What may come beyond the Wheeler–DeWitt approach?

The goal of this paper is to compare the Wheeler–DeWitt approach, understood in a broad sense, with an alternative one, the so-called extended phase space approach to quantization of gravity. By “the Wheeler–DeWitt approach”, I mean not only quantum geometrodynamics formulated by DeWitt in his seminal paper of 1967, but also any approach to quantization of gravity based on the Wheeler–DeWitt equation in some form. Since the Wheeler–DeWitt equation is a direct consequence of the Dirac formalism, its analysis requires examination of the latter and its application to gravity. In particular, I argue that there is a contradiction between canonical quantization and the idea put forward by founders of quantum gravity that, in this theory, all possible spacetime topologies should be taken into account. The path integral approach seems to be more adequate then the canonical approach. However, to derive the Wheeler–DeWitt equation from the path integral, most authors make the assumption about asymptotic states, that again contradicts the supposition of arbitrary spacetime topology. The extended phase space formalism is entirely based on the path integral approach. Note that if one refuses the assumption about asymptotic states, one cannot prove the gauge invariance of the path integral, and the Wheeler–DeWitt equation loses its sense. In the alternative approach, one derives the Schrödinger equation instead. Thus, the extended phase space approach is really beyond the Wheeler–DeWitt approach. The features of this alternative approach are explored with special emphasis on conclusions that cannot be obtained by using the Wheeler–DeWitt quantum geometrodynamics.