Fluctuations and renormalization of a field on a boundary

Abstract The renormalization of the ( ϕ 2 ) 2 n -vector field theoretical model in semi-infinite geometry, with a quadratic surface coupling, implies a derivative surface counterterm. This counterterm destroys the positive definiteness of the action, and introduces ambiguities and unacceptable infinities. We reexamine the mechanism which makes it indeed not needed, and show the role played by the fluctuations of the surface field, which build up the extra renormalization Z 1 1 2 of the surface field.