What is the meaning of a fundamental constant? Dimensioned or dimenionless

[nonequilibrium]

Gerenuk, that doesn't really seem to be the point? With that reasoning, you're basically saying k is equally fundamental as c, e, ...? We were talking about if k is less fundamental than the other fundamental constants. First you said k is not important because it depends on the existence of a superfluous unit. Then I argued that e or G could be of the same type, and then you conclude that all dimensioned constants are equal (and according to you, equal in their "unfundamentalness"). It seems like we're hopping around without knowing what the point is, or maybe that's me, I'm just a bit confused about what is actually being claimed. So are you still saying k is of a lesser nature than e or c? And is e of a lesser nature than c? Or are they all of the same nature?

 

[Gerenuk]

My first guess is that indeed all the constants you mention are on an equal footing. I mean the second is defined by an atomic transition, which is not better than defining temperature by the melting point of water.
So in the end only the dimensionless constants are really fundamental.
Does that make sense?

 

[nonequilibrium]

Well it depends on what you call fundamental, so to avoid arbitrary semantics, I'll define different ways of fundamentality:

prerequisite: constant must be independent of space and time (this assumes the units themselves can also be considered independent of space and time)

A) a constant is fundamental if its value doesn't arise out the theory [in principle, for as far as we know] but has to be put in (this means that what is considered fundamental can change in time, but I see no objection to that)

B) a constant is fundamental if it does not contain a superfluous unit

C) a constant is fundamental if and only if a change in its value corresponds with an observational change in our universe

It is clear that only dimensionless constants have c-fundamentality, and I assume this is what Gerenuk was talking about these lasts posts.

What constants have a- and/or b-fundamentality? If k follows out of chemistry which follows out of QM (which seems very plausible), then k does NOT have type 1-f. I would think c, e, G, h would have type 1-f. I suppose either epsilon or µ too, the other one is then determined by c = 1/sqrt(epsilon µ).

As for type 2-f: NOT e, G, k?

 

[Dale]

Temperature is, in a sense, another of those non-basic units. The basis for temperature is energy. If we instead measured temperature in units of joules/molecule or joules/mole there would be no need for Boltzmann's constant or the ideal gas constant.

This is not quite correct. It only works for an ideal monoatomic gas. The correct definition of temperature is dS/dU=1/T, it just happens to work out that for an ideal gas it becomes proportional to the energy per molecule. For other substances it can be more complicated.

However, your overall point is correct, if mass, length, time, and entropy are considered the basic units then temperature is a non-basic unit. Of course, I always feel like my brain is becoming disordered when I think too much about entropy.

 

[Rap]

Boltzmann's constant is just a conversion factor to convert temperature to an energy. You could have temperature defined in units of energy - every where you see kT, just replace it with T, every where you see T alone, leave it alone. The question of what does that energy represent is rather complicated. You could say that it is twice the energy per molecule per degree of freedom of that molecule, but this breaks down when quantum considerations are introduced which can partially "freeze out" some degrees of freedom.

Dimensionless constants are more fundamental than dimensioned constants and it is true that if the dimensionless constants are fixed, then the dimensioned constants could be varying all over the place, and we would have absolutely no way to detect it - which means that the whole idea of "what if I changed the speed of light" has no meaning if the dimensionless constants are kept the same.

There are many fundamental dimensionless constants - one for every force - the electromagnetic coupling constant (aka the fine structure constant), the gravitational coupling constant, the weak coupling constant, the strong coupling constant. Also the ratio of the mass of any elementary particle to the mass of an electron are fundamental dimensionless constants. Of course by "fundamental" I mean that we presently have no way of deriving them, not that they are by definition underivable. The dimensionless "numbers" (e.g. Reynolds number) of fluid dynamics are not constants, they are different for different situations.