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It's clear, from reading pop-science articles about the Higgs boson, that particle physicists have something very specific in mind when they say "mass". In classical physics the mass of a particle is just a given value, but in the context of particle physics you hear things like "computing the mass" or "such and such interaction gives some mass".

What does "mass" mean to a particle physicist? How does it relate-to/explain the classical notion of mass, and how does it differ?

I'm not sure what level of explanation I'm looking for. I don't know particle physics, obviously, so any grad-level equations are likely to go way over my head. But explanations like "bouncing back and forth all the time" from this minute physics video are too simplified. I'm left wondering, for example, how coupling with the Higgs field makes electrons bounce back and forth? And why wouldn't it just scatter them? And how is this process Lorentz invariant? And why does the bouncing resist/react-to forces?

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marked as duplicate by ACuriousMind, MAFIA36790, Gert, Craig Gidney, Community Jan 22 at 6:44

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
@ACuriousMind I found that before asking my question, read it, and didn't feel like I had an answer. It's definitely close. I think the difference is I want an answer that says why "the 'mass term' in the Lagrangian or Hamiltonian" is what we identify with mass, as opposed to just being told that in passing. I also gave several specific sub-questions which clearly don't line up with the question you linked, like "why is the 'bouncing' Lorentz invariant?" with the goal of prompting that kind of "how"/"why" answer. – Craig Gidney Jan 21 at 14:57
    
Note that mass is not just a "given value" in classical mechanics. It can be described as an objects' resistance to acceleration by a force. – jpmc26 Jan 22 at 5:16
    
@joshphysics that's much closer to what I have in mind, in terms of being a duplicate question. – Craig Gidney Jan 22 at 6:42
    
@Strilanc I figured, and I don't think that question ever received an answer that was particularly satisfying either. I tried adding some thoughts at the time in the form of an "answer" which you might find useful, but they're certainly not definitive by any means. – joshphysics Jan 22 at 6:45

The mass always means the same thing – but in different theories, one uses different equations and other tools to express the mass.

Inertial mass $m$ is the quantity expressing "resistance of the object with respect to acceleration", i.e. the coefficient that enters Newton's $F=ma$. The gravitational mass is what enters the formula for the gravitational force $F=GmM/r^2$. These two masses are the same, up to a universal ratio that may be chosen to be one in good units. This equality of the two masses is known as the "principle of equivalence".

In Newtonian physics, the mass of individual particles is basically conserved. Surely the total one is conserved. In special relativity, only the "total relativistic mass" which includes the enhancement from the kinetic energy is conserved. This conservation law becomes the same law with the energy conservation law; the mass may be "converted" to energy and vice versa. The total mass and the energy are related by $E=mc^2$ and this "unified" quantity is conserved whenever the laws of physics are time-translationally invariant (Noether's theorem).

For a given particle, in special relativity, we usually use the term "mass" for the rest mass $m_0$, the mass measured in the rest frame. It's related to the energy and momentum by $$ E^2 = m_0^2 c^4 + p^2 c^2 $$ The actual working theory used to describe practical particle physics is called quantum field theory (QFT). It's a combination of classical field theory (like Maxwell's electromagnetic fields) and the postulates of quantum mechanics.

In QFT, particles are created and annihilated by fields, they are quanta of these fields. Each particle has a value of the mass and it's encoded in the "mass term". For example, for a scalar boson, the Lagrangian is $$ L = \frac{1}{2} \partial^\mu \Phi \partial_\mu \Phi - \frac {m_0^2}{2} \Phi^2 $$ Why is it the same mass as I defined at the beginning? It's because the field equations resulting from the Lagrangian above are the Klein-Gordon equation $$(-\square - m_0^2 )\Phi =0$$ The so-called plane waves $$ \Phi = A\exp (ip_\mu x^\mu) $$ are the simplest solutions to the Klein-Gordon equation. The momentum must obey $p_\mu p^\mu =m_0^2$. In the rest frame, $\Phi$ is independent of space but it depends on time via $$\Phi = A\exp(-i m_0 c^2 t / \hbar) $$ where I restored the $c$ and $\hbar$ i.e. the usual units. But the angular frequency of this associated wave $\Phi$ is related to the energy via $E=\hbar\omega$, a basic formula of the quantum theory.

So the mass $m_0$ is measured as $E/c^2$ in the rest frame. The energy is calculated from the angular frequency $E=\hbar\omega$. And the angular frequency may be adjusted by adjusting the mass term $-m_0^2/2 \Phi^2$ in the Lagrangian. That's why this mass term controls the usual inertial mass.

For fermions, the box is replaced by the Dirac operator and the second power of $m_0$ becomes the first power. Different particle species have different values of the mass. Anna's answer is conveniently complementary and addresses this part of the question.

