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The Higgs field is responsible for the masses of all elementary particles. Including the Higgs particle. But doesn't this transfer the question of mass, or inertia, to the Higgs field itself? Why has the Higgs field inertia which can be transmitted to the massless elementary particles?

Deschele Schilder
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The mass of the Higgs is not due to the Higgs mechanism. There simply is a mass-term in the Higgs field Lagrangian.

The whole idea that Higgs is the originator of mass of otherwise massless particles is quite mistaken, at least as it stands in the collective psyche of the popular culture. ;) The correct statement is that coupling to Higgs gives mass to fermions of the Standard Model and they would have been massless otherwise. As such, in the framework of quantum field theory, a field can simply have a mass term -- there is nothing wrong with it. However, when a theory has certain symmetries, it can forbid the field from having the mass because the mass term might violate the symmetry. This is precisely what happens with the fermions of the Standard Model, the $SU(2)_L$ gauge symmetry forbids the mass term. However, due to the coupling of these fermions with the Higgs field, via the mechanism of spontaneous symmetry breaking, they acquire a mass.

In any case, the moral is that there is nothing that says that you need to somehow have a source of mass. The Lagrangian of a field can simply have a mass term. It is the particular case of some field theories where certain particles might not have mass due to symmetries but they nonetheless acquire mass due to spontaneous symmetry breaking. However, this need not be the case. Ironically, as I mentioned earlier, the mass of the Higgs does not come from its coupling with itself. It is massive regardless of spontaneous symmetry breaking.

  • Aren't elementary particles the ones of the SM? – Deschele Schilder Apr 01 '21 at 15:11
  • @DescheleSchilder I am not sure I understand your question but to the extent I do, I would like to point out two things: $1$. The Higgs boson is not a fermion (see, I wrote "fermions of the Standard Model"). $2$. The point of pointing out as to whether something is a fundamental aspect of the framework of quantum field theory or a rather accidental aspect of the standard model is to distinguish what is in principle forbidden and what just happens to be forbidden in a particular model. –  Apr 01 '21 at 15:21
  • Well, the quarks and leptons (and the intermediate gauge bosons) "acquire" their mass by interacting with the Higgs field. From where does the inertia of the Higgs field come? – Deschele Schilder Apr 01 '21 at 15:32
  • @DescheleSchilder What do you mean by intermediate gauge bosons? Gauge bosons are massless and they do not acquire their mass by interacting with the Higgs field -- they are simply massless. And again, the mass of the Higgs field comes from the mass term in its Lagrangian. As I tried to stress, a field can simply have a mass term, it is not a human rights violation. ;) –  Apr 01 '21 at 15:36
  • I mean the W- and Z-particles. But I feel I get you now. ;-) The Higgs field doesn't acquire its mass. It just has it (though its energy is zero and it pervades the whole universe, at present temperatures). All the fermions and quarks have zero mass and acquire mass. No human rights are stepped upon! But nevertheless, the origin of the Higgs particles mass is not explained in this way. Or is it? – Deschele Schilder Apr 01 '21 at 15:41
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    @DescheleSchilder Yes, sorry, my bad for forgetting the WZ bosons. But yes, the mass of the Higgs is not explained via spontaneous symmetry breaking because it does not need spontaneous symmetry breaking to acquire mass. If it helps, nobody (as far as I know) feels the need to somehow explain the mass of a field if the field can simply have a mass term (there is no need for an explanation, a field can simply have a mass term). It is when the mass term is forbidden due to symmetry, one needs to look for where the mass might come from if the said particle is known to be massive. –  Apr 01 '21 at 15:47
  • Gotcha! One more thing... Where do the self-interaction terms of the Higgs particle stand for (an interaction vertex with four lines)? – Deschele Schilder Apr 01 '21 at 15:53
  • Let us continue this discussion in chat. –  Apr 01 '21 at 15:54
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About mass and inertia.

The two are correlated, obviously, but they're not the same thing.

Example: the mass of each of the elements of the periodic system. The mass of a Helium nucleus is close to two times the mass of a proton plus two times the mass of a neutron, but not quite.

The inertial mass of a nucleus involves the energy state of that nucleus. A higher energy state comes with a correspondingly larger inertial mass.

The inertia of energy is unrelated to the Higgs mechanism.

My understanding is that the Higgs mechanism doesn't even address the question of inertia.

One author phrased this as follows (I'm quoting from memory): coupling to the Higgs field imposes an energy cost. The mass acquired due to coupling to the Higgs field is the inertial mass that corresponds to that energy.

Cleonis
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  • Do you think energy has inertia? – Deschele Schilder Apr 02 '21 at 07:31
  • @DescheleSchilder Special relativity leads to attributing inertial mass to energy content. This was first discussed in 1905 by Einstein in an article titled: "Ist die Trägheit eines Körpers von seinem Energieabhalt abhängig?" (Title of translated version: "Does the inertia of a body depend on its energy content?" I recommend the discussion Einstein on the inertia of energy by Kevin Brown. Using 'm' for inertial mass the expression for the amount of inertial mass corresponding to energy content E is $$m = \frac{E}{c^2}$$ – Cleonis Apr 02 '21 at 08:40
  • The famous equation only says that mass is equivalent to energy. Their values can be known from each other's values. Has a photon inertia? It can push other particles (when they absorb the photon). But can we push the photon? With other photons, yes (photon-photon scattering), but there is no three-photon vertex. There are always other virtual particles involved in the Feynman diagram – Deschele Schilder Apr 02 '21 at 09:12
  • @DescheleSchilder I recommend that you turn this into the subject of a physics.stackexchange question. For instance, according to the principle of equivalence a spinning flywheel has a larger gravitational mass than that same flyweel in non-spinning state. According to relativistic physics the rotational kinetic energy has inertial mass, and the PoE asserts equivalence of inertial and gravitational mass. For a practical experiment the mass difference is below detection limit, but the outcome of the thought experiment is unambiguious. – Cleonis Apr 02 '21 at 09:36
  • I was already typing the question, in fact! But then I saw the question was already asked:
    https://physics.stackexchange.com/questions/587347/do-photons-have-inertia     – Deschele Schilder Apr 02 '21 at 09:42
  • @DescheleSchilder It's not clear why you keep narrowing down to photons. Relativistic physics is a general theory. Electromagnetic radiation is a form of energy; there are many other forms. Relativistic physics extends to all forms of energy. That is the standard in application of relativistic physicss. There is no logical reason to narrow down to photons. – Cleonis Apr 02 '21 at 09:56
  • Well, I mean pure energy, to which also gluons and gravitons belong. In contrast to massive energy. – Deschele Schilder Apr 02 '21 at 10:01
  • @DescheleSchilder About the "in contrast" that you suggest: in physics no such distinction is made. Let's take the case of the mutual annihilation of a particle and an anti-particle. When a proton and an anti-proton annihilate each other then (according to physics as it stands today) the amount of energy that is transformed is described by the same equation: $E=mc^2$ The "contrast" that you suggest is not present in the current theories of physics. – Cleonis Apr 02 '21 at 10:28
  • But doesn't the equal sign in $E=mc^2$ imply a difference (contrast) between $E$ and $m$? Of course mass $M$ and energy $\frac{E}{c^2}$ are interchangeable. You can say the proton has mass 938($\frac{MeV}{c^2}$). But in doing so you imply that upon annihilation of a proton and anti-proton, double the amount of its massive energy is converted into pure energy. Maybe it's just an epistemic difference... – Deschele Schilder Apr 02 '21 at 10:44