The n-Category Café

I remember Eddington did a lot of fundamental thinking about dimensional analysis in his late years when he was working on his fundamental theory of physics. If I am right this theory wasn’t taken really seriously, but his thinking about dimensional analysis made sense. I will have to look it up but probably someone over here knows more about his work.

The first thing which springs to my mind is the introductory chapter to Nigel Goldenfeld’s book on the renormalization group. (Pauses to unpack book from box.) The example Goldenfeld takes is the relation between the phase speed of shallow-water waves and the depth of the water. In the limit that the depth is small compared to the wavelength, he derives that the phase speed $c$ scales as the square root of the depth $h$. Given that the physically significant variables are $h$, the gravitational acceleration $g$ and the fluid density $\rho$, juggling the units gives

$$c = \left(gh\right)^{1/2} f\left(\frac{h}{\lambda}\right),$$

where $f$ is some function which dimensional analysis cannot determine. Taking the aforesaid limit gives $c \propto \sqrt{h}$. However:

Actually, this happy state of affairs is an illusion — we made a strong assumption in going from eqn. (1.4) to eqn. (1.5), namely that the limit process was regular. Usually, this can only be justified properly by considerations other than dimensional analysis. This point is by no means obscure mathematical pedantry: in fact, the cases were the regularity assumption breaks down constitute the central topic of this book.

The first homework problem in the book is to prove the Pythagorean theorem knowing that the area of a right triangle “can be expressed in terms of the hypotenuse and (e.g.) the smaller of the acute angles”. Hint: construct a single, wisely chosen line and then apply dimensional analysis. Part (b) then asks what happens when the triangle is drawn in Riemannian or Lobachevskian geometry. The second problem gives a table the blast wave radius as a function of time, $R(T)$, for the first atom bomb explosion; the task is to deduce a scaling law for $R(T)$ assuming that the shock-wave motion depends only upon the blast yield $E$ and the air density $\rho$.

• Goldenfeld, Nigel. Lectures on Phase Transitions and the Renormalization Group (Westview Press: 1992), ISBN 0-201-55409-7.

Isn’t your ‘C’ not, well, fundamental? After all, 85.912384 is also a dimensionless number, but there’s nothing particularly interesting about it. How does that differ from your ‘C’? Neither show up in any theory that I know.

On the other hand, alpha shows up when one is doing QED. That seems to me to be the fundamental difference.

Indeed, unless I am mistaken (sorry for the bad format, it’s late
and I have to wake up very early tomorrow; some other day I’ll try itex),

[alpha] = [C^2]/([F/m] [J s] [m/s])

= [C^2]/([C^2/(m J)] [J s] [m/s])

= [C^2]/([C^2 s^2/(kg m^3)] [kg m^2/s] [m/s])

([] indicates units, C=Coulomb, F=Farad, J=joule, etc)

The only fundamental unit above is the electric charge unit. The others, kg, s and m are our inventions and are not fundamental. One for instance could set a new unit of time as being the meter, so that one meter of time is the time it takes light to travel one meter. (Then c=1, dimensionless). It could be set as any unit of length one wishes in their wildest dreams.

If that is logical, what if one uses the Planck length as a new unit of time? Or more reasonably yet, choose the Planck units altogether, and you have that e = sqrt(alpha) in these units. The Planck length, etc, is a fundamental property of free space, and does not depend on any object or elementary particle arbitrarily chosen. Hence, if you use the Planck units, which are defined independently of the elementary charge, then a variation in the value of alpha would have to be considered due to a variation in the elementary charge.

Please let me know if I am wrong somewhere or whether I did not understand your question; perhaps it is more subtle than I realize.

Best regards,

Christine
PS: BTW thanks for the references, I’ll take a look at them.

In a certain sense the dimensionless constant 299,792,458 depends on our choice of units. Of course this number is what it is, regardless of our units. But if we say c=Cm/s then the dimensionless quantity C depends on the definition of m and s.

The dimensionful constant c does not depend on our choice of units. If we double m, we halve C, but c stays the same.

