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I have read a passage in Wikipedia about the List of unsolved problems in physics and dimensionless physical constants:

Dimensionless physical constants: At the present time, the values of various dimensionless physical constants cannot be calculated; they can be determined only by physical measurement.[4][5] What is the minimum number of dimensionless physical constants from which all other dimensionless physical constants can be derived? Are dimensional physical constants necessary at all?

One of these fundamental physical constants is the Fine-Structure Constant. But why does Wikipedia say that these constants, such as the fine-structure constant, can be only measured and not theoretically calculated?

The fine-structure constant $α$ as far as I know for the electromagnetic force for example can be theoretically calculated by this expression:

$$ \alpha=\frac{e^{2}}{4 \pi \varepsilon_{0} \hbar c} \approx \frac{1}{137.03599908} $$

So why then does Wikipedia say that it can only measured but not calculated? I don't understand the meaning of this above-quoted Wikipedia text?

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  • $\begingroup$ IMHO, wiki may have in mind that it's hard to theoretically derive dimensionless constants based on some fundamental laws, albeit it may be possible, so strictly speaking wiki may be wrong this time. Dimensionless constants are hard stuff, because it can mean anything and is hard to verify, because it's not clear what methods to choose from. $\endgroup$
    – Agnius Vasiliauskas
    Mar 30, 2022 at 7:51
  • $\begingroup$ The sentence starts with "At the present time,..." which means that its not theoretically impossible to derive the value of the constant, just that we don't have any working model that can do so right now. $\endgroup$
    – Prahar
    Mar 30, 2022 at 10:34
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    $\begingroup$ Maybe an interpretation of this could be that the equations for these constants and calculated value cannot infer the physical origin and causality of these constants? How they came to be? What they physically represent? Opposite in known physics and mathematics we usually represent with a dimensionless constant a proportionality ratio between two same physical properties but different in quantity and with this ratio being fixed thus a constant. For example, radius of the Earth compared to its periphery. Both numerator and denominator are representing quantities of length. $\endgroup$
    – Markoul11
    Mar 30, 2022 at 13:54
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    $\begingroup$ It's a philosophical issue that natural physical 'laws' may be expressed as maths, but maths does not define our physical reality. The term 'dimensionless` should be seen as a distraction within the assertion about physics needing to measure the ratios between phenomena we observe. $\endgroup$
    – Philip Oakley
    Mar 30, 2022 at 21:57
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    $\begingroup$ In 1929 Eddington conjectured that the fine structure constant was 1/137, and that the integer 137 could be determined by "pure deduction". Of course he was wrong. $\endgroup$
    – Simon Crase
    Mar 30, 2022 at 23:07

6 Answers 6

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It may be helpful to point out that most dimensionless physical quantities (like $\alpha$ or $e$, under a suitable choice of natural units) are not unambiguously “calculable” or “non-calculable” entirely in and of themselves. Instead, only a whole set of independent physical quantities can pin down all the experimental parameters in a theory. E.g. you’re either free to calculate $\alpha$ from an experimentally measured value of $e$ or vice versa, but you can’t independently do both. While different people might choose different generating sets of independent experimentally measured constants, every such set should contain the same number of constants - the number of degrees of freedom for the theory.

In that sense, it’s kind of like a basis for a vector space - there may not be a unique natural basis, so no individual vector can be unambiguously categorized as “a basis vector” or “not”, but there is an unambiguous number of basis vectors, which is inherently a collective property of a whole set of vectors rather than a “pointwise” property of individual vectors.

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In "natural units", we set $\hbar=c=\epsilon_0=1$. In these units, the equation you wrote down becomes \begin{equation} \alpha = \frac{e^2}{4\pi} \end{equation} In this notation, it is perhaps more obvious that the equation you wrote down is not really a way to calculate $\alpha$, since $\alpha$ depends on another dimensionless constant $e^2$ that we do not know how to calculate. In fact, you can read this equation as defining $e^2$, given $\alpha$.

The dream of "calculating $\alpha$ from first principles" would be to have a formula where $\alpha$ was expressed purely in terms of mathematical constants like $2$ and $\pi$.

