Noncommutative Quantum Field Theory

Quantum field theory provides the best known description of the fundamental particles and forces in nature. It is based on the concept of classical, flat space-time. However, it is generally believed that this classical concept of space will break down at very small distances, where it should be quantized or "fuzzy`" due to quantum gravity. Therefore quantum field theory should take such quantum fluctuations of space into account. The UV divergences in quantum field theory are also an indication of our incomplete understanding of spacetime. Noncommutative quantum field theory (NCFT) has been designed to address these questions. It is based on the concept of a noncommutative or quantized space, where points no longer exist and positions can only be measured up to some minimal uncertainty, similar as in Quantum Mechanics. Many models of NCFT have been studied in recent years, and a coherent picture is beginning to emerge. One of the surprising features of NCFT is the so-called IR/UV mixing, where the usual divergences of field theory in the UV are reflected by new singularities in the IR. This is essentially a reflection of the uncertainty relation: determining some coordinates to very high precision (UV) implies a large uncertainty (IR) for others. This leads to a serious problem for the usual renormalization procedure of QFT, which has only recently been overcome for a 4- dimensional scalar field theoretical model on the canonically deformed Euclidean space, in a collaboration of the applicant. Furthermore, IR/UV mixing is also responsible for a new phase of NCFT, the so-called "striped phase". Much more work is required to understand the physical implications of these NCFT`s. The aim of this project is to improve the theoretical understanding of NCFT. One important but difficult goal is to establish renormalizability of NC gauge theories, i.e. to show that they are well-behaved and under control. Towards this goal, some simpler models will be investigated, starting in particular with the selfdual case of the scalar field theoretical model on noncommutative two and four dimensional Euclidean space. This can be studied using matrix-model techniques, which are very powerful nonperturbative methods which seem to be very useful for NCFT. In particular, it was proposed recently that such matrix methods provide the correct description of the phase transitions in NCFT. One part of this project is to refine and generalize this approach, to apply it to other models such as sigma models and the Higgs model, and to combine it with the perturbative results. Another important goal is to establish renormalizability for scalar field theories on other quantum spaces such as the fuzzy sphere, and hence to achieve a better understanding of NCFT in general and their physical significance.

Quantum field theory provides the best known description of the fundamental particles and forces in nature. It is based on the concept of classical, flat space-time. However, it is generally believed that this classical concept of space will break down at very small distances, where it should be quantized or "fuzzy" due to quantum gravity. Therefore quantum field theory should take such quantum fluctuations of space into account. The UV divergences in quantum field theory are also an indication of our incomplete understanding of spacetime. Noncommutative quantum field theory (NCFT) has been designed to address these questions. It is based on the concept of a noncommutative or quantized space, where points no longer exist and positions can only be measured up to some minimal uncertainty, similar as in Quantum Mechanics. Many models of NCFT have been studied in recent years, and a coherent picture is beginning to emerge. One of the surprising features of NCFT is the so-called IR/UV mixing, where the usual divergences of field theory in the UV are reflected by new singularities in the IR. This is essentially a reflection of the uncertainty relation: determining some coordinates to very high precision (UV) implies a large uncertainty (IR) for others. This leads to a serious problem for the usual renormalization procedure of QFT, which has only recently been overcome for a 4- dimensional scalar field theoretical model on the canonically deformed Euclidean space, in a collaboration of the applicant. Furthermore, IR/UV mixing is also responsible for a new phase of NCFT, the so-called "striped phase``. Much more work is required to understand the physical implications of these NCFT`s. The aim of this project is to improve the theoretical understanding of NCFT. One important but difficult goal is to establish renormalizability of NC gauge theories, i.e. to show that they are well-behaved and under control. Towards this goal, some simpler models will be investigated, starting in particular with the selfdual case of the scalar field theoretical model on noncommutative two and four dimensional Euclidean space. This can be studied using matrix-model techniques, which are very powerful nonperturbative methods which seem to be very useful for NCFT. In particular, it was proposed recently that such matrix methods provide the correct description of the phase transitions in NCFT. One part of this project is to refine and generalize this approach, to apply it to other models such as sigma models and the Higgs model, and to combine it with the perturbative results. Another important goal is to establish renormalizability for scalar field theories on other quantum spaces such as the fuzzy sphere, and hence to achieve a better understanding of NCFT in general and their physical significance.