Comment by rachofsunshine
3 days ago
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.
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In the unit analysis it appears as if it's just kinematic viscosity.
Kinematic viscosity, though, is an emergent property just as stiffness is. See my response to kridsdale1 upthread.
In the unit analysis that is most natural to quantum field theory, it's mass.