Chiral symmetry breaking and confinement in Minkowski space QED2+1

Without any analytical assumption we solve the ladder QED2+1 in Minkowski space. Obtained complex fermion propagator exhibits confinement in the sense that it has no pole. Further, we transform Greens functions to the Temporal Euclidean space, wherein we show that in the special case of ladder QED2+1 the solution is fully equivalent to the Minkowski one. Obvious invalidity of Wick rotation is briefly discussed. The infrared value of the dynamical mass is compared with other known approaches, e. g. with the standard Euclidean calculation. We have presented for the first analysis of the electron gap equation in Minkowski and Temporal Euclidean space. The dynamical generation of imaginary part of the fermion mass leads to the absence of Khallen-Lehmann representation, providing thus confining solution for all value of m. Apart very small κ the real pole in the propagator is absent as well. Similarly to Euclidean QED3 Minkowski QED2+1 exhibits spontaneous chiral symmetry breaking the mass function has nontrivial solution in the limit m = 0, however the mass is complex function. Furthermore, we compare with QED solved in similar approximation in spacelike Euclidean and Temporal Euclidean space. As a interesting results, although based on the simple ladder approximation, is the proof of the exact equivalence between the theories defined in Minkowski 2+1 and 3D Temporal Euclidean space. We expect large quantitative changes when the polarization effect is taken account, especially the existence of critical number of flavors can be different when compared to the known Euclidean space estimates. Opposite to naive belief we showed and explained that the Wick rotation -the well known calculational trick in quantum theory- provides continuation of Schwinger function of the Euclidean theory which do not correspond with the Greens function calculated directly in the original Minkowski space. We can note our finding has a little to do with the know usefulness of various extensions (largely anisotropic and nonrelativistic) of QED3 as an effective theories of the pseudogap insulator/superconducting phase transition. The finite temperature QED3 Euclidean action seems to be enough for the effective description of the phenomena. However the inequivalence between Euclidean and spacelike Minkowski subspace should be kept mind when trying to make some conclusion based on the naive continuation the real time metric. More interestingly, one can expect new outcomes when the similar ideology is applied to the strong coupling even dimensional theories, e.g. to the most successful theory of the strong interaction in the nature: QCD. Note, similar strategy in renormalizable theories is not so straightforward because of presented infinities. In this case, following the experience from QED2+1 here and proposing an approximations in Temporal Euclidean space can offer a simple guide which can be realistic enough to provide reasonable Minkowski space SDEs framework for low energy hadronic physics. (author)