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    $\begingroup$ Why can't we also set $e = 1$? I can ask that as a separate question if the answer is long and involved. $\endgroup$
    – Allure
    Mar 30, 2022 at 9:17
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    $\begingroup$ @Allure A glib answer is that if you did that, you would have $\alpha=1/4\pi$, but obviously $137 \neq 4\pi$. As a more serious answer, consider the formula in SI units $U=e^2/(4\pi \epsilon_0 r)$. In units where $\hbar=c=1$, energy has dimensions of 1/length. If we set $\epsilon_0=1$, then the formula becomes $U=e^2/(4\pi r)$. Since $1/r$ and $U$ both have dimensions 1/length in these units, it must be that $e^2$ is dimensionless. (In fact note in these units that $U=\alpha/r$). $\endgroup$
    – Andrew
    Mar 30, 2022 at 9:29
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    $\begingroup$ @Allure $\hbar \epsilon_0 c$ is, in this system, 1, and has dimension $M^{1-1+0} L^{2-3+1} T^{-1+4-1} I^{0+2+0}=T^2 I^2$ i.e. the units of charge squared. So the base unit for charge is already $\sqrt{\hbar c \epsilon_0}$ in this system, so you can't make it be $e$ as well. In general you have the freedom to define units such that with $n$ independent dimensions you specify the values of $n$ dimensionally independent quantities. $\endgroup$
    – Ian
    Mar 30, 2022 at 18:32
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    $\begingroup$ (Cont.) So because you have four dimensions ($M,L,T,I$) you might think you could specify the values of all four, but in fact charge is not dimensionally independent of $\hbar,c,\epsilon_0$. $\endgroup$
    – Ian
    Mar 30, 2022 at 18:33
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    $\begingroup$ @Markoul11 (a) You can always cook up a formula that will reproduce known digits of 𝛼. To be convincing, your formula must also predict digits of 𝛼 that have not yet been measured. (b) It doesn't do much to just have the formula, we also want a derivation/explanation from first principles. $\endgroup$
    – Andrew
    Apr 4, 2022 at 16:56
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Let's take a step back, and look at some physical constants. Take, for instance, $\Delta\nu_{Cs} = 9192631770\frac{1}{s}$. This constant is not measured, it is defined to be the exact value given. And via this constant, the unit $s$ is defined. I.e. $\Delta\nu_{Cs}$ does not tell us anything about the universe, it "only" tells us how much time has to elapse to call it $1s$.

Likewise, take the physical constant $c = 299792458\frac{m}{s}$. Again, this constant is not measured, it is defined to be the exact value given. And it defines relative scales of the units $m$ and $s$. (Since we have $s$ already defined, $c$ serves to define $m$, nothing more, nothing less.) I.e. $c$ does not tell us anything about the universe, it "only" tells us what it means to move at $1\frac{m}{s}$.

So what is different for dimensionless constants? And what does it mean that they can only be measured?
Since a dimensionless constant does not contain any units, it cannot tell us anything about our unit system. As such, it either tells us something about the math (like $\pi$ and $e$ do), or about the universe itself. If a constant (like $\pi$) tells us something about math, its derivation must be reducible to a purely mathematical derivation. That's definitely true for $\pi$ and $e$.

The fine structure constant is different because it must be measured. There is no purely mathematical equation that would define it, we must add information about the universe into the process. And that's precisely the point: Dimensionless constants that can only be measured actually tell us something about the universe itself. Not about our unit system, and not about our mathematical framework, but truly about the universe itself.

As such, it's an extremely important question what dimensionless, non-calculable constants exist, and whether they are independent of each other. Because each and every one such constants encodes some fundamental property of the universe we live in.

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    $\begingroup$ That's * really * nicely explained. May I be curious, what fundamental properties we know of, in this sense? $\endgroup$
    – Stilez
    Mar 31, 2022 at 22:04
  • $\begingroup$ @Stilez I'm not really into the current discussion about those constants. I learned about their significance by example of the fine-structure constant. But I can point you to the wikipedia article (en.wikipedia.org/wiki/Dimensionless_physical_constant) that has some more examples. As far as I can tell, the debate on which dimensionless constants exist, and which should be seen as the fundamental ones (and which should be seen as derivable from the fundamental ones), is far from closed. There is certainly not an exhaustive list of accepted fundamental constants yet. $\endgroup$
    – cmaster - reinstate monica
    Apr 1, 2022 at 6:17
  • $\begingroup$ Really, this whole answer should be added to en.wikipedia.org/wiki/Dimensionless_physical_constant :-D Great explanation, thanks. $\endgroup$
    – Martin Ba
    Apr 2, 2022 at 11:31
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Physics is not mathematics. It uses the strict self consistent mathematical theories with their axioms and theorems to find models that fit observations and data, and very important, are predictive of new situations, otherwise it would be just a mathematical map.

