Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
1997-05-10
Class.Quant.Grav. 15 (1998) 1463-1485
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
24p, LATEX
Scientific paper
10.1088/0264-9381/15/6/005
We quantize the generators of the little subgroup of the asymptotic Poincar\'e group of Lorentzian four-dimensional canonical quantum gravity in the continuum. In particular, the resulting ADM energy operator is densely defined on an appropriate Hilbert space, symmetric and essentially self-adjoint. Moreover, we prove a quantum analogue of the classical positivity of energy theorem due to Schoen and Yau. The proof uses a certain technical restriction on the space of states at spatial infinity which is suggested to us given the asymptotically flat structure available. The theorem demonstrates that several of the speculations regarding the stability of the theory, recently spelled out by Smolin, are false once a quantum version of the pre-assumptions underlying the classical positivity of energy theorem is imposed in the quantum theory as well. The quantum symmetry algebra corresponding to the generators of the little group faithfully represents the classical algebra.
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