Mathematically-Rigorous Path Integration??
in [Relativity]
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From: Jay R. Yablon on 23 Dec 2009 20:57 "Mike" <mjake(a)sirus.com> wrote in message news:5600d892-a19e-4107-8040-2cc209486537(a)j19g2000yqk.googlegroups.com... On Dec 21, 5:45 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... I looked at your file. And I searched it for terms like "measure" and "distribution" and did not find any reference to these. So I'm wonder how anyone can try to rigorously prove that the path integral is mathematical well behaved without reference to the measure used in it. Or maybe I'm not understanding what it is you're trying to prove. Try "probability" density and "element" of integration. And maybe some of the boxed equations. Jay.
From: Mike on 23 Dec 2009 21:38 On Dec 23, 8:57 pm, "Jay R. Yablon" <jyab...(a)nycap.rr.com> wrote: > > I looked at your file. And I searched it for terms like "measure" and > "distribution" and did not find any reference to these. So I'm wonder > how anyone can try to rigorously prove that the path integral is > mathematical well behaved without reference to the measure used in it. > Or maybe I'm not understanding what it is you're trying to prove. > > Try "probability" density and "element" of integration. And maybe some > of the boxed equations. Jay. I think what you need is an abstract in your paper to tell us what problem you're addressing and how you fix it.
From: Robert Israel on 23 Dec 2009 21:49 hrubin(a)odds.stat.purdue.edu (Herman Rubin) writes: > 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.pd >>>>f > > >> 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. Basically what has been done is to perform a "Wick rotation" so that time t becomes -it and the Schrodinger equation becomes the heat equation with a potential. The Feynman "integral", which was not well-defined, becomes a Wiener integral which is, and the resulting formula is the Feynman-Kac formula. -- 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 24 Dec 2009 15:20 In article <rbisrael.20091224021640$1cff(a)news.acm.uiuc.edu>, Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote: >hrubin(a)odds.stat.purdue.edu (Herman Rubin) writes: >> 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.pd >>>>>f >> >> 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. >Basically what has been done is to perform a "Wick rotation" so that >time t becomes -it and the Schrodinger equation becomes the heat equation >with a potential. The Feynman "integral", which was not well-defined, >becomes a Wiener integral which is, and the resulting formula is the >Feynman-Kac formula. -- >Robert Israel israel(a)math.MyUniversitysInitials.ca >Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia Vancouver, BC, Canada Even more has been done. The problem with this is that not all potentials cooperate with the analyticity needed for the Feynman-Kac approach. BTW, the internal integral in the Feynman approach does not exist even after the transformation is made. I am not sure that the subtraction of the infinity naturally induced will always work, as it might be different for different paths. However, it comes in a purely imaginary exponential, and the various phases need to be aligned. -- 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: Mike on 26 Dec 2009 12:56 On Dec 24, 3:20 pm, hru...(a)odds.stat.purdue.edu (Herman Rubin) wrote: > In article <rbisrael.20091224021640$1...(a)news.acm.uiuc.edu>, > Robert Israel <isr...(a)math.MyUniversitysInitials.ca> wrote: > > > > > > >hru...(a)odds.stat.purdue.edu (Herman Rubin) writes: > >> In article <7pcn86Fqo...(a)mid.individual.net>, > >> Axel Vogt <&nore...(a)axelvogt.de> wrote: > >> >Robert Israel wrote: > >> >> "Jay R. Yablon" <jyab...(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-intergati... > >>>>>f > >> >> 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. > >Basically what has been done is to perform a "Wick rotation" so that > >time t becomes -it and the Schrodinger equation becomes the heat equation > >with a potential. The Feynman "integral", which was not well-defined, > >becomes a Wiener integral which is, and the resulting formula is the > >Feynman-Kac formula. > > -->Robert Israel isr...(a)math.MyUniversitysInitials.ca > >Department of Mathematics http://www.math.ubc.ca/~israel > >University of British Columbia Vancouver, BC, Canada > > Even more has been done. The problem with this is that > not all potentials cooperate with the analyticity needed > for the Feynman-Kac approach. > BTW, the internal integral in the Feynman approach does not > exist even after the transformation is made. I am not sure > that the subtraction of the infinity naturally induced will > always work, as it might be different for different paths. > However, it comes in a purely imaginary exponential, and the > various phases need to be aligned. > -- > 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 > hru...(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558- Hide quoted text - > > - Show quoted text - I'm not sure I'm getting a definitive answer. Does the Feynman integral have a well defined measure or not? Does the path integral for quantum field theory have a well defined measure or not? I'm understanding that the infinite dimensional Lesbesgue measure, D[x] is undefined. But the Wiener measure, e^S[x]D[x], is well defined and serves as euclidean path integral measure. Is this right? But I'm not sure that this can be analytically continued into the complex plane to provide a measure for the Feynman path integral which has complex action. Any help would be appreciated.
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