This looks convincing at first, but I’m not quite sure if it’s right. If we define C as cs/m (as it seems that you do), then the value of C doesn’t depend on our choice of units. If we choose to measure things in feet instead of meters, then C retains its value, but c and m both “increase in value” by the same ratio, canceling out. When we change our units, we don’t change the dimensional quantity m, but just change the way we express it.

For exactly the reason you say that the dimensionful constant c doesn’t change its value when we change units, none of these constants (whether dimensionful or dimensionless) change their value when we change our units. Thus, this can’t really be the property that interests us in (traditional) dimensionless constants.

I think part of the problem is that we are accustomed to viewing scalar quantities as numbers, when in fact they are often (multiplicative) torsors instead. A torsor can be multiplied by a number, but isn’t actually a number itself. One can multiply and divide torsors together but then one gets a torsor in a different space. The units only come in as a kind of non-canonical functor from the category of torsors (or torsor spaces) to the category of numbers (or of R).

So long as one is content to work with torsors and not numbers, there is no distinction between dimensionless or dimensionful physical quantities. There is still, however, a meaningful distinction between being dependent on units and being independent of units, but the latter can be seen simply by inspecting the definition of the quantity. For instance, the speed of light c, when viewed as a torsor, is independent of units, which is obvious from definition. However, the speed of light, divided by the unit length, times the unit speed (which in SI units would be the number 299792458) is blatantly dependent on the choice of units.

A quantity is dimensionless if its torsor can be canonically identified with a number. This has nothing to do with whether it depends on units or not. But: if the torsor quantity is independent of units, and the torsor is dimensionless, then clearly the numerical quantity associated to the torsor is also independent of units. That’s why the numerical value of (say) the fine structure constant is independent of units.

It turns out that identifying some torsors (such as the torsors for mass, length, and time) uniquely generate the identification for many other torsors too (leaving aside the role of charge, etc. for now). In particular, the multiplicative group R_+^3, which acts on the identifications between the mass torsors, length torsors, time torsors, and numbers (otherwise known as the unit mass, unit length, and unit time), then naturally acts on the identifications of all other torsors as well. The weight of that representation is the dimension of those torsors. One could of course choose other torsors to be the generators (provided they are a basis), and one would get a slightly different looking, but equivalent, notion of dimension.

As for “we could also get any value of áby changing our definitions of e,ϵ 0 ,ℏ and c”, I don’t understand exactly what is meant. If you refer to the meaning of the symbols, well, in the formula they are just shorthands. You could say “the square of the elementary charge over the product of Dirac’s constant, the surface area of a unit sphere, the dielectric constant of free space and the speed of light”. And then you can go further noticing that 1/(4pi*epsilon_0) is Coulomb’s constant, so e^2/(4pi*epsilon_0) is “the electrostatic force between two electrons over their distance squared”, and so on, until you have no longer arbitrary definitions. For example, “the ratio between the energy needed to bring two electrons from infinite distance to an arbitrarily choosen distance, and the energy of a photon whose wavelenght equals the radius of a circle whose circumference equals the distance to which the electrons were brought”.

related perhaps to this discussion, I was amazed about the different meanings of the word “numerology” between laymen, esoterists, and physicists. For laymen it just mean some nice trick to argue for personality archetypes in the same way that the Zodiac is used nowadays. For esoterists it is about relationships between dimensionful quantities, for instance the length of the corridor in the great pyramid when measured using a lenght unit of “greatpyramid foots”. For esoterism, of course, such units have been choosen on purpose by ancient knowledge or something so. Last, for physicists, numerology applies for relationships involving dimensionless quantities. So the attempts to get 1/137 or the assertion of the quotient between two masses (an adimensional quantity) is related to some rational value. Or all these, ahem, great findings about the Dirac large Number hypothesis, sometimes even rediscovering it.

Of course, there is also another question about numerology between pure numbers, such as “is the sequence 10, 18, 26… related to Bott periodicity”? But this is beyond dimensional analysis (pun intended here).

I think units are data that specify isomorphisms of models (where I use model in a sense like here).

“Dimensionful constants” are quantities that do depend on a choice of representative in an isomorphism class of models.

“Dimensionless constants” are quantities that do not depend on a choice of representative in an isomorphism class of models.

