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The causality structure of two-dimensional manifolds with degenerate metricsis analysed in terms of global solutions of the massless wave equation. Certainnovel features emerge. Despite the absence of a traditional Lorentzian Cauchysurface on manifolds with a Euclidean domain it is possible to uniquelydetermine a global solution (if it exists), satisfying well defined matchingconditions at the degeneracy curve, from Cauchy data on certain spacelikecurves in the Lorentzian region. In general, however, no global solutionsatisfying such matching conditions will be consistent with this data.Attention is drawn to a number of obstructions that arise prohibiting theconstruction of a bounded operator connecting asymptotic single particlestates. The implications of these results for the existence of a unitaryquantum field theory are discussed.Comment: 27 pages LaTex (6 Figures), Journal of Mathematical Physics  (Accepted

Classical and quantum implications of the causality structure of two-dimensional space–times with degenerate metrics