Robert McNees
Physicist Ken Wilson was born OTD in 1936. He invented lattice gauge theory and pioneered the modern view of the renormalization group. This latter contribution informs everything from the physics of phase transitions to how we interpret quantum field theories.
Image: Cornell U.
Robert McNees
We now view almost every quantum field theory from Wilson's point of view, as what we call an "effective field theory." An EFT offers a useful description of Nature over some range of energies. But at sufficiently high energies it breaks down and loses predictive power.
Robert McNees
In a sense, an Effective Field Theory is just the stuff that matters for energies below some maximum "cutoff" above which that particular theory is no longer a useful description.
The real marvel is that Nature organizes itself so that physics at lower energies is largely independent of the details needed to describe phenomena at higher energies.
Robert McNees
Which is to say, we don't need a quantum field theory that works at the grand unification scale before we can do chemistry! We can make an astonishingly good prediction for the anomalous magnetic moment of the electron without understanding quantum gravity at the Planck scale.
Wilson's renormalization group approach shows precisely how this works for quantum field theories, how those high-energy details end up being irrelevant for predictions (and the ability to do science) at low energies.
Robert McNees
If this sounds interesting, I encourage you to read this short tribute that John Preskill wrote about Ken Wilson back in 2013:
https://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/
Though a bit more technical, you might also enjoy this 2021 article by Sean Carroll discussing the "Core Theory" – the standard model of particle physics, along with general relativity – as an EFT valid at all scales relevant to our everyday life.
https://arxiv.org/abs/2101.07884
Robert McNees
In honor of what would have been Ken Wilson's 87th birthday, let me try to use Effective Field Theory reasoning to motivate why the sky is blue and clouds are white.
First, Quantum Field Theory (QFT) is our framework for describing particle physics. It’s what we get when we combine quantum mechanics and special relativity.
Everything is fields. Particles like electrons or quarks are localized excitations of fields. These particles experience forces by exchanging excitations of other fields.
Robert McNees
There are many ways to construct a QFT describing various fields and interactions, but for "low energy" phenomena only some of these will contribute to the physics in a meaningful way.
Focusing on the most relevant possibilities gives what’s called an Effective Field Theory (EFT).
An EFT “parameterizes our ignorance” about the details of physics at higher energies. This lets us focus on stuff relevant at the lower energies where we conduct experiments or make our observations.
Robert McNees
(Obviously I am glossing over lots of stuff here.)
Every EFT comes with a cut-off Λ – the highest energy at which the EFT is a good framework for describing the physics. Physics at energies below Λ doesn’t much depend on the physics at higher energies.
This “separation of scales” is why EFT is so powerful.
We're lucky Nature is organized this way. If we needed to know all those high energy details to understand everyday phenomena, we'd have a hard time doing any sort of science!
Robert McNees
You can think up very complicated interactions of the fields in your theory, but these interactions contribute to physics at energy E with a factor of (E/Λ) to some power. This factor has to be there, to make the units come out right.
At energies E << Λ, the (dimensionless) number E/Λ will be very, very small. This means that complicated stuff gets suppressed by powers of a very small number.
Robert McNees
Of course there are other deep and important principles that constrain how the fields can interact. I’m glossing over lots of essential details so I can focus on one aspect.
So what does all this have to do with the color of the sky?
Robert McNees
Visible light covers a range of wavelengths λ from around 400-700 nm. Meanwhile, N₂ molecules in the atmosphere are at least 10,000 times smaller than that. In QFT, higher energies is the same thing shorter distances. So a good cut-off for our EFT will be the inverse size of the N₂ atoms. Visible light will involve energies E ~ 1/λ much smaller than this cut-off.
Robert McNees
Now we can write down an EFT that describes the physics of visible light interacting with N₂ molecules at energies below the cut-off.
The principles of EFT tell us we can ignore things like internal details of the N₂ molecules. Stuff internal to the molecule involves distances smaller than the size of the molecule, so it only gets probed at energies above Λ.
Robert McNees
Our EFT will use a complex scalar field to represent the N₂ molecules. We'll call this field ψ – it is a quantity whose value at any point in space and time is a complex number. Since the molecules are electrically neutral, they can only couple to electromagnetic fields (light) in certain ways.
The electromagnetic field is described by a “field strength tensor” Fᵤᵥ. It's an array of numbers containing all the info we need about the light that will interact with the molecules.
Robert McNees
In particle physics we write out all the interactions between ψ and Fᵤᵥ, consistent with a few other principles.
Usually we do particle physics experiments in a vacuum.
