Probing the Planckian Structure of Spacetime

Already Riemann contemplated the need for modifying our conventional notions of the metric properties of space on scales which are "immeasurably small", a need that should be driven empirically by new insights gained in physics. Great strides have been made since in understanding the theoretical foundations of the physical world, in the form of special and general relativity, quantum theory and quantum field theory. Taken together they strongly suggest the existence of a theory of quantum gravity, which should provide a consistent and quantitative description of the nature of "quantum spacetime" on ultrashort, Planckian length scales. After decades of research, the problem of finding this theory is still outstanding. I will report on recent, unprecedented progress in a new formulation of quantum gravity, called Causal Dynamical Triangulation. It is based on performing a "sum over histories" by using an intrinsically geometric way of regularizing this quantum superposition in terms of triangulated, piecewise flat spacetimes. In two dimensions, evaluating the sum takes the form of a combinatorial problem, which can be solved explicitly. In the physically relevant case of four spacetime dimensions, nontrivial properties of the sum over spacetimes can be extracted with the help of numerical "experiments", yielding some intriguing results which confirm the highly nonclassical nature of spacetime geometry at the Planck scale, and the emergence from it of classical geometry on large scales.