Exploring Naturalness: The Importance of Constraints in Theoretical Physics

[ChrisVer]

Why do physicists find naturalness as some important feature a theory should contain?
Since the choice of the theory's parameters is quite abstract (or can be experimentally determined), what's the reason a constant that you put by hand and can take values from 0 to "infinity", be constraint to take values near to some other constant of your theory?
I know they say that naturalness is an aesthetic argument, but some people seem to strongly believe in its power.
I hope my question makes sense? If not please allow me to retry.

 

[mfb]

If I present you a physical theory and its fundamental, dimensionless parameters are 1.000000023, 1.000000034, 1.000000046 and 1.000000084, would you believe this is really a fundamental theory, and not some effective model where the small contributions above 1 come from something else?

Or take 35.2, 15.9, 52.8, 25.5 and 3*10^(10^10) for parameters with a similar meaning (particle masses, coupling strenghts or whatever). It just looks odd. It could be true, but it would be surprising.

 

[Vanadium 50]

If you walked into a room with 10,000 pencils, all perfectly balanced on their points, would you think there was some reason why?

 

[ChrisVer]

@mfb Doesn't it seem more odd to try and impose some symmetry, break it and get an unobserved light/massless axion (as for example for the strong cp problem)?
The almost equal terms would imply a total unification of everything (all coupling constants= one coupling constant), right? So does unification need naturalness?@Vana, I would be indeed surprised by having 10,000 pencils all perfectly balanced (that's odd). But what if you entered the room and saw 100 pencils like that, 100 pencils like something else, 100 blah blah... wouldn't that seem more natural to you?

 

[kurros]

Giudice has a nice non-technical article on the subject here: http://arxiv.org/abs/0801.2562

It's quite a nice read.

 

[Haelfix]

There are about 6 or 7 not necessarily equivalent definitions of naturalness.

I'll list the three most convincing definitions:

Naturalness in the sense of 'T Hooft. "A parameter tends to be small only if it limits to a more symmetrical theory when it vanishes"

Naturalness in the sense of Dirac: Dimensional analysis works! A fundamental theory by Bayesian reasoning ought to have parameters that are O(1)

UV sensitivity: If there is a theory that has a cutoff, an unnatural theory involves a scenario where a small change in a parameter leads to very sensitive cancellations between uncontrolled quantities in the deep UV.

You can also think about naturalness in terms of phase transitions, and quite a few other ways.

 

[mfb]

@mfb Doesn't it seem more odd to try and impose some symmetry, break it and get an unobserved light/massless axion (as for example for the strong cp problem)?
The almost equal terms would imply a total unification of everything (all coupling constants= one coupling constant), right? So does unification need naturalness?

I would prefer additional fields to really odd constants.

@Vana, I would be indeed surprised by having 10,000 pencils all perfectly balanced (that's odd). But what if you entered the room and saw 100 pencils like that, 100 pencils like something else, 100 blah blah... wouldn't that seem more natural to you?

Sure, that's why 10,000 pencils on their tips are odd.

 

[Meir Achuz]

It is interesting that physicists (mostly atheists) derive theories as if they were created by a creator who liked neat theories. Every time a puzzle arises, and people try to solve it with all kinds of approaches, the final correct theory is more 'natural' than any of the first ones they tried.

 

[nrqed]

Why do physicists find naturalness as some important feature a theory should contain?
Since the choice of the theory's parameters is quite abstract (or can be experimentally determined), what's the reason a constant that you put by hand and can take values from 0 to "infinity", be constraint to take values near to some other constant of your theory?
I know they say that naturalness is an aesthetic argument, but some people seem to strongly believe in its power.
I hope my question makes sense? If not please allow me to retry.

It is more question of cancellation. For example, consider the quadratic divergences arising from a scalar field like the Higgs field. On one hand, treated as an effective field theory, we should cut off the momentum integration at an energy of the order of the square of the scale $\Lambda^2$ of new physics (around 1 TeV or so). This is the natural scale to use. So far, there is nothing strange. In addition, we add counter terms to the original lagrangian, as prescribed by renormalization, including a mass counter term proportional to $\delta m^2$ . In principle the scale of these counter terms (I am being very hand wavy here, there are of course different types of renormalization counter terms). There is a cancellation between those two terms which is incredibly fine tuned. They cancel almost exactly (I don't remember the precise values but we are talking about at better than 1 part in a million IIRC). One can indeed adopt the point of view that there is nothing to see there, that's what we get and that's the end of the story. But most people would think that there is some deeper reason explaining this impressive cancellation.

