Re: Third Draft, Possible Closed Form Solution to Gaussian-Based Integrals with High-Order Field Interactions


"Junoexpress" <MTBrenneman@xxxxxxxxx> wrote in message news:a587c25c-df74-45ab-9e35-c8e61d805ff2@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Nov 30, 2:32 pm, "Jay R. Yablon" <jyab...@xxxxxxxxxxxx> wrote:

I think there is a much easier way to do this. Clearly, there are some
coefficients for which my (and Zee's) integral will converge, and others
for which it will diverge. One can simply start by with the integral

$dx exp [ -Ax^4 -Bx^3 -.5Kx^2 +Jx ]

with coefficients "selected such that the function is convergent." Then
we can leave it to the mathematics gurus to tell us when this is and is
not convergent. But I think it is clear that there *exist* some
coefficients for which this will converge (for example, A=B=K=J=1 gives
you a convergent result), and others for which this will diverge (make A
negative). I am a physicist not a mathematician, but that fact that one
can show existence for convergent solutions seems sufficient.

Jay.

Jay,

I've looked over some of your material, and at risk of sounding
pedantic, I think you should reconsider Dono's advice, on account of a
couple of reasons:

1) Details in math matter. Too many amateurs want to find sweeping
theories of everything and are somewhat akin to a boxer looking for a
big knockout punch. Unfortunately, as someone who has done applied
math professionally and very successfully for 20 years, I know things
don't work that way. The reality is that when you go back and try to
account for every detail and get everything tacked down, that's when
you find the clues that help you actually solve the problem. I know
that you think these mathematical conditions are "details", and since
your method gives you what you want some of the time, you don't have
to really worry about it. But, I guarantee you, this is the only way
you are really going to get a deep understanding of anything.

2) A solid mathematical foundation is indispensable in a field like
theoretical physics, where the math IS the physics.

3) The firmer your foundation, the more certainty you have in what you
are doing. Right now you're like the guy who is building a bridge
across a river on a bunch of stones that aren't stable. Every step you
take out onto the river becomes less and less stable.

Bottom line: do ONE thing, do it completely, do it correctly, really
understand it. It takes time, it is painful, but you can't get to the
top of Everest sitting in your lounge chair.

I think what you are doing is cool, and I wholeheartedly encourage
you, but you have to learn how to do it right, and right, you aren't.

M

M--

I thank you for your serious feedback, and do take it seriously.

The ONE thing I am trying right now to do completely, correctly, and with full understanding, is work on Yang-Mills theory, linked at http://jayryablon.files.wordpress.com/2008/12/yang-mills-paper-151.pdf in its most current form. I also just started a new thread about this here and at SPF.

This IS my main line of research right now, and I have determined to forego all other physics work until this is fully developed in a rock solid manner. I will also point out that I was working on this 18 months ago, suspended this work having hit some conceptual roadblocks, and am now returning with much better perspective than I had 18 months ago.

The Gaussian integration discussions are really preparatory work to scout out some issues that I would need to understand correctly and completely, going downstream from here with the Yang-Mills work. Particularly, I think you will see that that Lagrangian (9.4) will eventually run into the questions raised in the Gaussian integration discussion, and so I was trying to stay a few steps ahead in understanding my main line of research, for when I arrive at that point.

Thanks again,

Jay.

PS: How did you know I like my lounge chair? ;-)



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