Applications: Linear Coupling, Damped Harmonic Oscillator

$\displaystyle H$	$\displaystyle =$	$\displaystyle H_S[q] + H_B[x] + H_{SB}[xq]= H_S[q] + \sum_{\alpha=1}^N \frac{p_\alpha^2}{2M_\alpha}+ \frac{1}{2}M_\alpha\Omega_\alpha x^2 + f_\alpha[q] x_\alpha$
$\displaystyle {\cal F}[q_{t'},q'_{t'}]$	$\displaystyle =$	$\displaystyle \exp \left\{-\Phi[q_{t'},q'_{t'}] \right\}$ Influence Functional
$\displaystyle \Phi[q_{t'},q'_{t'}]$	$\displaystyle =$	$\displaystyle \sum_{\alpha=1}^N\int_{0}^{t}dt'\int_{0}^{t'}ds \left\{ f_\alpha[... ...ft\{ S_\alpha(t'-s) f_\alpha[q_{s}] - S_\alpha^*(t'-s) f_\alpha[q'_{s}]\right\}$
$\displaystyle S_\alpha(\tau)$	$\displaystyle =$	$\displaystyle \frac{1}{2M_\alpha\Omega_\alpha} \left( \coth \frac{\beta\Omega_\alpha}{2} \cos \Omega_\alpha \tau - i \sin \Omega_\alpha \tau\right).$	(207)