Influence Functional for Coupling to Harmonic Oscillators

We start with the coupling of the system to a single harmonic oscillator,

$$H_B + H_{SB}[q] \equiv H_B(t)\equiv \frac{p^2}{2M}+ \frac{1}{2}M\Omega^2 x^2 + g(t) x,$$ (171)

where without loss of generality we set $f[q(t)]=g(t)$.