Mathematically-Rigorous Path Integration??

[Jay R. Yablon]

Dear Friends,

Following up some recent discussions in sci.physics.reseacrh with such
luminaries as Dr. Neumaier, Peter, J. Thornburg, X-Phy, P. Helbig, and
of course, the irrepressible Igor K., ;-) I have tried rolling up my
sleeves and diving into the problems that have been pointed out about
the ill-defined nature of the path integral, to see if I could make some
headway in cleaning things up. I have posted my efforts for review and
feedback at:

http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf

For sake of this discussion, I have also excerpted two pages from each
of Zee's QFT in a Nutshell, and Sakurai's Modern Quantum Mechanics, and
posted these in a single PDF file at:
http://jayryablon.files.wordpress.com/2009/12/sakurai-and-zee.pdf.

In summary, and seconding what Dr. Neumaier and Igor in particular have
been pointing out, it appears from my vantage point that the calculation
of the path integral in the form:

Z = ${-oo to +oo}Dq exp [iS] (1)

is really only "half" a calculation, in which the "ugly" terms are
gathered up and "swept under the rug" in Dq, and not ultimately dealt
with, including the mathematically-undefined infinite-dimensional
integral:

$...$$$ dq_0 dq_i dq_2 ... dq_oo, (2)

the pathology of which Igor has highlighted in prior discussion. In
particular, it seems very clear that Dq is a "faux" element of
integration, which really is a "rug" under which the ills of path
intergation are swept, and which does not have the rigorous calculus
meaning of, say, the usual integration element dq. The "handwaving"
which Dr. Neumaier has earlier referred to, appears to me, to occur when
one treats "D" as if it was "d" when doing integration, when is simply
is not a true, rigorous "d."

In essence, what I have attempted here, is to take everything back out
from under the Dq "rug," and complete the other "half" of this
calculation without sweeping anything "under the rug" into Dq, in a
mathematically rigorous fashion consistent with the limit-based
definition of Riemannian integration, and then redefined the transition
amplitudes W(J) in a way that places them as on a firm mathematical
footing of real integration based on properly taking limits and
resolving the nasty infinite products.

To summarize the "new" development, after taking everything "out from
under the rug" in Section 5, it is section 6 in which I carry through
the calculation with all of the "ugly" stuff from Dq included, and show
by a careful consideration of the infinitesimal limit, that in fact,
$Dq=1. Given that, a slight adjustment to the definition of the
transition amplitude W(J) is required, to place this as well on a
rigorous foundation. Section 1 is introductory, section 2 and 3 focuses
on integration in finite and infinite dimensional spaces based on
Sakurai's treatment, to ensure that even the single integral $dq in the
completeness relationship

I = ${-oo to +oo} dq |q><q| (3)

is introduced on a rigorous foundation. Section 4 carries through the
"customary" development of path integration.

I look forward to your comments, and to further discussion of these
foundational questions.

Happy holidays!

Jay
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm

[Ken S. Tucker]

Also to Jays Friends too.

On Dec 21, 2:46 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> Dear Friends,
> Following up some recent discussions in sci.physics.reseacrh with such
> luminaries as Dr. Neumaier, Peter, J. Thornburg, X-Phy, P. Helbig, and
> of course, the irrepressible Igor K., ;-) I have tried rolling up my
> sleeves and diving into the problems that have been pointed out about
> the ill-defined nature of the path integral, to see if I could make some
> headway in cleaning things up. I have posted my efforts for review and
> feedback at:
>

> http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergati...

Sure hope you'll comment on this example that uses a Convex
Lens to connect light ray paths from A to B like,

A +==== LENS ====+ B

For the moment let's assume a fixed frequency, with nil chromatic
and spherical aberrations, rather idealistic.
A light source at "A" focuses on point "B", due to the Lens.

Snell's Law,
http://en.wikipedia.org/wiki/Snell's_law
"Snell's law may be derived from Fermat's principle, which states
that
the light travels the path which takes the least time."

There are practically an infinite number of paths light can follow
to go from A to B, and I'll add (IMO) the 'phase' is synched at
B, otherwise the phase variance would cancel brightness, well
telescopes do not exhibit that to my knowledge.

I thought I'd mention that as a primitive example (?).

> Happy holidays!
> Jay

And same to you Jay.

> ____________________________
> Jay R. Yablon
> Email: jyab...@nycap.rr.com

> co-moderator: sci.physics.foundations
> Weblog:http://jayryablon.wordpress.com/
> Web Site:http://home.roadrunner.com/~jry/FermionMass.htm

Cheers
Ken