Perturbations of the Laplacian supported by null sets, with applications to polymer measures and quantum fields

We discuss Hamiltonians in L2(Rsup(d), dx) of the form H = -Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a Brownian motion b such that V(x) = ∫(1-0)lambda(x,ωdelta(x-b(s,ω)ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of -Δ when d <= 5. We must choose lambda to be an infinitesimal, negative function for d = 4,5, but for d <= 3 any bounded real-valued functions lambda will dol. The connection with Edwards' models of polymers as well as with quantum fields of the phisub(d)4-type is also discussed. The proofs use methods of nonstandard analysis. (orig.)