# Particle masses

## Main Question or Discussion Point

Is there any reason why the masses / sizes of all subatomic particles -- for instance, protons -- must necessarily be identical to each other? In other words, could individual protons or neutrons vary in their masses or sizes beyond our current ability to detect?

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That's a very good question. Before quantum field theory, we basically took this as a postulate.

Now we understand particles as excitations of fields. This is tied in Lorentz invariance. The mass of the electron for instance is one of the invariants which specifies the representation of the Lorentz group the electron belongs too. If two electrons had different masses, quantum field theory would not make sense, and quantum mechanics tells us that we could not apply the Pauli exclusion principle to electrons (so Mendeleev table would be left unexplained for instance).

Protons and neutrons are composite particles so it is a bit less obvious. Nevertheless, their constituents (quarks and gluons) must have the same mass for the same reasons. Besides, we can also apply effective quantum field theory to protons and neutrons, that is forget about their compositeness at low energy. Both schemes are very successful experimentally. Neither of them would make mathematical sense if protons and neutrons would come in different masses.

tom.stoer
Formally one can write down an eigenvalue equation

(H-E)|particle> = 0

In the rest frame we have E=M; each particle comes with its own quantum numbers, so we can rewrite the equation as

(H-M)|M, quantum numbers> = 0

The quantum numbers specify the particle (proton, neutron, Delta, ...). So different quantum numbers mean different particles. Same quantum numbers with different masses mean different particles as well.

So formally it's nothing else but solving an eigenvalue equation and applying some ordering scheme (usually derived from group theory, i.e. symmetries) to the solutions.

humanino,
I guess I don't really follow the logic. If electrons are excitations of some 'electron field', why must the excitations be the same rest energy (mass)? Why would the field theory not make sense if there were different excitations of different energies? For example the square well has non-equal spacing of excitation energies.

Does this just come down to something like: particles only make sense perturbatively, but we don't know how to handle non-perturbative field theory very well?

tom.stoer
particles only make sense perturbatively, but we don't know how to handle non-perturbative field theory very well?
No, not really. The eigenvalue equation is valid non-perturbatively; the symmetry argument from which the quantum numbers can be derived holds non-perturbately as well.

In perturbation theory it is rather easy to identify elementary particles like electrons, quarks etc. but it is impossible to identify protons and neutrons as they are bound by strong non-perturbative effects.

If electrons are excitations of some 'electron field', why must the excitations be the same rest energy (mass)? Why would the field theory not make sense if there were different excitations of different energies? For example the square well has non-equal spacing of excitation energies.
Electrons are fermions and thus there is only one spacing, only one excited level. In the case of boson fields, the Fock tower is that of the harmonic oscillator, which as you know does have equal spacing.

No, not really. The eigenvalue equation is valid non-perturbatively; the symmetry argument from which the quantum numbers can be derived holds non-perturbately as well.
Hmm... I'm not quite understanding.
How can particle number hold non-perturbatively? I thought we could only discuss particles as "free states" and that is why for calculations things come in from infinity, interact, and then go off to infinity. For instance, in pre-field theory the dirac equation described fixed number of particles. So it seems okay to say there are a certain number of electrons in the hydrogen atom. But with field theory, isn't it no longer well defined the number of electrons and positrons and photons that are in a hydrogen atom? Isn't that part of the idea behind the lamb shift? (otherwise Dirac's fixed particle number idea would have worked perfectly)

As I'm probably misunderstanding terribly (hopefully more along the lines of incorrect terminology use than incorrect concept), if you could explain a bit more in depth the concept of particles in field theory (especially in bound systems) it would be much appreciated.

Electrons are fermions and thus there is only one spacing, only one excited level. In the case of boson fields, the Fock tower is that of the harmonic oscillator, which as you know does have equal spacing.
Electrons have an internal degree of freedom, so there should be two spacings, no? And since electrons can interact, why necessarily would the "second" excitation add the same energy as the first? Also the harmonic spacing is just for free particles (ie. perturbative treatments), right?

I guess I just never really understood in what limits we can still discuss "particles" in field theory. I recently ordered Zee's "Field Theory in a Nutshell", since I tried reading Srednicki's book and was really struggling to get a hold of what the meaning of any of the mathematical manipulations were. Do you think that is an okay place to start, or can you suggest a better book to pedagogically bridge from particle quantum mechanics to quantum field theory?

