The Influence Functional

Let us assume that we can write

$$H_{SB}= H_{SB}[q]=f(\hat{q})\hat{X}$$ (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

$${\cal F}[q(t'),q'(t')] \equiv \int dx_0 dx_0' dx \langle x_0 \vert\rho_B\vert x_0'\rangle$$

$$\times \int_{x_0}^{x} {\cal{D}}x \exp\left[ i \left(S[x]+S[xq]\right)\ri... ... \int_{x_0'}^{x} {\cal{D}^*}x' \exp\left[ -i \left(S[x']+S[x'q']\right) \right]$$

$$= \int dx_0 dx_0' dx \langle x_0 \vert\rho_B\vert x_0' \rangle \lan... ...x \vert U_B[q] x_0 \rangle \left[\langle x \vert U_B[q'] x'_0 \rangle \right]^*$$

$$= {\rm Tr_B} \left(\rho_B U_B^{\dagger}[q'] U_B[q] \right),$$ (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 ${\cal F}[q(t'),q'(t')]$,

• $q(t')=q'(t')\leadsto {\cal F}[q(t'),q'(t')] =1$.
• $\vert{\cal F}[q(t'),q'(t')]\vert \le 1$.

The

$$\fbox{$ \begin{array}{rcl} \displaystyle & &\mbox{\rm Operator Fo... ...equiv {\rm Tr_B} \left(\rho_B U_B^{\dagger}[q'] U_B[q] \right), \end{array}$\ }$$ (170)

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