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arXiv · 2411.11078

Asymptotic Quantization of Palatini Action

A. P. Balachandran
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Abstract

The Palatini action is based on vector-valued one forms or frames and SL(2,C) connections on R^4. Using the spacetime split of R^4 as a direct sum of R^3 and R^1, the Gauss law in this paper is treated on a Hilbert space. This is achieved by noting that quantum operators act on a complex Hilbert space and SL(2,C) is just the complexification of the compact SU(2) in the self-dual (1/2,0) representation used for the Ashtekar variables. This observation enables a treatment of small and large gauge transformations and superselection sectors. An explicit representation of theta vacua and their attendant 'spin-isospin mixing' are also shown. It is argued that the Gauss law algebra replaces that of diffeomorphisms in the Palatini approach : operators implementing the latter with the correct algebraic relations do not seem available. (Those obtained by multiplying Gauss law operators with fields do not have the correct commutators.)

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A. P. Balachandran. 2024-11-17. Asymptotic Quantization of Palatini Action. https://arxiv.org/abs/2411.11078

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