Here is a simple example.

Assume for the moment that we are interested in the model of reality which says that the space we observe is a Euclidean 3-dimensional vector space

containing exactly two Newtonian point masses.

Back in the past, somebody came along and figured out an isomorphic model. Namely, he drew two coplanar ellipses into $\mathbb{R}^3$ and claimed that this is the right model for the observed two point masses.

Apart from saying what kind of models we consider, we need to say what a morphism is between two models. In the present case, we would want to agree that a morphism of models is a morphism of vector spaces which respects angles.

Now, $\mathbb{R}^3$, equipped with its standard scalar product, is a 3-dimensional Euclidean vector space.

In the above sense, $\mathbb{R}$^3 is isomorphic to $V$ - but not canonically so.

A choice of isomorphism

is precisely what we call a choice of unit of length.

Using the Euclidean scalar product on either $\mathbb{R}^3$ or on $V$, we may compute, say, the major diameter of our two ellipses. In general, the number we get for our ellipses in $\mathbb{R}^3$ and $V$ will differ. The difference (the quotient, actually) is measured by our isomorphism $\mathbb{R}^3 \stackrel{\simeq}{\to} V$, namely by our unit of length.

So the size of our ellipses is “dimensionful” (dimension of length in this case). It depends on a choice of isomorphism.

Other quantities of our model are independent of the choice of isomorphism. For instance the quotient of the major diameter of one ellipse with that of the other.

This quotient is hence a dimensionless quantity. In fact, it is the dimensionless quantity governing this model (encoding the relative mass of our two point masses).

So, in general, there is something called $R_o$, the observed reality. And there are models $M,\,M',\cdots$, of it in the sense that we have morphisms

Some properties on $M$ may change if we pass to a different, yet isomorphic model $M'$. These are the dimensionful properties of $M$. Other properties are invariant in a given isomorphism class of models. These are the dimensionless properties.

Here’s a related question that may or may not shed some light on the issues:

When you do physics, including GR and field theory, via the mathematics of differential geometry – where and what are the units?

For a specific example: When considering the Schwarzschild solution, do the coordinates in the various manifold charts have units, including length and time? Or are the coordinates unit free, with the metric (or frame) degrees of freedom carrying the units?

It turns out, as far as I can tell, that the machinery of differential geometry works just fine with whatever units (or lack of units) one chooses to attach to manifold coordinates. It is a matter of taste where to bring them in. Personally, I like leaving the coordinates free of units, and having the frame (and thus the metric) carrying the units of time (and/or length), since these are what I think of as “physical” variables. Once you bring them in, the units get carried along and attached to all the usual suspects.

Any one else have thoughts on this?

I don’t agree that c can be considered dimensionless. c is defined as ‘the speed of light in a vacuum’ - therefore it has the dimensions of L/T.

In my view dimensional analysis is a wonderful tool, but the risk is to take it too far. One of the best uses for dimensional analysis is a quick check on your math. If you derive an expression for velocity which has dimensions other than L/T you’ve made a mistake. You can also make educated guesses that can save a lot of time, especially when you only need to know how a quantity scales.

For example how does speed of sound in a gas scale with pressure? If it also depends on gas pressure a quick look at the units shows that you can combine these two quantities to get a velocity by making the speed of sound proportional to sqrt(pressure/density).

So what does it mean to say the fine structure constant is ‘more fundamental’? If you can’t define that, then it doesn’t mean much. Yes, beings from another planet would have a different number for c in their theories but the same numbers for fine structure constant, pi and e. So?

…since we could also get any value of α by changing our definitions of e, ε₀, hbar and c

A demonstration of this claim would add clarity.

What about topological measurements? For instance the euler characteristic of a surface? They are not related to any unit, in principle.

But… some of these topological invariants can be got from dimensionful operations, for instance integrating some local quantity along all the surface. The trick of units can be played again here: measure the local quantity in inverse of square foots but use square meters for the area of the surface… you will get Euler characteristic expressed in a consistent way as “xxx (meter/foot)^2”.

You can feel happy doing this kind of trick with non-topological quantities, as the fine structure constant. But here?

