Asymptotic quantization of Palatini action

The Palatini action is based on the vector-valued one-forms or frames and [Formula: see text] connections on [Formula: see text]. Using the spacetime split [Formula: see text], the Gauss law in this paper is treated on a Hilbert space. This is achieved by noticing that quantum operators act on a complex Hilbert space and [Formula: see text] is just the complexification of the compact [Formula: see text] in the self-dual [Formula: see text] representation used for the Ashtekar variables. This leads to a treatment of small and large gauge transformations and superselection sectors. An explicit representation of theta vacua and its 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 the Gauss law operators with fields do not have the correct commutators.)