B What is the analogy for EM waves traveling in vacuum?

[Dale]

But I *think* the QFT view is exactly that, that it views mass as a form of energy.

Well, with QFT there is the bare mass, the pole mass, the running mass, and the invariant mass, so QFT is simply more complicated. None of those correspond to the energy, although the running mass at least changes with changing energy scale.

 

[Delta2]

Well, with QFT there is the bare mass, the pole mass, the running mass, and the invariant mass, so QFT is simply more complicated. None of those correspond to the energy, although the running mass at least changes with changing energy scale.

Ok well I am sad that the QFT point of view doesn't see mass as a form of energy. I thought somehow that the energy of the Higgs field relates to mass.

 

[vanhees71]

In both classical and quantum field theory the "mass terms" are of course contributing to the total field energy. It's just as any other contribution to the energy, including the "kinetic term" and "interaction energy" between different fields.

In the Standard Model of elementary particle physics indeed all the fundamental mass terms in the description of the underlying quantum field theory are due to the coupling of the various fields with the socalled Higgs field. Due to the mathematical structure of this theory (a socalled gauge theory) you cannot simply write down "mass terms" for all the particles, because that would spoil the underlying mathematical structure (the socalled local gauge symmetry), which would destroy entirely destroy it. It would become mathematically inconsistent and couldn't be used to calculate anything reliably to describe the observed properties of the elementary particles and their interactions. That's why you have to introduce the Higgs field, and to tailor the theory in such a way that this Higgs field takes a non-vanishing "vacuum expectation value". The coupling of the other fields to this part of the Higgs field results in mass terms for all these fields and the particles described by them (i.e., the quarks, leptons, W- and Z-bosons; only the Higgs boson's own mass stems from its own generic mass term, which is allowed by the gauge symmetry).

One should, however, note that most of the mass of the matter around us is not due to this "Higgs mechanism" but "dynamically generated" by the strong interaction. Most of the mass is in the nucleons (protons and neutrons), which are bound states of quarks and gluons. Most of the mass of the nucleons (around $938 \; \text{MeV}/c^2$).

 

[Delta2]

One should, however, note that most of the mass of the matter around us is not due to this "Higgs mechanism" but "dynamically generated" by the strong interaction. Most of the mass is in the nucleons (protons and neutrons), which are bound states of quarks and gluons. Most of the mass of the nucleons (around 938MeV/c2).

So can we say that mass is energy in the strong nuclear field? I know you probably tell me it is an oversimplification...

 

[Orodruin]

I know you probably tell me it is an oversimplification...

It is an oversimplification.

What is rest energy really depends on how you draw the limits of your system that is at rest. To draw a classical analogy, consider gas in a container. If you heat the gas the total mass of the container goes up. If you look at the molecular level, the increase in energy is due to the increase in kinetic energy of individual molecules.

The great realisation of $E=mc^2$ in my opinion is that a system’s rest frame inertia is equal to its rest frame energy.

 

[vanhees71]

Well, the difference between this example and the question, where the mass of the nucleon comes from, is that the latter question is not completely answered yet. It's the notorious problem to understand "confinement" from first principles (in this case quantum chromodynamics, QCD). We are pretty sure about the statement that the nucleon mass (and the masses of the other hadrons) is due to the strong interaction is from lattice-QCD calculations, i.e., numerical evaluations of QCD, leading to a satisfactory description of the hadronic mass spectrum.

 

[Delta2]

@vanhees71 One final question, the mass that generates the gravitational field is the one due to Higgs field coupling, the one due to color confinement or both? Or we don't know yet?

 

[Orodruin]

@vanhees71 One final question, the mass that generates the gravitational field is the one due to Higgs field coupling, the one due to color confinement or both? Or we don't know yet?

The source in Einstein’s field equations (that govern spacetime curvature) is the stress-energy tensor. What generates gravity is therefore not only mass but energy, momentum, and stresses.

 

[Delta2]

The source in Einstein’s field equations (that govern spacetime curvature) is the stress-energy tensor. What generates gravity is therefore not only mass but energy, momentum, and stresses.

Yes ok, relativity says there are other sources of gravitational field as well, I am just interested to know regarding the mass as source.

 

[Orodruin]

Yes ok, relativity says there are other sources of gravitational field as well, I am just interested to know regarding the mass as source.

You cannot decouple mass as a source. It is part of the stress energy tensor.

(Also: I finally did it! When I write ”str” on my phone it suggests ”stress energy tensor” for completion)

 

[vanhees71]

@vanhees71 One final question, the mass that generates the gravitational field is the one due to Higgs field coupling, the one due to color confinement or both? Or we don't know yet?

The source of the gravitational field is not mass but the energy-momentum-stress tensor of matter and radiation.

 

[Hillbillychemist]

This equation states that there is a form of energy connected with mass, namely rest energy. And mass is not the same thing as matter, mass is yet another property of matter.

Yet nothing is at rest. Everything is in motion.

 

[weirdoguy]

Yet nothing is at rest. Everything is in motion.

Nonsense. There are infinitely many reference frames in which you are at rest.