C will change when we go to a cgs system as opposed to SI units. alpha won’t. It’s not that complicated.

I have an elementary grumble about a bit of terminological confusion here (which I see also bedevils the Wikipedia discussion). Let me get this off my chest:

Something that seems a bit peculiar in the original post is the business about “changing the definition” of the metre (or, in a later post, “changing the definition” of $\hbar$, $c$ etc). This isn’t normally how we change units. Normally, changing units means going from e.g. metres to inches. Different unit: different name. Obviously the ratio of the speed of light to the metre-per-second is different from the ratio of the speed of light to the inch-per-second. On the other hand, “changing the definition of the metre” means using the same name for a different unit. Obviously this is a recipe for confusion.

What about “changing the definition” of $\hbar$? Presumably this means changing the unit of action that we use to quote the size of the action $\frac{e^2}{4\pi\epsilon_0c}$. It obviously doesn’t change the actual physical quantity of action currently denoted by $\hbar$.

To give a linguistic parallel, consider that classic piece of love poetry:

Roses are red,
Violets are blue,
Sugar is sweet,
And so are you.

We are used to the fact that the word “you”, e.g. as it appears in the last line, has a context-dependent referent: who it refers to depends on who the speaker is talking to. There’s a technical linguistics term for this which I’m blanking on. Anyway, it means that if we “measure” the truth value of the last clause, it depends on our “unit”, i.e. which standard we are measuring sweetness against, i.e. who we are talking to.

This clearly makes the word “you” different from “roses”, “violets” and “sugar”. It is true that somebody could say, “Aha, but suppose you change the meaning of ‘roses’ to refer to violets, ‘violets’ to refer to roses, and ‘sugar’ to refer to manure? Then the truth-value of the other clauses would change, too.” Well, yeah, but that’s just being confusing for the sake of it.

Exactly the same confusion could be introduced into the definition of the fine structure constant. We could define a new $\alpha$ to be 2 times the old $\alpha$, or 2.9 times the old $\alpha$, or 299,792,458 times the old $\alpha$, which would apparently demolish one of the disputants on Wikipedia, but obviously wouldn’t say anything substantive about the issue of units. It we have a new definition, we should have a new symbol instead of $\alpha$, and use that new symbol in our equations.

Although if you did that, people would probably look at you funny and grumble about huge, meaningless numbers cluttering up their QED calculations.

Hey, this looks like fun, let me have a try…instead of talking about what is “fundamental”, somehow an emotionally charged issue always, we can concentrate on what it means to change dimensionful vs dimensionless quantity.

I think that mathematical theories with different values of the dimensionful constants are isomorphic: you can establish a mapping between all their observables simply by rescaling (a.k.a “dimensional analysis”). On the other hand theories with different values of “fundamental” dimensionless constants are really different from each other- if alpha of QED is any different the theory is different, possibly qualitatively different. It is then reasonable for me to call such constants, ones that parametrize the set of inequivalent theories, “fundamental”, but it really doesn’t matter.

Note that this criterion also distinguishes physical from mathematical constants, assuming one does not want to solve all cosmological problems by positing time variation of \pi or something…

Oh. This has grown enormously. Since my previous comment received no follow-ups, I am unsure whether I simply missed the point or I wrote something completely trivial, naive or (not?) even wrong! Please let me know!

My point is:

The Planck units reflect a fundamental property of free space, and do not depend on any object or elementary particle arbitrarily chosen (necessarily used to define, eg, the meter, the second, etc).

If you measure all quantities in Planck units, then they are all dimensionless, so what is the deal of being dimensionless or not?

We only say that alpha is “fundamental” because alpha = e^2 (in Planck units), and therefore the strength of the electromagnetic force relative to all other *fundamental* forces is proportional to alpha, a number. This is a number given by Nature that have to be explained. It is a fundamental number.

So the point is: alpha is not fundamental because it is dimensionless. It is so because it is a number that happens to set the relative strength of the electromagnectic force to the other fundamental forces.

Ok, I’ll no longer insist on my possibly naive understanding of all this. Perhaps there is much more to it. When you gentlemen find out all about this, please do post a summary on your conclusions!

