constant. No, I don't mean the (not so constant) value of gravitational acceleration on the Earth's surface, but the universal gravitational constant: G = 6.672·10–11 m3/kg/s2. Then there's Planck's constant: ħ = 1.0546·10–34 m2kg/s.

You'd expect that any physical theory worth its salt would be able to predict the values of these constants, derived from first principles. Right?

Wrong.

For starters, the speed of light is only 300,000,000 m/s (actually, more like 299,792,458 m/s) because we, humans (that is, the part of humanity that does not reside in countries still using an archaic system of measurement), choose to measure distances using the meter, and time using the second. Instead of the second, we could choose a different unit of time: for instance, the time it takes for light to travel exactly 1 meter. In that case, we would effectively make 1 second equal to 3·108 meters. In other words, whenever you see the symbol s as part of a quantity, you can replace it with 3·108 m. Let us give this a try in the value of the gravitational constant:

G = 6.672·10–11 m3/kg/s2 = 6.672·10–11 m3/kg/(9·1016 m2) = 7.413·10–28 m/kg

Meters per kilogram? That suggests a new possibility. We could make G equal to one* if only we started measuring mass not in kilograms, but in units of 7.413·10–28 meters. So now, instead of using three different fundamental units, meters, seconds, and kilograms, we're down to one: meters. It may sound a bit strange to measure time and mass using a unit of length, but it works**.

But we can go a step further. Substituting our new units into Planck's constant, we get:

ħ = 1.0546·10–34 m2kg/s = 1.0546·10–34 m2kg/(3·108 m) = 1.0546·10–34 m2(7.413·10–28 m)/(3·108 m) = 2.606·10–70 m2

What this tells us is that we can set Planck's constant to unity, but this will fix our unit of length: this fundamental unit will be the square root of the value above, or 1.614·10–35 meters.

In these "natural" units, all three so-called fundamental constants will be 1. The fact that we're not using "natural" units has nothing to do with physics. It is purely a human choice.

One constant we cannot do away with is the charge of the electron. Once we have set c = G = ħ = 1, we no longer have the freedom to set the charge of the electron to one. Why? Consider the Coulomb force law: F = q1q2/r2. In SI units, force is measured in Newtons, distance in meters, so charge would have to be measured in meters times the square root of Newtons. That's a bit weird, which is perhaps why the people who constructed the SI system introduced a new fundamental unit, the Ampere, as the measure of current, and A·s, or Coulombs, as the measure of charge. But if we used SI units, a new physical constant would be needed in the Coulomb force law, and what we want to do here is to get rid of unneeded constants, not to introduce new ones. So let us try and measure charges in N1/2m, or m3/2kg1/2s–1. Incidentally, this has been done before: in the cgs system of units, still preferred by many physicists, charge is measured in units of cm3/2g1/2s–1, also called an esu (electrostatic unit). The charge of an electron is 4.803·10–10 esu or 4.803·10–10 g1/2cm3/2s–1. Replacing g with (10–3 kg) and cm with (10–2 m) we get:

4.803·10–10 g1/2cm3/2s–1 = 4.803·10–10 (10–3 kg)1/2(10–2 m)3/2s–1 = 1.519·10–14 kg1/2m3/2s–1

Now we can go a step further and replace seconds with (3·108 m) and kilograms with (7.413·10–28 m):

1.519·10–14 kg1/2m3/2s–1 = 1.519·10–14 (7.413·10–28 m)1/2m3/2(3·108 m)–1 = 1.378·10–36 m

This is the charge of the electron, expressed in (what else?) units of length, i.e., meters.

But we already have a better fundamental unit of length: 1.614·10–35 meters. If we use this unit, the charge of the electron will be 0.0854 units. For reasons best known to quantum field theorists, it is not this value but its square that is viewed as a fundamental constant. This is the so-called "fine structure constant", roughly 7.294·10–3, or 1/137.

You cannot set the fine structure constant to 1 without changing c, G, or ħ. If the fine structure constant is one, either G or ħ must be set to 11.7 (the square root of 137) or c must be set to roughly 0.44 (the reciprocal of the sixth root of 137.)

That the fine structure constant is a relatively small number is a great blessing to quantum field theory: it allows so-called perturbative methods, which deal with ever increasing powers of 1/137, to converge quickly and yield accurate solutions. As to why this number is 1/137, that remains an unsolved physics mystery.

*We must be careful with this. The gravitational constant appears, among other places, in Einstein's equation, where it is a factor in a relationship connecting tensors of different weight. Thus arguably, G can never be viewed as a truly dimensionless constant.

**This stuff about measuring everything using units of length actually does make sort of sense intuitively. Consider: whatever measurement a physicist makes, in the end what is being measured is a distance. The distance a needle of an instrument or the hand of a clock travels, for instance. One way to measure time is to measure the distance a ray of light covers during that period of time. The way to measure mass is to let two identical masses, separated by a unit distance, attract each other gravitationally, and then measure their velocity after a unit of time has elapsed. Velocity, in turn, is distance per unit of time. In other words, once you have a yardstick, you can measure everything, and a natural standard for a yardstick is provided in the form of the Planck unit (for that's what it's called) of 1.614·10–35 meters (Planck length), equivalent to 5.38·10–44 seconds (Planck time), or 2.18·10–8 kilograms (Planck mass).