Is there is a reason for Pauli's Exclusion Principle?

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I think that while these "explanations" are all dancing around the same pole, they aren't created equal. I think the meat is in the fact that nature has a local Lorentz symmetry, so we expect to be able to decompose things into representations of the group $SO(3,1)$. It's a mathematical fact that this group (or it's algebra, rather) has integer and half-integer representations.

Once you have this structure, then a few meagre assumptions about causality and unitarity lead to the Spin-statistics theorem. In order to understand the proof you'll need to first dig deeper into the representations of the Lorentz group, and how they label single-particle states.

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Cheshire Cat
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Cheshire Cat

Updated on August 01, 2022

Comments

  • Cheshire Cat
    Cheshire Cat over 1 year

    As a starting quantum physicist I am very interested in reasons why does Pauli's Exclusion Principle works. I mean standard explanations are not quite satisfying. Of course we can say that is because of fermionic nature of electrons - but it is just the different way to say the same thing. We can say that we need to antisymmetrize the quantum wavefunction for many electrons - well, another different way to say the same. We can say that it is because spin 1/2 of electron - but the hell, fermions has by the definition half-integral spin so it doesn't explain anything. Is the Exclusion Principle something deeper, for example in Dirac's Equation, like spin of the electron? I think it would be satisfying.

    • FGSUZ
      FGSUZ almost 4 years
      Pauli's principle is less restrictive than symmetrization principle, they are not the same
  • Selene Routley
    Selene Routley almost 10 years
    @CheshireCat Perhaps add that the last step is that the spin-statistics theorem shows that for half integer spin representations the quantum state for two particles with quantum numbers $\vec{x}$ and $\vec{y}$ (I include "position" in the quantum number vector) is antisymmetric wrt swap of arguments $\psi(\vec{x}, \vec{y}) = -\psi(\vec{x}, \vec{y})$ so that now if two particles have the same quantum numbers $\psi(\vec{x}, \vec{x}) = - \psi(\vec{x}, \vec{x})$. A further piece of trivia which I like to dwell on here: when we represent the algebra by half integer representations, we're ...
  • Selene Routley
    Selene Routley almost 10 years
    ...actually representing the double cover $PSL(2,\mathbb{C})$ of the Lorentz group $SO(3,1)$, so you could say, with a slight strech, that the Dirac belt trick "proves" there are only bosons and fermions in the world.
  • Deschele Schilder
    Deschele Schilder almost 6 years
    All the answers explain Pauli's principle by mathematics, which is fine with me, but isn't there physics involved that lies behind the principle? You can say the math is the physics, but can't it be that on the physics of the principle (i don't know how the principle was found: by using math or by experiment) a mathematical net is thrown, making the "flesh" less visible? I know a lot of physical stuff is predicted by math before the stuff is found, but that doesn't mean the stuff is a mathematical thing. I suppose it is how you want to look at the stuff. To know how , for example,
  • Deschele Schilder
    Deschele Schilder almost 6 years
    two electrons (not too far apart of course) can't be in the same state is for some (the most, I guess) physicists mathematical knowledge, while others want to know the physical mechanism behind this principle.
  • Steven Sagona
    Steven Sagona almost 4 years
    Can we get a translation for those of us who aren't already experts in the subject?
  • Dan
    Dan over 1 year
    @StevenSagona It's a tough call. There are many gnarly math concepts. For example, in lionelbrits answer, in the link to spin-stats, if you scroll down to "Relation to representation theory of the Lorentz group" you will see that the Lorentz group has not finite-dimension unitary non-trivial representations. So seeing that, and since angular momentum is usually dealt with as a derivative, it is quite a job to get to the answer.