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started some remarks at physical unit. But I really need to stop with that now and do more urgent things…
Thanks for writing this, Urs! (There was a bit about units of measurement at unit, but I agree it’s probably better to have a stand-alone entry.)
Where you speak of constants of nature: one could get the impression that you mean just dimensionless constants like the famous “137”. I would have thought quantities like $c$, $h$, and the Planck length might also count: they are dimensionful, but the quantities to which they refer are natural and fundamental: God-given units of measurement so to speak. What would you say?
Right, that’s what I wanted to say, that a “constant of nature” strictly speaking should be a dimensionless number.
For quantities such as c, in isolation they are artifacts of our description of the world, not of the world. That’s why when you go visit somebody’s office at the LCH, you’ll never see them write any “c” or “h”. (The way they say is that “all these constants are set to 1”).
(This is something to shock the “man on the street” with: the man on the street thinks that a big insight of Einstein was that $E = m c^2$. But actually what hep physicists write is $E = m$. (Or rather: $E^2 = m^2 + p^2$).)
Of course if the speed of light would actually change over time or in space, and you could compare $c_1$ and $c_2$, then the quotient $c_1/c_2$ would indeed by an intrinsic property of nature.
Right, I am familiar with things like $E^2 - p^2 = m^2$.
The story I’ve always told myself is that “constants” like $c$, $h$, etc. allow one to make canonical identifications of dimensions, e.g., via $c = 1$ we have an explicit identification $[length] = [time]$, etc. But I’d never heard we should not call them constants. What would be the right term then?
It is entirely common to call them “constants”, but I am just suggesting that fundamentally “units” is really the right term. But I am not trying to stop people say that the speed of light is a “constant”.
So a meter is a physical unit, and a second is, and so is the quotient of a meter by a second, and so is any multiple of this.
But of course it is entirely common to say “constants” here. I don’t mean to fight about this. I just made a remark.
If one believes that a “constant” has to be a number, then I can certainly see the argument that something like the speed of light is more of a “unit”. But mightn’t we be able to make sense of calling it a “constant” if we allow a constant to be an element of some 1-dimensional real vector space that lacks a canonical basis? E.g. the speed of light would be an element of $Hom(T,D)$ where $T$ is the 1D real vector space measuring times and $D$ the 1D real vector space measuring distances (neither of which has a canonical basis either)? (The speed of light is a bad example here, because once we move to special relativity $T$ and $D$ are not canonically singled out as subspaces of spacetime, but maybe you get the point.)
Urs, I certainly accept what you say, but I think I’d like to slightly reword some of what appears at physical unit.
Physical units are often called physical constants. But by definition physical units are arbitrary choices made in the desciption of a physical system. Of course once made, one wants to keep these choices constant, such as to be useful.
Is it usual to refer to a meter as a “physical constant”? I thought what was meant in the looser meaning of physical constant is (quoting Wikipedia) some “quantity that is generally believed to be both universal in nature and constant in time”, e.g. the speed of light $c$. Of course it makes no sense to call $c$ a mathematical constant, precisely because it is dimensional. But still, the physical quantity $c$ comes from nature, and within physical theory as it has been developed, is less arbitrary (for lack of a better word) than say 1m./sec, whose significance is only historical.
So I guess what I propose is to replace the first cited sentence with “Some physical units, often called natural units (see below), are often called physical constants. […]” Then point to a section on natural units. Of course humans may choose between different systems of natural units, e.g., do we choose Planck units or something else.
I agree and tried to indicate right away,that the entry could do with a lot more editing, now that I have started it. If you feel energetic, I would like to invite you to just edit it. I don’t have the leisure right now. I am generally not dogmatic about this issue and will not start a fight whichever way you edit it. The only reason I started this entry was since I had been editing “Planck’s constant” and felt I needed a link to “physical unit”.
But while I am typing, two quick replies on the above:
re#7
you say that $c$ is given by nature but $1 m/s$ is given by human history. But $c$ is precisely that, too: a fact of nature expressed relative to human history, namely 299792458 times that historical accident.
