Effective equations are a powerful mathematical tool to extract physical phenomena from fundamental quantum theories. Once they have been derived, they show effects at an intuitive level such as in the balance of forces. This is much easier to visualize and compute than the underlying fundamental equations for, e.g., a quantum mechanical wave function. At a general level as well as in key examples, new effective equations will be analyzed and derived systematically in the areas of quantum mechanics, quantum cosmology and quantum gravity. Results will be integrated in the teaching of undergraduate quantum mechanics and of quantum aspects in cosmology such as the emergence of structure in the universe. In new research, the nature of the big bang singularity in quantum cosmology will be analyzed, providing insights into the state of the universe at and before the big bang. Moreover, potentially observable effects of quantum gravity in the early universe and new phenomena in the quantum physics of black holes will be investigated. By focusing on effects rather than the underlying mathematics, comparisons with other approaches to quantum gravity will become possible. The issues addressed underlie our understanding of the universe and typically attract wide interest. New methods to be introduced make implications of quantum gravity accessible for the non-specialist and bridge the traditional gap between fundamental research in this area and cosmological phenomenology. In this way, key infrastructure will be made available to other researchers, and manageable projects to guide students to the field will be generated. By integrating new methods and results into teaching at early stages, the understanding of quantum physics will be improved. Modern examples of applications in cosmology, an area of interest to many students, will provide additional motivation. Specific results will regularly be disseminated to the research community and the general public.
??Several deep problems of conceptual and computational nature surround the question of how to combine relativity theory with quantum physics. On the conceptual side, for instance, relativity deals with space-time as a physical and changing object, while quantum physics introduces quantum jumps, fluctuations and fundamental uncertainty. What do physical laws look like when even space and time may fluctuate, instead of being neatly aligned along the four dimensions? There are candidates for a combined theory of quantum gravity, but they are so complicated that it is difficult to see how they address the quantum nature of space-time, giving rise to important computational problems. This project attempted to strike a balance between controlled, doable calculations and access to deep quantum regimes of space time. It introduced and built a general class of approximations, called canonical effective descriptions, into the field of quantum gravity. In this way, one does not deal with the full, counter-intuitive wave function of quantum mechanics, but rather "effectively" computes the modifications to classical laws it implies. Several mathematical and computational questions have been addressed in this project, building a systematic way to derive effective descriptions of quantum systems in cosmology. Applications of these techniques have shed light on some long-standing questions in quantum gravity. For instance, the fluctuating and relative nature of time may require this familiar parameter to take complex rather than real values. Secondly, when quantum effects on space-time are very strong (at high density near the big bang or in black holes), time may change completely into a spatial direction and leave only 4-dimensional space without the possibility of change. Time and the expanding universe would emerge at a hypersurface in 4-dimensional space at which the density is sufficiently small. The project has contributed to different scientific fields, in addition to quantum gravity also mathematics and cosmology. In order to arrive at these results and to explore them further, the method of effective descriptions had to be extended significantly compared to familiar techniques used with much success in particle and condensed-matter physics. Especially the nature of relative space-time, on which particles move and with which they interact, poses challenging mathematical questions. The physical results of this project have led to new structures of mathematical interest. These structures provide an algebraic description of how the fundamental nature of space-time changes the familiar continuum, and they indicate how these consequences could be probed by cosmological observations. The project has in its last year trained and supported four graduate and two undergraduate students. Results have been communicated broadly, in scientific journals, in popular articles, and in books. Some of these contributions support teaching different aspects of quantum physics and relativity at an undergraduate level.
