Joan Estes
Oct5-04, 05:33 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 been running into subtle problems with the Polchinski\nbook treatment of conformal field theory. I cannot seem to figure\nout why the following statement is correct:\n\nSince the variation of the path integral w.r.t. the metric\nis given by an expectation value of the E-M-tensor as follows:\n\n\\delta <...>_g = \\int \\sqrt{g} h_{ab} <T^{ab}...>\n\nwhere h_{ab} = \\delta g_{ab},\nthe *second-order* variation (see 3.4.22, p 94) is given\nby a double integral with two insertions of T:\n\n\\int d^z \\sqrt{g(z)} \\int d^z\' \\sqrt g(z\')\nh_{ab}(z) h_{cd}(z\') <T^{cd}(z\') T^{ab}(z)>.\n\nI agree with the first formula when ... denotes operators\nthat do not explicitly depend on the metric.\nHowever, when I try to verify the second formula, it seems to me\nthat he has forgotten that\n\nT_{ab} = del_a X del_b X - half g{ab} g^{cd} del_c X del_d X\n\nreally *does* depend on the metric,\nso that one should really have an *extra* term in the second variation\n\n\\int \\sqrt{g} h_{ab} <\\delta T^{ab}>\n\nsince \\delta T^{ab} is not in general 0.\n\n[Moderator\'s note: T^{ab} is not zero in general, but have not you\nconsidered the possibility that h_{ab} T^{ab} *is* zero in general?\nIt\'s called "tracelessness" and holds for all conformal field theories.\nFor example, with the definition of T above, the two terms cancel. LM]\n\nAm I missing something? I suspect I am, since Polchinski\'s\n(incorrect?) formula for multiple insertions is actually\ntaken by some other authors as an axiomatic property of T.\nAny help would be appreciated.\n\nJ.E.\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>I have been running into subtle problems with the Polchinski
book treatment of conformal field theory. I cannot seem to figure
out why the following statement is correct:
Since the variation of the path integral w.r.t. the metric
is given by an expectation value of the E-M-tensor as follows:
\delta <[/itex]...>_g = \int \sqrt{g} h_{ab} <T^{ab}...>
where h_{ab} = \delta g_{ab},
the *second-order* variation (see 3.4.22, p 94) is given
by a double integral with two insertions of T:
\int d^z \sqrt{g(z)} \int d^z' \sqrt g(z')h_{ab}(z) h_{cd}(z') <T^{cd}(z') T^{ab}(z)>.
I agree with the first formula when ... denotes operators
that do not explicitly depend on the metric.
However, when I try to verify the second formula, it seems to me
that he has forgotten that
T_{ab} = del_a X del_b X - half g{ab} [itex]g^{cd} del_c X del_d X
really *does* depend on the metric,
so that one should really have an *extra* term in the second variation
\int \sqrt{g} h_{ab} <\delta T^{ab}>
since \delta T^{ab} is not in general .
[Moderator's note: T^{ab} is not zero in general, but have not you
considered the possibility that h_{ab} T^{ab} *is* zero in general?
It's called "tracelessness" and holds for all conformal field theories.
For example, with the definition of T above, the two terms cancel. LM]
Am I missing something? I suspect I am, since Polchinski's
(incorrect?) formula for multiple insertions is actually
taken by some other authors as an axiomatic property of T.
Any help would be appreciated.
J.E.
book treatment of conformal field theory. I cannot seem to figure
out why the following statement is correct:
Since the variation of the path integral w.r.t. the metric
is given by an expectation value of the E-M-tensor as follows:
\delta <[/itex]...>_g = \int \sqrt{g} h_{ab} <T^{ab}...>
where h_{ab} = \delta g_{ab},
the *second-order* variation (see 3.4.22, p 94) is given
by a double integral with two insertions of T:
\int d^z \sqrt{g(z)} \int d^z' \sqrt g(z')h_{ab}(z) h_{cd}(z') <T^{cd}(z') T^{ab}(z)>.
I agree with the first formula when ... denotes operators
that do not explicitly depend on the metric.
However, when I try to verify the second formula, it seems to me
that he has forgotten that
T_{ab} = del_a X del_b X - half g{ab} [itex]g^{cd} del_c X del_d X
really *does* depend on the metric,
so that one should really have an *extra* term in the second variation
\int \sqrt{g} h_{ab} <\delta T^{ab}>
since \delta T^{ab} is not in general .
[Moderator's note: T^{ab} is not zero in general, but have not you
considered the possibility that h_{ab} T^{ab} *is* zero in general?
It's called "tracelessness" and holds for all conformal field theories.
For example, with the definition of T above, the two terms cancel. LM]
Am I missing something? I suspect I am, since Polchinski's
(incorrect?) formula for multiple insertions is actually
taken by some other authors as an axiomatic property of T.
Any help would be appreciated.
J.E.