How the Higgs field gives mass to elementary particles

How the Higgs field gives mass to elementary particles
quantamagazine.org

The normal wave equation is (ignoring constant factors like mass and propagation velocity):

d^2/dt^2 f(x,t) = d^2/dx^2 f(x,t)

=

This says "if a point in the field is lower than its neighbors, it will be accelerated upwards. If a point in the field is higher than its neighbors, it will be accelerated downwards." This equation is the lowest-order description of most wave phenomena like sound waves, water surface waves, EM waves, etc. and it's usually pretty accurate.

If you look for solutions to this differential equation, you can get

f(x,t) = exp(i * w * (x±t))

w is the frequency of the wave

This tells you that the frequency and wavenumber of waves is determined by the same parameter (w), so they are proportional to each other

Now, what if we add a restoring force to this equation? This is a force that pulls the value of the field towards zero.

d^2/dt^2 f(x,t) = d^2/dx^2 f(x,t) - M^2 f(x,t)

M is just a parameter that tells you the strength of the restoring force. The force increases as the field gets farther from zero, like a spring.

Now, solutions to the equation look instead like

f(x,t) = exp(i*k*x ± i*w*t)

Where w^2 = k^2 + M^2

(or something like that, I need to re-derive this on paper, just going off memory, but I think if you plug it in it should work)

Notice that now, if you have a spacial frequency k, your temporal frequency is actually higher. In fact, if your spacial frequency k is 0 (corresponding to a stationary wave), your temporal frequency is still M!

This is what mass is. Having a non-zero frequency even if the wave is the same everywhere in space (which corresponds to no movement)

A field with no restoring force is e.g. the EM field, so photons are massless. The rate at which they oscillate in time is the same rate at which they oscillate in space. A massive particle has a restoring force, so its temporal frequency is higher than its spacial frequency.

In physics, this equation is often reordered like this:

d^2/dt^2 f(x,t) - d^2/dx^2 f(x,t) = - M^2 f(x,t)

(d^2/dt^2 - d^2/dx^2) f(x,t) = - M^2 f(x,t)

(d^2/dt^2 - d^2/dx^2) f(x,t) + M^2 f(x,t) = 0

◻ f(x,t) + M^2 f(x,t) = 0

(the d'alembert operator)

(◻ + M^2) f(x,t) = 0

Again, this is ignoring constant factors like c, h, etc.

The above equation is nice because it's relativistically invariant. The d'alembert operator is the contraction of the 4-momentup operator with itself, p^u p_u. This is a concept worth studying - tells you a lot about what mass, energy, velocity, and momentum actually are in a general sense

Compare that to a water wave, where gravity is trying to restore the particles to a "flat" position in space. If you cause a wave in water, the medium will return to the space it occupied before through the restoring force, even as the wave travels through it.

Is this really how it works, so that e.g. the EM field itself can move in space, whereas e.g. the electron field cannot move in space, it's "pinned" in some sense by the Higgs field?

A question for the more expert amongst you. Is the Higgs field unique in its interaction with other fields, or are there other similar fields which similarly change the way that other fields (and associated particles) behave?