ar X iv : g r - qc / 0 40 61 21 v 1 3 0 Ju n 20 04 Theory of Quantum Space - Time

A generalised equivalence principle is put forward according to which space-time symmetries and internal quantum symmetries are indistinguishable before symmetry breaking. Based on this principle, a higher-dimensional extension of Minkowski space is proposed and its properties examined. In this scheme the structure of spacetime is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time possesses a rich hierarchical structure. The natural extension of the Poincaré group to quantum space-time is investigated. In particular, we prove that the symmetry group of a quantum space-time is generated in general by a system of irreducible Killing tensors. When the symmetries of a quantum space-time are spontaneously broken, then the points of the quantum space-time can be interpreted as space-time valued operators. The generic point of a quantum space-time in the broken symmetry phase thus becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from quantum space-time to Minkowski space. It is shown that the general such map satisfying appropriate causality-preserving conditions ensuring linearity and Poincaré invariance is necessarily a density matrix.