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What prevents a particle travelling at a speed infinitesimally smaller than that of $c$ tunnelling a slightly further distance than it would have otherwise and thus have travelled faster than the speed of light?

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They can, however you can't use them to transfer information faster than light so no rules are broken.

(in simple terms they can go through the tunnel barrier faster than light but not be detectable until they have completely emerged on the other side, which slows the process down to less than c)

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