Classical and Quantum Implications of the Causality Structure of Two-Dimensional Spacetimes with Degenerate Metrics

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

The Extended Regularized Dual Averaging Method for Composite Optimization

We present a new algorithm, extended regularized dual averaging (XRDA), for solving composite optimization problems, which are a generalization of the regularized dual averaging (RDA) method. The main novelty of the method is that it allows more flexible control of the backward step size. For instance, the backward step size for RDA grows without bound, while XRDA the backward step size can be kept bounded