Consider this: there is no sense in asking “What would a universe be like in which the speed of light were twice of what it is in ours, but otherwise all the laws of nature are the same.”
Instead, all that makes sense are relative statements, such as “The typical speed scale of those self-replicating amino-acids on that blue planet is roughly 300000000 times smaller than the speed of light”.
re #6:
I think the point here is that, yes, the speed of light in our theory will be a fixed (and in this sense constant, yes) element in some abstract vector space, but to apply the theory to nature we next need to identify this abstract vector space with the one “in nature”. It is that second identification which makes the unit.
But, as I said, I won’t try hard to make anyone not say “constant of nature” for “speed of light”.
Another attempt, to make this more pronounced:
consider a spacetime given by a pseudo-Riemannian manifold. Just that. What would be the “physical quantity” that should be called the speed of light, independent of a further choice of chart? Consider two spacetimes, hence two pseudo-Riemannian manifolds. Would there be a way to say that in one of these worlds the physical quantity which is the speed of light is different from that in the other?
Similarly for $\hbar$: consider a circle bundle with connection with curvature a given symplectic form. What would that “physical quantity” be which should be called “Planck’s constant”, independent of a choice of identification of the circle with a quotient of the real line?
Is there some wisdom to mine from Cafe posts, such as Dimensional Analysis and Dimensional Analysis and Coordinate Systems? In the latter, we have you posing questions similar to those above.
Thanks to you both, Urs and David. I’ll spend some time today looking at those Café posts.
I still have a question which I don’t think has been answered. It’s a question about language practice, and it should be pretty easy for people like Urs and David.
Let us grant that “constants” such as $c$, $G$, $h$, taken in isolation, are not meaningful; that only dimensionless quotients which compare $c$ to some chosen standard like 1 m./sec. are meaningful. Still, one hears these referred to as “physical constants”. As far as this language practice goes, I only seem to hear these things like $c$, $G$, $h$ (which have a theoretical significance) referred to as physical constants, and not ordinary SI units like 1 m./sec. Or am I wrong – do various circles refer to SI units like “meter” as “physical constants” as well? (Is the question clear?)
Again, this is not a question of what people ought to do, it’s a question of what people in fact do do (for better or worse). And it’s not a rhetorical question – I honestly am not sure of the answer.
Thanks, Urs!
The entry physical unit remained a bit non-concrete. I have jotted down a paragraph concretely on units of length in Lagrangian field theory: here.
Together with the rest of the entry this deserves more attention. Hopefully later…
Should the section “Units of measurement” at unit have a pointer to physical unit for further discussion? I would assume so, but I don’t quite understand the technical business that’s going on at physical unit, so I thought I’d check.
added pointer to:
re #16: Yes. I have now hyperlinked “unit of measurement” (there) and made that redirect to physical unit
Added:
For a mathematical description of physical units and the associated “physical dimensions”, including a discussion on how densities can be used to define real and complex powers of physical quantities, see
Changed the examples (a wavelength, and the meter) to use the positive reals rather than nonzero reals.
I don’t see that it makes a lot of sense to consider either of these quantities potentially being negative. For example, the meter is the unit of length, and many lengths are fundamentally positive or non-negative numbers: the prototypical length is basically the norm of a displacement vector, and being a norm is non-negative. If we were to consider the torsor from which the meter is chosen to include negative reals, and if we were to choose the meter to have the opposite sign from what it actually has, then we’d have to say that the bond length of dihydrogen is minus so-and-so many picometers, the radius of the Earth is minus 6,370 kilometers, and so on.
If one wants an example of a physical quantity whose torsor’s group really is the nonzero reals, both positive and negative, then electrical charge could provide a good example. That is, the coulomb (a physical unit) and the charges of the electron and proton (physical constants, but not dimensionless ones) belong to the same torsor; and we unknowingly chose the coulomb to have the sign of the proton’s charge, the opposite of the electron’s charge, which has been an inconvenience ever since but fundamentally works just fine, demonstrating that the relevant torsor really is the whole nonzero reals.
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