The Renormalizability Of Noncommutative U(1) Gauge Field Theories

The concept of noncommutative spacetime was suggested long time ago, it was sug-gested to give a natural cutoff to UV divergence in quantum field theories. But due to the success of renormalization theory, noncommutative spacetime did not attract much atten-tion. From last decade, with the development of string theories, the research of noncom-mutative spacetime and noncommutative theories become a focus of theoretical physics. This is because noncommutative spacetime and noncommutative theories appear naturally in some low energy limits of string theories. For example, the coordinates of a D-brane are noncommutative when there is a background field on the D-brane. In a certain superstring compactification, noncommutative gauge theories will appear naturally on D-brane world volume. For the first example, we have discussed in this thesis.In the context of string theories, noncommutative theories are certain limits of string theories. But in the context of field theories, we can treat noncommutative theories as fundamental theories. Noncommutative field theories have already been applied to phe-nomenology of particle physics. For example, the collision of high energy particles, etc. But we must admit that the consistence and correctness of noncommutative field theories are not yet proved. Noncommutative field theories also have divergences, so the renormal-ization of noncommutative field theories is an important problem. This is the problem we discuss in this thesis.For general noncommutative gauge field theories, noncommutative fields can be ex-pressed by ordinary fields which is called Seiberg-Witten map. Based on this map, an ap-proach to study the renormalization of noncommutative gauge theories is calledθ-expanded approach. When we discuss the renormalization of fields, we can use Feynman diagrams or path integral methods. Because the noncommutativity yields many additional interaction terms, so the number of Feynman diagrams increase rapidly, especially when we discuss loop diagrams. Thus, using the path integral method is a good choice. In this thesis, we use the background field method to discuss the renormalization problems of noncommutative U(1) gauge field theories. It is already shown that when the gauge field is coupled with fermion field and the action is expanded toθorder, the theory is one loop renormalizable toθorder except a four fermions vertex. It is said that the gauge sector has good one loop renormalization properties, but the matter field sector spoils the whole renormalizability of the theory. But at 92 order, the gauge field propagator is not renormalizable when the matter field is massive. When gauge field is coupled with scalar fields, with the Feynman diagrams method, it is shown that the case is similar with the above. It is suggested that if the gauge sector is completely renormalizable, we might find a new symmetry in gauge sector and according to this symmetry, we might modify the action of matter fields so that the matter sector could be renormalizable. So the first thing is to prove the gauge sector is renormalizable at higher order inθ. This the main point of this thesis. We use the background field method to calculate the case when gauge fields couple with scalar fields. When the action is expanded toθorder, it is shown that the gauge field propagator is one loop renormalizable toθorder. But there is an unrenormalizable terms ofθ2 order. This result is consistent with other papers. But we use a simpler method and need not to consider many Feynman diagrams. Furthermore, we expand the action toθ2 order and consider all divergent contributions ofθ2 order to the gauge field propagators, we find that the gauge field propagators are renormalizable even when the fermion field is massive. We think this result is true for noncommutative scalar QED. This result is very important for further consideration of gauge fields renormalization.