How Many Fundamental Constants Are There?

John Baez

You might at first think that the speed of light, Planck's constant and Newton's gravitational constant are great examples of fundamental physical constants.

But in fact, in fundamental physics these constants are so important that lots of people use units where they all equal 1. The point is that can choose units of length, time and mass however we want. That's 3 independent choices, so with a little luck we can use them to get our favorite 3 constants to equal 1. Planck was the first to notice this, so these units are called "Planck units".

Planck units are great for quantum gravity. They are not so convenient for other purposes, however. The Planck length, for example, is ridiculously small: about 2 x 10-35 meters. The Planck time looks even worse: about 5 x 10-44 seconds. The Planck mass is 2 x 10-8 kilograms. In ordinary life, and even in nuclear physics, Planck units can be a real nuisance.

Luckily units are not very important! They are arbitrary human conventions. As long as you stick with *some* choice or other you will do okay.

Many constants involves units of length, time, mass, temperature, charge and so on. The numerical value of these constants depend on the units we use. The numbers would change if we used different units. Thus, though they certainly tell us something about nature, to some extent they are human artifacts.

On the other hand, certain constants don't depend on the units we use - these are called "dimensionless" constants. Some of them are numbers like pi, e, and the golden ratio - purely mathematical constants, which anyone with a computer can calculate to as many decimal places as they want. But others - at present - can only be determined by experiment. These tell us facts about nature that are completely independent of our choices of units.

The most famous example is the "fine structure constant", e2/ hbar c. Here e is the electron charge, hbar is Planck's constant, and c is the speed of light. If you work out the units involved you'll see it's dimensionless, and experiments show that it's about 1/137.03599. Nobody knows why it equals this. At present, it's a completely mysterious raw fact about the universe!

Constants that aren't dimensionless can be regarded as relating one sort of unit to another. For example, the speed of light has units of length over time, so it can be used to turn units of time (like years) into units of length (like lightyears), or vice versa. People who are interested in fundamental physical constants usually start by doing this as much as possible - leaving the dimensionless constants, which are the really interesting ones.

How many of these dimensionless fundamental constants are there? Somewhere between 18 and 22, depending on your opinion on some new developments. All other dimensionless constants (aside from those built into the initial conditions) can in principle be derived from these, if our best theories of physics are correct - by which I mean general relativity, which covers gravity, and the Standard Model, which covers all the other forces. Of course, "in principle" means "not necessarily by any simpler method than by simulating the whole universe"!

General relativity and pure quantum mechanics have NO dimensionless constants, because the speed of light, the gravitational constant, and Planck's constant merely suffice to set units of mass, length and time. Thus, all the dimensionless constants come in from our wonderful, baroque theory of all the forces *other than* gravity: the Standard Model.

For starters, we have 6 quarks, one positively charged and one negatively charged of each generation: up, down; charm, strange; top, and bottom. The masses of these quarks, divided by the Planck mass, give 6 dimensionless constants. We also have 3 kinds of massive leptons --- electron, muon, tau. Then there is the Higgs, which while not yet detected, is very much part of the *theory*, so we get another mass. The W and Z bosons are also massive.

This gives us 6 + 3 + 1 + 2 = 12 dimensionless constants so far.

Then we have two coupling constants: the electroweak coupling constant and the strong coupling constant. People used to think of the fine structure constant as a measure of the strength of the electromagnetic force. But since Weinberg and Salaam came up with a unified theory of the electromagnetic and weak nuclear forces, it's more fundamental to work with the electroweak coupling constant, which measures the strength of this unified "electroweak" force. Similarly, the strong coupling constant measures the strength of the strong nuclear force which binds quarks together into baryons and mesons.

That gives 2 more, for a total of 14.

But, alas, it ain't that simple. The W particle interacts with quarks in a complicated way that depends on a bunch of parameters called the Kobayashi-Maskawa matrix. The point is that the W is charged, and any positively charged quark can emit a W+ and turn into any negatively charged quark, not necessarily of the same generation. (Thus while a top can turn into a bottom, it can also turn into a strange or a down; this is how funky exotic hadrons are able to decay into the usual boring stuff we see around us, which is made of ups and downs.) We need a 3 x 3 matrix of numbers to describe the amplitude for any positively charged quark to turn into any negatively charged one by this mechanism. There is however some room to simplify this matrix by multiplying the quark fields by phases, and there are some constraints this matrix has to satisfy, too, so there are not really 9 independent numbers but only 4.

That's 4 more, for a total of 18.

Recent experiments have suggested that perhaps neutrinos have mass. There are 3 kinds of neutrinos (the electron neutrion, muon neutrino and tau neutrino), so if they all have masses that makes 21 fundamental constants.

Also, recent astronomical observations have suggested that maybe the vacuum has a nonzero energy density - this is called the "cosmological constant". If this is true, we have a total of 22 fundamental constants.

There is also a parameter in the Standard Model which measures how much the strong force violates parity - the symmetry between right and left. It's sometimes called "theta". If we count this too, we get 23 fundamental constants. However, as far as experiments can tell so far, this parameter is zero. As I said, I'm not counting "zero" or any other number you can crank out on a computer as a fundamental *physical* constant. So until we get a glimmering of evidence that this parameter might be nonzero, I'll say there are at most 22 fundamental physical constants.

22 constants is not too many - but most physicists would prefer to have none. The goal is to come up with a theory that lets you calculate *all* these constants, so they wouldn't be "fundamental" any more. However, right now this is merely a dream.