# Is it possible to speak about changes in a physical constant which is not dimensionless?

Every so often, one sees on this site* or in the news or in journal articles a statement of the form "we have measured a change in such-and-such fundamental constant" (or, perhaps more commonly, "we have constrained the rate of change of..."). These are often shrouded in controversy, with people often stating quite loudly that statements of the sort are meaningless and even unverifiable, but the end result is often quite confusing. Thus:

• Is it possible to speak about changes to a physical constant that contains dimensional information? If not, why not?

* These are some examples but there's more.$\ \$† example.$\ \$‡ example.

The problem is that any statement that a dimensionful physical constant has changed is meaningless if you do not state what it is you are keeping constant. When you do state it, and you work it out in full, it turns out you're measuring the change of a dimensionless physical constant.

Suppose, for example, that I claim "the speed of light has increased by 10% since last year". My experimental evidence for this is simple: I have a ruler and a clock, and every day I use them (the exact same ruler and clock) to measure the speed of light. I then calculate the time $T$ - as a quantity, in seconds, given by my clock - taken by light to cross the ruler, and I divide it into the length $L$ of the ruler. Over a year, this quantity $L/T$ increases by 10%.

For the metrologically minded, let's take a look at my experimental apparatus:

• My ruler consists, essentially, of a bunch of atoms in a linear arrangement. For example, I could make my ruler by putting a bunch of hydrogen atoms in a line, separated from each other by one Bohr radius (which can be experimentally determined, at least in principle, as the distance at which the ground state charge density decreases to $e^{-2}$ of the maximum). If my ruler is 1m long, it will have $N=1\:\mathrm m/a_0\approx1.89\times10^{10}$ atoms in it.

Moreover, since it's the same ruler, and because I'm lazy, I don't recalibrate it to the SI every day before I measure (which would have catastrophic consequences for my result!). Instead, what I do before I measure is look at all the bonds to ensure they're still one Bohr radius long, and re-count the atoms to ensure the ruler still has the exact same number $N$ of atoms.

I should also note that, despite sounding esoteric, this ruler is close to the best possible model for an actual physical ruler made of platinum or whatever, since its length is governed by the same physics (nonrelativistic QM plus electrostatics, a.k.a. chemistry) that governs the size of all everyday objects. It's been distilled into a form that's clearly definable, but the essence of the definition is along the lines of "this ruler here, so long as the ends don't get worn out and it doesn't get bent and there's no monkey business with thermal expansion or whatever".

• My clock is a caesium clock as per the SI second. That is, it has a bunch of caesium atoms which are placed in a specific state (technically, a coherent superposition of their ground state at and the first hyperfine excited state) which emits microwave radiation. The clock measures this radiation and counts the number of maxima; every 9,192,631,770 cycles it increases the counter by 1 second.

With this apparatus, then, I observe the measured speed of light to change. What does this result mean? The easiest interpretation is simple:

• The speed of light has changed. Light simply travels faster than it did last year. That's pretty mysterious, but then the result is pretty weird too.

However, there are other possible interpretations for the result. For example,

• the size of the hydrogen atoms might have changed. That is, the speed of light is still the same, but for some mysterious reason all my hydrogen atoms are 10% bigger than they were. This is not that crazy at all: the Bohr radius is determined by the Schrödinger equation under the electrostatic force, so their constants $\hbar,m_e$ and $e^2$ determine $a_0$ to be $a_0=\hbar^2/m_ee^2$. If any of those changed - say, all electrons are suddenly 10% lighter, which is as mysterious as light being faster - then I'd see exactly the same result I do observe.

It's important to note that this is the reading of my result that would come from a strict application of the current SI system of units, since the SI meter is defined as the distance covered by light in 1s as given by my clock. However, if my ruler has "shrunk" then so have I (as I'm made of atoms) and so has every piece of equipment in my lab and elsewhere on Earth. I could then speak of the "mysterious growth of the SI meter" equally meaningfully.

Alternatively,

• the caesium atoms could be getting progressively more sluggish. They could simply no longer be giving out microwave peaks and troughs as fast as they used to (or as I'd see if I teleported myself to the past), so even though the speed of light is the same, the length of the ruler is the same, and the traversal time is the same, the counter on the clock now reads 10% less seconds than it would have a year ago.

