Meir Achuz said:
A free parameter must be dimensionless. Only mass ratios are free, not individual masses.
c is not a free parameter, but is a definition of the ratio m/s.
alpha is a free parameter.
A
free parameter does not have to be dimensionless:
A
free parameter is a variable in a
mathematical model which cannot be predicted precisely or constrained by the model and must be
estimated experimentally or theoretically. A mathematical model, theory, or
conjecture is more likely to be right and less likely to be the product of
wishful thinking if it relies on few free parameters and is consistent with large amounts of data.
For example, in the Standard Model, while the mass ratios are dimensionless, if you express the Standard Model fundamental particle masses in that form, you need a mass scale for the Standard Model as a whole that does have a dimension of mass to fully describe it, or you have to describe them as ratio of a dimesionful constant such as the Planck mass.
Some theories in physics (like general relativity) don't even have any dimensonless physical constants, although you could define related parameters that are dimensionless in a variety of ways using Planck mass and Planck time (which are based upon combinations of other experimentally measured fundamental constants).
There is a special subcategory of
dimensionless physical constants (in the Standard Model, these include the coupling constants of the Standard Model, and the ratio of the fundamental particle masses):
In
physics, a
dimensionless physical constant is a
physical constant that is
dimensionless, i.e. a pure number having no units attached and having a numerical value that is independent of whatever
system of units may be used.
For example, if one considers one particular
airfoil, the
Reynolds number value of the
laminar–turbulent transition is one relevant dimensionless physical constant of the problem. However, it is strictly related to the particular problem: for example, it is related to the airfoil being considered and also to the type of fluid in which it moves.
On the other hand, the term
fundamental physical constant is used to refer to some
universal dimensionless constants. Perhaps the best-known example is the
fine-structure constant,
α, which has an approximate value of 1⁄137.036. The correct use of the term
fundamental physical constant should be restricted to the dimensionless universal physical constants that currently cannot be derived from any other source. This precise definition is the one that will be followed here.
However, the term
fundamental physical constant has been sometimes used to refer to certain universal dimensioned
physical constants, such as the
speed of light c,
vacuum permittivity ε0,
Planck constant h, and the
gravitational constant G, that appear in the most basic theories of physics.
NIST and
CODATA sometimes used the term in this way in the past.
But a mere parameter of a model need not be a dimensionless fundamental physical constant, and as the examples quoted above illustrate, many core physics physical constants are not dimensionless.
Also, describing dimensionless physical constants as "dimensionless" is honestly not something to be too hung up on, because many superficially dimensionless physical constants still have some implicit dimension "hidden in the footnotes" so to speak.
For example, since the masses of the fundamental particles in the Standard Model run with energy scale, a ratio of masses implicitly includes a usually unstated footnote regarding the energy scale at which the masses compared to each other are measured.
Similarly, the "dimensonless" physical constant of the Standard Model which is the strong force coupling constant, is subject to the same limitation. The most commonly cited value of the strong force coupling constant is 0.118(1) is the value at the Z boson mass energy scale, but at other energy scales it is different (in a manner that goes up to a peak value and then down again).
Likewise, the fine structure constant's value at the Z boson mass energy scale is about 1/127 rather than 1/137.
There is no truly "dimensionless" physical constant in the Standard Model for which there is not an implicit energy scale footnote that has a dimension in some sort of units. All Standard Model physical constants run with energy scale.
The numerical value of the speed of light is used to defined the meter in the current version of the SI standard of units, but that doesn't mean that it isn't an experimentally measured free parameter of the Standard Model, general relativity and special relativity. If it had a different physical value (regardless of the numerical value we assign to it in arbitrary human made units determined by committee in a political process), the world would behave differently. You need to know its dimensioned value to do physics.
Since 1983 (with the wording updated in 2019), the meter has been internationally defined as the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second, where the second is defined in terms of the caesium frequency ∆ν (which, in turn, was determined by calibrating this to the previous definition of 86,400 seconds in an average Earth day as exactly as was possible at the time it was defined).
The current definition of the meter was the product of ultraprecise measurements as of 1983 that were compared to a physical exemplar which was the previous definition (if the meter had been defined in terms of the speed of light earlier, the defined speed of light would have defined to be 300,000,000 meters per second, would be about 0.1% shorter, and physics students everywhere would have been thankful, but back compatibility issues demanded adherence to the physical exemplar already used to make ultraprecise measurements in 1983). The physical exemplar, in turn, was
established with state of the art 18th century science based upon the distance from the north pole to the equator:
The metre was initially defined as one ten-millionth of the distance on the Earth's surface from the north pole to the equator, on a line passing through Paris. Expeditions from 1792 to 1799 determined this length by measuring the distance from Dunkirk to Barcelona, with an accuracy of about 0.02%.
The numerical value of the speed of light isn't an axiom of general relativity, special relativity, or the Standard Model, all of which utilize this parameter. It is a property of Nature which we measure, either directly, or indirectly using other measurements which are functionally related to it.
Aesthetically, there is something nice about defining physical constants in a way that is independent of particular arbitrary human defined units, as this is a more "universal" expression of them. But lack of any dependence on scale (which is what a truly dimensionless physical constant would have as a property) may simply be inconsistent with the nature of what is being described by a physical constant. Nature may be scale dependent.
