Re: [Axiom-mail] Dirac Delta

[C Y]

--- Ondrej Certik <address@hidden> wrote:

> > It looks like the place to start would be the work of Laurent
> > Schwartz.

Incidentally it looks like the key papers are in French - does anyone
know if there are "authoritative" translations anywhere?  If not it
looks like some quality time with Babelfish and a French->English
dictionary are somewhere in the future...  Once a proper pamphlet is
written that should contain all necessary background and theory but
someone has to write it...

> I was also thinking how to implement delta functions. I know the
> "distribution" way of doing it, however, I don't see any advantage of
> using the
> 
> <delta, phi> = phi(0)
> 
> formalism instead of writing directly delta(0)phi(x) and of course
> understand that in order to give a precise meaning to it, one needs
> to integrate it and thus arriving at the <delta, phi> formalism, 
> which is mathematically precise, but to me it is as precise as
> writing the delta(0)phi(x) directly. 

Indeed, I found a paper looking at the formalization of the Dirac Delta
and how it had (or hadn't) impacted physics.  An interesting question,
but another part of that question from my standpoint is whether the
lack of tools to handle it "properly" with a minimum of trouble would
let the formally correct way pick up more steam.

Plus, from an Axiom standpoint, being rigorously correct seems like the
only reasonable way to proceed.  Scalability and all that.  How much
work it will be is a different question, of course...

> For a research in mathematics it can be maybe useful to use <delta,
> phi> formalism, but I don't know of any application of delta 
> functions (for example in quantum mechanics or the quantum field
> theory) where the simpler formalism delta(0)phi(x)
> would give less precise (or less well-defined) results.

That's probably true, supported by the fact that physics as a whole
seems to be able to proceed on those assumptions without running into
trouble.  However, while it may be that nature happens to restrict
problems to those where non-formal approaches aren't a problem, I think
it would be better practice for Axiom to not make this assumption and
instead implement the correct formalism.

> So the way it is done in Mathematica is the way I like and I'll do
> the same in SymPy, when I find some time.

For Axiom I think the logical approach is to implement the theory of
distributions/general functions, with Dirac Deltas being built on the
framework.  There may be a paper or two in verifying that major physics
calculation results of the past half century hold up under full formal
mathematical treatment, although I doubt the physics community itself
would be terribly worried.  (It works, its descriptive, we're going
with it.)  The benefits of going fully formal in Axiom might not appear
in current practice, but it is not known whether they may appear in the
future and the safe bet is to put in the work and make it rigorous.

Cheers,
CY


 
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