# HUP and Rest Mass of Electrons

1. Feb 21, 2014

### anorlunda

Electrons emit and absorb photons all the time. I heard that each electron is surrounded by a cloud of 10$^{20}$ photons. That suggests to me that the rest mass of an electron must fluctuate, and that raises the prospect of uncertainty.

My questions:
Is the rest mass of an electron really constant?

If not, is rest mass an observable?

If so, is there an observable that does not commute with rest mass such that the pair of observables are subject to HUP?

If so, which one?

2. Feb 21, 2014

### ZapperZ

Staff Emeritus
Where did you hear such a thing? Please cite your source (this is something that you should learn to do in this forum).

https://www.physicsforums.com/blog.php?b=2703 [Broken]

Secondly, it would be silly if the mass of the electron varied so much that we can no longer identify what it is. After all, how would we know what particle we have in a particle collider? If you open the Particle Data Book, you'll see that the mass of the electron is well-defined. So who is wrong here?

And just for your information, we DO know when the effective mass of an electron changes when compared to the bare mass. This happens all the time in condensed matter physics, which is inside conductor and semiconductors that you are using in your electronics. But this certainly doesn't mean that the actual mass (bare mass) of the electron is fluctuating.

Zz.

Last edited by a moderator: May 6, 2017
3. Feb 21, 2014

### anorlunda

I apologize for not giving my source. It is a video course on QM taught by Leonard Susskind. http://www.newpackettech.com/Resources/Susskind/PHY25/QuantumMechanics_Overview.htm. It was Susskind who quoted the 10$^{20}$ number. Unfortunately it is not a paper you can quickly check, but a series of two hour lectures.

My reasoning is this. Say in one interval of time that the electron emits or absorbs photons with a net energy change in photon energy of E. I further suppose that the energy and momentum of the electron-cloud system remain constant. Then to conserve energy, the electron's rest mass must change by E/c$^{2}$. Granted we are talking a tiny change.

I suppose an alternative explanation is that the energy of the electron-cloud system is not conserved. Energy-time uncertainty says that ΔHΔt≥$\hbar$. In that case, the rest mass could remain unchanged while total energy fluctuates. I think that would correspond to the photons in the cloud being virtual particles.

4. Feb 21, 2014

### vanhees71

Well, that's a quite misleading statement. What you are discussing here is the electron self-energy correction due to quantum fluctuations. The point is that in relativistic QFT you use perturbation theory to calculate physical quantities as a formal power series in the electromagnetic coupling contant $\alpha \approx 1/137$. At 0th order you start with an "uncharged" non-interacting electron, and then you correct for this doing an expansion in powers of $\alpha$, which is nicely organized in Feynman diagrams.

Doing the radiation corrections, i.e., Feynman diagrams with loops naively, the corresponding integrals are divergent. E.g., the self-energy contritubtion is divergent, leading to an apprantly infinite mass of the electron, which is of course nonsense. The reason is that we never measure an uncharged non-interacting electron but an electron carrying its charge and with it its electromagnetic field, which has energy and this contributes to the mass of the electron. In classical electrodynamics the energy of the field of a point charge is infinite, because to the diverging electric Coulomb field at the point charge's position.

Now, in quantum electrodynamics the problem is solved by renormalization theory, i.e., you subtract systematically the infinities in the calculation of loop diagrams by lumping them to the unobservable bare quantities of the theory (wave-function renormalization for electrons and photons, the electron mass, and the coupling constant). In this way you adapt the observable quantities, like the electron mass and its charge to the measured physical values. This procedure can be mathematically proven to work to any order of perturbation theory, and it is astoningly successful, providing an accuracy of several significant digits in the agreement between the theoretical calculations and the experimental values of the pertinent quantities (like the magnetic moment of the electron, the lamb shift of the hydrogen atom, etc.).

In some sense Susskind is right: It's the electromagnetic field of the electron, and thus in some sense to "photons", which leads to the renormalization of its mass, but the mass is a parameter we have to put into the theory from experiment. Today, there's no way to predict the masses of the elementary particles in the standard model, but they are all tuned to their experimental values. There is no indication that the mass or charge of an electron are fluctuating, when considered as these fundamental constants of the standard model of elementary particles.

5. Feb 21, 2014

### anorlunda

Thanks much Sci Advisor. I learned from your answer. I never thought of masses as fine tuned values before.