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Comment by yccs27 | Hacker News Reader

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.

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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#...

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