A. Neumaier
2019 Award
Hmm... I'm not quite understanding.
How can particle number hold non-perturbatively? I thought we could only discuss particles as "free states" and that is why for calculations things come in from infinity, interact, and then go off to infinity.
What you describe is the general principle of scattering, which is a nonperturbative notion. Perturbation theory comes in only when you want to calculate scattering cross sections numerically.

But with field theory, isn't it no longer well defined the number of electrons and positrons and photons that are in a hydrogen atom?
A neutral hydrogen atom consists of exactly one proton and one electron, and an indeterminate number of (virtual) photons and additional (virtual) particle-antiparticle pairs. The latter, being unobservable, are usually not counted but occasionally talked about (often only by people like kexue who mistakenly believe that that would explain a lot).

Isn't that part of the idea behind the lamb shift?
No. That comes from the contribution of a single photon to the energy levels of the hydrogen atom.

if you could explain a bit more in depth the concept of particles in field theory (especially in bound systems) it would be much appreciated.
Particles are the asymptotic states of definite momentum. Elementary particles are those particles that can be related to the fields occurring in the action.

In QED, the electron number operator N is just a particular operator in field theory that happens to have integral eigenvalues only. An eigenstate of N with the eigenvalue n>0 is interpreted as an n-electron state, an eigenstate of N with the eigenvalue n<0 is interpreted as an n-positron state, and an eigenstate of N with the eigenvalue zero
as a neutral state. An example of a neutral state is that formed by a positron colliding
with an electron (with a small chance of annihilation); such a state is nonlocal since in the past the positron and the electron were far apart.

Sometimes, especially by people like kexue, localized neutral states are identified with the fluctuating vacuum, in which virtual particles can pop in and out of existence for a
extremely short time. But serious people call vacuum only the ground state of a field theory (which is neutral). That state is completely inert. Nothing ever happens.

Electrons have an internal degree of freedom, so there should be two spacings, no?
The number operator - unlike the Hamiltonian - has a purely discrete, integral, equispaced spectrum, in which the internal degrees of freedom are degenerate.

I guess I just never really understood in what limits we can still discuss "particles" in field theory.
The right limit is usually called the limit of geometric optics, although it is usually discussed only for photons (hence the name).

I recently ordered Zee's "Field Theory in a Nutshell", since I tried reading Srednicki's book and was really struggling to get a hold of what the meaning of any of the mathematical manipulations were. Do you think that is an okay place to start, or can you suggest a better book to pedagogically bridge from particle quantum mechanics to quantum field theory?
No single source is good to start. You need to try reading at different levels, and often go in cycles. It takes a long time to understand QFT, as there are deeper and deeper levels of understanding. So if you have understood something on some level, step back and review the more elementary stuff to see whether what you learnt sheds new light on what you knew already - it often will.

If you are not afraid of the math involved, Volume 1 of Weinberg is the clearest.
But probably you need other books on the side....

No single source is good to start. You need to try reading at different levels, and often go in cycles. It takes a long time to understand QFT, as there are deeper and deeper levels of understanding. So if you have understood something on some level, step back and review the more elementary stuff to see whether what you learnt sheds new light on what you knew already - it often will.
Thanks for the advice. I'll keep that in mind.

In QED, the electron number operator N is just a particular operator in field theory that happens to have integral eigenvalues only. An eigenstate of N with the eigenvalue n>0 is interpreted as an n-electron state, an eigenstate of N with the eigenvalue n<0 is interpreted as an n-positron state, and an eigenstate of N with the eigenvalue zero as a neutral state.
It seems like you are purposely defining an operator, so that any pairs don't count, and therefore sidestep the problem. This means of considering "real" vs "virtual" seems flawed. For example it wouldn't work in bound states like positronium, or for photons which are their own anti-particles.

For example here:
A neutral hydrogen atom consists of exactly one proton and one electron, and an indeterminate number of (virtual) photons and additional (virtual) particle-antiparticle pairs.
Can you specify how many photons are in the ground state of the hydrogen atom? How about the first excited state? I don't think particle number makes sense in non-perturbative cases like bound states.

