Covariant quantum spacetimes, higher spin, and gravity

It is established beyond doubt that nature is governed by quantum mechanics. This is well understood for all fundamental interactions, except for gravity. Despite much work over several decades, reconciling gravity with quantum mechanics remains to be one of the major open problem in theoretical physics. One of the underlying difficulties is that quantum gravity arguably entails a modification of space-time itself, which requires a broadening of the established framework to formulate physical theories. It is widely expected that space-time should have some kind of quantum structure at short distances. On the other hand, relativity requires (local) Lorentz invariance, to a very high precision. Reconciling these two requirements is a very non-trivial task. Progress in recent years has led to the remarkable insight that space-time and geometry can emerge from simple models with little or no geometry. This happens in particular in certain matrix models models which are related to string theory, notably the so-called IKKT model. Certain solutions of this model, which were found recently, have indeed the required geometrical properties of a quantized cosmological space-time geometry which is exactly homogeneous and isotropic, with a discrete quantum structure at very short distances. This quantum geometry behaves as an expanding universe originating from a Big Bang, and might play the role of physical space-time. The present project studies the physical properties of these quantized space-time solutions within the IKKT model, and in particular the gauge theory of the spin 2 modes which arise on these spaces. These naturally provide the degrees of freedom for gravity. This resulting gravity will be studied and elaborated in detail, along with its higher spin extensions, and compared with similar theories. This requires a systematic organization and analysis of all fluctuation modes on this space-time, and a description of their dynamics in the IKKT model. Other topics to be studied include quantum corrections beyond the leading classical contributions, stability issues, and properties of the Big Bang in this scenario.

One of the great outstanding problems in theoretical physics is to establish a consistent quantum theory of fundamental interactions including gravity. It is generally expected that this requires a quantum structure of space-time and geometry. The present project studies a specific such quantum geometry, which describes an expanding cosmological space-time, featuring a Big Bang or rather a Big Bounce. This geometry arises as a solution of the IKKT matrix model which is known as "Matrix Theory", and is viewed as a constructive variant of string theory. This project studies mathematical and physical aspects of that quantum space-time as governed by the IKKT model. One focus was on the spin 2 fluctuation modes, which constitute the degrees of freedom of gravity. The resulting theory of gravity was examined in detail. Moreover, a more general higher-spin gauge theory arising on this space-time was studied in detail, which can be interpreted as generalized theory of gravity. Via a complete analysis of all fluctuation modes it was established that this theory is stable, and free of pathological fluctuation modes. Moreover, a suitable mathematical description of the non-linear geometry and dynamics was developed. This description allowed for the first time to establish the emergence of the Einstein-Hilbert action within matrix theory, which is underlying Einstein's theory of general relativity. This is an important step towards understanding the origin of gravity within a fully quantum framework. In a further publication, the time evolution of elementary particles on the cosmic background under consideration was examined. In particular, it was possible to establish that the usual rules of time evolution in quantum field theory do apply in the new framework, consistent with established physics. Finally, an alternative formulation of the theory in terms of spinors was given, which is useful for the computation of scattering processes. Besides structural and mathematical foundations and consistency checks, this project also examined physical properties of the emergent gauge theory under consideration. For example, the standard Big Bang is replaced by a Big Bounce. Remarkably, it was possible to show that information can be transmitted across the Big Bounce. This could have significant implications on our understanding of cosmology, which however requires a more detailed understanding of the model. Finally, the implications of the novel gravitational theory on fermions was studied. An extra term was found in the gravitational coupling of fermions, which is not present in Einstein's theory. However, we have demonstrated that this term does not affect the dynamics of fermions on curved backgrounds at leading order. This provides further support for the physical viability of the model.