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It is often said that one of the drawbacks of the standard model is that it has many free parameters. My question is two-fold:

  1. What exactly is a free parameter? My understanding is that the free parameters of a model/theory are the ones that cannot be predicted by the theory and need to be measured and put in the theory 'by hand' so to speak. Are all constants of nature free parameters then? Also, can you give me an example of a non-free parameter in a theory?

  2. Why is it bad for a theory to have free parameters? Couldn't be that some quantities in nature such as the mass of the electron just 'happen' to have a certain value that cannot be predicted by a theory?

Floyd
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7 Answers7

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A parameter is a number that does not "come out" of the theory, but needs to be input into the theory by hand. The electron mass is a good example -- no one can (currently) calculate the electron mass, it is simply something we measure.

There are no absolute rules about things like "how many parameters is too many parameters," though of course many people have different opinions. From the point of view of fundamental physics, typically progress is made by synthesis or unification, which is the development of a deeper theory that simultaneously explains two phenomena you originally thought were different. Typically, a synthesis of two apparently different physical theories leads to realizing that what was originally a free parameter in one of the theories can actually be calculated in the unified theory.

There are a few very famous examples of this in undergraduate physics. Here are a few examples (admittedly the last is not really an undergraduate level example)

  1. The acceleration due to gravity on the surface of the Earth, $g$, can be calculated from the mass $M_\oplus$ and radius $R_\oplus$ of the Earth and Newton's gravitational constant, $g = G\frac{M_\oplus}{R_\oplus^2}$. In Galileo's time, gravity on the Earth was not known to be connected to astronomical quantities like the mass and radius of the Earth, but Newton showed that gravity is one force underlying apples falling from trees and moons orbiting planets.

  2. The speed of light $c$ can be calculated from the electric and magnetic field coupling constants, which in SI units are $\epsilon_0$ and $\mu_0$. In SI units, we have $c^2 = (\mu_0 \epsilon_0)^{-1}$. This follows from Maxwell's realization that light, electricity, and magnetism -- three apparently different subjects -- are all actually one unified theory of electromagnetism.

  3. The gas constant $R$ is related to Avogadro's number $N_A$ and the Boltzmann constant $k$, $R = k N_A$. Realizing the atomic structure of matter explained a lot of phenomena in thermodynamics by reducing them to the statistical motion of individual atoms.

  4. The peak wavelength of light emitted from a black body at temperature $T$, was known in the 1800s to be given by Wien's displacement law, $\lambda_{\rm peak} = b/T$. The constant $b$ is known as Wien's displacement constant. With the development of quantum mechanics and the Planck distribution, it was shown that $b=h c / x k$, where $h$ is Planck's constant, $c$ is the speed of light, $k$ is Boltzmann's constant, and $x$ is a calculable order 1 factor that drops out of some annoying algebra ($x=4.965...$ solves $(x-5)e^x+5=0$).

  5. While this is not as "basic" as the others, the unification of the electric and weak forces showed that the Fermi constant $G_F$ controlling the lifetime of the muon, is closely related to the Higgs vacuum expectation value $v$, via $v = (\sqrt{2} G_F)^{-1/2}$ (and therefore also to the mass and coupling of the W boson).

Note that these examples of unification leading to a deeper understanding of an input parameter are correlated with major developments in physics -- gravity, electromagnetism, statistical mechanics, quantum mechanics, and the electroweak theory.

There is a belief in theoretical physics that the Standard Model is not a fundamental theory (indeed we know it can't be because it doesn't contain gravity and dark matter). By extrapolating the logic of unification that has always been associated with progress in physics, one hopes that at least some of the numbers that are currently input parameters to the Standard Model, will actually be shown to be calculable outputs using a deeper theory.

