Homotopic Classification of Yang–Mills Vacua Taking into Account Causality

Abstract

Existence of theta-vacuum states in Yang--Mills theories defined over asymptotically flat space-times examined taking into account not only the topology but the complicated causal structure of these space-times, too. By a result of Galloway apparently causality makes all vacuum states, seen by a distant observer, homotopically equivalent making the introduction of theta-terms unnecessary. But a more careful analysis shows that certain twisted classical vacuum states survive even in this case eventually leading to the conclusion that the concept of ``theta-vacua'' is meaningful in the case of general Yang--Mills theories. We give a classification of these vacuum states based on Isham's results showing that the Yang--Mills vacuum has the same complexity as in the flat Minkowskian case hence the general CP-problem is not more complicated than the well-known flat one. We also construct the theta vacua rigorously via geometric quantization.