To do this, it introduces extra axiomatic proposals , called laws, postulates, principles, plus specific measurements that are not predicted but assumed. For the latter an example is the table of elementary particles in the standard model of particle physics. The postulates of quantum mechanics is what allows to compare results of experiments with the quantum mechanical calculations.

In general that is the way that physical dimensions are identified with the result of calculations, cross sections are in $cm^2$.

The complexity of the calculations and methods of studying data have brought forth what are dimensionless physical constants because the physical dimensions factor out. In your example the numerator and denominator are in physical units, and can only be predicted if the units are used consistently.

If a theory of everything is discovered, it will still need the extra axiomatic definitions to connect with data, but it is possible that the dimensionless numbers stop being dependent on physical units and could appear through the mathematics, but we are not at that point yet,

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    $\begingroup$ Indeed. You may measure π by constructing a circular object and measuring it. But we have a mathematical model of circles, and the value of π is a necessary consequence of the properties of the abstract circles of the model, so it is considered a dimensionless mathematical constant, not subject to measurement. But the value of α is not a necessary consequence of the properties of a known mathematical abstraction, so it must be measured. It is thus a dimensionless physical constant. $\endgroup$
    – John Doty
    Mar 30, 2022 at 19:37
  • $\begingroup$ @JohnDoty a better link mathworld.wolfram.com/ArchimedesRecurrenceFormula.html $\endgroup$
    – anna v
    Sep 27 at 8:28
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The point is that these dimensionless numbers are independent of how we chose unit systems and sometimes even which quantities are really measured.

(Disclaimer: My personal worldview is that still these number don't a priori take a special place in observations)

The general idea is like this: These numbers are very often independent of how you specifically measure. And very often noting down such ratios will tell you something. You can divide the charge of ions by the charge of electrons and you get a dimensionless number, in this case close to an integer. If you would find something like that, it severely constrains your theories, and sometimes finding these regularities allows you to extrapolate and form new theories.

However, there are dimensionless numbers which appear - like the fine-structure constant which are not explained by a current theory, even if it would be tempting and has been attempted very often. Many of these attempts cross the line from forming theory to guessing and sometimes discarding contradicting evidence. And while if for sure would make a nice good-night story if this constant was indeed e.g. 1/137 (or some other wild numbers) and all experiments on the world don't get it right, it isn't covered at all by observations or any real theoretical base.

I would restate the wikipedia statement to be: As of now there is no emergent theory available to predict these numbers

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If you take the statement from Wikipedia as an axiom to mean that a dimensionless physical constant is not possible to calculate or infer, whether now or anytime, it would would be wrong, or at least a bet that has been proven wrong. In fact, senior theoreticians strongly believe the contrary, they are actively engaged in the search of the one law for all laws, a Unification Theory that can explain all experimental results in physics at all scales, and that includes all the experimental physical constants. The laws of physics have the stature of laws precisely because they are ultimately based on these invariants.

So the first (high-level) principles of a unification theory are necessarily of philosophic nature in that they must represent a view of the natural world, in particular its intimate composition. Call it a hypothesis. This view must then be formulated in a mathematical language in order to constrain it for rigor and precision. At this point it becomes a mathematical physics model or axiomatic, yet devoid of any physical or experimental quantity. When this framework is virtuous enough that it is able to cast thru its own logical proceedings and mechanisms numbers or quantities that are identical or construable per closeness as the experimentally known physical constants, at any or all scales of matter, then we have a sure fire, assertive framework of unified physical law for the understanding and interpretation of the natural world. It’s also called the “new physics”.

Singh’s paper published last year, though calling upon a maze of variables and notions, is nevertheless heading in that direction. There is an even larger framework with impressive all-scale results, that has been published since 2015 out there. This paper, which I published at ResearchGate a while ago, offers a broad preview of this programme: https://www.researchgate.net/publication/313114545_Quanto-Geometric_Tensors_and_Operators_on_Unified_Quantum-Relativistic_Background

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