Andrew
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  • For example 2, how does that relate to the speed of light being chosen as a convention? – Galen Oct 27 '21 at 16:03
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    @Galen That video drives me crazy! The speed of light is not a convention -- light travels at a finite, measurable speed. That video blows way out of proportion a small technicality about how you synchronize clocks in special relativity. There is a free choice in how you agree to synchronize clocks, but no physical results depend on this choice, and there is one clear choice that makes the math much simpler and the physics much clearer -- no one in my professional physics life has ever even considered making a different choice. – Andrew Oct 27 '21 at 16:07
  • An analogy I've seen on this site is measuring the duration of a plane trip from the UK to the US by subtracting the arrival time in the US time zone from the departure time in the UK time zone. You can get zero for the duration of the plane trip if you do this, and conclude the plane traveled instantaneously from the UK to the US. But that's just because you used a crazy convention. If you use the actual time measured by someone on the plane as the way to measure the duration, you'll always get a positive answer. The one way speed of light "controversy" has as much content as this example. – Andrew Oct 27 '21 at 16:10
  • I'm having difficulty parsing your comments.It sounds like you're saying there is a free choice and that the mathematically convenient one is always used by physicists. Is this an accurate summary of your points? – Galen Oct 27 '21 at 16:23
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    @Galen It's hard to summarize this in a chat box, but there are a lot of questions about this topic on this site. You have to decide how to synchronize clocks between observers in special relativity. This is a convention, and the sensible convention is to assume light travels at the same speed on the way from observer A to observer B, as it does from observer B to observer A, for all pairs A and B. If you decide to use a different convention, there are no observable consequences that are different. But now you have built in that somehow light travels at different speeds on different paths... – Andrew Oct 27 '21 at 17:50
  • ...and in different directions. Your convention has introduced special paths on which you assume light goes faster one way vs the other. The set of paths and "one way speeds" cannot be justified by any observation or physical principle, and has no observable consequences. Since, ultimately, the issue was synchronizing your clocks, it is essentially the same thing as defining speed as $\Delta X / \Delta T$, where $\Delta T$ is the difference between two times measured on clocks that are not synchronized; you are defining $\Delta T$ as the difference of times measured in different time zones. – Andrew Oct 27 '21 at 17:53
  • My very strong opinion is that this is a total red herring (mathematically, physically, philosophically) and it's really a shame that that video has led people to spend so much time thinking about it. It's worth understanding the issue, but I personally very strongly feel that it is very easy to spend too much time on this without getting any actual physical insight in return. – Andrew Oct 27 '21 at 17:55
  • Okay. Thanks for sharing and clarifying your perspective. – Galen Oct 27 '21 at 17:59
  • @Galen I'm sorry for harping on this, and I'll stop after this message. While I agree with the way you summarized my thoughts, I would actually go a step further. I think it is actually a deep misunderstanding of the physics to use the "one-way speed of light" thing to claim that light does not travel at the speed of light. The free choice you have is not really a free choice in the speed, it's a free choice in how you are comparing measurements of clocks at different places. – Andrew Oct 27 '21 at 18:05
  • The point is that you always can choose to synchronize your clocks so the speed of light is the same. The only physically sensible meaning of the sentence "the speed of light is different in different directions", is that it's impossible to choose measurement conventions where the speed of light is the same. In other words, different physicists with different conventions will all agree on the invariant statement that the speed of light varies. That's clearly not the case in special relativity. – Andrew Oct 27 '21 at 18:11
  • @Andrew: Maybe a mention of Occam's Razor would be prudent in this context, since it gives us a clear criterion to decide which of those assumptions is the "right" one, using "right" not in the sense of "right or wrong", but in the sense of "the right tool for the job". Clearly, the assumption that light travels at the same speed in all directions has fewer "entities" (in the Occam's Razor sense of entia non sunt multiplicanda praeter necessitatem) and is thus the "right" one, and that's not just an opinion but an objectively measurable fact. – Jörg W Mittag Oct 27 '21 at 19:58
  • @JörgWMittag That's certainly one way to frame the issue. I agree with you, but I also feel appealing to Occam's Razor is overkill in this situation and perhaps gives the impression that there is there is room for a reasonable scientific disagreement. For me, it's not that the one-way speed of light introduces a plausible but more complicated-than-needed-empirically set of models. Rather, it is a confused approach leading to a uselessly muddled formalism. It's like putting dirt on a telescope lens intentionally and publishing images with a note to the reader to subtract the dust particles. – Andrew Oct 27 '21 at 20:21
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    There's something that irks me about this answer. It's not wrong, but there's constants and there's constants. Ten free parameters for the cross sections of the fundamental particles isn't going to fit an elephant. Four free parameters in the equations of force will fit said elephant. – Joshua Oct 28 '21 at 02:55
  • @Joshua I read your comment as saying we should be impressed the Standard Model has so few parameters given the vast number of phenomena it describes, rather than complaining it has too many. If that's a fair summary, I actually pretty much agree with you (and I also hate the elephant quote). But, one can still hope to do better. I tend to be pessimistic that humanity will be able to do better than the SM in the next 100 years, but if it turns out that someone figures out some relationship between the Yukawa couplings (for example) I would be thrilled. – Andrew Oct 28 '21 at 03:17
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  1. Yes, a free parameter is a parameter that is not fixed by the theory and has to be measured in an experiment. Examples are the electron mass or its charge. No, not all constants of nature are free parameters, in these cases, we can find expressions to represent the quantity by a number of truly free parameters. An example would be the $g$ factor. It is predicted by QFT very accurately solely in terms of the SM free parameters.

