Quantum Mathematical Physics : A Bridge between Mathematics and Physics Preface Christopher J. Fewster. On the Spin Statistics Connection in Curved Spacetimes Daniel Becker Martin Reuter. Is there a C Function in 4D Quantum Einstein Gravity José M. Gracia Bondía.Systematic Renormalization at all Orders in the DiffRen and Improved Epstein{Glaser Schemes Michael Dütsch. Higgs Mechanism and Renormalization Group Flow:Are They Compatible Claudio Dappiaggi. Hadamard States From Null Infinity Michael Gransee.Local Thermal Equilibrium States in Relativistic Quantum Field Theory Frédéric Paugam. Categorical Methods in Quantum Field Theory Harald Grosse Raimar Wulkenhaar.A Solvable Four Dimensional QFT Rainer Verch. Wave Equations with Non Commutative Space and Time Gandalf Lechner Jan Schlemmer.Thermal Equilibrium States for Quantum Fields on Non Commutative Spacetimes Christian Fleischhack.Kinematical Foundations of Loop Quantum Cosmology Gerhard Börner.Cosmic Puzzles Dark Matter and Dark Energy Israel Michael Sigal.Radiation and Scattering in Non Relativistic Quantum Electrodynamics Stefan Teufel Roderich Tumulka. Avoiding Ultraviolet Divergence by Means of Interior{Boundary Conditions Felix Finster.Causal Fermion Systems An Overview Dirk André Deckert Franz Merkl. A Perspective on External Field QED Enno Keßler.Super Riemann Surfaces and the Super Conformal Action Functional Stefan Waldmann. Recent Developments in Deformation Quantization Michael K. H. Kiessling.Diracs Point Electron in the Zero Gravity Kerr Newman World Christoph A. Stephan. Noncommutative Geometry and the Physics of the LHC Era Klaus Kröncke. Variational Stability and Rigidity of Compact Einstein Manifolds Index. Quantum physics has been highly successful for more than 90 years. Nevertheless a rigorous construction of interacting quantum field theory is still missing. Moreover it is still unclear how to combine quantum physics and general relativity in a unified physical theory. Attacking these challenging problems of contemporary physics requires highly advanced mathematical methods as well as radically new physical concepts. This book presents different physical ideas and mathematical approaches in this direction. It contains a carefully selected cross section of lectures which took place in autumn 2014 at the sixth conference ``Quantum Mathematical Physics A Bridge between Mathematics and Physics in Regensburg Germany. In the tradition of the other proceedings covering this series of conferences a special feature of this book is the exposition of a wide variety of approaches with the intention to facilitate a comparison. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up to date overview of a fascinating active research area.