Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This chapter derives the conditions under which spin-lowering and spin-raising operators preserve these local boundary conditions on a three-sphere for fields of spin 0, 1/2, 1, 3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a three-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains.
The second part of this chapter studies the two-spinor form of the Rarita-Schwinger potentials subject to local boundary conditions compatible with BRST invariance and local supersymmetry. The Rarita-Schwinger field equations are studied in an arbitrary four-real-dimensional Riemannian background with boundary. Gauge transformations on the potentials are shown to be compatible with the field equations providing a set of second-order partial differential equations hold. An equivalent, first-order form of the compatibility conditions is also obtained. The boundary conditions do not restrict severely the choice of background four-geometries, as it happens in the case of Dirac’s potentials with reflective boundary conditions on field strengths. The recent construction by Penrose of secondary potentials which supplement the Rarita-Schwinger potentials is then extended from Ricci-flat space-times to arbitrary curved backgrounds. Remarkably, the traces of such secondary potentials are linearly related to the independent spinor fields appearing in the Rarita-Schwinger equations. The resulting set of equations for these secondary potentials is hence ontained.
KeywordsGauge Transformation Twistor Space Spinor Field Preservation Condition Gauge Freedom
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