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Distribution theory has lots of applications in physics. The charge density of a point particle is the delta function. Also when Fourier transforming over the
by cjfd 2y ago
Distribution theory has lots of applications in physics. The charge density of a point particle is the delta function.
Also when Fourier transforming over the whole real line (not just an interval where the function is periodic), one has identities that involve delta functions.
E.g. \int dx e^(i * k1 * x) e^(-i * k2 * x) = 2 * pi * delta (k1 - k2).
- elcritch 2y agoThat's fascinating about charge density of a particle being a dirac delta function. Is that a mathematical convenience or something deeper in the theory?
- cjfd 2y agoWell, if we assume that a point particle is an infinitely small thing with all of its charge concentrated in one point, the dirac delta function is obviously the correct way to describe that. Of course, there is not really a way to find out whether that is true. Still, the delta function makes sense if something is so small that we do not know its size. This idealization has, however, led to problems in classical electrodynamics: https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force. Search for 'preacceleration' in this page. This particular problem was ultimately solved by realizing that in this context quantum electrodynamics is the theory to applies. But, then again, using point particles also causes problems, namely the need to renormalize the theory which is or is not a problem depending on your point of view.
- ogogmad 2y agoThe article showed that Dirac deltas could be defined WITHOUT distributions. You ignored the article when answering my question. The question is why distribution theory is a particularly good approach to notions like the Dirac delta.