Re: Dirac-Delta function

From: Randy Poe (poespam-trap_at_yahoo.com)
Date: 10/03/04 

Date: 3 Oct 2004 03:26:24 -0700

"Adam" <addam@rogers.com> wrote in message news:<sZydnV7TO_0X_8LcRVn-jg@rogers.com>...
> "A N Niel" <anniel@nym.alias.net.invalid> wrote in message
> news:021020042211149692%anniel@nym.alias.net.invalid...
> > It is a "generalized function" or a "distribution". Mathematicians
> > cover these in a course known as "Functional Analysis". Maybe you
> > will take that course one day.
> >
> I'm not sure what you mean. It has been expressed multiple ways. In each
> course, it is taught slightly differently.

Yes. You have seen how it is introduced and used in physics
courses. But you asked how mathematicians see it, and the answer
is that from the point of view of mathematics, it isn't a function.
In order to work rigorously with the Dirac delta, first you have
to define a thing called a "distribution".

> I searched online and there are
> so many forms of it

There are lots of non-rigorous ways to develop it and get an intuition
about the non-rigorous way it is used to do non-rigorous calculations
in physics. Your first post didn't ask about those.

> that it tends to make me think not so nice things about
> "it." I say "it" because it seems that many of the things that are said to
> be the same dirac delta function are not the same at all.

They have the properties which are important to physics. Most
importantly

  integral(x = -oo to oo) f(x) delta(x-x0) dx = f(x0)

The Dirac delta "picks out" the value of f(x) at x0.

> There is also the kronecker delta function taught in relation to tensor
> notation.
> delta_a_b = { 1 if a = b, 0 if a != b } where a and b are einstein index
> notation indices.
>
> That seems to me to be the same object but for discrete cases. When
> continuous cases occur, the object becomes the dirac delta function.

I could see that connection. Yes, it has the same property that
when it appears under a summation, it "picks out" specific values.

> The dirac delta is said to have a limit over infinity of 1 but be
> infinite at 1 and 0 everywhere else, or some other nonsense.

It is the limit of sequences of functions. The limit has the property
that it is 0 away from the origin, and the integral is 1. The article
at Mathworld has a long list of different choices for these sequences.

  http://mathworld.wolfram.com/DeltaFunction.html

>
> I'm starting to think that this is some sort of object that when used in
> specific ways and specific situations, just happens to result in desired
> outcomes.

Yes. It really makes sense only under an integral sign. But that's
OK, because that's when you really want to use it. Now there will
be many cases where something has a "solution" that is a delta
function, such as a commutator in quantum mechanics. But the
physically meaningful quantity will be the result of an integration.

> It doesn't seem logical at all.

It's self-consistent in the way it is used. If you want rigor,
don't look too hard at physics calculations.

If you really are bothered by non-rigorous calculation being used
to make predictions, perhaps you want to see the rigorous development,
and then understand the specific circumstances under which the
physics manipulations are valid ("OK, this integral works out in
the limit if f(x) has the property X, Y and Z"). Then you might
also see that physics works in the well-behaved arena where
those properties hold. Usually.

> > True. Perhaps you can convince the theoretical physicists to stop using
> > it.
> > But I wouldn't count on it.
> >
> More and more areas of physics are using it as time goes on. Dirac gave
> it to us for good. I'd rather take longer to do mathematics, but have an
> actual understanding of it and have it use common logic, than to use some
> strange thing in strange ways to save time or simplify notation. However,
> that's just me.

Then I'd suggest working through a book on functional analysis.

If you really want to see non-rigor, try going through some
engineering textbooks.

                - Randy

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