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?
Best Answer
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 property of a whole set of vectors rather than a “pointwise” property of individual vectors.