## Elementary: on the vertex (cont'd)

<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 Urs,\n\nI tried to understand the formula you gave me from Peskin Schroeder\ni.e., &lt;Q, phi(x) , phi(y) Q&gt; = lim_{T -&gt; oo (1 -i epsilon)}\n\n&lt;0| U(T, x_0) phi_I(x) U (x_0, y_0 ) phi_I(y) U(y_0, -T) |0&gt;\n----------------------------------------------------------------\n&lt;0| U(T, -T) |0&gt;\n\nwhere Q is what they call capital omega i.e., the groundstate of\nthe full interacting Hamiltonian.\n\nI examined it, and honestly I wasn\'t able to make sense of it.\n\nso 1) either it\'s wrong,\nor 2) I have a serious misunderstanding of some term here.\n\nI guess I tried to get what they get given the assumption that:\n\n****let\'s first change some notation quickly : Let m = epsilon\n\nlet Q = capital omega, HQ=LQ , let g = |0&gt; i.e., the groundstate\nof the\nFree Hamiltonian H_0, Let &lt;Q|0&gt;^{-1} = &lt;Q, g&gt;^{-1} = K .\n\nthen the initial formula in 4.2 says\n\nlim_{T -&gt; oo (1 -i epsilon)} e^{ i E_0T} e^{-iHT} &lt;Q|0&gt;^{-1} |0&gt;\n= Q\n\nor in terms of my shorter "email readable notation"\n\nlim_{T -&gt; oo (1 -im) } e^{i L T} e^{-iHT} K g = Q\n\nNow, since I find all the complex T notation weird,\nI rewrote it as\n\nlim_{T -&gt; oo}\nK e^{i L T} e^{mLT} e^{-iHT} e^{-mHT} g = Q\n\nSo, then given my interpretation of that is right,\n\n&lt;Q, phi(x) phi(y) Q&gt;\n= lim_{T -&gt; oo } e^{2mLT} KK*\n\n&lt;g , e^{mHT} e^{iHT} phi(x) phi(y) e^{-iHT} e^{-mHT} g &gt;\n\n= .....\n\n( assuming phi(x) = e^{iH_0 x_0} e^{-iH x_0} phi_I(x) e^{iH x_0}\ne^{-iH_0 x_0}\n\n...... Then upon doing the calculation using what I listed here, I\ndon\'t get the result in Peskin Schroeder that I stated before.\nSomething\'s wrong but I don\'t know what.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi Urs,

I tried to understand the formula you gave me from Peskin Schroeder
i.e., <Q, $\phi(x) , \phi(y) Q> = lim_{T -> oo (1 -i \epsilon)}<0| U(T, x_0) \phi_I(x) U (x_0, y_0 ) \phi_I(y) U(y_0, -T) |0>$
----------------------------------------------------------------
$<0| U(T, -T) |0>$

where Q is what they call capital $\omega i$.e., the groundstate of
the full interacting Hamiltonian.

I examined it, and honestly I wasn't able to make sense of it.

so 1) either it's wrong,
or 2) I have a serious misunderstanding of some term here.

I guess I tried to get what they get given the assumption that:

$$****let's[/itex] first change some notation quickly : Let $m = \epsilon$$ let Q = capital [itex]\omega, HQ=LQ$ , let $g = |0> i$.e., the groundstate
of the
Free Hamiltonian $H_0,$ Let $<Q|0>^{-1} =$ <Q, $g>^{-1} =$ K .

then the initial formula in 4.2 says

$$lim_{T -> oo (1 -i \epsilon)} e^{ i E_{0T}} e^{-iHT} <Q|0>^{-1} |0>[/itex] = Q or in terms of my shorter "email readable notation" $lim_{T -> oo (1 -im) } e^{i L T} e^{-iHT}$ K g = Q Now, since I find all the complex T notation weird, I rewrote it as $lim_{T ->$ oo} K $e^{i L T} e^{mLT} e^{-iHT} e^{-mHT}$ g = Q So, then given my interpretation of that is right, <Q, $\phi(x) \phi(y) Q>= lim_{T -> oo } e^{2mLT} KK*$$ <g , [itex]e^{mHT} e^{iHT} \phi(x) \phi(y) e^{-iHT} e^{-mHT} g$ >

= .....

( assuming $\phi(x) = e^{iH_0 x_0} e^{-iH x_0} \phi_I(x) e^{iH x_0}e^{-iH_0 x_0}$

...... Then upon doing the calculation using what I listed here, I
don't get the result in Peskin Schroeder that I stated before.
Something's wrong but I don't know what.