How many fundamental constants do we know of? (Or at least, constants we think are fundamental. I suppose it’s possible we might discover some of them are actually determined by the others.)
I’d rather not double count anything. E.g., if we count both the permeability and the permittivity of free space, it’s not necessary to also count the speed of light in a vacuum. Perhaps the best way to go would be to count how many dimensionless ratios of fundamental constants the universe depends on, since these are in some sense more fundamental (being independent of our particular choice of units.)
Yup, that list of 26 that John Mace posted pretty much covers it.
The speed of light c isn’t really fundamental, since it depends on the units I choose. I could decide to measure lengths in flurkles and times in schnitts, and find that the speed of light is 2396.284 flurkles per schnitt; it’s just as good a number as 3 x 10[sup]8[/sup]. As for the rest, from Baez’s article:
An interesting aside: In the early days of string theory, it was hoped that once we understood it well enough we’d be able to figure out why these constants have the values they do, in terms of some small number of even more fundamental constants (one or two, hopefully.) In other words, we would be able to derive the electron mass, say, from first principles. However, in recent years there’s been a growing suspicion that it’s possible to derive any values of what we know as the fundamental constants from string theory, or at least an awful lot of possible sets of values (I’ve seen estimates as high as 10[sup]500[/sup].) Debate is ongoing.
Pi, e, and the Golden Ratio are all mathematical constants, not physical ones. If one accepts them as “fundamental”, then one must accept every other real number (or at least every other definable number) as fundamental.
Some things which are usually considered fundamental constants don’t show up on Baez’s list, such as the speed of light, Newton’s constant, Planck’s constant, and Boltzmann’s constant. This is because, to folks working in the appropriate fields, all of these are generally considered to equal 1. That is to say, for instance, that space and time are two different aspects of the same thing, and c is just a conversion factor since we stubbornly use two different sorts of units for them. But 3*10[sup]8[/sup] meters per second is really no more fundamental than 2.54 centimeters per inch.
The cosmological constant is debateable; there’s a lot of work in trying to relate it to the others in some way. Admittedly, that work contains the single most spectacular failure ever in the history of science, but nonetheless, there’s still a strong suspicion that some relationship exists, and the cosmological constant may in fact not be a constant at all, much less a fundamental one.
I’m also curious about the listing of four constants each for the two mixing matrices; I had understood that there were only three independant ones. It’s basically a problem (in each case) of orientation of a coordinate system in three dimensions, and that only requires 3 Euler angles.
Oh, one other thing to add: One could, in principle, define Newton’s constant (capital G) using units based on (say) the electron mass, Planck’s constant, and c. If one did so, then one would consider the electron mass to be just a mass scale, and call G a fundamental constant. But this would not change the total number of fundamental constants; it would just be a relabelling. Which ones you label as fundamental constants is largely a matter of aesthetics, but how many there are is not.
And I hadn’t read Baez’s full page before writing my other post, but he says more or less the same thing as I. His explanation is perhaps a bit clearer, though, so you might want to read that instead.
When Baez refers to those masses, he really means those masses divided by the Planck mass, which is (h-bar*c/G)[sup]1/2[/sup]. This gives a dimensionless (i.e., unit-less) value, which can thus be considered truly fundamental.
In other words, the dimensionless value for the electron mass is m[sub]e[/sub]G[sup]1/2[/sup] / (h-barc)[sup]1/2[/sup]. Which is why there’s no need to count c, h-bar, and G as separate fundamental constants. How much of the fundamental value is contributed by m[sub]e[/sub] and how much is contributed by G (and so forth) is just a matter of your choice of units.
It’s a unitary matrix, not an orthogonal one. You have the three Euler angles corresponding to a rotation, and six phase angles. You can then redefine each of your six quark fields by six independent phase rotations, but one of these will be redundant (since if you multiply your whole wavefunction by a phase, you shouldn’t be able to tell the difference.) So there’s one non-trivial phase angle left over in the matrix.
Note that back when they thought there were only two generations of quarks, they could make the mixing matrix real: a 2x2 unitary matrix has one rotation angle and three phase angles, and you have enough non-trivial phase redefinitions in your four quark fields to be able to eliminate all three.
Yep, I used the tried and true method of picking a username by combining my first name and my favorite number. (Well, my favorite number times 100 rounded to the nearest integer.)
Regarding the fine structure constant, as Baez notes you can either include
the expectation value of the Higgs field
the U(1) coupling constant
the SU(2) coupling constant
or
the fine structure constant
the mass of the W boson
the mass of the Z boson
in the list of fundamental constants, and derive the other three (using also the Higgs mass). So there’s some flexability about what you want to call fundamental. The first three appear in the theory in a more basic way, but the latter three are easier to measure experimentally.
Speaking of the Higgs, here’s another question: If the LHC is successful at detecting the Higgs, is it thought that this will lead to a better understanding of why the other particles have the masses they do? Or would it simply be confirming that the Higgs mechanism is indeed how mass arises, with us needing some other theory (M-theory?) to explain the values of mass we see?
Feynman of course knew that. The author of that website apparently does not.
I personally have never seen much significance in comments like the following:
Isn’t this just another example of using h and c to make things dimensionless? It’s basically just an observation that h times c has units of charge squared (for 4[symbol]pe[/symbol][sub]0[/sub] = 1).
I also don’t really understand why the fine structure constant is so much more talked about than, say, the strong coupling constant.
Sure, you can look at it that way. But the point is, the ratio of electron mass to Planck mass would be the same regardless of how we defined our units, whereas, the ratio of electron mass to gram depends on the meaning of the word “gram”.
I suppose one could say, “Well, in the first case it depends on the meaning of the words ‘Planck mass’.” But the difference is that the Planck mass is defined in terms of measurable properties of the universe, which presumably are the same everywhere. Whereas a gram is currently defined as 0.001 times the mass of a particular platinum-irridium object located in France.
So, given suitable instructions, aliens from another galaxy with no knowledge of our terrestrial units of mass could compute the ratio of the electron mass to the Planck mass and get the same value as we do here on Earth. Whereas, they couldn’t do the same for the ratio of the electron mass to one gram without us telling them the conversion factor.
Tim314
Exactly. I have no idea why 137 (the reciprocal of the fine structure constant) is given such significance either.
Personally, I’ve always wondered why the proton to electron mass ratio is 1,836.15267261
The atom is one of the most fundamental “structures” in the Universe. You would think that the mass of the objects at its center would be some “neat” multiple of the mass of the objects which orbit it. (Something like 2,000 electrons = 1 proton mass) as opposed to 1,836.15267261 electrons = mass of 1 proton.
Well, (in the words of Sir Arthur Eddington), “Not only is the universe stranger than we imagine, it is stranger than we can imagine”
Except the fact that “2000” seems like a nice round number is just an artifact of our counting system. Granted, it could do us the favor of at least being an integer, but no one consulted me on the matter.