Christine

PS: But what happens if you change c, etc? Would we notice? (Read the section on Planck units and the invariant scaling of nature here: http://en.wikipedia.org/wiki/Planck_units).

John wrote:

But with this definition of m, the dimensionless quantity C=c/(m/s) seems to tell us nothing about our universe!

As long as you think about one constant at a time it looks like they tell us nothing because you can redefine their value by changing units. But as soon as you your physical theory has more constants than dimensions you no longer have complete freedom and you’re making meaningful statements.

John, you just said:

I should emphasize that this is a complete turnaround from the position I’ve had for years. Until Kehrli emailed me, I’d never noticed that the dimensionful quantity c doesn’t depend of our choice of units, and that the dimensionless quantity C does.

first, i agree that it is apparent that this is a complete turnaround and am somewhat astounded by this since i was trying to represent the position taken by you in sci.physics.research as well as Barrow, Duff (both of whom i had email chats with, and weirdly, it seems that Barrow has flipped his position from the quote of his “Constants of Nature” book - Duff is still very consistent in his position). nothing Kehrli has written is the least bit persuasive. as an engineer, i feel pretty confident that i understand the meaning of dimension of physical quantity and of units. i think i even understand the difference between the terms “unitless” and “dimensionless”. measureing an angle in degrees or sound intensity in dB are not unitless, but they are dimensionless. measuring the same quantities in radians and nepers is both unitless and dimensionless. anyway, all Kehrli is doing is repeating a fact we all know (that a dimensionful physical quantity is represented as a “product” of a dimensionless value with a unit) and then some less coherent ramblings where she says some clearly false things (which you now seem to give credence to) and makes appeals to some “IUPAC green book” which i think is non sequitur, it doesn’t speak to this issue in the least.

the reason that human beings measure $c$ to be 299792458 m/s is because (before they redefined the meter to fix $c$ to 299792458 m/s) the distance between the two little scratch marks of the platinum-iridium International Prototype Metre was, to the same precision, very nearly 6.18718916 x 10³⁴ Planck lengths and the number of Planck times in one cycle of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom is, to the same precision, very nearly 2.01778195 x 10³³ and that humans have anthropometrically decided that 9192631770 of those cycles make up one second (and that $c$ is always 1 Planck length per Planck time). that is a human construct. and that is solely why $c$ = 299792458 m/s.

we could get more historical about it when we remember that the meter was chosen so that there would be exactly 10 million of them on an arc of the earth from the north pole to the equator going through Paris (making the meridional circumference 40 million meters - they got that wrong by 0.02%, but it’s not bad for the 18^th century). and, given that the mean solar day was originally 86400 seconds (before the atomic clock and leap seconds), the human POV of the measure of $c$ is that light is fast enough to make 647551.7 revolutions around our planet at the mean surface radius and over the poles in the period of time between consecutive instances that the sun is directly above the same given meridan. that is what $c$ = 299792458 m/s means to humans, and that is purely anthropometric. the aliens on the planet Zog could give a rat’s ass about it.

now we could communicate to the Zogs a dimensionful quantity by referencing to another like-dimensioned quantity (call it a standard, if you choose). but how do we communicate the magnitude of the standard to them? we could use something like Planck units as a standard to communicate the size of anything to them. so we say that our meter is 6.1872 x 10³⁴ Planck lengths and that our second is 1.85487 x 10⁴³ Planck times, but as soon as we say “we measured the speed of light to be 299792458 m/s” to them, that means, from our definitions of the meter and second, precisely the same as “we measured the speed of light to be 1 Planck length per Planck time.” in other words, it says nothing (new) at all.

however we can ask them to measure and report to us what the value of $\alpha$ is and if the number the report is substantively different, we know they live in a different world. i believe this is the basis of the dispute at Wikipedia about Physical constants. there is nothing profound i can see in Kehrli’s insistence that there is no qualitative difference between the dimensionful physical constants like $c$, $G$, $\hbar$ and the dimensionless constants like $\alpha$ or $m_p/m_e$ (essentially that dimensionless is just another dimension). it’s just wrong.

