Noncommutative quantum field theory. (English) Zbl 1338.81354
Summary: We summarize our recent construction of the \(\phi^{4}\)-model on four-dimensional Moyal space. This is achieved by solving the quartic matrix model for a general external matrix in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of that matrix. The \(\beta\)-function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. The Schwinger 2-point function is reflection positive iff the diagonal matrix 2-point function is a Stieltjes function.
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 33B15 | Gamma, beta and polygamma functions |
| 45E05 | Integral equations with kernels of Cauchy type |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
quartic matrix modelReferences:
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