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The Influence Functional

Let us assume that we can write
$\displaystyle H_{SB}= H_{SB}[q]=f(\hat{q})\hat{X}$     (168)

with some given bath operator $ \hat{X}$ and some given function $ f(\hat{q})$ of the system coordinate $ \hat{q}$. The influence functional can then be written as
$\displaystyle {\cal F}[q(t'),q'(t')]$ $\displaystyle \equiv$ $\displaystyle \int dx_0 dx_0' dx \langle x_0 \vert\rho_B\vert x_0'\rangle$  
  $\displaystyle \times$ $\displaystyle \int_{x_0}^{x} {\cal{D}}x \exp\left[ i \left(S[x]+S[xq]\right)\ri...
...
\int_{x_0'}^{x} {\cal{D}^*}x'
\exp\left[ -i \left(S[x']+S[x'q']\right) \right]$  
  $\displaystyle =$ $\displaystyle \int dx_0 dx_0' dx \langle x_0 \vert\rho_B\vert x_0' \rangle
\lan...
...x \vert U_B[q] x_0 \rangle \left[\langle x \vert U_B[q'] x'_0 \rangle \right]^*$  
  $\displaystyle =$ $\displaystyle {\rm Tr_B} \left(\rho_B U_B^{\dagger}[q'] U_B[q] \right),$ (169)

where $ U_B[q]$ is the unitary time-evolution operator for the time-dependent Hamiltonian $ H_B + H_{SB}[q]$ with a given $ q(t'), 0\le t' \le t$. Note that $ q(t')$ and $ q'(t')$ are independent paths, they enter as `external' parameters into the influence functional which then in the final expression for $ \langle q\vert\rho(t)\vert q'\rangle$ is integrated over all paths $ q(t')$ and $ q'(t')$. This form is useful to recognise general properties of $ {\cal F}[q(t'),q'(t')]$, The
$\displaystyle \fbox{$ \begin{array}{rcl} \displaystyle
& &\mbox{\rm Operator Fo...
...equiv {\rm Tr_B} \left(\rho_B U_B^{\dagger}[q'] U_B[q] \right),
\end{array}$\ }$     (170)

is particularly useful for discussing the coupling to other baths (spin-baths, Fermi baths etc.)


next up previous contents index
Next: Influence Functional for Coupling Up: Feynman-Vernon Influence Functional Theories Previous: Double Path Integrals   Contents   Index
Tobias Brandes 2004-02-18