Duality between operator ordering factor and massless scalar field

Abstract

Abstract In order to investigate the role of quantum effects in the evolution of the universe, one can either use the Wheeler–DeWitt equation (WDWE) that contains an operator ordering factor, or add an item called massless scalar field to WDWE. In this paper, we study the relationship between operator ordering factor and massless scalar field, by applying de Broglie–Bohm quantum trajectory approach to WDWE. In theory, the evolution of the universe is determined by action, i.e., the phase part of the wavefunction of the universe. For the case of operator ordering factor and the case of massless scalar field, the functions that determine the phase part of the wavefunction of the universe satisfy the same differential equation, both in the minisuperspace model and in the Kantowski–Sachs model. This shows the equivalence of using operator ordering factor or massless scalar field to study evolution of the universe. Since there is no accelerating solution of WDWE with operator ordering factor for a grownup universe in the minisuperspace model, the equivalence of the operator ordering factor and the massless scalar field rules out the possibility of a massless scalar field as the candidate for dark energy, if the current universe is indeed homogeneous and isotropic.