May be $f(\vec{x}), \vec{g}(\vec{x})$ an arbitrary functions dependent on the coordinates $\vec{x}=(x,y,z)^T$. Defining the following function dependent on a 3-dimensional curve $\vec{\gamma(t)}$ parametrized by $t \in [0,1]$:
$S := \int_{R^3}\int_{R^3} (f(\vec{x})exp(\int_0^1 \vec{g}(\vec{\gamma(t)}) \vec{\gamma'(t)}dt)f(\vec{x'}))d^3x d^3x'$.
It was defined $\vec{x}=\vec{\gamma(0)}, \vec{x'}=\vec{\gamma(1)}$ and it was integrated over all possible endpoints of the curve $\vec{\gamma(t)}$. How I can compute
$I := \int Sd[\vec{\gamma(t)}]$ ($[\vec{\gamma(t)}]$ denotes that every possible path with endpoints $\vec{x}, \vec{x'}$ that is connected must be integrated up)?
This Integration is similar to Feynman's path-integral over all possible paths in Quantum field theory. The exponential I can expand into a taylor series, but how can I evaluate the products $\int \vec{\gamma'(t_1)} \otimes ... \otimes \vec{\gamma'(t_n)}d[\vec{\gamma(t)}]$ where every $\vec{\gamma(t_i)},i \in \{ 1,...,n \}$ has the same endpoints (there occur both open and closed paths)?
Every help would be greatly appreciated.