# Distributions (e.g., Dirac Delta): confused and unhappy [closed]

I am sorry that the following set of questions is very fuzzy and ill informed.

I am a trained mathematician and now studying an undergraduate theoretical physics course. We use distributions. I have no clue of the How and Why, which makes me unhappy.

So far I have some tiny idea about the physical intuition; and I have read the definition (and some properties) in http://en.wikipedia.org/wiki/Distribution_(mathematics) .

## Very fuzzy questions

Sorry for the vagueness. If you do not want to be bothered but would kindly be willing to answer more concrete questions please jump to the next section.

1. Distributions make most sense to me in the following mindset: We have a situation A too complicated to calculate, we approximate it as B using step functions or delta distributions or similar stuff, which we can solve. Then we somehow claim that this solution (e.g., equations of motion) of B is close to the solution of the original problem A. Question: Is there any good argument for this claim?
2. Continuing the previous question: My (very poor) physical intuition would rather be: a delta distribution is the limit of stuff which has integral 1 and smaller and smaller support. The derivatives have no role in this intuition, which seems incompatible with even the most basic mathematical contents of distributions: The topology of the test function space uses (higher order) derivatives. Question: Is the simple intuition just nonsense? Is there a better working intuition?
3. Specific rant: It makes me particularly uneasy that some QM texts (cf.\, for instance, the introduction of Cohen-Tannoudji et al's) often use as an argument in a proof that wave functions have to be continuous, bizarrely arguing that non-continuous things make no physical sense because of measurement precision(!), while in the same proof happily using non-continuous potentials, distributions (and, of course, ignore the fact that the basic operators are non-continuous). Question: Is there any valid point in Cohen-Tannoudji et al's argument (that wave functions have to be continuous, even when appearing in solution for artificial problems that where constructed using distributions)? Which one?
4. I am completely lost about which calculus concepts/constructions/arguments are still OK with distributions and which ones are not. Well, obviously you can differentiate, and you can generally not multiply. But this leaves open the vast area of vector calculus, where it is not at all clear to me what is OK and what is not. I see two ways to rectify this: Either study mathematical physics for 5 years and then arrive at a very complicated mathematical understanding of simple stuff that physicists already knew anyway; or just continue to stumble through the horrors of distributions, without a clue, and somehow get used to it and learn simply by example what you can do and what not. Both sounds depressing. Question: Is there any better way to learn about the practical use of distributions in a consistent way, including some mathematical justification?

## Slightly less fuzzy questions

1. (Classical) functions that are not locally integrable, such as $\frac{1}{x}$, can not be interpreted as distributions. Is that correct?
2. In particular, the formula $\Delta\frac{f(r)}r=-4 \pi f(0) \delta^{(3)}(\vec x)+\frac{f''(r)}r$ does not seem to make much sense if $\frac{f''(r)}{r}$ is not integrable?
3. Speaking of which: $\Delta\frac{1}r=-4 \pi \delta^{(3)}(\vec x)$. What does this even mean? (It seems that $\frac{1}r$ is not a distribution, how can I differentiate it?) And, once it is clarified what it means: How is it proven?

• Way too many questions here, try trimming it down to 1. Or maybe 2. Also, Cohen-Tannoudji is one person (and a Nobel laureate at that), your comment in Q3 suggests he is more than 1. – Kyle Kanos Dec 27 '13 at 21:45
• – Qmechanic Dec 27 '13 at 21:48
• @KyleKanos : thanks (et al because his introduction has coauthors. I now added the missing Tannoudji, though). Would you recommend deleting the question and reposting it as several new ones? (However, I think I cannot delete the question myself; and it already has some comments...) – Jakob Dec 27 '13 at 21:52
• Here's a two second comment that helps me when I'm thinking about delta functions. Delta functions only make sense under integrals. Of course everyone uses the notation that the integrals aren't there. But to compute any genuine physical quantity there has to be an integral involved. As you're a mathematician, I'd say the place to start understanding delta functions is Green's functions. With some linear analysis they be made completely rigorous. – Edward Hughes Dec 27 '13 at 23:24
• For the meaning of distributions in quantum mechanics, I try to explain this here – Selene Routley Dec 28 '13 at 0:22

As to $\frac{1}{x}$, see the answer by @5891user; you can define it using Cauchy's principal value and show that it has the expected propeties, e.g. that $\text{pv} \frac{1}{x} \cdot x = \mathbb{1}$ (the identity). Thus, your last three questions pose no particular difficulty apart from acquiring an intuitive understanding of these equations (which can be a challenge).