Theory of quantum space-time

Abstract

A generalized 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 space-time is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time (QST) possesses a rich hierarchical structure. The natural extension of the Poincare group to QST is investigated. In particular, we prove that the symmetry group of this space is generated in general by a system of irreducible Killing tensors. After the symmetries are broken, the points of the QST can be interpreted as space-time valued operators. The generic point of a QST in the broken symmetry phase then becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from QST 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. © 2005 The Royal Society.