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Difference between the Rarita-Scwinger field and the gravitino?

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The Rarita-Scwhinger field is a spin 3/2 field $\psi_{\mu}^{\alpha} $ but this looks a lot like the gravitons found in string theories, for example it looks like the gravitinos in Ibanez's and Uranga's book of string theory. Therefore why we need two names for the same thing (if the same)? Is its supersymmetry global or local?
asked Jan 4, 2015 in Theoretical Physics by Outlander (95 points) [ no revision ]
recategorized Jan 6, 2015 by dimension10

1 Answer

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The graviton we find in string theory and in supergravity in a specific limit is the Rarita-Schwinger field. Specifically in order to get supergravity (let us forget string theory for now) we need to gauge the supersymmetry transformation parameters, i.e. $\epsilon_{\alpha} \to \epsilon_{\alpha}(x)$. Then the associated gauge field is the vector-spinor $\psi_{\alpha \, \mu}$ where $\alpha$ is the internal symmetry index and $\mu$ is the Lorentz index. Now gauging this parameter, which is a constant spinor, we get supergravity which is an interacting theory. The Rarita-Schwinger field is nothing more than the non-interacting or free limit. At this limit the various fields of the theory (which depend a lot on the supersymmetry(ies)) do not interact and we can consider them case by case. Then the vector-spinor field $\psi_{\alpha \, \mu}$ is the Rarita-Schwinger field which transforms in the $$ \Big( (1/2, 0) \oplus (0,1/2)\otimes (1/2,1/2) \Big) $$ of the Lorentz group. Just note that general supergravity theories restrict a lot the type of spinor $\psi_{\alpha \, \mu}$ is normally but for a free theory and for any dimension $d$ one can use a complex spinor with $2^{d/2}$ components. A nice review I found was this one where you will find much more (and accurate) info. 

answered Jan 4, 2015 by conformal_gk (3,535 points) [ no revision ]

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