Mathematically-Rigorous Path Integration??
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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. |