At any rate, the mass is in no way "redefined" in particle physics. It exactly agrees with everything we expected from the "mass" in the past. The equations needed to describe the creation and annihilation of elementary particles are given by QFT and QFT has to change the value of the masses – and it's still the same concept of "masses" – through the "mass terms".

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Particle physics has a set of elementary particles , some of which have zero mass.

elempart

In this table the general public has heard of the electron and maybe the photon. The mass of the electron has been measured consistent with classical definition of mass.

The mathematical theory of elementary particles is called the standard model and the table has the masses that enter in the lagrangian. These are the only masses that are generated by the Higgs mechanism.

Due to the fact that special relativity reigns at the particle domain, the invariant mass defined as this mass.

invarmass

In the case of the particles in the table this mass and the mass in the table coincide. When though there are combinations of particles, or composite ones, as the proton and the neutron which are composed of quarks and gluons, the mass of the composite is the invariant mass, i.e. the energy momentum vectors of all components added up in the invariant mass formula define the mass of the composite. It is evident that in this case the intrinsic mass generated by the Higgs mechanism has a small part to play. The mass of the proton is also measured by classical methods

I'm left wondering, for example, how coupling with the Higgs field makes electrons bounce back and forth? And why wouldn't it just scatter them? And how is this process Lorentz invariant? And why does the bouncing resist/react-to forces?

This is really another question, on how the Higgs mechanism works. Have a look here and here . The popularized explanations can only be approximations.

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Seeing how the other answers mention things like Lagrangian and quite a few details of the Standard Model, I feel that a more popular level explanation is needed. This comes with the pitfall of not being accurate, but for that you have other sources. I will pull a rabbit out of my hat and answer without a single (real) formula.

What does "mass" mean to a particle physicist? How does it relate-to/explain the classical notion of mass, and how does it differ?

Without going into definitions (you've received a link in you comments section), it is important to distinguish between two notions when talking about mass.

  1. Rest mass: this is the mass of an object at rest, that is, with 0 velocity.
  2. Measured mass (energy): this is the mass of an object which will be measured. It is composed of the rest mass plus a contribution we'll call "motion mass" produced by the non-zero velocity of the object.

There is some caveat relating to frames of reference in relativistic theory, but we'll skip that.

In classical physics, objects don't move fast enough to cause a distinction. Here is a graph to give you the idea of the change of measured mass with velocity (as a fraction of the speed of light):

enter image description here

When we talk about mass in particle physics, we mean the rest mass. Because the measured mass depends on the current (possibly changing) velocity, it is of little use as a defining property.

A note about "massless" particles

When you read/hear about a "massless" particle, it means that its rest mass is 0. You can now ask "Since the measured mass is comprised of the rest mass and the 'motion mass', and the rest mass is 0, what happens if the 'motion mass' is also 0?" Turns out particles with 0 rest mass behave a bit differently and must move at the speed of light. A photon's (a "particle of light") measured mass will be dependent on its frequency instead. Yes, light has a measured mass. You might be surprised because you never feel it, but look into solar sails to see that light can push other objects.

in the context of particle physics you hear things like "computing the mass"

Unfortunately, you can't put a particle on a scale and read its mass like you do with potatoes in the supermarket. A basic idea in computing the mass of a particle is using the law of energy-momentum conservation ("conservation of matter and energy", A.K.A. "law of equivalent exchange"). The law basically states that what comes in must come out.

By perform enough experiments in a particle collider, you will have enough data of what comes in and what comes out (you measure the measured mass) and will be able to compute the rest mass.

"such and such interaction gives some mass"

This and all the Higgs-related part is a bit more difficult to explain because it mostly arises from mathematics. I wouldn't go too much into the bouncing stuff in the video. I will try to approach this from the view point of interactions.

Not all particles can interact with the rest. For example, the aforementioned photon interacts with particles if and only if they have a (non-zero) electric charge. For the Higgs, you can make a similar argument only with mass instead of electric charge. Please note that the Higgs is not the equivalent of the photon when it comes to mass - see the graviton and this question. However, you can say that a particle has mass only if it interacts with the Higgs. In this sense, the Higgs "gives" mass to particles.

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things are not very clear about the Higgs mechanism that comes from mathematics and in the last paragraph ... – igael Jan 21 at 22:40
    
@igael "things are not very clear about the Higgs mechanism that comes from mathematics and in the last paragraph". Then why not state which part is not clear or why? Otherwise your comment is hardly constructive. – user1803551 Jan 21 at 22:45
    
@user1803551 : it came from maths but was found. Many other particles were also predicted by theories. For the Higgs, you can make a similar argument only with mass instead of electric charge. : this is not clear because a reader may think that the mass comes always from the Higgs field. The comparison with the photon requires more knowledge to be well understood. Fair critics are constructive and never authoritative. – igael Jan 21 at 23:14
    
Indeed I do owe you an apology. reacting then writing in a hurry between tasks is no excuse but it is how that happened. – dmckee Jan 22 at 1:50

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