In most introductory QFT treatments, it's stated early on (and without proof) that particles with even integral spin are always attractive, while those with odd integral spin can be repulsive; sometimes this is even cited as evidence that the graviton must be spin 2 (I think Feynman's Gravitation lectures were one such reference).(adsbygoogle = window.adsbygoogle || []).push({});

However, I've not been able to find an explanation (let alone a proof) of this anywhere; the closest I've found is this answer on another site to basically the same question: http://www.quora.com/Quantum-Field-...s-particles-of-even-integer-spin-only-attract

Now, that goes a long way to giving an answer; it also clarifies something I'd always been confused about -- namely, that even spin just means that *like* charges attract; unlike charges would repel (though in e.g. gravity all "charges" are positive).

However, there is still some hand-waving...I can swallow that the factor of i in the spin-zero case is required "for unitarity", but where I get lost is here:

Now, if you take as given that the spacelike components of the propagator must be positive, that follows...but, why must the spacelike components be positive? I'm guessing it's related to relavistic requirements, but I don't see how it could be Lorentz covariance alone, since the propagator is equally covariant with or without the minus sign. More generally, polarization tensors have nonzero spacelike components because they are transverse to the motion of the force-carrier. When you sum over them, you find that the numerator of the propagator must have positive spacelike components. Since the metric is diag(1,-1,-1,-1), this means that whenever you have an odd number of metrics, you need an extra minus sign to make the spacelike components positive.

An interesting thing happened here: the sign of the propagator was fixed by consistency with propagating external states, which have spacelike polarizations

So, can anyone help clear this up for me? Or, even better -- point me towards an early proof (or even reference) of this in the literature? I don't recall it being in Pauli's Spin/Statistics paper, and it's definitely not in his earlier Particles obeying Bose-Einstein statistics paper (which I just finished translating), and Google isn't being helpful (some terms tried: "even spin" "odd spin" attractive repulsive, "spin parity" attractive repulsive, "spacelike components" propagator)

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# Spin parity and attractive/repulsive forces

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