Mathematically-Rigorous Path Integration??
in [Math]
|
Prev: Kind of Godel numbering
Next: Subgroup Inclusion
From: Jay R. Yablon on 21 Dec 2009 17:45 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: jyablon(a)nycap.rr.com co-moderator: sci.physics.foundations Weblog: http://jayryablon.wordpress.com/ Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
From: Robert Israel on 21 Dec 2009 19:31 "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. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Androcles on 21 Dec 2009 19:42 "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 > > Glimm and Jaffe, "Quantum Physics: A Functional Integral Point of View", > Springer-Verlag 1981, and > > Simon, "Functional Integration and Quantum Physics", Academic Press 1979. > -- "Mathematical physicist" -- Bwhahahaha! That's funny, Robert.
From: Larry Hammick on 22 Dec 2009 01:53 "Robert Israel" > You might look at > Glimm and Jaffe, "Quantum Physics: A Functional Integral Point of View", > Springer-Verlag 1981 I have that book. Thanks for mentioning it, Robert. I feel a little less old now. :) I do a little a tutoring and my kids weren't even born when I was reading some of these scholarly tomes. How about this one: Prugovecki, "Quantum Mechanics in Hilbert Space", Springer-Verlag 1981 It's a careful and thorough mathematical backgrounder on integration and spectral theory, among other things. I should drop by UBC again one of these years. LH
From: Axel Vogt on 22 Dec 2009 15:07
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? |