SPECTRAL GEOMETRY AND CAUSALITY

For a physical interpretation of a theory of quantum gravity, it is ne cessary to recover classical space-time, at least approximately. Howev er, quantum gravity may eventually provide classical space-times by gi ving spectral data similar to those appearing in noncommutative geomet ry, rather than by giving directly a space-time manifold. It is shown that a globally hyperbolic Lorentzian manifold can be given by spectra l data. A new phenomenon in the context of spectral geometry is observ ed: causal relationships. The employment of the causal relationships o f spectral data is shown to lead to a highly efficient description of Lorentzian manifolds, indicating the possible usefulness of this appro ach. Connections to free quantum field theory are discussed for both m otivation and physical interpretation. It is conjectured that the nece ssary spectral data can be generically obtained from an effective fiel d theory having the fundamental structures of generalized quantum mech anics: a decoherence functional and a choice of histories.