How many fundamental constants are there?

[tim314]

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.)

[John Mace]

According to John Baez, there are 26. (http://math.ucr.edu/home/baez/constants.html)

# the mass of the up quark
# the mass of the down quark
# the mass of the charmed quark
# the mass of the strange quark
# the mass of the top quark
# the mass of the bottom quark
# 4 numbers for the Kobayashi-Maskawa matrix

# the mass of the electron
# the mass of the electron neutrino
# the mass of the muon
# the mass of the mu neutrino
# the mass of the tau
# the mass of the tau neutrino
# 4 numbers for the Maki-Nakagawa-Sakata matrix

# the mass of the Higgs boson
# the expectation value of the Higgs field

# the U(1) coupling constant
# the SU(2) coupling constant
# the strong coupling constant

# the cosmological constant

[Jpeg Jones]

Okay, never heard of those.

How about Pi, e, C, the Golden Ratio, etc?

[MikeS]

Yup, that list of 26 that John Mace posted pretty much covers it.
How about Pi, e, C, the Golden Ratio, etc?
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 108. As for the rest, from Baez's article:
Some [constants] 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.
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 10500.) Debate is ongoing.

[Jpeg Jones]

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;Say what? And "the mass of the up quark" isn't measured in some kind of unit? How can something be measured without units?

[Chronos]

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*108 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.

[Chronos]

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.

[wolf_meister]

Well, according to the National Institute of Standards and Technology (NIST) here is a rather thorough listing of fundamental constants:
http://physics.nist.gov/cuu/Constants/Table/allascii.txt

Physicist Richard Feynman thought that the fine structure constant 137 http://www.137.com/137/ had a special significance.
The fine structure constant is not in the NIST list posted above but is in this list also at the NIST http://physics.nist.gov/cgi-bin/cuu/Category?view=html&All+values.x=53&All+values.y=9

And not to correct the late Richard Feynman, but 137 is the reciprocal of the fine structure constant. (and the more precise value is 137.0359991)

Tim314 - would 50% of your SDMB name have something to do with a circle's cirumference to diameter ratio ?

[tim314]

Say what? And "the mass of the up quark" isn't measured in some kind of unit? How can something be measured without units?
When Baez refers to those masses, he really means those masses divided by the Planck mass, which is (h-bar*c/G)1/2. 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 me*G1/2 / (h-bar*c)1/2. 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 me and how much is contributed by G (and so forth) is just a matter of your choice of units.

[MikeS]

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.
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.

[tim314]

Well, according to the National Institute of Standards and Technology (NIST) here is a rather thorough listing of fundamental constants:
http://physics.nist.gov/cuu/Constants/Table/allascii.txt

Physicist Richard Feynman thought that the fine structure constant 137 http://www.137.com/137/ had a special significance.
The fine structure constant is not in the NIST list posted above but is in this list also at the NIST http://physics.nist.gov/cgi-bin/cuu/Category?view=html&All+values.x=53&All+values.y=9

And not to correct the late Richard Feynman, but 137 is the reciprocal of the fine structure constant. (and the more precise value is 137.0359991)

Tim314 - would 50% of your SDMB name have something to do with a circle's cirumference to diameter ratio ?
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?

[tim314]

Physicist Richard Feynman thought that the fine structure constant 137 http://www.137.com/137/ had a special significance.And not to correct the late Richard Feynman, but 137 is the reciprocal of the fine structure constant. (and the more precise value is 137.0359991)
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:Now, alpha is nothing more, nothing less than the square of the charge of the electron divided by the speed of light times Planck’s constant. Thus this one little number contains in itself the guts of electromagnetism (the electron charge), relativity (the speed of light), and quantum mechanics (Planck’s constant). All in one number!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 4pe0 = 1).

I also don't really understand why the fine structure constant is so much more talked about than, say, the strong coupling constant.

[gazpacho]

When Baez refers to those masses, he really means those masses divided by the Planck mass, which is (h-bar*c/G)^.5. This gives a dimensionless (i.e., unit-less) value, which can thus be considered truly fundamental.
I see this as giving the electron mass in units of Planck mass as appose to say grams.

[tim314]

I see this as giving the electron mass in units of Planck mass as appose to say grams.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.

[wolf_meister]

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"

[Excalibre]

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.

[wolf_meister]

Excalibre
:smack:
Yes, excuse my "base-10-centric" mentality.
But as you said it would be nice if the ratio were at least an integer.

[Fish]

"The bureaucratic mentality is the only constant in the universe."

--Leonard McCoy, MD