Summary
In any quantum theory, in which the metric tensor of Einstein’s gravitational theory is also quantized, it becomes meaningless to ask for an initial space-like surface on which to specify the conventional field commutators. The covariant quantum formalism, in which all fields either commute or fail to do so only when the field’s points coincide, is proposed as being suitable to quantize gravity. The extension of the covariant quantum formalism to general boson fields that interact in an intrisically nonlinear way with external fields is analysed in some detail. This formalism is applied to the case of the free gravitational field. In a functional representation, the measure on metrics is found to be that proposed by Misner. A basic state of the quantized gravitational theory is proposed, which involves a summation over all permissible metrics in the entire space-time manifold.
Riassunto
In ogni teoria quantistica, nella quale sia anche quantizzato il tensore metrico della teoria gravitazionale di Einstein, diventa privo di senso richiedere una superficie spaziale iniziale su cui specificare i commutatori di campo convenzionali. Si designa il formalismo quantistico covariante, in cui tutti i campi o commutano o non commutano solo quando i punti dei campi coincidono, come adatto a quantizzare la gravità. Si analizza un po’ dettagliatamente l’estensione del formalismo quantistico covariante a campi bosonici generali che interagiscono in modo intrinsecamente non lineare. Si applica questo formalismo al caso del campo libero gravitazionale. Si trova che, in una rappresentazione funzionale, la misura sulle metriche è quella proposta da Misner. Si propone uno stato fondamentale della teoria gravitazionale quantizzata, che comporta una sommatoria estesa a tutte le metriche possibili in tutto il complesso spaziotempo.
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References
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Klauder, J.R. Covariant quantization of the gravitational field. Nuovo Cim 25, 542–564 (1962). https://doi.org/10.1007/BF02733694
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DOI: https://doi.org/10.1007/BF02733694