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Here is a picture from Padmanabhan's book on quantum field theory.

enter image description here

Also described in the first part of this lecture within $17-25$ minutes.

He says that a relativistic particle can follow a path like the one shown below in the spacetime diagram (in contradiction to the answer here). At this point, he doesn't use any funny quantum mechanics. After saying that such paths are possible in relativity, he goes on saying that if we want to construct the path integral of a relativistic particle, such paths must be included.

Now my concern is that if special relativity really forbids such backward moving paths in time, why should we consider those paths in the path integral (if we want to combine special relativity with quantum mechanics)? So my question is why does he consider such paths if relativity forbids those?

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Professor Padmanabhan is in OP's linked video talking about virtual paths in the path integral formalism. They are needed to construct the full amplitude. They are not actual paths and do not necessarily respect causality.

See also e.g. this related Phys.SE post.

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