Noncommutative QFT and Renormalization

Abstract

Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled.In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator.We review these ideas, show the application to φ 3 models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θ-deformed space and derive noncommutative gauge field actions.

On the problem of renormalizability in non‐commutative gauge field models – a critical review

“…Historically, (an incomplete list of references is given by [28,29,30,31,32]) the IR divergences have been neglected in the discussion of renormalization. Instead, direct correspondences between known commutative results and the outcome of planar part calculations of the non-commutative counter parts have been sought.…”

Heat Trace Asymptotics on Noncommutative Spaces

“…In the presence of a potential which grows at infinity the heat kernel expansion is modified essentially even in the commutative case. In the NC case, the heat kernel expansion was constructed in [27,28] and used to study an induced gauge field action (in the spirit of the spectral action principle, see Section 2).…”

“…More examples of applications of the heat kernel expansion to renormalization and anomalies in NC theories, as well as comparison to the results obtained by other methods and further references can be found in [18,19,20,23,27,39,40,46,47,48].…”

Differential Calculus and Gauge Transformations on a Deformed Space

“…The uncertainty relation, however, brings us in conflict with Einstein's law of gravity if we assume continuity in the space variable for arbitrary small distances [2]. From the uncertainty relation…”

“…This is only one of several arguments that we have to expect some changes in physics for very small distances. Other arguments are based on the singularity problem in Quantum field theory and the fact that Einstein's theory of gravity is nonrenormalizable when quantized [2].…”