Schrödinger representation and Casimir effect in renormalizable quantum field theory

Abstract

We show that the Schrödinger representation exists in renormalizable quantum field theory to all orders in the perturbation expansion. In this sense, completeness of the Schrödinger states also holds. However, the field operator that is being diagonalized on a smooth three-dimensional hypersurface differs from the usual renormalized one by a factor that diverges logarithmically if the distance from the hypersurface goes to zero. This requires a limit procedure to be employed if expectation values of the renormalized field operator are to be computed in this representation. The Schrödinger functional differential operator involves point splitting Δ and has coefficients depending logarithmically on Δ, and also some by factors Δ −1, Δ −2, Δ −4. Details are given for the massless φ 4 4 theory, but the extension to other models, in particular with spin- 1 2 fermions, is outlined. The Casimir potential for disjoint surfaces is shown to be finite to all orders in the perturbation expansion, and computed for a pair of parallel plates to first order in massless φ ν 4.