Matrix Wheeler-DeWitt Equation, Quantum Cosmological Tunneling and Three-Dimensional Space

Abstract

To give a possible explanation of the 3-dimensionality of space, a matrix version of the Wheeler-DeWitt equation for quantum gravity is developed. In this equation, the number of spatial dimensions D is associated with a discrete variable. In this framework, a toy quantum-cosmological model for the birth of the universe from nothing is constructed. Here, nothing means a precosmic state with no classical space-time, but constituted by a collection of potential universes in different dimensions. The tunneling probability for the birth of a closed (D + 1)-dimensional Friedmann-Lemaitre-Robertson-Walker space-time is calculated, and two particular cases are distinguished: (1) In the interpretation of parallel universes, we find that there is no preferred number of spatial dimensions. Then, the 3-dimensionality of space is explained by the anthropic principle. (2) In the interpretation of a single universe, we find a solution showing that, under certain assumptions, a 3-dimensional space is preferred.