  2. You have to understand that at the very core of every theory we are trying to connect some measured quantities to some prediction. For example, we want to compute the crosssection for $e^- e^+ \longrightarrow \mu^- \mu^+$ and we do so by using $e, m_e$ and $m_\mu$. A less predictive theory than QED (like classical QM) would not be able to make this prediction and would need to make the crosssection a new free parameter. So the more predictive and hence useful a theory is, the more free parameters are connected to observables. It would therefore be great to have no free parameters and have an unlimited amount of observables. This would be the ultimate theory but whether it exists is a completely different matter (but I don't think so).

tomtom1-4
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  • Could you please elaborate a bit more on 2? I understand that the more free parameters a theory has, the less predictive, and hence less useful it becomes. But how is it possible to have a theory with no free parameters at all? Aren't observables like cross section necessarily always dependent on at least some free parameters? Basically, if I understand correctly, all the parameters of the ''ultimate theory'' would be determined exclusively from the theory, but I can't see how the value of a parameter can be determined without any inputs from experimental data. – Floyd Oct 25 '21 at 19:09
  • @Floyd In Planck units, all quantities are adimensional numbers. A theory could in principle give mathematical formulas for all the relevant parameters, and hence predict their values in Planck units. – Javier Oct 25 '21 at 19:53
  • I can't imagine it either xD. As I said I don't think that we can find or that there exists such a theory. But I could imagine that at some point we find that the CKM matrix for instance is parametrized by only one or two parameters. But who knows? Maybe I am naive and we can predict much more in the future or I am a dreamer and we can't reduce the number any further. – tomtom1-4 Oct 25 '21 at 19:56
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    @Floyd Such a theory would be some fundamental assumptions, from which you unambiguously get a certain equation and values for certain quantities, from which you make further assumptions and equations, until you eventually derive the value of fundamental constants like $m_e$, $c$, $h$ and so on. I also find this unlikely, but it is plausible. For example, you might could assume some certain x-dimensional manifolds to correspond to something, and use their topological numbers and mathematical constants like $\pi$ to arrive to some specific constant via some topological equation magicary. – Neinstein Oct 26 '21 at 07:32
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Theory: It's true that free parameters are bad for a theory. For a certain value of true.

A useful theory doesn't simply encode all existing observations. It predicts new ones. With enough free variables the theory isn't predicting. It becomes a record of what we've seen, presented in a different way.

It's hard to find a deeper meaning if the theory is just an infinitely distortable funhouse mirror held up to what we already know.

candied_orange
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An ideal theory can perfectly explain all of the observed data, and could not at all explain anything that was not observed.

For example the theory

[Theory] : "stuff just happens, except when it doesn't"

can be used to explain any experimental observation (Q: "Why did the electron move when I dialled the voltage?" A: "stuff just happens, except when it doesn't", Q:"Why..." A: "Same".) But this is a bad theory, because it could equally well explain the things that did not happen. (Q: "Why did a dragon materialise when I dialled the voltage?" A: "same", Q: "... But I was joking about the dragon!")

Free parameters are a problem, because they always give you more power to twist the theory to match things, whether those things have been observed or not. A theory with too many such parameters starts to get closer and closer to the "stuff just happens" theory. Of course a theory with too few parameters might fail at explaining some real physics. If adding a new parameter one ideally should weigh up how much the theory gains in its ability to explain the real (which we want to maximise) against how capable it becomes at explaining the fictitious (which we want to minimise).

Dast
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I think that "free parameter" and "adjustable parameter" and "parameter" are essentially synonyms. I would call the $g$ factor (mentioned in the earlier answer) a prediction of the Standard Model, not a parameter. It could be a free/adjustable parameter of a different model.

The values of the parameters may be random and meaningless, but we don't know that they are, and as long as we don't know, there is a motivation to keep looking for explanations. If a physicist says that free parameters are bad, they mean that it's bad, in some sense, that we don't know everything yet. But in another sense it's good, because it gives them something to do.

benrg
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  • Adjustable parameter and free parameter are not the same for mathematical ( most sciences) model. Adjustable parameter has no relationship to experimental data- whatever needs to fiddles to predictions whereas free parameter comes from data. – user2617804 Oct 26 '21 at 08:58
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    @user2617804 Language is descriptive, not prescriptive. I agree that the philosophical distinction you're trying to make is useful, but I don't think most scientifically literate English speakers would infer it from the phrases "adjustable parameter" and "free parameter" without your linguistic conventions being spelled out in the document in question. But I would be interested to see some kind of scientific dictionary that made you think that, if there is one. – Ian Oct 26 '21 at 13:17
  • I think there's a historical context. At an early time we may have measured g. Then we develop more understanding, and discover that it can be predicted from other parameters. So it goes from being a free parameter to a prediction. But we can still use it as a parameter in formulas that use it. – Barmar Oct 26 '21 at 15:10
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Theory/model is essentially a mathematical framework, which predicts the data observed in an experiment as a function of a set of parameters. Some parameters are specific to the experiment, others are considered as known, and yet others may have to be fit from the results of the experiment (or to better match the data). It is these latter that we call free parameters.

Having many parameters is not necessarily bad: the question is that of overfitting (having too many parameters and fitting "noise") vs. underfitting (having too few of them, and missing important physics). There are standard statistical procedures for validating and comparing models, as discussed in this answer.

Urb
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Roger V.
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In short: suppose for a phenomena, there are $n$ quantities that describes it completely. Your model take $a$ number of free parameters as inputs (these needs to be measured and plugged into the theory) and outputs $b$ number of quantities as predictions such that $$a + b = n.$$

As a result, the higher $a$ is, the less $b$ becomes, hence the less predictive the theory becomes.

Our
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