Stochastic quantization of the linearized gravitational field∗

Quantum systems which are described by a Schrödinger equation allow a stochastic interpretation. The Fenyes-Nelson model [1]–[3] and its generalization [4]–[6] provide such a stochastic description of quantum mechanics. Several field theories have been considered in this context [7]–[11], and the results have so far been of some interest. In the present paper, the generalized Fenyes-Nelson model is applied to the weak field approximation of Einstein’s general theory of relativity, the so-called linearized gravitational field [12]–[14]. It is worthwhile to attempt a stochastic model of quantum gravity for several reasons. First, since the gravitational field in the classical theory has the interpretation of a metric tensor, it is disturbing that in the usual quantum formulation this geometric interpretation is completely obliterated because the field and its derivatives become abstract operators on a rigged Hilbert space. Because of this difficulty it has become fashionable to play down the geometric role of gravity when dealing with quantization, as in the formulation of Weinberg [14]. A stochastic formulation of the quantized gravitational field may revitalize the geometrical interpretation of quantum gravity. Second, the stochastic formulation of quantum mechanics, at least in spirit, is a progeny of Einstein’s profound discomfiture with the complementarity vision of Bohr. Since Einstein’s views were based at least in part on considerations of the general theory of relativity, it seems fitting to pursue a stochastic model of quantum gravity. Third, the great successes which gauge theories have had in the theory of elementary particles suggest that fundamental efforts such as the stochastic reformulation of quantum mechanics be concentrated in this general area.