Abstract
We review a sequence of papers in which we show that the quartic matrix model with an external matrix is exactly solvable in terms of the solution of a non-linear integral equation. The interacting scalar model on four-dimensional Moyal space is of this type, and our solution leads to the construction of Schwinger functions. Taking a special limit leads to a QFT on \(\mathbb{R}^{4}\) which satisfies growth property, covariance and symmetry. There is numerical evidence for reflection positivity of the 2-point function for a certain range of the coupling constant.
Mathematics Subject Classification (2010). 81T16, 81T08, 39A99, 45E05
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Notes
- 1.
For each assignment \(N\mapsto f_{N} \in \mathcal{S}^{N}\) of test functions, one has
$$\displaystyle{\sum _{M,N}\int \!dx\,dy\;S(x_{1},\ldots,x_{N},y_{1},\ldots,y_{M})\overline{f_{N}(x_{1}^{r},\ldots,x_{N}^{r})}f_{M}(y_{1},\ldots,y_{M}) \geq 0\;,}$$where \((x^{0},x^{1},\ldots x^{D-1})^{r}:= (-x^{0},x^{1},\ldots x^{D-1})\).
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Acknowledgements
We would like to cordially thank Jürgen Tolksdorf and Felix Finster for invitation to Quantum Mathematical Physics and hospitality during the conference in Regensburg. We submit our best wishes to Eberhard Zeidler.
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Grosse, H., Wulkenhaar, R. (2016). A Solvable Four-Dimensional QFT. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds) Quantum Mathematical Physics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26902-3_8
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