# What are the mathematical problems in introducing Spin 3/2 fermions?

Can the physics complications of introducing spin 3/2 Rarita-Schwinger matter be put in geometric (or other) terms readily accessible to a mathematician?

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What complications? Could you elaborate? Spin 3/2 fermions are completely standard fermions and also occurring naturally (some nucleons, gravitinos, etc.). Rarita-Schwinger equation is standard equation on par with Proca equation, Maxwell equations or Dirac equation. I never heard of spin 3/2 particles being special in any regard (perhaps except being encountered less often in standard physics). – Marek Jul 23 '11 at 15:56
@Marek, I suppose this refers to higher spin theories not being renormalizable when interactions are introduced. As such, it is the renormalization group that is the issue. A relatively mathematical reference that discusses the renormalization group for higher spin would perhaps be sufficient Answer (but I do not know of one straight off). – Peter Morgan Jul 23 '11 at 16:37
@Peter: I never heard about higher spins not being generically renormalizable. Are you referring to spin 2 gravity (this is non-renormalizable because of the form of the GR Lagriangian, not because of spin; at least AFAIK) or something else? – Marek Jul 23 '11 at 16:40
Feel free to interpret the question as a request for mathematical understanding of conditions whereby such fields are unproblematic. However, there certainly appears to be a sense within physics that these fields raise problems different from spinors in a Dirac equation: en.wikipedia.org/wiki/Velo%E2%80%93Zwanziger_problem – Chet Marone Jul 23 '11 at 17:07
I would advice OP to try to read works by Massimo Porrati and his collaborators, e.g., arxiv.org/abs/0906.1432 – Qmechanic Jul 24 '11 at 19:23

Free spin 3/2 fields cause no problems; see Weinberg's QFT book, Volume 1.

The problem with elementary spin 3/2 fields is the difficulty of accounting for the interaction with the electromagnetic field. The Rarita-Schwinger field equations with the standard minimal coupling via the covariant derivative violate causality, as they allow superluminal signalling - already on the single particle level.

Nonrenormalizability is another issue, but could be handled in the sense of effective field theories if the other defect were absent.

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@UrsSchreiber Yes, but this is only the external field problem. Further problems appear when one tries to define a full QFT. – Arnold Neumaier Sep 12 '13 at 9:51
I think whatever statement you have in mind, for a reply it would be nice to supply a minimum of substantiation. A pointer to a reference for instance. If not an actual discussion of some details here. – Urs Schreiber Sep 12 '13 at 9:59
@UrsSchreiber: Well, I no longer post new answers this forum, but I still get email reports about comments; so I sometimes answer these - briefly. If you want more info, write me an email. – Arnold Neumaier Sep 12 '13 at 10:24

You may be interested in following article: Thomas-Paul Hack, Mathias Makedonski "A No-Go Theorem for the Consistent Quantization of the Massive Gravitino on Robertson-Walker Spacetimes and Arbitrary Spin 3/2 Fields on General Curved Spacetimes" http://arxiv.org/abs/1106.6327

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Physics complications of "introducing" spin 3/2 matter are the same as for spin 1/2 and spin 0 - the initial approximation in the corresponding interaction theory is physically wrong and calculations give too big (= just wrong) perturbative corrections. It is a complete failure of physics description and it cannot be casted in "geometric terms". Most people, however, does not see it.

Edit for downvoters: While in case of Rarita-Schwinger equation the solution violates even causality and it cannot be repaired with the constant renormalizations, this feature is still not considered as a failure of coupling. Indeed, we cannot be wrong. It is nature who is wrong, especially at short distances.

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If you considered only posting the relevant part of this about R-S equation and not the omnipresent blabbering about QFT not working at all, you might receive no downvotes and many upvotes. Just saying... – Marek Jul 24 '11 at 9:31
@Marek: I said something I really believe to be relevant and omnipresent. You just do not want to recognize the failures to be failures. In QFT and even in CED we couple coupled already things. Can it be right? Think of it. – Vladimir Kalitvianski Jul 24 '11 at 10:25
For the last time, I am not going to argue with you as this is pointless, so this is the last comment here. I am the first to admit that QFT is not mathematically complete; it will take a lot more time to formalize the theory. On the other hand, as a physical theory it is perfectly adequate as it predicts everything that can be predicted about the microscopic world and has had inumerable applications in the past 50 years most of which are the most sophisticated gadgets humankind created. So next time try to think of this before you utter your nonsense again... – Marek Jul 24 '11 at 10:50
@marek you have to bear in mind that Vladimir is 51 and you're 25 and unfortunately people's brains develop inertia as they get older. No doubt you're learning new stuff daily, whereas Vladimir goes over the same stuff again and again... and again. – Larry Harson Jul 24 '11 at 14:58
@user2146: Vladimir is not 51 but nearly 53, i.e., he is much more experienced than Marek. He's seen a lot of efforts in overcoming conceptual difficulties in his life and he has found the ultimate explanation why renormalizations may work. So bear in mind he is teaching you something essential. – Vladimir Kalitvianski Jul 24 '11 at 19:10