Undergrad What is (rest) mass for a particle?

[calinvass]

The energy equation for a particle contains the rest mass and momentum. If the momentum is zero, all the energy comes from the term

mc^{2}

. That means the particle still holds some energy. What is the form of that energy? For example string theory explains particles as vibrating strings, and I suppose we can think that the energy is stored as the vibration.
Another thing I can imagine is when we look at the electron- positron annihilation. The energy stored in the mass of these particles turns to a form of energy specific to motion, we can call kinetic. Can we say that these particles already had this form of energy within themselves? If not how do we call it and how do we explain it.
There is also a problem regarding the uncertainty principle that I was told it is related to this subject. If the position and momentum of a particle cannot be determined with absolute precision, does it mean that no particle can be at rest? If so, should it influence the rest mass? I think not.

 

[Drakkith]

That means the particle still holds some energy. What is the form of that energy? For example string theory explains particles as vibrating strings, and I suppose we can think that the energy is stored as the vibration.

If we think of something like a free electron, which is an elementary particle, then it has no vibrational or any other internal modes in which to store energy, so it cannot be any internal process like some atoms or molecules have. The 'form' of the energy is simply the mass of the particle.

Another thing I can imagine is when we look at the electron- positron annihilation. The energy stored in the mass of these particles turns to a form of energy specific to motion, we can call kinetic. Can we say that these particles already had this form of energy within themselves? If not how do we call it and how do we explain it.

We explain it using the mass-energy relationship equation you already know of. Two particles with a combined mass of $m_{t}$ can annihilate and the combined energy of the photons is equal to $m_{t}c^2$ plus any additional energy the two particles may have had, such as the combined kinetic energy (contained within the momentum term in the equation).

Don't get too caught up in 'labels' for energy. They are mostly there for convenience. We could very well get rid of the term 'kinetic energy' and nothing would change except it would be much more of a pain in the butt to talk about the work that a moving object can perform. For the energy capable of being liberated from a particle upon annihilation, or the energy required to create a particle from a collision or decay event, we must include a mathematical term whose value is equal to the mass of the particle times the square of the speed of light, or $mc^2$. The only label I know of for this is 'rest energy', but I don't know if that's an official term or not. But whatever you choose to call it, that term still has to be there in order for the math to work and the physics to make sense.

There is also a problem regarding the uncertainty principle that I was told it is related to this subject. If the position and momentum of a particle cannot be determined with absolute precision, does it mean that no particle can be at rest? If so, should it influence the rest mass? I think not.

Unfortunately I can't answer that.

 

[PeterDonis]

If the position and momentum of a particle cannot be determined with absolute precision, does it mean that no particle can be at rest?

Not really. But it does mean you have to be careful how you define "at rest", and in finding quantum states that satisfy a workable definition of that term. (These states are called "coherent states".)

should it influence the rest mass?

No. In quantum field theory, the rest mass (or invariant mass) is an inherent property of the field.

 

[bhobba]

Not really. But it does mean you have to be careful how you define "at rest", and in finding quantum states that satisfy a workable definition of that term. (These states are called "coherent states".)

That's correct.

You can measure zero momentum, but you can't infer from that it has a position until you actually measure it.

Thanks
Bill