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The nonrelativistic Schrodinger field allows spin independent of statistics, so that you can imagine a nonrelativistic Schrodinger scalar field with Fermionic statistics, or a Schrodinger spinor field with Bose statistics. These models are mathematically consistent, but they are not the nonrelativistic limits of any consistent relativistic fields.

But this doesn't mean that these fields with wrong spin/statistics are realized as effective fields at long-distances in any physical system. Is there an argument that starting from relativistic fields that obey spin statistics, which reduce at low velocities to nonrelativistic fields which obey spin statistics, whenever you make a system with long distance rotational invariance (so that the effective spin of particles makes sense), and translational invariance (so that it's a normal field theory with scattering states), then all the long-distance composite effective fields obey spin/statistics? Is there any nontrivial statement which is true which is spin-statistics in the nonrelativistic context?

There are papers by Berry and collaborators which describe how the nonrelativistic spin/statistics is supposedly natural using a peculiar structure that relates rotations are related to exchanges. I didn't buy these arguments at all, because I couldn't see the point of proving spin/statistics in situations where it clearly isn't true. But perhaps there is a nontrivial correct statement.

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While writing the question, I realized there is an example of a system of this sort. Consider a gas of free neutrons at very low energies, where the spin and orbit are decoupled, in a strong constant magnetic field. The low energy dynamics is for the low-energy spin configuration, and it is the ordinary Schrodinger dynamics. So the resulting low-energy action is exactly the fermionic scalar Schrodinger equation. It is translationally invariant (the fundamental configuration is translationally invariant), accidentally rotationally invariant (at low energy the spin is completely decoupled from the orbit), and has the wrong connection between spin and statistics (it's a fermionic scalar). So the answer is no.

I don't know if there is a nontrivial statement along the lines of spin/statistics which is ever nonrelativistically true.

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  • $\begingroup$ I like the question, but what you answered here seems just to be an example of what you wrote in the first paragraph of your question, which is something you already knew: that you can have fermionic scalars. How does this example answer your question? can you explain please. $\endgroup$
    – kηives
    Commented Jun 26, 2012 at 17:00
  • $\begingroup$ @kηives: This is an example of Fermionic scalars that emerge physically--- it's a realization of the mathematical system. I wanted to make sure that there was an actual physical example, not just mathematical examples, and this provides it. Perhaps there is no bosonic spinor, but I think you can make theories whose low energy effective theory is a bosonic spinor. $\endgroup$
    – Ron Maimon
    Commented Jun 26, 2012 at 19:33
  • $\begingroup$ okay, I think I get it, and so you were asking: can we make an argument that says "if we start with a relativistic field theory that obeys spin-statistics, and move to the low-energy limit while maintaining long range rotational and translation invariance, that theory must obey spin-statistics." But, low energy neutron gas is the low energy limit of some relativistic theory that itself obeys spin-statistics, yet as an effective field theory, is long-range rotationaly and translationaly invariant, yet fails to obey spin-statistics. Is this how you answer your own question? $\endgroup$
    – kηives
    Commented Jun 26, 2012 at 19:57
  • $\begingroup$ @kηives: Yes. I am confused, because there should be something of the sort which is true, considering the condensed matter cases where exchange and rotation are related, like the anyon gas you can make in the 3d Witten-Chern-Simons theory. $\endgroup$
    – Ron Maimon
    Commented Jun 26, 2012 at 20:28
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    $\begingroup$ @drake: The neutrons' equation of motion only allows one spin direction, where the magnetic moment is aligned by the magnetic field) at energies less than the magnetic splitting. The low energy limit is one component obeying the Schrodinger equation. These neutrons no longer feel the magnetic field--- this is a one component projection of the Pauli equation which reduces to the Schrodinger equation. The effective rotation at energies lower than the B-splitting is rotating the neutron spatial wavefunction and not rotating the spin, and this is an effective scalar. $\endgroup$
    – Ron Maimon
    Commented Jul 27, 2012 at 2:36

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