On Free Parameters

In the last post I talked about a measurement we did in our lab to characterize some properties of two parallel laser beams. The theory, if you want to dignify that equation with such a title, gave the power of the two beams as a function of mow much of the beams we cut off with a moving opaque block at position x.

The equation had 6 constants - the power of each beam, the width of each beam, the separation of the beams, and background power with no beams. Compare this to Newton's law of gravitation:


For two masses m1 and m2 separated by a distance r, this equation gives you the force. If you have two known masses and you know their separation and the force of their attraction, you can solve for the single constant G. That G is the only free parameter in the theory. Once G is known for one set of masses, it will be the same regardless of what other m's and r's you may plug into the equation. Every single instance of masses producing a gravitational force has to satisfy that equation or the theory is wrong. The fact that is does is impressive.

Our toy theory of blocked laser beams has to fit similar requirements: once those 6 free parameters are set by fitting methods, the equation has to give the right answer for however many data points we choose to measure. And it does, to reasonable accuracy.

The number of free parameters in a theory is something that physicists prefer to keep low. You could, for instance, write down a 365-term polynomial equation that perfectly described the last year worth of closing prices on the NASDAQ, but it would be worthless as a predictive theory because you have so many free parameters you could perfectly fit any 365 data points. A theory is more likely to be right (and less likely to be wishful thinking) if it fits large amounts of data with few free parameters.

The standard model in particle physics, for instance, has something like 19 free parameters. That's very small compared to the torrents of experimental data that it manages to fit quite well, but it's a much larger number than most successful physical theories. Many physicists spend a lot of time trying to think of deeper theories which would relate those parameters to each other such that in fact maybe there would be only a few (or one! or zero!) free parameters which still manage to fit all the data. So far, no dice.

Bonus question: Maxwell's equations in SI units appear to have two free parameters, μ0 and ε0. In CGS units they appear to have just one - he vacuum speed of light c. In CGS with "natural" units (i.e., c = 1), they appear to have zero. How many free parameters do Maxwell's equations actually have?

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Bonus question answer: 1. Even in SI units, the permeability and permittivity are not independent (they're related by the speed of light), so they don't count as two separate free parameters.

"Natural" units still have c in them, just with an ad hoc definition.

Maxwell's equations have two free parameters: μ0 and ε0. The speed of light is c = 1/âε0μ0 so the one free parameter actually depends on two things.

The one free parameter in SI units is actually the definition of the meter. c is defined to be 299792458 m/s, and the second also has an official definition, so the meter is whatever it needs to be to satisfy those two definitions (the reason for this standard is because the second and the value of c can be measured to greater precision than the meter as a length standard per se). Recall that μ_0 also has a defined value of 4π*10^-7 henrys. Since ε_0μ_0c^2 =1, that gives a de facto definition of ε_0.

By Eric Lund (not verified) on 11 Mar 2011 #permalink

It's a subtle point that I didn't appreciate until just a couple weeks ago, but this is all closely tied to the fact that units of measurement are much more arbitrary than intuition might lead you to think. This'll be a post pretty soon, but it boils to a generalization of Eric Lund's point - the number of "fundamental" units (meters, kilograms, seconds, etc) in physics is very arbitrary, and this arbitrariness affects the way constants are written.

"(19...) it's a much larger number than most successful physical theories."

It's a much smaller number of free parameters than any other theory that explains the same huge set of data.

Great post! I don't comment often, but I enjoy your blog. thanks.

I think it's sensible to say that Maxwell's equations have two free parameters. If gravity has one, there must be two for electromagnetism, namely the strength of electric forces and the strength of magnetic forces. If you're making these things appear to go away, you're just stuffing the free parameter into your definition of one of the units or another.