Let me see if I understand the main conflict in this thorny thread, and how I think it’s resolved.

r b-j is saying (from the “Physical constant” wikipedia talk page) [square brackets mine]:

“You [Kehrli] are fully mistaken in thinking that dimensionless physical constants (like [the fine structure constant] α) are not more fundamental than the dimensionful physical constants [like the speed of light, c]. The former are numbers that are properties of the universe (and the aliens on the planet Zog will come up with the same number) and the latter are human constructs.”

Kehrli is saying (I paraphrase):

“Thhhhpppppttt.”

I have to side with Kehrli on this one. The physical constants with dimensions are just as fundamental as the dimensionless ones. Take the speed of light, c. We conventionally express c in terms of meters per second. And r b-j says these units are arbitrary and c is thus physically meaningless. But we could go measure (via scattering or something) the expected radius of a hydrogen atom, and measure the expected decay time for a 2p -> 1s transition, and then convert c to these speed units of hydrogen radii per hydrogen decay time. This way hydrogen atoms (the most common atoms in the universe) could act as standard rulers and clocks. And we could indeed compare our value for c, in these hydrogen units, with those found by the Zogians. The value will be the same if the speed of light is the same and hydrogen atoms are the same. In this way we can understand c to indeed be a physical constant.

But here is where things get tricky. What if the speed of light at planet Zog and hydrogen atoms at planet Zog were both different in just such a way as to make the Zogian’s measurement of c in Zog hydrogen units the same as our measurement of c in our hydrogen units? We’d get a false impression that our c’s were the same! Does this mean r b-j was right, and α is more fundamental? No, because it’s just as possible to imagine the physical measurements behind the definition of α to change and balance in a similar way to give the same numerical value. This brings us to the key revelation that John Baez made a couple of posts ago:

Physical constants, even dimensionful ones, are fundamentally about relations between physical measurements.

I’ll go ahead and call this “dimensional relativity.” It means that what really matters, as I think Urs has been saying, is whether one whole shebang of physical measurements is the same (isomorphic) as another. The physical constants, even dimensional ones, provide information in the form of relationships between measurements. The speed of light means nothing by itself – to be meaningful it has to be related with the rest of the universe, such as to hydrogen atoms, or to meters which are compared to hydrogen atoms. That’s pretty neat.

From this point of view, we can draw an analogy between physics and economics. Without money, you could always barter (trade) some items for other items. There are value relationships, expressible as ratios, between items. Money makes things easier – by introducing an item of arbitrary but universal value, you get an intermediary by which you can value and compare many items, including new ones. Now it should be clear: physical units are the money used to describe physical quantities.

To drive this home: If you have a hydrogen atom, you can describe its radius and decay time in terms of (arbitrary) meters and seconds. Now, to describe the speed of light, you can describe it in hydrogen units or you can describe it in meters per second – it’s the same information, and the same fundamental physical constant, either way. It makes no difference, economically, whether you say “a banana is worth a dollar and an apple is worth a dollar” or you say “a banana is worth an apple” – either way you’re defining the banana value.

You can also always use some physical measurements to stand in for units – this (nondimensionalization) is like going to the barter system and getting rid of money. What’s physically important is the web of relations between measurements.

I hope I’ve helped make this more clear instead of confusing things.

Tim Silverman wrote:

Something that seems a bit peculiar in the original post is the business about “changing the definition” of the metre (or, in a later post, “changing the definition” of $\hbar, c$ etc). This isn’t normally how we change units. Normally, changing units means going from e.g. metres to inches. Different unit: different name. Obviously the ratio of the speed of light to the metre-per-second is different from the ratio of the speed of light to the inch-per-second. On the other hand, “changing the definition of the metre” means using the same name for a different unit. Obviously this is a recipe for confusion.

That’s a good point, though I don’t think it should paralyze us.

In fact, when mathematicians first invented letters for variables, they ran into just this confusion. They had real arguments about how a single letter $x$ could stand for different values. They had serious fights about how a “variable quantity” differed in kind from an ordinary number, yet could equal it! Precisely what act are you performing when you “set $x$ equal to $1$”? What are the rules governing such acts?

