Joan Estes
Sep14-04, 12:23 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>\nI have a question about a simple formula in 2-d conformal field\ntheory of a scalar field, given for example by Gawedzki in\n"Quantum Fields and Strings - A Course for Mathematicians".\nI also posted the question to sci.physics.strings, so far without\nresponse.\n\nIt is stated that the holomorphic-antiholomorphic product of\nE-M tensors for a scalar has the following contact term\n\n<T_{zz} T_{\\bar w \\bar w}>\n= - (\\pi c / 12) \\partial_z\\partial_{\\bar z}\\delta^2(z-w) + ...\n\nwhich Gawedzki derives from CFT axioms. However, when I try to\nreproduce this formula by a direct field theory computation,\ntaking double Wick contractions of\n\nT_zz = \\partial_z X \\partial_z X\n\nT_{\\bar w \\bar w} = \\partial_{\\bar w} X \\partial_{\\bar w} X\n\nand using\n\nX(z)X(w) = ln |z-w|^2,\n\\partial\\bar\\partial ln |z-w|^2 = \\delta^2(z-w),\n\nI get instead an answer proportional to the ill-defined product\nof distributions\n\n\\delta^2(z-w) \\delta^2(z-w)\n\nI am at a loss to explain the discrepancy.\nDoes anybody know how to carry out this computation correctly?\nJ.E.\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>I have a question about a simple formula in 2-d conformal field
theory of a scalar field, given for example by Gawedzki in
"Quantum Fields and Strings - A Course for Mathematicians".
I also posted the question to sci.physics.strings, so far without
response.
It is stated that the holomorphic-antiholomorphic product of
E-M tensors for a scalar has the following contact term
<T_{zz} T_{\bar w \bar w}>= - (\pi c / 12) \partial_z\partial_{\bar z}\delta^2(z-w) + .[/itex]..
which Gawedzki derives from CFT axioms. However, when I try to
reproduce this formula by a direct field theory computation,
taking double Wick contractions of
[itex]T_{zz} = \partial_z X \partial_z XT_{\bar w \bar w} = \partial_{\bar w} X \partial_{\bar w} X
and using
X(z)X(w) = ln |z-w|^2,\partial\bar\partial ln |z-w|^2 = \delta^2(z-w),
I get instead an answer proportional to the ill-defined product
of distributions
\delta^2(z-w) \delta^2(z-w)
I am at a loss to explain the discrepancy.
Does anybody know how to carry out this computation correctly?
J.E.
theory of a scalar field, given for example by Gawedzki in
"Quantum Fields and Strings - A Course for Mathematicians".
I also posted the question to sci.physics.strings, so far without
response.
It is stated that the holomorphic-antiholomorphic product of
E-M tensors for a scalar has the following contact term
<T_{zz} T_{\bar w \bar w}>= - (\pi c / 12) \partial_z\partial_{\bar z}\delta^2(z-w) + .[/itex]..
which Gawedzki derives from CFT axioms. However, when I try to
reproduce this formula by a direct field theory computation,
taking double Wick contractions of
[itex]T_{zz} = \partial_z X \partial_z XT_{\bar w \bar w} = \partial_{\bar w} X \partial_{\bar w} X
and using
X(z)X(w) = ln |z-w|^2,\partial\bar\partial ln |z-w|^2 = \delta^2(z-w),
I get instead an answer proportional to the ill-defined product
of distributions
\delta^2(z-w) \delta^2(z-w)
I am at a loss to explain the discrepancy.
Does anybody know how to carry out this computation correctly?
J.E.