Prior to the Dirac delta function, what other distributions functions where physicists using? I find it hard to motivate the theory of generalized functions with just the delta function alone.
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Distributions occur in many areas in physics. Here are two interesting cases: In general relativity, the Einstein equations admit distributional (and even more general (Colombeau algebras)) solutions. The following review article by: Steinbauer and Vickers describes some of these solutions. In relativistic quantum mechanics, the wave functions belong to distributional valued Hilbert spaces, for example the Sobolev space $\mathfrak{S}_{-1/2}$ in the case of the massive Klein-Gordon equation. See, for example the the following lecture notes by Arthur Jaffe. This phenomenon is characteristic to unitary representations of non-compact groups (the Lorentz group in the case of the Klein-Gordon equation). |
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I think Green's function equations had something like a "point source", maybe not written analytically as $\delta(x-x')$, but it was meant. As P. Dirac said in an interview, he just naturally wrote a formal kernel of an integral equation with the desirable effect. The other distributions ("concentrutions") followed from working with delta-functions and Green's functions, I guess. |
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It is not that simple. Sometimes it is tricky to work with "normal" functions like constant or exponent because they are not integrable. However, you may work with them properly in terms of distributions. When you try to differentiate (especially few times) continuous function, if you consider arbitrary limit of functional serie, if you do anything which is not guarantted to leave you inside function space you start from, you need to use distribution. The idea is similar to real numbers. In principle, you might stay confined to rational numbers. But it is not convinient and does not allow you to use analysis methods. To do anything you want you need to fill the holes in space of rationals. The result is called real numbers and in many ways more convinient. The same story is with distributions. Any functional space you may construct is full of holes. If you fill them you end up with distributions (generalized functions). AFAIK, this was not well understood prior to Dirac. Actually, it was not well understood by Dirac as well. |
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In Quantum Mechanics the eigenfunctions of the continuous spectrum are not square integrable but lie in a bigger space of distributions. One can formalize this using rigged Hilbert spaces or Gelfand triples, for the example of QM in 1D one can take $\mathcal S(\mathbb R) \subset L^2(\mathbb R) \subset \mathcal S'(\mathbb R)$, where $\mathcal S$ are the Schwartz functions and $\mathcal S'$ is the topological dual vector space of tempered distributions, aka (tempered) generalized functions. Further, distributions are needed if one wants to differentiate a functions which is not differentiable. Actually all generalized functions are of this type, namely there is some structure theorem saying that every tempered distribution can be written as a finite derivative of some polynomial bounded continuous function (cf. Theorem V.10. Reed Simon I). But one should be careful to see them as functions, because in general two generalized functions cannot be multiplied. |
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