Solvable 4D noncommutative QFT: phase transitions and quest for   reflection positivity
  We provide further analytical and first numerical results on the solvable $\lambda\phi^4_4$-NCQFT model. We prove that for $\lambda<0$ the singular integral equation has a unique solution, whereas for $\lambda>0$ there is considerable freedom. Furthermore we provide integral formulae for partial derivatives of the matrix 2-point function, which are the key to investigate reflection positivity.   The numerical implementation of these equations gives evidence for phase transitions. The derivative of the finite wavefunction renormalisation with respect to $\lambda$ is discontinuous at $\lambda_c \approx -0.39$. This leads to singularities in higher correlation functions for $\lambda<\lambda_c$. The phase $\lambda >0$ is not yet under control because of the freedom in the singular integral equation.   Reflection positivity requires that the two-point function is Stieltjes. Implementing Widder's criteria for Stieltjes functions we exclude reflection positivity outside the phase $[\lambda_c,0]$. For the phase $\lambda_c<\lambda \leq 0$ we show that refining the discrete approximation we satisfy Widder to higher and higher order. This is clear evidence, albeit no proof, of reflection positivity in that phase. 
Harald Grosse (Vienna) and Raimar Wulkenhaar (M\"unster)
hep-th math-ph math.MP