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

Chet Marone
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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
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    @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
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    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: http://en.wikipedia.org/wiki/Velo%E2%80%93Zwanziger_problem – Chet Marone Jul 23 '11 at 17:07
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    I would advice OP to try to read works by Massimo Porrati and his collaborators, e.g., http://arxiv.org/abs/0906.1432 – Qmechanic Jul 24 '11 at 19:23
  • @Qmechanic: That is an interesting paper indeed. – BebopButUnsteady Jul 25 '11 at 01:40
  • Some of the complications are covered in this answer: http://physics.stackexchange.com/questions/14932/why-do-we-not-have-spin-greater-than-2 – Ron Maimon Oct 08 '11 at 04:25
  • @Marek: I have heard that there is a proof that interactions involving spin higher than 1 are all generically non-renormalizable in the abscence of supersymmetry, but I'm not sure of a reference for it (or even if this is an actual result, rather than folklore) – Zo the Relativist Nov 21 '11 at 22:34

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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.

Arnold Neumaier
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  • http://ncatlab.org/nlab/show/Velo-Zwanziger+problem – Urs Schreiber Sep 10 '13 at 17:07
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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 09:51
  • 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 09:59
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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

Marcel
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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.

Abhimanyu Pallavi Sudhir
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Vladimir Kalitvianski
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