mandro
Sep27-04, 09:27 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>Hi Urs,\n\nI tried to understand the formula you gave me from Peskin Schroeder\ni.e., <Q, phi(x) , phi(y) Q> = lim_{T -> oo (1 -i epsilon)}\n\n<0| U(T, x_0) phi_I(x) U (x_0, y_0 ) phi_I(y) U(y_0, -T) |0>\n----------------------------------------------------------------\n<0| U(T, -T) |0>\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> i.e., the groundstate\nof the\nFree Hamiltonian H_0, Let <Q|0>^{-1} = <Q, g>^{-1} = K .\n\nthen the initial formula in 4.2 says\n\nlim_{T -> oo (1 -i epsilon)} e^{ i E_0T} e^{-iHT} <Q|0>^{-1} |0>\n= Q\n\nor in terms of my shorter "email readable notation"\n\nlim_{T -> 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 -> 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<Q, phi(x) phi(y) Q>\n= lim_{T -> oo } e^{2mLT} KK*\n\n<g , e^{mHT} e^{iHT} phi(x) phi(y) e^{-iHT} e^{-mHT} g >\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"> 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 \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>
= 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, [itex]\phi(x) \phi(y) Q>= lim_{T -> oo } e^{2mLT} KK*
<g , 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.
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 \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>
= 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, [itex]\phi(x) \phi(y) Q>= lim_{T -> oo } e^{2mLT} KK*
<g , 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.