How the Higgs Field (Actually) Gives Mass to Elementary Particles

PBS Spacetime has a fantastic video on the Higgs Field that explains it about one level deeper that typical pop science, and answers some of the questions I'm seeing in this thread, include "why did the field switch on suddenly?" and "Why is the Higgs Field different from other fields"

link: https://www.youtube.com/watch?v=G0Q4UAiKacw

If anyone wants to dig deeper, there is an excellent lecture on YouTube by Leonard Susskind. This goes into some details on how fields in general give mass to (composite) particles, and how the Higgs field has certain properties that allow it to give mass to elementary particles. It goes only into a tiny bit of math, absolutely intelligible at the high-school or at least undergraduate level.

https://youtube.com/watch?v=JqNg819PiZY

> Once upon a time, there came into being a universe. Searingly hot, it swarmed with elementary particles. Among its fields was a Higgs field, initially switched off. But as the universe expanded and cooled, the Higgs field suddenly switched on, developing a nonzero strength.

Any particular reason/mechanism why the Higgs field suddenly (gradually?) switched on?

Imagine some preindustrial scientist being awakened in the modern era to find that the aether has been first debunked for more than a century and then rediscovered, but with different rules.

As a lay person, I found that a clear and understandable explanation, which in my experience suggests it is a wild wild over simplification - but enjoyable nonetheless

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?

Layman trying to wrap my head around this: the Higgs field causes other fields to stiffen by giving them a resonant frequency, with higher frequencies meaning more mass.

So to conceptualize the difference between fields with and without restoring forces, I imagine that, for a field that doesn't have a restoring force, the medium itself can move permanently. For example if you have just a bunch of ball bearings lying on the surface of a table, you can cause a wave to go through the balls by hitting one. One bumps into the next, which bumps into the next, etc. There's no restoring force, so the wave is moving through the balls, and the balls are actually moving into a new position and they stay there.

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 common approach has been to tell a tall tale. Here’s one version: There’s this substance, like a soup, that fills the universe; that’s the Higgs field. As particles move through it, the soup slows them down, and that’s how particles get mass.

Is that really so? I've never heard this analogy, so the whole premise seems a bit of a straw man...

> By suggesting that the Higgs field creates mass by exerting drag, they violate both Newton’s first and second laws of motion.

Personally, I've wondered why theoretical physicists don't dive into Newton's laws more. Ever since I was a kid and first learned about the Voyager probes continuing to move through space forever, my question was why??

All matter is energy, and energy is vibrations in quantum fields, and that vibration never stops (you can never reach absolute zero). From the smallest gluon bouncing between quarks to galaxies to the expansion of the universe itself, matter never stops moving. Where does this infinite source of energy come from?

I understand that physics simply describes how reality works, not why, but I think it'd be valuable to know the reason fields continue to vibrate forever.

I studied wave mechanics in college, but the origin of mass didn't click for me until several years later (and in fact I don't believe it was every brought up in the context of wave mechanics, which seems like a problem in retrospect). The conceptualization that worked for me is this:

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)

<acceleration> = <pulled towards neighbors>

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

> Quantum field theory, the powerful framework of modern particle physics, says the universe is filled with fields. Examples include the electromagnetic field, the gravitational field and the Higgs field itself. For each field, there’s a corresponding type of particle, best understood as a little ripple in that field. The electromagnetic field’s ripples are light waves, and its gentlest ripples are the particles of light, which we call photons.

What are these fields made of? Are all fields made of the same thing(s), or is each field made differently?

And "the Higgs field suddenly switched on" is analogous to the pendulum's random vibrations slowing down enough that they no longer overwhelm its pendulum behaviour?

Very nice explanation by Matt Strassler. I am not sure it is possible to do better without getting into the details of quantum field theory.

For those who know quantum mechanics I would add that the oscillations mentioned in the article are just the familiar exp( i E t ) of any wave function that is an eigenfunction of the Hamiltonian. For a particle at rest in a relativistic theory (and in units where c=1), we of course have E = m.

Does anyone know the genesis of the Higg’s field as mud explanation?

I remember reading that since I first heard about the “God Particle” in the Science Times maybe 20 years ago.

Have journalists been using that deeply flawed analogy since Higg’s hypothesis was first published?

How does this idea mesh with the other model given to laymen that the Higgs field causes charged particles to flip helicity extremely rapidly?

This article is suspect as it mentions a "stationary electron". Such an electron would have precisely known momentum, and so exist throughout all of spacetime. This is a common starting point for solving the (e.g. Dirac) equations, but it's not physical.