Mathematically-Rigorous Path Integration??
in [Math]
|
Prev: Kind of Godel numbering
Next: Subgroup Inclusion
From: Jay R. Yablon on 22 Dec 2009 17:04 "Robert Israel" <israel(a)math.MyUniversitysInitials.ca> wrote in message news:rbisrael.20091222002353$0dbd(a)news.acm.uiuc.edu... > "Jay R. Yablon" <jyablon(a)nycap.rr.com> writes: > >> 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 > > Rather than re-inventing the wheel, why don't you look at what > mathematical > physicists have already done? You might look at Thank you for these references. However, from what I understand after some extensive discussion in sci.physics.research with some folks who have extensive knowledge about this (you should check those threads), there remains a mathematical problem which has not yet been solved despite years of effort by many, of giving a rigorous calculus limit foundation to the path integral, because of the infinitely-dimensional $...$$$dqdqdq...dq integral that is swept into the "integral over paths" Dq. I strongly suspect that this problem is not solved by these references, otherwise people would not still be writing textbooks and papers 25 or 30 years later with this problem still not resolved, and the knowledgeable folks at sci.physics.foundations would not be talking about how path integration, while useful, "couldn't be made logically consistent, in spite of many attempts by some of the best mathematicians and physicists." (quote from A. Neumaier) Jay > > Glimm and Jaffe, "Quantum Physics: A Functional Integral Point of > View", > Springer-Verlag 1981, and > > Simon, "Functional Integration and Quantum Physics", Academic Press > 1979. > -- > Robert Israel israel(a)math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada
From: Jay R. Yablon on 22 Dec 2009 17:06 In my second paragraph, extracted below, I meant sci.physics.research. "Jay R. Yablon" <jyablon(a)nycap.rr.com> wrote in message news:7pcu43F2n3U1(a)mid.individual.net... > .. . . > > I strongly suspect that this problem is not solved by these > references, otherwise people would not still be writing textbooks and > papers 25 or 30 years later with this problem still not resolved, and > the knowledgeable folks at sci.physics.foundations would not be > talking about how path integration, while useful, "couldn't be made > logically consistent, in spite of many attempts by some of the best > mathematicians and physicists." (quote from A. Neumaier) > > Jay > >> >> Glimm and Jaffe, "Quantum Physics: A Functional Integral Point of >> View", >> Springer-Verlag 1981, and >> >> Simon, "Functional Integration and Quantum Physics", Academic Press >> 1979. >> -- >> Robert Israel israel(a)math.MyUniversitysInitials.ca >> Department of Mathematics http://www.math.ubc.ca/~israel >> University of British Columbia Vancouver, BC, Canada >
From: Herman Rubin on 23 Dec 2009 12:37 In article <7pcn86FqoqU1(a)mid.individual.net>, Axel Vogt <&noreply(a)axelvogt.de> wrote: >Robert Israel wrote: >> "Jay R. Yablon" <jyablon(a)nycap.rr.com> writes: >>> 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 >> Rather than re-inventing the wheel, why don't you look at what mathematical >> physicists have already done? You might look at >> Glimm and Jaffe, "Quantum Physics: A Functional Integral Point of View", >> Springer-Verlag 1981, and >> Simon, "Functional Integration and Quantum Physics", Academic Press 1979. >Just (a naive, of course) question: is that (meanwhile) settled in a >rigorous mathematical sense? This has been looked at for a long time; Feynman's intuitions in his presentation caused him to believe that the necessary mathematical objects for his approach existed, and not only did they not, they cannot. I believe that this should be apparent to anyone who knows the mathematics claimed. This does not mean that it cannot be done; I do not believe it has been done. What has been done, I believe, is to handle special cases by showing that they agree with what other approaches yield. However, I do not believe that a rigorous mathematical meaning has been given to the path integral. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
From: Ken S. Tucker on 23 Dec 2009 13:43 Also to Jays Friends too. On Dec 21, 2:46 pm, "Jay R. Yablon" <jyab...(a)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... > > 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. 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...(a)nycap.rr.com > co-moderator: sci.physics.foundations > Weblog:http://jayryablon.wordpress.com/ > Web Site:http://home.roadrunner.com/~jry/FermionMass.htm Cheers Ken PS:Also posted to SPF.
From: master1729 on 23 Dec 2009 06:08
"Jay R. Yablon" <jyablon(a)nycap.rr.com> writes: > >> 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 > > > > Rather than re-inventing the wheel, why don't you > look at what > > mathematical > > physicists have already done? You might look at > > Thank you for these references. > > However, from what I understand after some extensive > discussion in > sci.physics.research with some folks who have > extensive knowledge about > this (you should check those threads), there remains > a mathematical > problem which has not yet been solved despite years > of effort by many, > of giving a rigorous calculus limit foundation to the > path integral, > because of the infinitely-dimensional > $...$$$dqdqdq...dq integral that > is swept into the "integral over paths" Dq. > > I strongly suspect that this problem is not solved by > these references, > otherwise people would not still be writing textbooks > and papers 25 or > 30 years later with this problem still not resolved, > and the > knowledgeable folks at sci.physics.foundations would > not be talking > about how path integration, while useful, "couldn't > be made logically > consistent, in spite of many attempts by some of the > best mathematicians > and physicists." (quote from A. Neumaier) > > Jay > in 2 words : epic fail. yet they pretend QM is brilliant , complete and consistant. especially towards the media and education. and that is not the responsability of the media or the education system *alone*. tommy1729 |