# Does the potential energy related to a particle determines its rest mass?

Would it be possible to determine the rest mass of a particle by computing the potential energy related to the presence (existence) of the particle, if this potential energy could be determined accurately enough?

I noticed from the answers to a recent question that I always assumed this to be true, without even thinking about it. However, it occurred to me that this concept was at least unfamiliar to the people who answered and commented on that question, and that it's even unclear whether this concept is true or meaningful at all.

Let me explain this concept for an idealized situation. Consider an idealized classical spherical particle with a charge $q$ and a radius $r$ at the origin. Assume that the particle generates an electrostatic field identical to the one of a point charge $q$ in the region outside of radius $r$ and vanishing inside the region of radius $r$. Now let's use a point charge $-q$ and move it to the origin in order to cancel this field in the region outside of radius $r$. Moving the point charge to the origin will generate a certain amount of energy, and that would be the energy which I mean by the potential energy related to the presence (existence) of this idealized classical spherical particle.

I'm well aware that really computing the potential energy related to the presence (existence) of any real particle is not practically feasible for a variety of reasons, but that never worried me with respect to this concept. What worries me now is whether this notion of potential energy is even well defined at all, and even if it is, whether it really accounts for the entire rest mass (not explained by other sources of kinetic, internal or potential energy) of a particle. After all, the rest mass of a particle might simply be greater than the mass explained by any sort of potential energy.

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The answer is ultimately no, but this is a reasonable idea, although old. This idea was floating around in the late 19th century, that the mass of the electron is due to the energy in the field around the electron.

The concept of potential energy is refined in field theories to field energy. The fields have energy, and this energy is identified with the potential energy of a mechanical system, so that if you lift a brick up, the potential energy of the brick is contained in the gravitational field of the brick and the Earth together.

This is important, because unlike kinetic energy, it is difficult to say where the potential energy is. If you lift a brick, is the potential energy in the brick? In the Earth? In Newton's mechanics, the question is meaningless both because things go instantaneously to different places, and also because energy is a global quantity with no way to measure the location. But in relativistic physics, the energy gravitates, and the gravitational field produced by energy requires that you know where this energy is located.

The upshot of all this is that potential energy is field energy, and you are asking if all the mass-energy of a particle is due to the fields around it.

This model has a problem if you think of it purely electromagnetically. Using a model where the electron is a ball of charge, and all the mass is electromagnetic field, you would derive, along with Poincare, Abraham and others that the total mass is equal to 4/3 of the E/c^2. The reason you don't get the right relativistic relation is because of the stresses you need to hold a ball of charge from exploding. The correct relation really needs relativity, and then you can't determine if the mass is all field.

The process of renormalization in quantum field theory tells you that part of the mass of the electron is due to the mass of the field it carries, but there are two regimes now. There is a long-distance regime, much longer than the Compton wavelength of the electron, where you get a contribution to the mass from the electric field which blows up as the reciprocal of the electron radius, and then there is the region inside the Compton wavelength, where you get the QED mass correction from electrons fluctuating into positrons, which softens the blowup to a log. The compton wavelength of the electron is 137 times bigger than the classical electron radius, so even with a Planck scale cutoff, not all the mass of the electron is field, because the blow-up in field energy is so slow at high energy.

So in quantum field theory, the answer is no--- the field energy is not the entire mass of the particle. But in another sense it is yes, because if you include the electron field too, then the total mass of the electron is the mass in the electron field plus the electromagnetic field.

Within string theory, you can formulate the question differently--- is there a measure of a field at infinity which will tell you the mass of the particle? In this case, it is the gravitational field, so that the far-away gravitational field tells you the mass.

But you probably want to know--- is the mass due to the combination of gravitational and electromagnetic field together? In this sense, since this is a classical question, it is best to think in classical GR.

If you have a charged black hole, there is a contribution to the mass of the black hole from the field outside, and a contribution from the black hole itself. As you increase the charge of the black hole, there comes a point where the charge is equal to the mass, where the entire energy of the system is due to the external fields (gravitational and electromagnetic together), and the black hole horizon becomes extremal. The extremal limit of black holes can be thought of as a realization of this idea, that all the mass is due to the fields.

Within string theory, the objects made out of strings and branes are extremal black holes in the classical limit. So within string theory, although it is highly quantum, you can say the idea that all the mass-energy is field energy is realized. This is not very great in giving you what the mass should be, because in the cases of interest, you are finding particles which are massless, so that all their energy is the energy in infinitely boosted fields. But you can take comfort in the fact that this is just a quantum regime of a system where the macroscopic classical limit of the particles are classical gravitational systems where your idea is correct.

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No. The rest mass is determined by the kinetic energy, not by the potential energy. Indeed, one can move a particle of arbitrary rest mass $m$ in a potential with arbitrary potential energy $V(q)$, using the Hamiltonian H=$\frac{p^2}{2m}+V(q)$ (or, relativistically, $H=\frac{p^2}{m+\sqrt{p^2+m^2}}+V(q)$).
To be honest, I don't see how your answer is related to my question. In my question, the potential $V(q)$ was caused by the particle of interest, and one of the unclear points is whether the potential energy related to this potential is well defined. I don't see the relation to moving a particle of arbitrary rest mass $m$ in this potential. Perhaps my question is just: "Does the internal energy of a particle determines its rest mass, if we define internal energy suitably and really take all sources of internal energy into account". –  Thomas Klimpel Jul 30 '12 at 11:59
OP is asking if the potential energy can equal the total energy. The question is the old one of "can the electron's mass be purely electromagnetic?" This doesn't answer it. Also what's up with your Hamiltonian? The normal relativistic H is $\sqrt{p^2+m^2} + V(q)$. –  Ron Maimon Jul 30 '12 at 21:09
@drake: it shows without series expansion that for $p^2\ll m^2$ we get the right nonrelativistic limit. –  Arnold Neumaier Aug 11 '12 at 9:12