![]() ![]() But this means that the ball’s gravitational field is in a superposition, too. The superposition of the particles is thereby transferred to a superposition of the ball being simultaneously at two positions. He then imagines a connection of the counters to a ball of macroscopic dimensions. He considers a Stern-Gerlach type of experiment in which two spin-1/2 particles are put into a superposition of spinup and spindown and is guided to two counters. In the 1957 Chapel Hill Conference, Richard Feynman gave the following argument, see also Zeh. It is, in fact, the superposition principle that points towards the need for quantizing gravity. Despite the ongoing discussion about its interpretational foundations (which we shall address in the last section), the concepts of states in Hilbert space, and in particular the superposition principle, have successfully passed thousands of experimental tests. It is rather a general framework for physical theories, whose fundamental concepts have so far exhibited an amazing universality. ĭespite its name, quantum theory is not a particular theory for a particular interaction. ![]() This poses the question of the connection between gravity and quantum theory. Independent of the problems with ( 4), one can try to test them in a simple setting such as the Schrödinger-Newton equation it seems, however, that such a test is not realizable in the foreseeable future. They may nevertheless be of some value in an approximate way. They spoil the linearity of quantum theory and even seem to be in conflict with a performed experiment. These “semiclassical Einstein equations” lead to problems when viewed as exact equations at the most fundamental level, compare Carlip and the references therein. A straightforward generalization would be to replace by its quantum expectation value, For then we have operators in Hilbert space on the right-hand side and classical functions on the left-hand side. 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 that is identically zero in GR (see e.g., ).Īs one recognizes from ( 2), these equations can no longer have exactly the same form if the quantum nature of the fields in is taken into account. The right-hand side displays the symmetric (Belinfante) energy-momentum tensor From the sum of these actions, one finds Einstein’s field equations by variation with respect to the metric, In the presence of nongravitational fields, ( 1) is augmented by a “matter action”. 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 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. Where is the determinant of the metric, is the Ricci scalar, and is the cosmological constant. It can be defined by the Einstein-Hilbert action Let us have a brief look at Einstein’s theory, see, for example, Misner et al. Whereas the Standard Model is a quantum field theory describing an incomplete unification of interactions, GR is a classical theory. ![]() From a theoretical (mathematical and conceptual) point of view, however, the situation is not satisfactory. From a pure empirical point of view, we thus have no reason to search for new physical laws. Gravity is described by Einstein’s theory of general relativity (GR), and no empirical fact is known that is in clear contradiction to GR. Except for the nonvanishing neutrino masses, there exists at present no empirical fact that is clearly at variance with the Standard Model. 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. Quantum Theory and Gravity-What Is the Connection?Īccording to our current knowledge, the fundamental interactions of nature are the strong, the electromagnetic, the weak, and the gravitational interactions. For this purpose, the main current approaches to quantum gravity are briefly reviewed and compared. Here, I address the main conceptual problems, discuss their present status, and outline further directions of research. The obstacles are of formal and of conceptual nature. The search for a consistent and empirically established quantum theory of gravity is among the biggest open problems of fundamental physics. ![]()
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