Operator product question [Archive]

Joan Estes

Sep11-04, 01:16 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>I have a question about a simple formula, given by Gawedzki in\n"Quantum Fields and Strings - A Course for Mathematicians".\n\nHe states that the E-M tensor satisfies\n\n<T_{zz} T_{\\bar w \\bar w}>\n= - (\\pi c/ 12) \\partial_z\\partial_{\\bar z}\\delta^2(z-w) + ...\n\nwhich he derives from CFT axioms. However, when I try to\nreproduce this formula by a direct computation, taking Wick\ncontractions of\n\nT_zz = \\partial_z X \\partial_z X\nT_{\\bar z \\bar z} = \\partial_{\\bar z} X \\partial_{\\bar z} X\n\nand using X(z)X(w) = ln |z-w|^2, I get instead an answer proportional\nto the ill-defined distribution product\n\n\\delta^2(z-w) \\delta^2(z-w)\n\nDoes anybody know how to carry out this computation correctly?\nJ.E.\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>I have a question about a simple formula, given by Gawedzki in
"Quantum Fields and Strings - A Course for Mathematicians".

He states that the E-M tensor satisfies

<T_{zz} T_{\bar w \bar w}>= - (\pi c/ 12) \partial_z\partial_{\bar z}\delta^2(z-w) + .[/itex]..

which he derives from CFT axioms. However, when I try to
reproduce this formula by a direct computation, taking Wick
contractions of

[itex]T_{zz} = \partial_z X \partial_z XT_{\bar z \bar z} = \partial_{\bar z} X \partial_{\bar z} X

and using X(z)X(w) = ln |z-w|^2, I get instead an answer proportional
to the ill-defined distribution product

\delta^2(z-w) \delta^2(z-w)

Does anybody know how to carry out this computation correctly?
J.E.

Joan Estes

Sep16-04, 05:55 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>Joan Estes <joanestes2000@yahoo.com> wrote\n\n> I have a question about a simple formula, given by Gawedzki:\n\n> <T_{zz} T_{\\bar w \\bar w}>\n> = - (\\pi c/ 12) \\partial_z\\partial_{\\bar z}\\delta^2(z-w) + ...\n\nStrangely enough, after a week\'s literature search, still the only\nauthor I could find who writes down a <T \\bar T> product is Gawedzki.\nPolchinski does note, in a footnote on p48 of his book, that such\nholomorphic-antiholomorphic correlators may have delta-function\ncontact terms, with the cryptic and mysterious comment that these "may\ndepend on definitions, so one must be careful".\n\nStill, while I can follow the axiomatic derivation of the above formula,\nI remain unable to compute it from a direct field theory\ncalculation. If anyone knows of any references that mention\nsuch T-T-bar correlators, I would be grateful.\n\nJ.E.\n\n[Moderator\'s note: I was unable to give any explanation. I seem to agree\nwith Joan that because T and Tbar for a single boson, for example,\nare squares of something (del X), the completely contracted term seems\nto be a square of something - namely of the 2D delta function. A kind\nof distribution that does not make much sense. In some sense,\nPolchinski\'s meaningful second (mixed) derivative of a single delta\nfunction must be a "corrected" definition of the squared delta function,\nbut I am not able to go beyond this handwaving right now. LM]\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>Joan Estes <joanestes2000@yahoo.com> wrote

> I have a question about a simple formula, given by Gawedzki:

> <T_{zz} T_{\bar w \bar w}>
> = - (\pi c/ 12) \partial_z\partial_{\bar z}\delta^2(z-w) + ...

Strangely enough, after a week's literature search, still the only
author I could find who writes down a <T \bar T> product is Gawedzki.
Polchinski does note, in a footnote on p48 of his book, that such
holomorphic-antiholomorphic correlators may have \delta-function
contact terms, with the cryptic and mysterious comment that these "may
depend on definitions, so one must be careful".

Still, while I can follow the axiomatic derivation of the above formula,
I remain unable to compute it from a direct field theory
calculation. If anyone knows of any references that mention
such T-T-bar correlators, I would be grateful.

J.E.

[Moderator's note: I was unable to give any explanation. I seem to agree
with Joan that because T and Tbar for a single boson, for example,
are squares of something (del X), the completely contracted term seems
to be a square of something - namely of the 2D \delta function. A kind
of distribution that does not make much sense. In some sense,
Polchinski's meaningful second (mixed) derivative of a single \delta
function must be a "corrected" definition of the squared \delta function,
but I am not able to go beyond this handwaving right now. LM]

Lubos Motl

Sep19-04, 07:18 AM

<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>Joan Estes <joanestes2000@yahoo.com> wrote:\n\n> ... <T_{zz} T_{\\bar w \\bar w}>\n> = - (\\pi c/ 12) \\partial_z\\partial_{\\bar z}\\delta^2(z-w) + ...\n\nI know a different CFT in which you can realize this formula easily.\nTake the linear dilaton CFT, with the term proportional to\nV\\partial\\partial X included in T_{zz}. You know that this contributes\nsomething like 6 V^2\\alpha\' to the central charge, but on the other\nhand, you can also see that it contributes your desired second (mixed)\nderivative of the two-dimensional delta function to the correlator of\nT_{zz} and T_{\\bar w\\bar w} (fourth derivative, the 2+2 mixed one, of\nthe logarithm), and I believe that the coefficient will match (the\nfactors of \\pi and 3 work out correctly; I have not checked the powers\nof two and the sign). Well, I still do not know how to get this nice\nresult from the free boson, but at least one CFT seems to give the\nformally derived contact term explicitly.\n\nBest wishes, Lubos\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>Joan Estes <joanestes2000@yahoo.com> wrote:

> ... <T_{zz} T_{\bar w \bar w}>
> = - (\pi c/ 12) \partial_z\partial_{\bar z}\delta^2(z-w) + ...

I know a different CFT in which you can realize this formula easily.
Take the linear dilaton CFT, with the term proportional to
V\partial\partial X included in T_{zz}. You know that this contributes
something like 6 V^2\alpha' to the central charge, but on the other
hand, you can also see that it contributes your desired second (mixed)
derivative of the two-dimensional \delta function to the correlator of
T_{zz} and T_{\bar w\bar w} (fourth derivative, the 2+2 mixed one, of
the logarithm), and I believe that the coefficient will match (the
factors of \pi and 3 work out correctly; I have not checked the powers
of two and the sign). Well, I still do not know how to get this nice
result from the free boson, but at least one CFT seems to give the
formally derived contact term explicitly.

Best wishes, Lubos