Hi, I read in the Feynman Lectures Vol II that the assumption that the electron is a point leads to an infinite energy in its field, and that the difficulty hasn't been resolved. Has there been any progress on this since Feynman gave his lectures?
The diverging self energy of a point charge is indeed still a very open problem in physics today. The problem arises when we consider the formula for the energy stored in an electromagetic field: [tex]W=\frac{\epsilon_o}{2}\int_{all space}E^2d\tau[/tex] For a point charge, such as an electron, this reduces to: [tex]W=\frac{q^2}{8\pi\epsilon_o}\int_0^{\infty}\frac{1}{r^2}dr[/tex] This integral diverges, leading to the infinite value for the energy contained in the field of a point charge. From a qualitative perspective: The formula derived above is derived by adding up all the energy needed to assemble a charge distribution. Well, to assemble a point charge, we would need to take a finite amount of charge and "cram" it into a point, a space infinitely small, having no dimension. Considering this, it makes sense that to "assemble" a point charge will require an infinite amount of energy, since we would need to create a distribution with infinite an charge density. Thus, the problem is that classical electromagnetism predicts that the energy required to "create" a point charge distribution is infinite, which makes no sense. This is a great example of a "singularity" appearing in a physical theory. At this singularity, the laws of physics don't work and predict ridiculous infinite answers. This problem is not just present in classical electrodynamics but is also present in the quantum theory as well. Fixing or explaining this problem is an open area of theoretical research.
It seems to me that infinite energy is a problem only if electron is separable. If it is not (and this is what is currently assumed), then this is just a constant addition to energy: are there any troubles with that?
Yes, if we consider that an electron is inseparable, then this diverging integral does not effect the rest of the calculations needed in electrostatics, but this infinite "self energy" is still embarrassing from the point of view of many physicists. This is not the only point particle inconsistency in Classical E&M. There are also other problems caused by point particles in classical electrodynamics. For instance, consider in radiation theory the Abraham-Lorentz formula which describes a "radiation reaction force" that will be experienced by a radiating point charge. This formula describes ridiculous accelerations for point particles in radiative situations, such as accelerations that happen a short time before the force causing them acts, or accelerations that spontaneously increase exponentially! Again, what this means is beyond my current knowledge, but it goes to show that electrodynamics has some problems with point charges that still need to be dealt with.
[tex]\sum_{n=1}^{n=\infty}\frac{1}{n^2}[/tex] converges [tex]\int_{0}^{\infty}\frac{1}{r^2}dr[/tex] diverges. Big difference between the two as the latter includes a singularity.
The problem is not at [itex]r\rightarrow \infty[/itex] , it's at [itex]r\rightarrow 0[/itex]. The summation you're thinking of starts at "1". Starting the integral at 0 is what is problematic.
What about the possibility that electron is not a point particle? Can it have a very small positive radius? It is possible to calculate "classical electron radious" with the assumption that electron is a uniformly charged ball and that only electrostatic energy contributes to its mass. Does anybody know whether the experimental upper limit for electron radius has already crossed classical electron radious?
in quantum mechanics and QED the electron is treated as a point charge--a point particle. You can consider other quanties like strings instead of points if you like, but it's probably not worth the trouble in my opinion.
That extrapolation applied to the mass of a electron results in a black hole. A black hole with the mass of an electron is unstable. Perhaps an electron is a stable coexistence of mass and charge. The closeness of Electron / positron annihilation to energy of mass, indicates a limit on an electron's charge energy.