I've been looking at the 1+1 dimensional Dirac equation recently and am wondering if the Heisenberg uncertainty principle would allow application of said model to spin-1/2 fermions in 3+1 dimensions if the system were required to be localized within 3+1 D spacetime?

My reasoning for this doubt is as follows: If the uncertainty principle applies to 3 spatial dimensions then setting, say, the $p^1$ and $p^2$ momentum components equal to zero would reduce their corresponding uncertainty to 0. This then seems to require that the positions $x^1$ and $x^2$ be completely unknowable meaning that one could not confine the 1+1 D system within any finite region inside 3+1 D spacetime... This appears to contradict the localizability assumption.

Yet, there are clear situations where 2+1 D or 1+1 D models apply and make successful predictions... So I am wondering if I have made a mistake somewhere and, if so, where?


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