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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. – Pricklebush Tickletush 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
up vote 5 down vote accepted

There are no forbidden trajectories in Feynman' path integral approach to quantum mechanics. That's really the main point of the approach – we have to sum over all histories of the system. Every trajectory whose degrees of freedom (coordinates) obey the local constraints that should be obeyed has to be summed over.

We certainly don't exclude non-differentiable trajectories – the differentiable trajectories are measure-zero and their contribution to the path integral may, in fact, be completely neglected! The dominance of the non-differentiable trajectories is how Feynman's path integral obeys the uncertainty principle. This also means that in the language of the history of point-like particles, we don't forbid superluminal trajectories. In fact, the trajectories that contribute the bulk (effectively all) to the path integral are superluminal almost everywhere.

We don't exclude trajectories in which particles-that-will-become-fermions are overlapping, either.

The special relativity postulates including the speed-of-light limit on the velocity, differentiability of the expectation values as a function of time, and the Pauli exclusion principle all arise from the interference of many trajectories that contribute to Feynman's path integral. Physics resembles classical physics in the classical limit because the trajectories near the classical solution constructively interfere. Physics restores the limit on the velocity because of a cancellation between the trajectories of particles and antiparticles, if we choose their coordinates as the primary degrees of freedom. And the Pauli exclusion principle arises from the interference of a trajectory and another one in which the particles are permuted.

There could be confusion: we are talking about "classically forbidden trajectories". They're trajectories that would be forbidden in classical physics. But in quantum physics – and Feynman's approach defines quantum physics, not classical physics – they are allowed although they may end up having an exponentially small probability. That's also why e.g. the tunneling effect exists in quantum mechanics. In this effect, a particle emerges on the opposite side of a wall that would be impenetrable in classical physics. Quantum mechanically, the probability amplitude for penetration contains contributions from classically forbidden but quantum allowed trajectories going through the wall.

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