What gives particles mass? Also why are particles like photons considered massless when they have energy and momentum?

Question

Interested in knowing the quantum explanation for what gives particles mass, and why particles like photons are considered massless when they have energy and momentum.

Answer 1

The idea that a photon is massless is a pedagogic convention, that has not always held sway. When I was taught special relativity in the '70s, there were two kinds of mass that were distinguished. Rest mass and relativistic mass. Relativistic mass was simply and always equal to momentum divided by velocity. Photons have momentum and velocity, and so have relativistic mass.

An advantage of that approach to SR is that mass is conserved. Always. When for example, a radioactive atom emits a gamma ray, the atom becomes lighter, but the lost mass is found in the gamma ray. If one were somehow able to confine the explosion of a nuclear bomb within a sealed container, that permitted no loss of matter or energy, then the container and its contents would weigh exactly the same both before and after the bomb was detonated. (Or the mass would be the same as measured by a non-gravitational spring balance).

The modern approach seems to be that a particle with no rest mass has no mass. Conservation of mass is violated in some cases, and one must replace that law with a law of conservation of mass-energy.

It isn't a big deal which convention is used, but one should be aware that it is a convention, and there are multiple ways of defining terms such that, in the end, the math done under whatever convention, gives the same empirical results.

Answer 2

Mass is energy that a particle has when it is at rest. Photons are never at rest, and so don't have mass. A related fact is that in the limit that the photon's momentum goes to zero, its energy also goes to zero.

Fundamental particles like electrons, quarks, and the $W$ and $Z$ bosons receive mass by interacting with the Higgs field, via the Higgs mechanism.

However, most of your mass actually comes from nucleons (protons and neutrons). The mass of nucleons predominantly comes from interaction energy among the constituent quarks and gluons due to the strong interactions.