When I teach calculus, I run into this issue when students compute

$${d \over d x} x^2$$

at

$$x = 5.$$

The good ones compute

$${d\over d x} x^2 = 2x$$

and then set $x = 5$ to get $10$. The bad ones set $x = 5$ to get

$$x^2 = 25$$

and then compute

$${d\over d x} 25 = 0.$$

The work of early 20th-century logicians provided very precise answers to all these puzzles. Later work by Lawvere and others on topos theory provided different answers, which take the concept of “variable quantity” more seriously. But, most people live merrily on without ever reading any of this stuff. They either get it instinctively and do well in math and physics, or don’t get it and flunk out.

It’s true that normally when we change units we give them a different name. But sometimes we give them the same name! For example, the meter has been given 8 different definitions so far. Most of us don’t remember when the meter was the length of a pendulum whose half-period was one second, or when it was $10^{-7}$ times the Earth’s meridian along a quadrant from the equator to the North Pole through Paris, or the length of a brass rod, or 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the $2p^{10}$ and $5d^{5}$ quantum levels of the krypton-86 atom. If we were in a certain fussy mood we would give each of these meters a separate name, like $m_1, m_2, \dots, m_8$. But usually we call them all $m$.

If we allow ourselves the notion of a variable unit of length $m$, and a variable unit of time $s$, we get a dimensionless variable

$$C = c/(m/s)$$

which changes as we change $m$ and $s$. If instead we demand a separate name for each unit of length and time, say $m_i$ and $s_i$ for the $i$th one, we get a bunch of dimensionless constants

$$C_i = c/(m_i/s_i)$$

Either approach is okay. Either we say $C$ depends on $m$ and $s$, or we say $C_i$ depends on $m_i$ and $s_i$.

Where, of course, $i$ is a variable.

Christine Dantas wrote:

Since my previous comment received no follow-ups, I am unsure whether I simply missed the point or I wrote something completely trivial, naive or (not?) even wrong! Please let me know!

Sorry to take so long to reply. Last week I flew back from Shanghai, and yesterday I taught my first course in this fall’s Quantum Gravity Seminar. So, blogging had to take a back seat for a while.

Nothing you said seemed wrong to me. So, I’ll make up some very small objections, just to annoy you , or maybe give you something to think about.

You write:

The Planck units reflect a fundamental property of free space, and do not depend on any object or elementary particle arbitrarily chosen (necessarily used to define, eg, the meter, the second, etc).

That’s true, but it’s worth noting that the Planck units treat gravity as special, and many modern physicists like to question that special role. For example, string theorists like to treat gravity as part of a package including other fields. Sakharov and Jacobson have considered the possibility that gravity is just a small side-effect of other fields.

Even without any theory like these, one might argue that the electron field is just as “fundamental” as the metric tensor. Sure, the way gravity couples to energy-momentum is very special! But, it’s not as if god made the metric tensor very carefully on Monday, and made the electron field by accident on Wednesday. They’re both fields defined on all of spacetime. So, it’s a bit funny to declare that gravity is a “fundamental property of free space”, while the electron is not.

If we used units where the electron charge, the electron mass, the permittivity of free space, and the speed of light were all equal to 1, then we’d think $\alpha$ was telling us something about Planck’s constant. This seems stupid, but it’s interesting to ponder why it’s stupid. Somehow it’s a less enlightening way to slice the pie.

You conclude:

So the point is: alpha is not fundamental because it is dimensionless. It is so because it is a number that happens to set the relative strength of the electromagnetic force to the other fundamental forces.

I agree that being dimensionless is not a sufficient condition for $\alpha$ to be “fundamental”. It’s fundamental because it says something about the strength of the electromagnetic force - at least if we pick units where $c = -e = \hbar = 1$. If we pick units where $c = 4\pi\epsilon_0 = \hbar = 1$, it says something about the charge of the electron. Either way, it’s saying something about electromagnetism. There are other ways to slice the pie (see above), but these are stupider.

(I wouldn’t say $\alpha$ sets the strength of electromagnetism “relative to the other forces”, since we aren’t bringing them in here. For example, we don’t need to set $G = 1$ in this discussion.)