On the third quantization of general relativity

The author studies the possibility of defining a Hilbert space for n=(d+1)-dimensional general relativity in some asymptotic regions of Wheeler's superspace. The author distinguishes two asymptotic regions: (i) the 'classical asymptotic region', which contains geometries with a volume much larger than the Planck volume and (ii) the 'quantum asymptotic region', which contains geometries with a volume much smaller than the Planck volume. It is shown that for n>or=4 one can define a Hilbert space only in the classical asymptotic region of superspace, while for n(4 one can define a Hilbert space in the quantum asymptotic region or in the classical asymptotic region, but not both. It is argued that in a good theory of quantum gravity one should be able to define a Hilbert space in the two asymptotic regions. Therefore it seems that Einstein's general relativity is not a good candidate for a quantum theory of gravitation. But it (the 3+1)-dimensional case) can be a good classical limit of that theory. The above criterion can serve in the search for a quantum theory of gravitation.