# Does a non-relativistic proof of spin-statistics theorem exist?

There are lots of questions here related to the spin-statistics theorem, though none of them answer this question directly.

I had the notion that one can only prove the theorem on relativistic grounds and for example the Wikipedia page on the subject list Lorentz invariance as one of the assumptions needed to prove the theorem.

I got confused when I was reading Preskill's Lectures notes on quantum computation. He said on the subject:

All that is essential for a spin-statistics connection to hold is the existence of antiparticles. Special relativity is not an essential ingredient.

Then he proceeds to give an argument about why this is so. The argument is somewhat convincing, albeit a little hand-wavy.

Looking into the literature, there seem to be some back and forth arguments about this. For example see here, here, here, here, here, and here. So I have an idea as to where to start with this, however, I thought maybe I'm missing something obvious, and can save myself some time by asking here first.

Does one need relativity to proof the spin-statistics theorem? I'd also appreciate yes or no comments with reference to maybe later work on the subject that I might have missed.

• Does it matter that antiparticles are a prediction of special relativity (via the Dirac equation)?
– rob
Aug 11, 2020 at 14:33
• I guess it does. But the question then is if the converse is true. Whether the existence of antiparticles implies special relativity. Aug 11, 2020 at 14:40
• I have to say I'm at a bit of a loss as to why the existence of antiparticles requires SR. It's easy enough :-) to run experiments showing positrons whacking electrons, for example. Now, as to the how and the underlying theory/rules behind Feynman Diagrams may depend on SR, but that seems to be a separate question. Aug 11, 2020 at 14:50

You can consider the case of "fat" solitons like skyrmions in non-relativistic solid states systems. There are skyrmions and anti-skyrmions and if you quantize them via a path integral in which you give a minus sign weight for a path in which a skyrmion is rotated by 360 degrees, you can give a homotopy argument involving the antiskyrmion so show that you must also give a minus sign when a skymion is exchanged with a skyrmion. Exactly what logical input is needed to precisely define "fat" is perhaps unlear. To me it means that there is some geometrical notion of rotating the thing of interest, and not just an internal (mathamtical) spin degree of freedom. The source of these ideas is surely Finkelstein and Rubenstein Connection between Spin, Statistics, and Kinks Journal of Mathematical Physics 9, 1762 (1968).

• Would you say that one can in general use the braiding properties of the particles as to say something about their angular momentum? Or does this only apply for "fat" excitations, or excitations with strings? This seems like a nice way of approaching the problem. Aug 11, 2020 at 16:17

The spin-statistics theorem is derived by evidencing the terrible consequences of the wrong statistics. At least three criteria can be used: Lorentz invariance of the S-matrix, stability, causality.

Lorentz invariance of the S-matrix
The S-matrix is constructed from Lorentz covariant fields, however the time-ordered product, to be Lorentz invariant, requires anticommutation relations for half-integer spin particles.

Stability
The total energy of a system should be bounded from below. For free particles, if the wrong statistics are used, antiparticles will have arbitrarily negative energies. It would mean that crazy things like $$p^+ \to p^+ e^+ e^-$$ would not be forbidden.

Causality
The operators corresponding to observables should commute at spacelike separation, otherwise they would influence each other. If so, one could use them to communicate faster than the speed of light. However this is a weaker requirement and can only prove that integer spin particles commute, but not that half-integer spin particles anticommute.

Out of the above three criteria, while Lorentz invariance of the S-matrix and causality are relativistic, stability is non-relativistic.

Requiring stability is a necessary and sufficient condition for the spin-statistics theorem. This is important for instance in condensed matter systems in which Lorentz invariance is irrelevant.

• If you say stability is sufficient, does that include the existence of antiparticles? I'd have thought that that is (something lik) the CPT theorem, which again is relativistic. In other words, a nonrelativistic theory of two anticommuting scalars would be fine, or wouldn't it? Aug 12, 2020 at 10:43
• An interesting claim. Would you have some reference I can read about this as a necessary and sufficient condition in more detail? Aug 12, 2020 at 12:37
• @A. Jahin. A reference is "Schwartz, Quantum field theory and the Standard model". Aug 12, 2020 at 14:41
• @Toffomat. The topic is subtle. It is assumed that antiparticles exist, if so, stability requires that spinors anticommute. However antiparticles do exist! Aug 12, 2020 at 15:01
• @MicheleGrosso We can write down a non-relativistic quantum field theory of spin-zero fermions whose Hamiltonian is bounded below, and that theory is perfectly healthy, so stability can't be sufficient. Can you clarify the context of the statement that stability is sufficient? Aug 12, 2020 at 16:41