I wish to evaluate the following path integral - to find a conditional probability.

$$P(x(T) | x(0)) = \int Dx(t) \exp(S[x]) = \int Dx(t) \exp \Big(\int -\frac{1}{2}m(x)\ddot{x} + b(x)\dot{x} + V(x) \ dt\Big)$$ With the condition that $m(x) > 0$. I use the saddle point approximation:

$$ P(x(T) | x(0)) \approx e^{S[x_{cl}]} \text{Det}\Big[\frac{\delta S}{\delta x(t) \delta x(s)}\vert_{x_{cl}} \Big]^{-\frac{1}{2}}$$ Where $x_{cl}$ extremises the functional $S[x]$. The fluctuation operator is; $$ \frac{\delta S}{\delta x(t) \delta x(s)}\Bigg|_{x_{cl}} = \frac{d}{dt}m(x_{cl})\frac{d}{dt} + m'(x_{cl}) \ddot{x}_{cl}+ \frac{1}{2}m'' \dot{x}_{cl} + V''(x_{cl}).$$ My issue now is that this differential operator is not positive definite. So I am unsure of how to proceed with evaluating the fluctuations? Ignoring this possible issue and just applying Gelfand-Yaglom theorem results in fluctuations that sometimes become imaginary, which cannot work with a conditional probability! Would anyone be able to point me in the right direction? Would performing a wick rotation, making the operator positive definite, then transforming back work?


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