The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. Locality is fulfilled. The Schwinger 2-point function is reflection positive in special cases. | Vietnam Journal of Mathematics (2019) 47:93–112 How Prof. Zeidler Supported Our Research on Exact Solution of Quantum Field Theory Toy Models Harald Grosse1 · Raimar Wulkenhaar2 Received: 27 December 2017 / Accepted: 6 April 2018 / Published online: 6 September 2018 © The Author(s) 2018 Abstract Over many years, we developed the construction of the φ 4 -model on four-dimensional Moyal space. The solution of the related matrix model Z [E, J ] = d exp(tr(J −E 2 − λ 4 4 )) is given in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of E. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. Locality is fulfilled. The Schwinger 2-point function is reflection positive in special cases. Keywords 4D noncommutative quantum field theory · Renormalisation · Non-perturbative solution · Integral equation PACS 2010 · · 1 Introduction Professor Zeidler supported and influenced our common work over many years: During the one-semester stay in 2000/2001 of one of us (HG), we enjoyed the friendly hospitality at MPI Leipzig. On the other hand, from 2002 to 2005, RW was post-doc at the MPI. During this time, we achieved the perturbative renormalisation proof of the noncommutative φ44 model. Prof. Zeidler was constantly interested in the progress of this work, financed our mutual visits in Leipzig and Vienna and, most importantly, introduced Vincent Rivasseau to Dedicated to the memory of Prof. Eberhard Zeidler (1940–2016). Harald Grosse Raimar Wulkenhaar raimar@ 1 Fakult¨at f¨ur Physik, Universit¨at Wien, Boltzmanngasse 5, A-1090, Wien, Austria 2 Mathematisches Institut der Westf¨alischen Wilhelms-Universit¨at, Einsteinstraße 62, D-48149, M¨unster, Germany 94 H. Grosse, R. Wulkenhaar RW in December 2003. This changed life of all three of us. Vincent