Here, though, we're interested in what happens in our atmosphere! The presence of the atmosphere changes how we apply an EFT principle – the Lorentz invariance of special relativity. This opens up a few possibilities for the fields and their interactions that we might not consider in a particle accelerator experiment.
Robert McNees
So we think of all the ways our scalar field and electromagnetic field might interact, allowing for the fact that the field is electrically neutral and the atmosphere gives us a preferred frame of reference. Two possibilities are more important than all the others.
Schematically, for those of you who know a little QFT, they look like |ψ|² F^{μν} F_{μν} and |ψ|² u^{μ} u^{ν} F_{μ}^{λ} F_{νλ}, where u^{μ} is the four-velocity singled out by the atmosphere's frame of reference.
Robert McNees
Both combinations of fields come with a factor of (1/Λ)³, to get the units right.
The second term is relevant for the scattering of light we’re interested in.
Using Feynman’s rules for computing the effect of the interactions, we find the “amplitude” for a photon (the particle excitation of the electromagnetic field) being scattered by an N₂ molecule is A ~ f² / Λ³, where f is the photon's frequency. Energy ~ f ~ 1/λ for a photon!
Robert McNees
(Technical aside: The field strength F_{μν} involves a derivative of the electromagnetic potential, so we get one factor of the photon’s momentum k for each factor of the field strength in the interaction. And k~f for a photon, so A ~ f² / Λ³.)
Robert McNees
The likelihood of a photon being scattered by a N₂ molecule is described by a “cross-section” we usually denote σ. It is proportional to the amplitude squared, so we get σ ~ f⁴/Λ⁶. The frequency of light times its wavelength is equal to the speed of light c, so f = c/λ and we get σ ~ (1/λ)⁴.
Robert McNees
This is known as "Rayleigh scattering." It occurs when the wavelength of light is much longer than the size of the scattering particles (when f ~E << Λ).
Blue light has a shorter wavelength than red light, so it is more strongly scattered. That is, the cross section for scattering of blue light is larger than the cross section for scattering of red light.
(1/λ_blue)⁴ >> (1/λ_red)⁴
Robert McNees
So EFT tells you that if you look at the sky in a direction away from the sun, it should look blue.
Why? Because looking away from the sun, you're seeing light that has scattered around a bunch and is finally headed towards you. The stuff that is most strongly scattered (blue light) has the most reason to be coming at you from a direction other than sunward!
Robert McNees
Now, the cut-off in our EFT was inversely proportional to the size of N₂ molecules, which is much shorter (higher energy) than the wavelength of visible light.
What if the light interacts with something bigger? A collection of water molecules in a cloud droplet may have size similar to the wavelength of visible light.
In that case, the energies associated with visible light would be pretty similar to the inverse size of the scattering particles – the natural cut-off for the EFT.
Robert McNees
If we build an EFT to describe light scattering off water in a cloud, we find there are many more interactions that must be considered. In the previous scenario we ignored all but a few interactions because powers of f/Λ made most possibilities irrelevant. That’s not the case here!
The result is something called “Non-selective scattering.” All wavelengths of visible light get scattered more or less the same way, because EFT principles tell us many previously ignored interactions are relevant.
Robert McNees
Since all colors scatter about the same, you're likely to see all of them coming from all directions. All colors coming from all directions looks white.
So Effective Field Theory tells us that the sky is blue, and clouds are white.
~The End~
Eric Carroll
@mcnees
That's very impressive, thank you.
Robert McNees
@EricCarroll Thanks!
Debbie Goldsmith 🏳️⚧️🏳️🌈
@mcnees It’s a double-edged sword: it makes it easier to develop theories at lower energies but harder to figure out what’s going on at higher energies!
Level 98
@mcnees Is it good, or sad, that I double-checked your indices? 😀
Level 98
@mcnees I did something vaguely analogous with my high school students the other day... showing them how the relativistic expression for kinetic energy reduces to the Newtonian expression by using a series expansion where smaller terms can be ignored for v << c (low energy) - Newtonian mechanics, "an EFT".
I'm currently struggling to get students through the curriculum, otherwise, If I had students who were at a more sophisticated level, I would take them through your tweet... love it!
Robert McNees
@HaveyDeckAstro Sure, but the point here was to talk about it in terms of EFT since it’s Ken Wilson’s birthday.
HarveyDeckAstro
@mcnees would this scattering be more simply understood as dipole radiation, for which the power is frequency^4? Classical theory seems to be more straightforward in the case of why sky is blue. Rayleigh did not know quantum mechanics, let alone the effective field theory.
RD Hall
@mcnees , thank you. I have never heard this explanation. I have never had higher courses in maths nor science, therefore do not understand physics etc. I think with you as an instructor, I could grasp the basic concepts.
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