Comment by Timwi 3 days ago
What makes me skeptical here is that the author claims that fields have a property that is necessary to explain this, and yet physicists have not given that property a name, so he has to invent one (“stiffness”). If the quantity appears in equations, I find it hard to believe that it was never given a name. Can anyone in the field of physics elucidate?
I think perhaps the 'maths' at the bottom is a bit of a retelling of the Yukawa potential which you can get in a "relatively understandable" way from the Klein-Gordon equation. However, the KG equation is very very wrong!
Perhaps an approach trying to actually explain the Feynman propagators would be more helpful? Either way, I agree that if someone wanted to understand this all properly it requires a university education + years of postgrad exposure to the delights of QED / electroweak theory. If anyone here wants a relatively understandable deep dive, my favourite books are Quantum Field Theory for the Gifted Amateur [aka graduate student] by Stephen Blundell [who taught me] and Tom Lancester [his former graduate student], and also Quarks and Leptons by Halzel and Martin. It is not a short road.
The Yukawa potential is also just a more "classical" limit of an inherently quantum mechanical process. Sure you can explain things with it and even do some practical calculations, but if you plan on going to the bottom of it it'll always fail. If you want to explain Feynman propagators correctly you basically have to explain so many other things first, you might as well write a whole book. And even then you're trapped in the confines of perturbation theory, which is only a tiny window into a much bigger world. I really don't think it is possible to convey these things in a way that is both accurate (in the sense that it won't lead to misunderstandings) and simple enough so that people without some hefty prerequisites can truly understand it. I wish it were different. Because this is causing a growing rift between scientists and the normal population.
IIRC, Feynman said something like "I can't explain magnetism to a layperson in terms they can understand."
> ...causing a growing rift between scientists and the normal population.
True.
I haven't read the other two, but I'll second 'Quarks and Leptons'. I do believe it's Halzen though, rather than Halzel...
The biggest glaring issue with it (eg in the form (square^2+m^2)\psi=0) is that it is a manifestly Lorentz invariant equation in which particle number is conserved (which is highly unlikely for any relativistic interaction). I know that you can extend it into a scalar field theory proper, quantize it, and sidestep around those issues (and use it as a model in cmp!), but the bigger problem I think is that you really need spin -- and ideally all other interactions...
Fortuitously the author of the posted article also has a series on the Higgs mechanism (with the math, but still including some simplifications): https://profmattstrassler.com/articles-and-posts/particle-ph...
He addresses this in the comments. The term that corresponds to "stiffness" normally just gets called "mass", since that is how it shows up in experiments.
Roughly put:
- A particle is a "minimum stretching" of a field.
- The "stiffness" corresponds to the energy-per-stretch-amount of the field (analogous to the stiffness of a spring).
- So the particle's mass = (minimum stretch "distance") * stiffness ~ stiffness
The author's point is that you don't need to invoke virtual particles or any quantum weirdness to make this work. All you need is the notion of stiffness, and the mass of the associated particle and the limited range of the force both drop out of the math for the same reasons.
This is it. Typically in a QFT lecture, you'd include a "mass term" (in the article: stiffness term) in your field equations, and later show that it indeed gives mass to the excitations of this field (i.e. particles). So you temporarily have two definitions of "mass" and later show that they agree.
For this discussion it makes sense call the "mass" of a field "stiffness" instead, since it's not known a priori that it corresponds to particle mass.
I think mathematically "stiffness" is well-defined, but the interpretation varies substantially depending on the context. For example, in chemistry or plasma physics, one writes down Poisson's equation for a collection of positive and negatives charges in thermal equilibrium and linearise the Boltzmann factors. The result is called the Debye–Hückel equation and is identical to the one shown in the "with math" section.
Here the "stiffness" is interpreted as the effect of nearby charges "screening" a perturbing "bare" charge of the opposite sign. If you solve the equation you find the that effective electric field produced by the bare charge is like that of the usual point charge but with a factor exp(-r/λ). So, the effect of the "stiffness" term is reducing the range of the electric interactions to λ, which is called the Debye length. see this illustration [1].
Interestingly, if you look at EM waves propagating in this kind of system, you find some satifying the dispersion relation ω² = k²c² + ω_p² [2]. With the usual interpretation E=ℏω, p=ℏk you get E² = (pc)² + (mc²)², so in a sense the screening is resulting in "photons" gaining a mass.
[1]: https://en.wikipedia.org/wiki/File:Debye_screening.svg
[2]: https://en.wikipedia.org/wiki/Electromagnetic_electron_wave#...
> The term that corresponds to "stiffness" normally just gets called "mass", since that is how it shows up in experiments.
Then why not just call it "mass"? That's what it is. How is the notion of "stiffness" any better than the notion of "mass"? The author never explains this that I can see.
Undergrad-only level physics person here:
I think stiffness is an ok term if your aim is to maintain a field centric mode of thinking. Mass as a term is particle-centric.
It seems these minimum-stretching could also be thought of as a “wrinkle”. It’s a permanent deformation of the field itself that we give the name to, and thus “instantiate” the particle.
Eye opening.
> I think stiffness is an ok term if your aim is to maintain a field centric mode of thinking.
"Stiffness" to me isn't a field term or a particle term; it's a condensed matter term. In other words, it's a name for a property of substances that is not fundamental; it's emergent from other underlying physics, which for convenience we don't always want to delve into in detail, so we package it all up into an emergent number and call it "stiffness".