See also Haelfix's post, which is excellent.

 

[KL7AJ]

We live in a coherent, consistent universe. That's a good thing...otherwise we could never learn anything! It is only reasonable that we would see consistency through all of nature. I like to look at the natural logarithm...it permeates EVERY conceivable system. In a sense, it may seem sort of strange...but it would be stranger still if the natural log wasn't around!

The fact that we can even use Math for every physical subject also points this out.

Now you know. :)

Eric

 

[mfb]

It is interesting that physicists (mostly atheists) derive theories as if they were created by a creator who liked neat theories. Every time a puzzle arises, and people try to solve it with all kinds of approaches, the final correct theory is more 'natural' than any of the first ones they tried.

The ideal fundamental theory would be one without any dimensionless constants. Then there are no constants to explain :D.

 

[KL7AJ]

It is interesting that physicists (mostly atheists) derive theories as if they were created by a creator who liked neat theories. Every time a puzzle arises, and people try to solve it with all kinds of approaches, the final correct theory is more 'natural' than any of the first ones they tried.

I would question the assertion that most physicists are atheists, especially if you go back a few hundred years. Consider Newton, Liebniz, Maxwell, Faraday, and even Einstein who was more of an agnostic than anything.

Intelligent design doesn't require that things are particulary simple, but it does imply a certain clean elegance. :)

Eric

 

[mfb]

I would question the assertion that most physicists are atheists, especially if you go back a few hundred years. Consider Newton, Liebniz, Maxwell, Faraday, and even Einstein who was more of an agnostic than anything.

Most physicists live today.

Atheist and agnostic can be hard to separate, but most physicists I know fall in this spectrum.

 

[atyy]

Naturalness is motivated by the belief that our theories are not the final theory. If our theories are not the final theory, then there should be some sort of "robustness" to how we coarse grain to arrive at effective theories that are useful to us. The UV sensitivity criterion that haelfix listed is, I believe, in this spirit.

An alternative is that we have discovered some fundamental aspect of "the final theory", but it is still unlikely that we have exactly the final theory. So we explain the robustness by a "symmetry", which is a way of saying if this is a fundamental principle of the final theory that we can see, even though we only have effective theories, then we'll use that to constrain our next and better effective theory. The 't Hooft sense of naturalness that haelfix listed is, I believe, in this spirit.

In both cases, it comes about from not believing that we have the final theory, which is quite reasonable.

The aesthetic part comes from how reasonable it is to believe that we don't have the final theory. If by some chance the standard model is the final theory of everything, then there is no reason for it to be natural.

 

[ChrisVer]

Sure, that's why 10,000 pencils on their tips are odd.

I would continue that by saying "doesn't it seem odd to have all coupling constants equal [but I want to look at the paper sent in a previous post before, because I'm also having a problem with understanding the UV arguments which seem more fundamental for me] ? If they are equal then of course something is hiding behind (GUTs maybe or other extensions as SuSy).
Some happen not to be equal, and we are trying to make them. That's what makes me so cautious with the naturalness argument.

 

[arivero]

I would continue that by saying "doesn't it seem odd to have all coupling constants equal [but I want to look at the paper sent in a previous post before, because I'm also having a problem with understanding the UV arguments which seem more fundamental for me] ? If they are equal then of course something is hiding behind (GUTs maybe or other extensions as SuSy).
Some happen not to be equal, and we are trying to make them. That's what makes me so cautious with the naturalness argument.

I think that the most serious naturalness arguments are the ones that suspect of all mass being nearly equal... to zero. Of course, it is very suspicious to see yukawa's top equal to one, but this is never considered a naturalness issue (IMO, it should flash some alert lights, but it is just IMO). But it is even worse to see all the other fermion masses very small respect to Higgs scale, and to see Higgs scale very small respect to GUT or Planck scale.

Also, a primitive version of naturalness argument was the one of the relationship between the mass of a (composite, or extended) particle and its size. The size being a measurement of the magnitude of the force that keeps the particle components together. So the tension of a string is taken also as a measurement of the size of the string. Thus we need arguments, if the particles are not postulated to be singularly point-like, to justify why a particle compton length is orders of magnitude greater than its size.EDIT: I was just reading this article of 1983 where Veneziano calls "'t Hoftons" to the particles whose mass is protected to be near zero via 't Hoft naturalness :-D