Joan Estes
Oct6-04, 08:03 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>Organization: (no organization specified)\nI 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\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</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>Organization: (no organization specified)
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 .
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.
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 .
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.