These three explanations are all equally mysterious, or equally reasonable, as each other. Moreover, from my experiment I have no way to test which is right, as I have no way of teleporting myself back to the past to compare my (possibly) engorged hydrogens or my (allegedly) sluggish caesiums with their previous versions, in the same way that I cannot set up a race between my (supposedly) faster light and the light of yesteryear. It is clear that something in physics has indeed changed, but you can pan the change into different factors depending on your perspective.

As it turns out, of course, the thing in physics which has changed, and which is the only thing that can unambiguously be said to have changed, is the fine structure constant. This $\alpha$, as any atomic physicist will tell you, is one over the speed of light in atomic units, which is exactly what we're measuring. More specifically, $$\alpha=\frac{e^2}{\hbar c},$$ and $e^2/\hbar$ is easily seen to be the atomic unit of velocity. This abstract 'atomic unit' is of course a very physical quantity: it is, within a constant, calculable factor, the mean velocity of any electron around its atom.

Now, as it happens, this crazy experiment I proposed is indeed being performed. The actual experimental realization does not rely on meter-long rulers or on external clocks, but it relies instead on the natural length and time scales of ytterbium ions, which of course are completely determined by the same length and time scales as all atomic physics.

Atoms, it turns out, have natural built-in velocimeters for measuring the speed of light: their kinetic energies $p^2/2m$ change by relativistic corrections on the order of $p^4/8m^3c^2$, and the different electronic and spin currents have magnetic interactions on the order of $p/c$. Different states respond differently, and even in different directions, to these perturbations, so it's possible to monitor $c$ by observing the precise location of the different energy levels. (For more information, see NPL | Physics | PRL | arXiv.)

There is, so far, no observed change in $\alpha$. But if there is, we simply won't be able to tell whether a change in $\alpha=e^2/\hbar c$ is because of a change in the speed of light $c$, the size $\hbar$ of the fundamental phase-space cell, or the strength $e^2$ of the electrostatic interaction, or of the finely-tuned joint changes of these constant that would implement the three implementations stated above. Those individual changes have no meaning by themselves.

Finally, some useful references for further reading are

How fundamental are fundamental constants? M.J. Duff, Contemp. Phys. 56 no. 1, 35-47 (2014), arXiv:1412.2040,

which supersedes arXiv:hep-th/0208093 (Comment on time-variation of fundamental constants, 2002), and

Trialogue on the number of fundamental constants. M.J. Duff, L.B. Okun and G. Veneziano, J. High Energy Phys. 03 (2002) 023, arXiv:physics/0110060.

• I'm amused by the idea that you're too lazy to recalibrate your ruler, choosing to count $10^{10}$ atoms each day instead. – Michael Seifert Jul 30 '20 at 19:37
• @Michael I'm just weird that way =). – Emilio Pisanty Jul 30 '20 at 22:08

It's not always meaningless. For example, we may talk about the change of the mass of a human (in kilograms) after a diet.

But the point is that the value of a dimensionful quantity – and its change or constancy – depends on the magnitude of the units and they're matter of conventions which may change, too. So an increasing numerical value of $\hbar$ could be just to a decreasing value of what we call one kilogram. We define e.g. one kilogram using the international prototype and consider it a "constant mass" while the mass of a human on diet is more variable.

In this comparison, the constants $\hbar,c,G$ are even more naturally constant than the international prototype. In fact, "adult" physicists use units where $\hbar=c=1$ (quantum relativistic units) and sometimes also $G=1$ (general relativistic units or Planck units when all conditions are imposed). So $\hbar,C,G$ can't really change when natural units are used because they're always equal to one.

Even if we use more everyday units in which the numerical value isn't one, it's still better to define such units in such a way that $\hbar,c,G$ are constant. In fact, one meter is already defined so that $c=299,792,458$ m/s at all times. It's not quite the case for $\hbar$ and $G$ yet but this may change in the future.

Because $\hbar=c=k_B=1$ etc. are so natural, it makes sense to consider the constancy or variability of all other dimensionful constants by the constancy or variability of these constants converted to the natural units.

Is it possible to speak about changes in a physical constant which is not dimensionless?

