The self-intersecting paths are not included in the Feynman path integral (FPI) approach because by definition a curve is defined as $\gamma:\mathbb{R}\rightarrow \mathbb{R}^3$ and since we take the independent variable of $\gamma$ itself (which is simply time-coordinate) in the definition of propagator $\int\mathcal{D}x(t)e^{iS[x(t)]}$ we don't include self-intersecting curve in the propagator. Another way of saying the same thing is: by including self-intersecting curve we are providing Close-timelike-curve to seep in, in our calculation which might be problematic.
So my question is does there exist any toy model which takes these curves into consideration and do the further calculation using resulting propagator to show the implication of including self-intersecting paths?
If it is not done how should I proceed to do this calculation?
Edit: When I say the path is not self-intersecting I refer to the corresponding path in M(1,3) which is what is usually meant when the word "path" is used in FPI approach or to be a bit sloppy here one can call them the worldline