constants
How Many Fundamental Constants Are There?
John Baez
April 22, 2011
You might at first think that the speed of light, Planck's constant<br>and Newton's gravitational constant are great examples of fundamental<br>physical constants.
But in fundamental physics, these constants<br>are so important that lots of people<br>use units where they all equal 1! The point<br>is that we can choose units of length, time and mass however we want.<br>That's three independent choices, so with a little luck we can use them to<br>get our favorite three constants to equal 1. Planck was the<br>first to notice this, so these units are called "Planck units".
Planck units are great for quantum gravity. They are not so<br>convenient for other purposes, however. The Planck length, for example,<br>is ridiculously small: about 2 × 10-35 meters. The Planck<br>time looks even worse: about 5 × 10-44 seconds. The Planck<br>mass is 2 × 10-8 kilograms. In ordinary life, and even in<br>nuclear physics, Planck units can be a real nuisance.
But in the grand scheme of things, units are not very important.<br>They are arbitrary human<br>conventions. As long as you stick with some choice or other<br>you will do okay.
Many constants involves units of length, time, mass, temperature,<br>charge and so on. The numerical value of these constants depend<br>on the units we use. The numbers would change if we used different<br>units. Thus, though they certainly tell us something about nature,<br>to some extent they are human artifacts.
On the other hand, certain constants don't depend on the units<br>we use - these are called "dimensionless" constants. Some of them<br>are numbers like pi, e, and the golden ratio - purely mathematical<br>constants, which anyone with a computer can calculate to as many<br>decimal places as they want. But others - at present - can only<br>be determined by experiment. These tell us facts about nature<br>that are completely independent of our choices of units.
The most famous example is the "fine structure constant", e2/ℏc. Here e is the electron charge, ℏ is Planck's constant, and c<br>is the speed of light. If you work out the units involved you'll see<br>it's dimensionless, and experiments show that it's about 1/137.03599.<br>Nobody knows why it equals this. At present, it's a completely<br>mysterious raw fact about the universe!
Constants that aren't<br>dimensionless can be regarded as relating one sort of unit to another.<br>For example, the speed of light has units of length over time, so it can<br>be used to turn units of time (like years) into units of length (like<br>light-years), or vice versa. People who are interested in fundamental<br>physical constants usually start by doing this as much as possible -<br>leaving the dimensionless constants, which are the really interesting<br>ones.
How many of these dimensionless fundamental constants are there?<br>This depends on your opinion on some new developments, but my best<br>guess is 26. All other dimensionless constants (aside from<br>those built into the initial conditions) can in principle be derived<br>from these, if our best theories of physics are correct - by which<br>I mean general relativity, which covers gravity, and the Standard Model,<br>which covers all the other forces. Of course, "in principle"<br>means "not necessarily by any simpler method than by simulating the<br>whole universe"!
General relativity and pure quantum mechanics have no dimensionless<br>constants, because the speed of light, the gravitational constant, and<br>Planck's constant merely suffice to set units of mass, length and time.<br>Thus, all the dimensionless constants come in from our wonderful,<br>baroque theory of all the forces other than gravity: the<br>Standard Model.
For starters, we have a bunch of masses. There are 6 kinds of quarks,<br>one positively charged and one negatively charged of each generation:<br>up, down; charmed, strange; top, and bottom. The masses of these<br>quarks, divided by the Planck mass, give 6 dimensionless constants. We<br>also have 3 kinds of massive leptons --- electron, muon, tau. The W and<br>Z bosons also have their masses. Then there is the Higgs, which while<br>still not detected, is very much part of the theory, so we get<br>another mass.
This gives us 6 + 3 + 2 + 1 = 12 dimensionless constants so far.
Then we have two coupling constants: the electromagnetic coupling<br>constant and the strong coupling constant. The electromagnetic<br>coupling constant is just another name for the fine structure constant;<br>it describes the strength of the electromagnetic field.<br>Similarly, the strong coupling<br>constant describes the strength of the strong force - the force transmitted<br>by gluons, which binds quarks together into baryons and mesons.
You may wonder why I'm not listing a coupling constant for the weak<br>force here. The reason is that you can calculate this from<br>the numbers I've already listed.
I should warn you here: there are different ways of slicing the<br>pie. Instead of the electromagnetic coupling constant<br>together with the masses of the W, Z, and Higgs, we could have used<br>4...