Summary
A nonperturbative approach to the quantization of the canonical algebra of pure gravity is presented. The problem of factor ordering of operators in the constraints μ Ψ=0 is resolved by invoking Hermiticity under the invariant inner product in hyperspace—the space of all three-dimensional metricsg ij (x)—and covariance under co-ordinate transformations. The resulting operators μ receive corrections of order\( / \) and\(h / ^2 \) only and the algebra closes up to a conformal anomaly term. If the algebra is enlarged by the inclusion of the anomalous operator, it can be shown that, by some suitable choice of the gauge parameter corresponding to this unphysical symmetry, the integrated form of the algebra can be made to close.
Riassunto
Si presenta un approccio non perturbativo alla quantizzazione dell'algebra canonica della gravità pura. Il problema dell'ordinamento in fattori di operatori nei vincoli μ Ψ=0 è risolto invocando l'hermiticità sotto il prodotto interno invariante nell'iperspazio—lo spazio di tutte le metriche tridimensionalig ij (x)—e covarianza sotto trasformazioni di coordinate. I risultanti operatori μ ricevono solo correzioni d'ordine\( / \) e\(h / ^2 \) e l’algebra chiude fino ad un termine ad anomalia conforme. Se l'algebra è allargata con l'inclusione dell'operatore anomalo, si può mostrare che, con adeguata scelta del parametro di gauge che corrisponde a questa simmetria non fisica, la forma integrata dell'algebra può esser fatta chiudere.
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Christodoulakis, T., Zanelli, J. Operator ordering in quantum mechanics and quantum gravity. Nuov Cim B 93, 1–21 (1986). https://doi.org/10.1007/BF02728299
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DOI: https://doi.org/10.1007/BF02728299