%  tf2.tex         Heisse QCD , WS 1993/94
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%  Response function from thermal Greens

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\vskip -.1cm \nz
\hskip .7cm {\gr Response function from thermal Greens}
             \qquad \lower 5pt\vbox{
  \hbox{\kl ( Kritik$\,$: \ Quelle, Antwort}  \vskip -.2cm
  \hbox{\kl \ \ und response function sind}     \vskip -.2cm
  \hbox{\kl \ \ hier leider eichabh\"angig.)} } 
\bigskip
Consider the QCD Lagrangian (including massless quarks,
gauge fixing and ghosts) and add a source term ${\cal
L}_{\rm ext} =J^{\nu b} \, A^b_\nu$. We study $\li
A^a_\mu (x) \re _\bullet$ as one of the possibly nontrivial
responses of the system ($H$). The index $_\bullet$ refers
to the full system $H_\bullet \gll H + H_{\rm ext} \, , \;
H_{\rm ext} = - \int \! d^3 r \, J^{\nu b} A^b_\nu$; $\;
A(x)$ are Heisenberg operators with respect to $H$, but
are in the interaction picture with respect to $H_\bullet$
; $\li ... \re _\bullet \! \gll \tr (\sigma ... ) \; , \;
\li ... \re \! \gll \tr (\rho_o ... ) \; , \; \rho_o \gll
e^{\beta (F - H)}, \; F=-T \, {\rm ln}(Z)$. \smallskip 
    $\p \rho = -i \!\; [H_\bullet, \rho]$
(Sch\"odinger picture), $\sigma = e^{iHt} \rho \!\> e^{-iHt}
\;\; \Rightarrow \; \p \sigma = -i \!\; [H_{\rm ext} ,
\sigma] \; , \; \sigma (- \infty ) = \rho_o \; : $ 
\smallskip \nz
$$ \eqalign{ 
   \li A^a_\mu (x) \re _\bullet & = \tr (A^a_\mu (x) \, \sigma )
    = -i \!\int_{- \infty }^t \!\!\!\!\! dt^\prime \; \tr
   ( A^a_\mu (x) [H_{\rm ext} (t^\prime ), \rho_o])
   + O(J^2) \hskip 3cm \cr
\noalign{\vskip -.2cm}
   & = i \!\int_{-\infty}^t \!\!\!\!\! dt^\prime 
   \int \! d^3 r^\prime \!\li [ A^a_\mu (x) , A^b_\nu
   (x^\prime) ] \re \! J^{\nu b} (x^\prime)
   + O(J^2)  \cr } $$ \nz
Let $J(x)=e^{\epsilon t} j(x)$. The system answers by
$\li A(x) \re = e^{\epsilon t} a(x)$. $j(x)$ and $a(x)$ are
the quantities "in the presence", the relation between we
are really interested in. Thus we define the response
function $\chi$ by $a(x) = \int d^4x^\prime \chi (x,
x^\prime) j(x^\prime)$ and obtain  
\smallskip \nz
$$ \chi^{a b}_{\mu \nu} (x, x^\prime )
   = i \theta (t-t^\prime) e^{-\epsilon (t-t^\prime)} \li
   \; [ A^a_\mu (x) , A^b_\nu (x^\prime) ] \;\re \glr
   \chi^{a b}_{\nu \mu} (x-x^\prime) \; , 
$$ \nz 
where the last relation is due to translational
invariance. Fourier transforming by \nz
$(2 \pi )^{-4} \int d^4 K e^{-iKx} ...$ we have
$\schl a(K) = \schl \chi (K) \, \schl j (K) \;\; ( K \gll
(\omega , \vc k)$, metrics $+ - - -$) and 
\smallskip \nz
$$  \schl \chi ^{a b}_{\mu \nu} (K) \; = \int \! dt
    \int \! d^3r e^{i \omega t - i \vcsm k \vcsm r} i
    \theta (t) e^{- \epsilon t} \li \,
    [ A^a_\mu (x) , A^b_\nu (0) ] \, \re \hskip 5.6cm $$
