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This monograph presents a review and analysis of the main mathematical, physical and epistemological difficulties encountered at the foundational level by all the conventional formulations of relativistic quantum theories, ranging from relativistic quantum mechanics and quantum field theory in Minkowski space, to the various canonical and covariant approaches to quantum gravity. It is, however, primarily devoted to the systematic presentation of a quantum geometry framework meant to deal effectively with these difficulties by reconsidering the foundations of these subjects, analyzing their epistemic nature, and then developing mathematical tools which are specifically designed for the elimination of all the basic inconsistencies. The deficiencies of the conventional approaches are not usually discussed in the mainstream literature on these subjects, even though many of them were uncovered, and the need for their elimination was emphasized, by many founders of quantum theory, such as Born, Dirac, and Heisenberg. Hence, a historical survey is included, and additional extensive notes containing quotations from original sources are incorporated at the end of each chapter, so that the reader will be brought up-to-date with recent developments in quantum field theory in curved spacetime, quantum gravity and quantum cosmology. The survey further provides a backdrop against which the new foundational and mathematical ideas of the present approach to these subjects can be brought out in sharper relief. These ideas reflect the legacy of the founders of relativity and quantum theory, as they are attempting to find a common ground in the apparently irreconcilable points of view of Bohr and Einstein. In developing them, as much attention is devoted to incorporating them into a mathematically sound framework for relativistic quantum theory, advocated by Dirac, as to securing their epistemological soundness, reflected by a clear enunciation of all the underlying physcial principles, whose need was stressed by Heisenberg. This work is aimed at theoretical and mathematical physicists, mathematicians interested in differential geometry, fibre bundles, and global analysis, philosophers and historians of sciences interested in general relativity and quantum theory. Despite the advanced level of many of the areas of mathematics and theoretical physics treated, every effort has been made to produce a text which is also accessible to graduate students in physics and mathematics.
