The search for a consistent and empirically established quantum theory of gravity is among the biggest open problems of fundamental physics. The obstacles are of formal and of conceptual nature. Here, I address the main conceptual problems, discuss their present status, and outline further directions of research. For this purpose, the main current approaches to quantum gravity are briefly reviewed and compared. 1. Quantum Theory and Gravity—What Is the Connection? According to our current knowledge, the fundamental interactions of nature are the strong, the electromagnetic, the weak, and the gravitational interactions. The first three are successfully described by the Standard Model of particle physics, in which a partial unification of the electromagnetic and the weak interactions has been achieved. Except for the nonvanishing neutrino masses, there exists at present no empirical fact that is clearly at variance with the Standard Model. Gravity is described by Einstein’s theory of general relativity (GR), and no empirical fact is known that is in clear contradiction to GR. From a pure empirical point of view, we thus have no reason to search for new physical laws. From a theoretical (mathematical and conceptual) point of view, however, the situation is not satisfactory. Whereas the Standard Model is a quantum field theory describing an incomplete unification of interactions, GR is a classical theory. Let us have a brief look at Einstein’s theory, see, for example, Misner et al. [1]. It can be defined by the Einstein-Hilbert action where is the determinant of the metric, is the Ricci scalar, and is the cosmological constant. In addition to the two main terms, which consist of integrals over a spacetime region , there is a term that is defined on the boundary (here assumed to be space like) of this region. This term is needed for a consistent variational principle; here, is the determinant of the three-dimensional metric, and is the trace of the second fundamental form. In the presence of nongravitational fields, (1) is augmented by a “matter action” . From the sum of these actions, one finds Einstein’s field equations by variation with respect to the metric, The right-hand side displays the symmetric (Belinfante) energy-momentum tensor plus the cosmological-constant term, which may itself be accommodated into the energy-momentum tensor as a contribution of the “vacuum energy.” If fermionic fields are added, one must generalize GR to the Einstein-Cartan theory or to the Poincaré gauge theory, because spin is the source of torsion, a geometric quantity
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