# Stationary phase approximation with a non-positive definite fluctuation operator

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?

• – Qmechanic Mar 4 at 17:45