RELATION BETWEEN EINSTEIN AND QUANTUM-FIELD EQUATIONS

We show that there exists a choice of scalar field modes such that the evolution of the quantum field in the zero-mass and large-mass limits is consistent with the Einstein equations for the background geometry . This choice of modes is also consistent with zero production of thes e scalar particles and thus corresponds to a preferred vacuum state pr eserved by the evolution. In the zero-mass limit, we find that the qua ntum field equation implies the Einstein equation that determines the scale factor for a radiation-dominated universe; in the large-mass cas e, it implies the corresponding Einstein equation for a matter-dominat ed universe. Conversely, if the classical radiation-dominated or matte r-dominated Einstein equations hold, there is no production of scalar particles in the zero and large mass limits, respectively. The suppres sion of particle production in the large mass limit is over and above the expected suppression at large mass. Our results hold for a certain class of conformally ultrastatic background geometries and therefore generalize previous results by one of us for spatially flat Robertson- Walker background geometries. In these geometries, we find that the te mporal part of the graviton equations reduces to the temporal equation for a massless minimally coupled scalar held, and therefore the resul ts for massless particle production hold also for gravitons. Within th e class of modes we study, we also find that the requirement of zero p article production of massless scalar particles or gravitons is not co nsistent with a non-zero cosmological Constant. Possible implications are discussed.