# Forbidden trajectories in path integrals

In Feynman's path integral formulation we add all the possible trajectories of a particle to get the probability amplitude.

What are forbidden trajectories? Not differentiable?

Is this related to the forbidden states of Pauli exclusion?

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The trajectories in the PI formulation of standard quantum mechanics have to be continuous, but they don't have to be smooth or even differentiable i.e. they can zigzag quite happily. –  twistor59 Jun 22 '13 at 6:24
I'm not qualified to answer but I'll say two things. Differentiability isn't an issue -- in Feynman's thesis he constructed the path integral from continuous, non-differentiable trajectories. The "possible paths" are the paths which satisfy the constraints of the problem -- though these can be tricky to define. –  santa claus Jun 22 '13 at 6:28
For the last part of your question, the answer is no. But, for fermionic paths, you have to use Grassmann variables. –  Trimok Jun 22 '13 at 9:57
@Trimok So even if we have two electrons, we would have to consider the possibility when they are in the same place? –  jinawee Jun 22 '13 at 18:07
You cannot interpret Grassmann numbers as space variables, or something measurable. They are anti-commuting quantities. The interesting thing, is, while these quantities are not measurable, if you take an integral with "gaussian" exponential, you get a result, roughly $det A$, instead of $(det A)^{-1}$ for standard bosonic (commuting) quantities (which can be interpreted as positions). So Grassmann numbers are strange objects, but you can extract exploitable information. –  Trimok Jun 22 '13 at 18:25