Abstract
The two-point function for spinors on maximally symmetric four-dimensional spaces is obtained in terms of intrinsic geometric objects. In the massless case, Weyl spinors in anti de Sitter space can not satisfy boundary conditions appropriate to the supersymmetric models. This is because these boundary conditions break chiral symmetry, which is proven by showing that the “order parameter”\(\left\langle {\bar \psi \psi } \right\rangle \) for a massless Dirac spinor is nonzero. We also give a coordinate-independent formula for the bispinor\(S(x)\bar S(x')\) introduced by Breitenlohner and Freedman [1], and establish the precise connection between our results and those of Burges, Davis, Freedman and Gibbons [2].
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These are the same γ-matrices given by [5] in the long footnote on p. 221 except for a factor of −i. That factor is necessary since [5] assumes {γ a , γ b }=−2g ab and [2] assumes {γ a , γ b }=2g ab
The quickest way to establish that the solutions are the same is to show that they have the same asymptotic fall off asZ→∞, and the same singular behavior atZ=1. Since one may show that they both satisfy the same second-order ordinary differential (3.6), they are therefore equal
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Freedman, D. Z.: A fond farewell to anti de Sitter space. In: Proceedings of the Cambridge supersymmetry workshop 1985. Gibbons, Hawking, Townsend (eds.) Cambridge: Cambridge University Press 1986
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Communicated by A. Jaffe
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Allen, B., Lütken, C.A. Spinor two-point functions in maximally symmetric spaces. Commun.Math. Phys. 106, 201–210 (1986). https://doi.org/10.1007/BF01454972
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DOI: https://doi.org/10.1007/BF01454972