It is possible to express a general Green's function in momentum-space as a sum of (individually non-covariant) time ordered graphs. These are obtained by carrying out the energy integrals of the loop momenta in a Feynman diagram. In this representation internal lines travel on-shell but energy is not conserved at each vertex, while in a Feynman graph energy is conserved at each vertex but lines are off the mass shell. This picture is in deep connection with the Coleman-Norton representation and gives a nice interpretation of the Feynman parameters and a causality viewpoint.
We present here a derivation of the general form of time-ordered perturbation theory in coordinate-space representation for a cubic scalar non-gauged theory to all orders. The results are easily extended to diagrams with fermionic lines and they offer a clear picture on what space-configurations of internal vertices will give unavoidable divergencies. The expected factorization ''pinch'' surface soft, collinear and hard configurations of vertices appears in TOPT and the correct degree of divergence is reproduced.