\vskip -.3cm \nz
$$ \eqalign{
  & \hskip 7.6cm \hbox{$A^a_\mu (x) = e^{iHt} (2 \pi)^{-3}
    \int \! d^3q e^{i \vcsm q \vcsm r} \schl A^a_\mu
    (\vc q) e^{-iHt} $} \cr
  & = i \int_0^\infty \! dt e^{(i \omega - \epsilon)t}
    (2 \pi)^{-3} \int \! d^3p \tr [ e^{\beta (F-H)}
    ( e^{iHt} \schl A^a_\mu (\vc k)
    e^{-iHt} \schl A^b_\nu (\vc p) - \schl A^b_\nu (\vc p)
    e^{iHt} \schl A^a_\mu (\vc k) e^{-iHt} ) ] \cr
  & \hskip 11.5cm  \hbox{$\omega_{fg} \gll E_f - E_g \; :$} \cr
  & = i \int_0^\infty \! dt e^{(i \omega - \epsilon)t}
     (2 \pi)^{-3} \int \! d^3p  \sum_{f, g} e^{-i
     \omega_{f g} t} \; [ e^{\beta (F-E_g)} - e^{\beta
     (F-E_f)} ] \li f \vert \schl A^b_\nu (\vc p) \vert
     g \re \li g \vert \schl A^a_\mu (\vc k) \vert f \re
     \hskip .2cm \cr
  & = \int \! dx \; {1 \over x- \omega -i \epsilon} \;\; 
   {\cal A}^{a b}_{\mu \nu} (x, \vc k) \qquad 
           {\rm with} \hskip 8.3cm \! \cr } $$ 
\nz
$$ {\cal A}^{a b}_{\mu \nu} (x, \vc k) \; = \;
   (2 \pi)^{-3} \int \! d^3p  \sum_{f, g} e^{\beta (F-E_f)}
   \; [ e^{\beta \omega_{f g}} - 1 ] \;\; \delta (x
   - \omega_{f g}) \li f \vert \schl A^b_\nu (\vc p) \vert
   g \re \li g \vert \schl A^a_\mu (\vc k) \vert f \re 
$$ \vskip -.1cm \nz
We might compare this spectral representation of the
response function with that of the Matsu- \nz
bara thermal Greensfunction $G^{a b}_{\mu \nu} (x)
= \li {\cal T} A^a_\mu (x) A^b_\nu (0) \re$ where here
$x \gll ( -i \tau , \vc r)$ and (in the following)
$K \gll (i \omega_n , \vc k) \; , \; \omega_n \gll
2\pi n T$. ${\cal T}$ orders larger $\tau$ to the left. 
\smallskip \nz
$$ \schl G^{a b}_{\mu \nu} (K) \; 
   = \int \! d^3r \int_0^\beta \! d\tau e^{iKx} 
      G^{a b}_{\mu \nu} (x) \hskip 9cm $$ 
\vskip -.2cm \nz
$$ \eqalign{ 
  & = \int_0^\beta \!\!\! d\tau e^{i \omega_n \tau} \int
      \! d^3r e^{-i \vcsm k \vcsm r} \tr [ e^{\beta (F-H)}
      e^{H \tau} (2 \pi)^{-3} \int \! d^3q e^{i \vcsm q
      \vcsm r} \schl A^a_\mu (\vc q) e^{-H \tau}
      (2 \pi)^{-3} \int \! d^3p \schl A^b_\nu (\vc p) \, ] \cr
  & = \sum_{f g} {1 \over \omega_{f g} - i \omega_n}
      e^{\beta (F-E_f)} \; [ e^{\beta \omega_{f g}} - 1 ]
      \; (2 \pi)^{-3} \! \int \! d^3p \li f \vert
      \schl A^b_\nu (\vc p) \vert g \re \li g \vert
      \schl A^a_\mu (\vc k) \vert f \re \cr
  & = \int \! dx \; {1 \over x - i \omega_n} \;\;
      {\cal A}^{a b}_{\mu \nu} (x, \vc k) \cr } 
$$ \nz
The relation between the response function \nz
and the thermal Green is now obvious: \nz \vskip -1.4cm
$$ \hskip 6.9cm \schl \chi^{a b}_{\mu \nu} ( \omega ,
   \vc k) \; = \; \schl G^{a b}_{\mu \nu} ( \; i \omega_n
   \longrightarrow \omega + i \epsilon \; , \, \vc k) $$
\ende

\nz {\fiverm ( hschulz@itp.uni-hannover.de ) }
\end