A microscopic Liouville arrow of time

We discuss the treatment of quantum-gravitational fluctuations in the space-time background as an 'environment', using the formalism for open quantum-mechanical systems, which leads to a microscopic arrow of time. After reviewing briefly the open-system formalism, and the motivations for treating quantum gravity as an 'environment', we present an example from general relativity and a general framework based on non-critical strings, with a Liouville field that we identify with time. We illustrate this approach with calculations in the contexts of two-dimensional models and D branes. Finally, some prospects for observational tests of these ideas are mentioned. (C) 1999 Elsevier Science Ltd. All rights reserved.