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

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    $\begingroup$ Nothing in the definition says that $\gamma(t)$ has to be one-to-one, so self intersections are allowed --- and in fact sometimes mandated (for example when the target space is the circle $S^1$ instead of ${\mathbb R}^3)$. $\endgroup$ – mike stone Mar 12 at 16:36
  • $\begingroup$ @mikestone but the corresponding wordline of the paths even when $S^1$ is considered will never intersect for ex. if the particle goes at a constant speed on $S^1$ the corresponding wordline will be a helix. $\endgroup$ – aitfel Mar 12 at 16:51
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    $\begingroup$ Two paths intersect when they are at the same place at different times. That's what intersecting paths means. By your definition no path can ever intersect. $\endgroup$ – mike stone Mar 12 at 17:04
  • $\begingroup$ @mikestone ooh! I see. So does that mean FPI never includes paths(information), even for a very short duration, going backward in time. $\endgroup$ – aitfel Mar 12 at 17:14
  • $\begingroup$ That's true for the non-relativistic Feynman integral. The Euclidean relativistic many body PI is a rather different beast as one maps $\gamma:{\mathbb R}\to {\mathbb R}^4$ where the ${\mathbb R}^4$ includes Euclidean time $\tau$ and the $s$ in $\gamma(s)$ has nothing to do with time. $\endgroup$ – mike stone Mar 13 at 11:50

Keldysh formalism for treating non-equilibrium systems is a widely known case that uses a closed time contour, and which also exists in path integral formulation, e.g., see here.

For the sake of justice it is worth mentioning that a version of this approach is also known as Kadanoff-Baym ansatz.

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