Nelson's symmetry and all that in the Yukawa2 and (φ4)3 field theories

We prove Guerra's theorem, φ bounds and Fröhlich bounds in the Y₂ and φ₃⁴ field theories. Among our technical results of interest is a proof that Z ≠ 0 in φ₃⁴ and that the spatially cutoff vacuum in Y₂ has a charge zero component. The two main inputs are Osterwalder-Schrader positivity in the spatial direction as well as in the time direction, and a finite renormalization of the “usual” partition function and Hamiltonian so that Euclidean and Hamiltonian counterterms match exactly.