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Martin Chang

Apr22-04, 02:42 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi, my question is: is Scharf\'s formulation of causal QFT defined\nnonperturbatively? I have read that it reproduces the same expansion as QED\nusing renormalization, which many people think is an asymptotic expansion.\nI think that the reason for the expansion being asymptotic is because of\nStokes phenomena in the Maxwell equations and Dirac equation coupled to the\nEM field, when they are solved simultaneously. This leads me to another\nquestion: are there methods to extract convergent expressions for S-matrix\nelements (I am think of methods different from resumming the series such as\nin Borel summation although I am also interested in methods that tell us\nwhether it is Borel summable; I suspect we would have to find a stokes line\nin the nonperturbatively defined(?) differential equation for the quantum\nfields. I would think that a nonperturbative formulation would give a\nmethod that gives a convergent expression for the S-matrix. Do my questions\neven make sense?\n\nThanks, martin\n\n________________________________________ _________________________\nMSN Toolbar provides one-click access to Hotmail from any Web page – FREE\ndownload! http://toolbar.msn.com/go/onm00200413ave/direct/01/\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi, my question is: is Scharf's formulation of causal QFT defined
nonperturbatively? I have read that it reproduces the same expansion as QED
using renormalization, which many people think is an asymptotic expansion.
I think that the reason for the expansion being asymptotic is because of
Stokes phenomena in the Maxwell equations and Dirac equation coupled to the
EM field, when they are solved simultaneously. This leads me to another
question: are there methods to extract convergent expressions for S-matrix
elements (I am think of methods different from resumming the series such as
in Borel summation although I am also interested in methods that tell us
whether it is Borel summable; I suspect we would have to find a stokes line
in the nonperturbatively defined(?) differential equation for the quantum
fields. I would think that a nonperturbative formulation would give a
method that gives a convergent expression for the S-matrix. Do my questions
even make sense?

Thanks, martin

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Arnold Neumaier

Apr23-04, 03:13 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Martin Chang wrote:\n> Hi, my question is: is Scharf\'s formulation of causal QFT defined\n> nonperturbatively?\n\nYes for electrons in a classical electromagnetic field,\nno for full QED with photons. There they only compute the coefficients\nof the asymptotic expansion (in facto only the first few, but give\ninstructions how to do them all).\n\n> I have read that it reproduces the same expansion as\n> QED using renormalization, which many people think is an asymptotic\n> expansion. I think that the reason for the expansion being asymptotic\n> is because of Stokes phenomena in the Maxwell equations and Dirac\n> equation coupled to the EM field, when they are solved simultaneously.\n> This leads me to another question: are there methods to extract\n> convergent expressions for S-matrix elements\n\nNo. At present, nothing rigorous exists nonperturbatively about any\n4D interacting quantum field theory. This is a hard unsolved problem.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Martin Chang wrote:
> Hi, my question is: is Scharf's formulation of causal QFT defined
> nonperturbatively?

Yes for electrons in a classical electromagnetic field,
no for full QED with photons. There they only compute the coefficients
of the asymptotic expansion (in facto only the first few, but give
instructions how to do them all).

> I have read that it reproduces the same expansion as
> QED using renormalization, which many people think is an asymptotic
> expansion. I think that the reason for the expansion being asymptotic
> is because of Stokes phenomena in the Maxwell equations and Dirac
> equation coupled to the EM field, when they are solved simultaneously.
> This leads me to another question: are there methods to extract
> convergent expressions for S-matrix elements

No. At present, nothing rigorous exists nonperturbatively about any
4D interacting quantum field theory. This is a hard unsolved problem.

Arnold Neumaier