It seems you feel photons can be counted in this case, as your comment on the lamb shift states:
That comes from the contribution of a single photon to the energy levels of the hydrogen atom.
I don't think you can make such a clear distinction between real and virtual particles. Griffith's intro text makes a similar statement, and suggests the best we can do is say how close to "on shell" a particle is. I think one example was something like: a photon comes from a scatter event in the sun's corona and comes to earth and hits a molecule in our eye. Should this be described as a virtual photon in a large interaction diagram? Or is it a "real" photon? How far does a photon have to go between two events before we can decide to draw the line between real/virtual? The point is such a line is arbitrary. It is taxonomy at best.

Correct me if I'm mistaken, but Feynman diagrams are just a means of representing perturbation theory. While we can talk about the "number" of particles in a diagram, it seems like we can only definitively discuss the physical (after summing over diagrams) "number" of particles in the intial or final states (when the particles are free).

You seem to make a similar comment yourself:
Particles are the asymptotic states of definite momentum.
Is there a more encompassing definition which can handle bound states? For example is it meaningful to ask how many gluons are in the lowest energy glueball?

It seems like going from dirac or klein-gordon -> field theory, we would have to give up the concept of particles for the concept of a field in all but the nice harmonic level like free particle states.

A. Neumaier
2019 Award
It seems like you are purposely defining an operator, so that any pairs don't count, and therefore sidestep the problem. This means of considering "real" vs "virtual" seems flawed. For example it wouldn't work in bound states like positronium, or for photons which are their own anti-particles.
The reason I do this is because lepton number is easily observable as charge, while electron number and positron number separately are not.

Positronium is not a bound state but a resonance. Both positronium and photons are neutral states, consistent with my explanation.

In practice, one almost never has a definite number of photons; the mean photon number is essentially the intensity of the transversal e/m field.

For example here: [AN: hydrogen atom]
Can you specify how many photons are in the ground state of the hydrogen atom?
No. There is no definite such number. It is like asking for the position of a plane wave.

How about the first excited state? I don't think particle number makes sense in non-perturbative cases like bound states.
Particle number makes sense only if it is (at least approximately) conserved. This is why one discusses baryon and lepton number on the subatomic level, and about number of electrons and number of nuclei of each kind in case of atoms.

It seems you feel photons can be counted in this case, as your comment on the lamb shift states:
Actually there are contributions of any number of photons, and all contributions are needed to get agreement with experiment. But the 1-photon contribution is by far the largest, and is the one typically discussed in the textbooks, so I was talking about it. One can separate the contributions only in the perturbative computations, not in reality.

One cannot count photons bound inside an object. One can count them only the moment they die in a photoelectric cell. (According to the conventional view. Even that is disputable since it can be proved that classical light also produces clicks in a photodetector made of quantum matter.)

a photon comes from a scatter event in the sun's corona and comes to earth and hits a molecule in our eye. Should this be described as a virtual photon in a large interaction diagram? Or is it a "real" photon? [...]
Correct me if I'm mistaken, but Feynman diagrams are just a means of representing perturbation theory.
Your last statement is correct. And virtual particles are just internal lines in such a diagram. Not particles traveling from the sun to our eye.

[AN: Particles are the asymptotic states of definite momentum.]
Is there a more encompassing definition which can handle bound states?
Bound states are particles in this sense. For example, the proton is a particle though it is a bound state of three quarks. It possesses asymptotic states of definite momentum.

While we can talk about the "number" of particles in a diagram, it seems like we can only definitively discuss the physical (after summing over diagrams) "number" of particles in the intial or final states (when the particles are free).
In a scattering experiment, we can only talk about incoming and outgoing directions of particles, and measure the traces of the outgoing ones.

Talking about the number of particles going in along a particular direction is moot. In a typical collider experiment, a huge, indefinite number of particles cross each other's path, and by accident, a number of them collide and their debris leaves traces. From conservation laws and selection rules we then infer what went in. Before doing this analysis, we don't know.

Single particles can be easily prepared only when they are quite heavy (gold atoms or buckyballs, say), so that quantum uncertainties can be neglected.