I'm somewhat curious about how the Pauli Exclusion Principle functions when relativistic time becomes a significant factor. Just to clear things up, my (possibly poor) understanding goes like this.

The PEP basically states that no two fermions can be in the same state at the same time.

Relativity states that in certain conditions, time is no longer really objective but can be quite subjective- passing at different rates depending on who is the observer, etc.

I'm curious, therefore, how this notion of "time" relates to the "time" required by PEP. For example, would it be possible from the perspective of one observer that two fermions could be in the same state at the same time from their perspective, but not from another perspective? And if the fermions in question are distinct particles, then logically they should be experiencing time at a different rate. So what would occur if fermion A tries to occupy the same state as fermion B in a position in time that from A's perspective is not the same position in time that B occupied it, but from B's perspective, it's still that position in time?

In short, the PEP seems to require that there's some sort of objective timekeeper that rules who can be in what states and when, but relativity seems to indicate that no such thing exists. How are these two ideas of time reconciled?

  • $\begingroup$ Is there ever a conflict about the time at a single location? The issue of an observer is irrelevant as the two fermions are also at the same location, so the observer is observing them both in the same place. $\endgroup$ – Jiminion Apr 7 '15 at 21:14
  • $\begingroup$ They could be changing their location. $\endgroup$ – DeadMG Apr 7 '15 at 21:16
  • $\begingroup$ Just one thing, when you think about those things you have to be careful when you say state: state means "$|n l m_l m_s\rangle$" and not "position". There is no ambiguity in the "$|n l m_l m_s\rangle$"-state, since the electron doesn't, e.g., change it's energy state just because the observer is moving faster. $\endgroup$ – user42076 Apr 7 '15 at 21:57
  • $\begingroup$ QM + SRT -> need QFT. In a QFT, you get a Hamiltonian by slicing spacetime into equal-time slices and then one quantizes on these equal time slices by imposing equal-time (anti-)commutation relations. The concept of what equal time is observer-dependent, so one can quantize on a different slicing. The Pauli exclusion principle is inherent in the Hilbert spaces from quantizing a fermionic theory. Is there a unique isomorphism between the two Hilbert spaces? Yes, one just has to perform the appropriate Lorentz transformation as Lorentz covariance is inherent in the construction. $\endgroup$ – physicus Apr 7 '15 at 22:58

The reason why there is no contradiction is that the PEP states, essentially, that no two particles can be at the same place at the same time. Without going into difficult QFT concepts mentioned above, a simple answer to your question would be that "at the same place at the same time" is equivalent to saying "at the same point in space-time." While simultaneity may vary between observers, there is no ambiguity among observers as to whether two events occupy the same point in space-time. Leaving out the "at the same place" portion of the PEP would be, in a relativistic sense, like saying no two particles could be at the same place at different times, which is clearly false.

If you want a more precise answer to this, you of course need to work with QFT. Here, the PEP is inherent with quantized Fermionic fields, and the anticommutation relation implies that the fields of two fermions cannot take on the same value at all points in space-time. In this formalism, the space-time framework is inherent, and there is no trouble with contradictions.


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