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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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Heaviside step function maybe ? –  twistor59 Sep 11 '11 at 20:05
In the first sentence, do you really mean "prior to"? Or do you actually want "other than"? –  Willie Wong Sep 11 '11 at 20:30
I am more interested in "prior to", but "other than" is OK. –  echoone Sep 12 '11 at 0:09

4 Answers 4

up vote 5 down vote accepted

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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It was very well understood by Dirac. He invented the theory. –  Ron Maimon Sep 12 '11 at 15:58
@Ron Maimon Dirac invented functional analysis? Banach spaces and all that stuff? Heh. Dirac was a great guy and knew mathematics better than mathematicians, but "invented the theory" is not what he did. Inspired maybe. –  Misha Sep 12 '11 at 16:15
You're talking about formal nonsense, names of things. People knew about $L_2$ back then, Dirac certainly did, they just called it square integrable functions. They knew what a normed function space is. Dirac invented distribution theory, and he understood all the test-function business normally associated with Schwarz. But distribution theory is a rare case where the mathematicians did a good job of taking physics stuff and making mathematics out of it, so I won't complain. –  Ron Maimon Sep 12 '11 at 16:39
@Ron Maimon Wiki says "Generalized functions were introduced by Sergei Sobolev in 1935. They were re-introduced in the late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions." so Dirac hardly knew them. Dirac did a good job, but it was non-formal and could not be freely used by normal people, because "Quod licet Iovi, non licet bovi": trying to follow Dirac intuitive approach one soon usually finds nonsense results. There is no nonsense in the knowledge that Dirac delta is not a function. –  Misha Sep 12 '11 at 17:29
Except that Dirac never used the delta function as a function, and neither has anyone else, except students, rarely. The delta function paradoxes are and were always relatively trivial, unlike, say, the algebra of quantum field theory. Dirac always used the delta-function in the modern smeared-out test-function sense. The only thing he wasn't doing is using mathematicians language, or checking that everything works within Lebesgue's integration. Sobolev tried, and Schwarz succeeded, to put the results into mathematics, but distributions were invented by Dirac. –  Ron Maimon Sep 12 '11 at 20:04

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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sorry, most of this was already said in different words... –  Marcel Sep 14 '11 at 16:45

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