The λφ₂⁴ quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian

We introduce an inductive method to estimate the shift δE in the vacuum energy, caused by a perturbation δH of the ℘(φ) ₂ Hamiltonian H. We prove that if δH equals the field bilinear form φ(x, t), then δE is finite. We show that the vacuum expectation values of products of fields (Wightman functions) exist and are tempered distributions. They determine, via the reconstruction theorem, essentially self-adjoint field operators φ(f), for real test functions f∈S(R ²). We also bound the perturbation of the ℘(φ)₂ Hamiltonian by a polynomial (℘₁(φ))(h) = δH, so long as ℘ + ℘₁ is formally positive. In that case, and with ||h||_∞ ≤ 1, δE is bounded by const(1 + diam supp h).