Communications in Mathematical Physics

, Volume 329, Issue 3, pp 1069–1130 | Cite as

Self-Dual Noncommutative \({\phi^4}\) -Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory



We study quartic matrix models with partition function \({\mathcal{Z}[E, J] = \int dM}\) exp(trace\({(JM - EM^{2} - \frac{\lambda}{4} M^4)}\)). The integral is over the space of Hermitean \({\mathcal{N} \times \mathcal{N}}\) -matrices, the external matrix E encodes the dynamics, \({\lambda > 0}\) is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function.

As the main application we prove that Euclidean \({\phi^4}\) -quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for \({\mathcal{N} \to \infty}\) the same spectrum as the Laplace operator in four dimensions. Using the theory of singular integral equations of Carleman type we compute (for \({\mathcal{N} \to \infty}\) and after renormalisation of \({E, \lambda}\)) the free energy density (1/volume) log\({(\mathcal{Z}[E, J]/\mathcal{Z}[E, 0])}\) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem.

The derivation of the non-linear integral equation relies on an assumption which in subsequent work is verified for coupling constants \({\lambda \leq 0}\) .


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Fakultät für PhysikUniversität WienWienAustria
  2. 2.Mathematisches Institut, WestfälischeWilhelms-UniversitätMünsterGermany

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