Re: What contributes to the -ve and infinite bare energy of electron

Date: Fri Aug 8 14:27:35 2003
Posted By: Benjamin Monreal, Grad student, Physics, MIT
Area of science: Physics
ID: 1059729690.Ph

Message:

Hello Jagmeet,

An electron does not have minus infinity bare energy! Actually, nothing is known to have negative energy - matter, antimatter, electromagnetic fields, etc., always have positive energy or energy density. Something straightforward like an electron - well, we can measure its mass, we can measure its charge, and they are finite. If something in the mathematics tells you that the mass is infinite ... well, it is the mathematicians who need to fix it, because the electrons are not listening!

These sort of infinities do appear, and we have found ways to fix them - in other words, we have uncovered mistakes or confusions in our calculations. Let me explain a few of these infinities.

A quantum field theorist might try to draw a picture of a massless, pointlike particle with charge e'. Based on the laws of quantum electrodynamics, if you try to calculate the mass m'of this particle, including the electromagnetic energy swarming tightly around it, the answer appears to be infinity! But this infinity is just an anomaly of the mathematics; if we start our calculation with e' being infinitesimally small, and use it to generate the infinitely large "self-energy" m', then the two infinities cancel out! The actual mass-energy m, then, can be finite rather than infinite (as is observed) and the actual charge e can be finite rather than infinitesimal, as is observed. You can read about this mathematical trick, called "renormalization", at this web page

There is also the "classical electron radius" issue. Based on the (non-quantum) laws of electromagnetism, Maxwell's equations, you can calculate the electromagnetic energy of a sphere of charge. This energy sort of tells you how much work you must do to assemble a bunch of charges in one place. If I give you ten electrons, and ask you to bring put them into a 1-meter box together, you will have to do work - pushing on the charges, since they are repelling one another - to get them all in the box. If I ask you to put them into a 1-centimeter box, you'll have to push even harder; even harder for a 1-micron box; etc. All of the work you do ends up being stored inside of the box, in the electric fields. Anyway, if you take this to the extreme - imagine trying to "build" a pointlike electron, in a box of size zero, out of a handful of charged, pointlike components - the energy required would be infinite. This is essentially a classical analogue to the quantum-field theory problem above. As a sort of solution, since the electron is observed to have a finite mass, we can sort of imagine it having a finite radius, just so that this calculation works out. This is called the "classical electron radius:. But what this really tells us is that classical electrodynamics does not work at really small scales. See the calculations here. Maybe this has something in common with your "glue" idea?

Finally, there is always some confusion about "bare" versus "dressed" particles in quantum field theory. Basically, when doing field theory calculations - for example, you are calculating the probability that a neutrino will collide with an electron - sometimes it is appropriate to use something like e', the "bare charge", and sometimes it is appropriate to use e, the physical charge (1.6 x 10^-19 Coulombs). The same is true for the mass. You see, sometimes your calculations already include (in addition to whatever else you're working on) that swarm of electromagnetic energy which is responsible for the electron's mass; the calculation would be incorrect if you then carelessly add in the electron's entire mass again. This is especially important for quarks, for which you have to add up the electromagnetic energy as well as the gluon fields. The "bare mass" of an up or down quark, ignoring its gluon fields, is something like 5 MeV; put three of them together, add up the gluons, and the mass is 1000 MeV! The difference between "bare mass" and "dressed mass" is sort of like the difference between entering 5 MeV, or 1000 MeV x (1/3), into your equations. (And, the bare mass, I think, is always smaller than the dressed mass!)

Hope this helps,

-Ben

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