On this view, "stiffness" is a worse term than "mass", which does have a fundamental meaning (see below).
> Mass as a term is particle-centric.
Not to a quantum field theorist. :-) "Mass" is a field term in that context; you will see explicit references to "massless fields" and "massive fields" all over the literature.
In the unit analysis it appears as if it's just kinematic viscosity.
It does have a name, it's called "coupling." A spring (to physicists all linkages are springs :-) ) couples a pair of train cars, and a coupling constant attaches massive fields to the higgs field.
Even capacitors and thermal models in solids are springs.
The longer I read the article, the more "stiffness" feels like mass. In Lagrangians, the quantity saying how stiff it is is precisely the mass term, vide https://en.wikipedia.org/wiki/Scalar_field_theory.
At the same time, the author does not give any different definition; he says it's "stiffness". In the comment, he writes:
> The use of a notion of “stiffness” as a way to describe what’s going on is indeed my personal invention. Physicists usually just call the (S^2 phi) part of the equation a “mass term.” But that’s jargon, since this thing doesn’t give mass to the field; it just gives mass to its particles, which exist only in the context of quantum physics. The word “mass term” also doesn’t explain what’s going on physically. My view is that “stiffness” conveys the basic physical sense of what is happening to the field, an effect it has even without accounting for quantum physics.
So well, it is mass. Maybe not mass one may think about (in physics, especially Quantum Field Theory, there are a few notions of mass, which are not the same as what we set on a scale), but I feel the author is overzealous about not calling it "mass (term)".
So, I am not convinced unless the author shows a way to have massive particles carrying a long-term interaction (AFAIK, not possible) or massless particles giving rise to short-term interactions (here, I don't know QFT enough so that it might be possible). But the burden of proof is on the inventor of the new term.
> If the quantity appears in equations, I find it hard to believe that it was never given a name.
It does have a name: mass!
What I'm skeptical of is that this "stiffness" is somehow logically or conceptually prior to mass. Looking at the math, it just is mass. The term in the equation that this author calls the "stiffness" term is usually just called the "mass" term.
> it's not really just "mass", it's "characteristic mass of stationary minimal possible wrinkle in a given field"
If you are referring to the claim in the article that goes along with the equation E = m c^2, that claim is the author's personal interpretation, which I don't buy. The mass appears in the dispersion relation whether the particle is at rest or not. "Rest mass" is an outdated term for it; a better term is "invariant mass", i.e., it's the invariant associated with the particle's 4-momentum. Or, in field terms, it's the invariant associated with the dispersion relation of the field and the waves it generates.
It's nonsense. The fact that the particle is massive is a direct cause of the fact that the interactions are short ranged.
The nuance is this: Naturally, in a field theory the word "particle" is ill-defined, thus the only true statement one can make is that: the propagator/green function of the field contains poles at +-m, which sort of hints at what he means by stiffness.
As a result of this pole, any perturbations of the field have an exponential decaying effect. But the pole is the mass, by definition.
The real interesting question is why Z and W bosons are massive, which have to do with the higgs mechanism. I.e., prior to symmetry breaking the fields are massless, but by interacting with the Higgs, the vacuum expectation value of the two point function of the field changes, thus granting it a mass.
In sum, whoever wrote this is a bit confused and just doesn't have a lot of exposure to QFT
> whoever wrote this ... just doesn't have a lot of exposure to QFT
Incredible.
https://scholar.google.com/citations?user=19WGkFsAAAAJ&hl=en
be sure to check past the first 20 papers or so, like, oh, say his 1990 paper with Michael Peskin (438 citations), a copy of which can be found at <https://www.slac.stanford.edu/pubs/slacpubs/5250/slac-pub-53...>.
Actually upon further reading I realize that the author actually goes deeper into what I thought, so it's not nonsense, it's actually a simplified version of what I tried to write.
But I don't particularly like the whole "mass vs not mass" discussion as it's pointless
The author isn't inventing anything. He's just dumbing it down in an extreme way so that non-physicists could have the faintest hope of understanding it. Wich seems odd, because if you actually want to understand any of this you should prepare to spend two or three years in university level math classes first. The truth is that in reality all this is actually a lot more complex. In the Higgs field (or any simple scalar field for that matter) for example, there is a free parameter that we could immediately identify as "mass" in the way described in the article. But weirdly enough, this is not the mass of the Higgs boson (because of some complicated shenanigans). Even more counterintuitive, fermionic (aka matter) fields and massive bosonic fields (i.e. the W and Z bosons mentioned in the article) in the Standard Model don't have any mass term by themselves at all. They only get something that looks (and behaves) like a mass term from their coupling to the Higgs field. So it's the "stiffness" of the Higgs field (highly oversimplified) that gives rise to the "stiffness" of the other fields through complex interactions governed by symmetries. And to put it to the extreme, the physical mass you can measaure in a laboratory is something that depends on the energy scale at which you perform your experiments. So even if you did years of math and took an intro to QFT class and finally think you begin to understand all this, Renormalization Group Theory comes in kicks you back down. If you go really deep, you'll run into issues like Landau Poles and Quantum Triviality and the very nature of what perturbation theory can tell us about reality after all. In the end you will be two thirds through grad school by the time you can comfortably discuss any of this. The origin of mass is a really convoluted construct and these low-level discussions of it will always paint a tainted picture. If you want the truth, you can only trust the math.