First to note is of course that the very idea of "a constant having changed" smacks of a (semantic) absurdity. In the present context, however, the notion of "physical constant" is not necessarily understood so strictly as to rule out any "change" outright, as a matter of definition. Rather, the task is implied to distinguish and categorize in the first place for any "physical quantity" or "physical value" whether it is indeed fixed by definition; or instead the result of a genuine measurement (with several distinct values in the range of the measurement operator, to be evaluated from given observational data) where obtaining different values in several trials might at most be considered "surprising" or "curious".

And this distinction may be considered for real numbers and even Boolean values as well as for "dimensionful" quantities. Still, the reference to "dimensionality" in the OP question is not necessarily in vain. It points to the natural presumption and concession that "we all" understand and accept real numbers as "outright definite and unambiguous"; and of course Boolean values, too.

So the focus of addressing the OP question should be on definitions what/how to measure (genuinely), vs. whatever might be defined unambiguously fixed outright; because "speaking of change" surely relates to the possibility of measurement (experimental test) instead of changing definitions. The question at hand may therefore be interpreted as asking:

Which definitions of "how to measure" can assure that the result values are and remain as definite and unambiguous and communicable as real number values?,

especially with regard to dimensionful quantities.

Now, looking foremost to measurements of geometric (incl. kinematic) relations, such as determinations of

• whether or not the ends of a given "ruler" had been and remained "at rest" to each other,

• whether or not a given oscillator took "equal duration" for each oscillation period

• whether or not the period duration of one oscillator had been equal to the the period duration of another oscillator (at least if they had been at rest to each other), etc.

one guiding principle to consider in the construction and selection of suitable measurement definitions is certainly the "point coincidence principle"; as Einstein put it (1916):

This surely also includes the presumption/concession that each participant may judge (at least in principle) which observations she had collected in coincidence ("together", "at once") and which not; as used explicitly for instance in Einstein's coordinate-free defintion of (how to determine) "simultaneity".

MTW Boxes 10.2 ("Schilds Ladder") and 16.4 ("Ideal clocks and rulers") as well as my sketch of how to determine "mutual rest" (PSE/a/70646) give an idea on how to employ the point coincidence principle for the required definitions.

These definitions imply that participants who were (measured as having been) at rest to each other also have equal ping durations wrt. each other. (Always in reference to observations of signal fronts, of course). Such pairs of participants (e.g. two "ends" of a given "ruler") can therefore be assigned a (mutually agreed) value of "distance between each other", as

$$c_0~\frac{\text{ping duration}}{2}$$,

where "$c_0$" is simply and plainly a distinctive fixed symbol (which is not to be treated as the number "Zero"); which is attached as a prefix to the half ping duration just for the purpose of signifying that the resulting quantity (a distance value) refers to the geometric relation of two participants at rest to each other.

(That's called the "chrono-geometric" (or "chronometric") definition of "distance. Inserting this in turn into the familiar definition of "(average) speed" as "the ratio of distance between starting block and finish line to the duration of the course having been occupied", the (average) speed at which a signal front is exchanged between two participants at rest to each other is obviously, necessarily evaluated as "$c_0$".)

And it is of course absurd for a mere symbol, such as "$c_0$" (i.e., as subsequently identified, the value of "signal front speed") to "change by 10 %", for instance.

What may be asked and (in general) be measured is instead, for instance,

• whether two ends of a ruler, which had been at rest to each other in some trial, might have been at rest to each other in another (e.g. subsequent) trial.

Similar (but perhaps more involved) arguments may be made regarding the plainly distinctive fixed symbol "$\frac{c^2}{G}$" (which is employed to distinctly denote some quantity of dimension distance as "mass"; signifying that the resulting mass value is characterized by and derived from "its" Schwarzschild radius), and
the plainly distinctive fixed symbol "$\hbar$" (which is employed to distinctly signify "derivative with respect to" any particular quantity; which is subsequently identified e.g. as "quantum of angular momentum").

These symbolically introduced and subsequently identified quantities therefore constitute stable "natural units"; the Planck units.

What may correspondingly be asked and (in general) be measured is, for instance,

• whether some particpant had "kept its mass" from one trial to the next,

• whether two particpants had equal mass,

• whether some particpant had integer or half-integer spin, in any particular trial,

• whether some particpant had "kept its charge" from one trial to the next,

• whether two particpants had equal charge, ...