Sahlmann H, Seeger R (2024)
Publication Language: English
Publication Type: Journal article
Publication year: 2024
Book Volume: 41
Article Number: 075011
Journal Issue: 7
We consider the quantization of gravity as an S L ( 2 , C ) gauge theory in terms of Ashtekar’s selfdual variables and reality conditions for the spatial metric (RCI) and its evolution (RCII). We start from a holomorphic phase space formulation. It is then natural to push for a quantization in terms of holomorphic wave functions. Thus we consider holomorphic cylindrical wave functions over S L ( 2 , C ) connections. We use an overall phase ambiguity of the complex selfdual action to obtain Poisson brackets that mirror those of the real theory. We then show that there is a representation of the corresponding canonical commutation relations in the space of holomorphic cylindrical functions. We describe a class of cylindrically consistent measures that implements RCI. We show that spin networks with S U ( 2 ) intertwiners form a basis for gauge invariant states. They are still mutually orthogonal, but the normalisation is different than for the Ashtekar-Lewandowski measure for S U ( 2 ) . We do not consider RCII in the present article. Work on RCII is ongoing and will be presented elsewhere.
APA:
Sahlmann, H., & Seeger, R. (2024). Revisiting loop quantum gravity with selfdual variables: Hilbert space and first reality condition. Classical and Quantum Gravity, 41(7). https://doi.org/10.1088/1361-6382/ad2ceb
MLA:
Sahlmann, Hanno, and Robert Seeger. "Revisiting loop quantum gravity with selfdual variables: Hilbert space and first reality condition." Classical and Quantum Gravity 41.7 (2024).
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