Self-Dual Noncommutative Φ~4-Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory

摘要

We study quartic matrix models with partition function Z[E, J] = S dM exp(trace(JM ? EM~2 ? λ/4M~4)). The integral is over the space of Hermitean N × Nmatrices, the external matrix E encodes the dynamics,λ > 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 Φ~4-quantum field theory on fourdimensional 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 N → ∞the same spectrum as the Laplace operator in four dimensions. Using the theory of singular integral equations of Carleman type we compute (for N → ∞and after renormalisation of E, λ) the free energy density (1/volume) log(Z[E, J ]/Z[E, 0]) exactly in terms of the solution of a nonlinear 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 λ ≤ 0.