Generalized Operators

A formal expression T in creation and annihilation operators (e.g., the Hamiltonian for a field theory model) is generally not a densely defined bona fide Hilbert space operator but is usually a densely defined sesquilinear form; as such it is convenient to consider it as a linear map from a dense domain Φ− of a Hilbert space Φ0 to a still larger space Φ+ of antilinear functionals on Φ−; that is, T:Φ−→Φ+⊃Φ0. We give here the basis of a mathematical structure theory of such generalized operators. The idea which we explore is that, associated with T, there is a (not necessarily unique) analytic family Rλ of generalized operators called the resolvent of T. Formally, Rλ = (λ − T)−1, an equation to which we give more precise interpretations. The ambiguities in determining Rλ are associated with the arbitrary adjustments that are characteristic of renormalization programs. When appropriate conditions are met, we can construct from Rλ a new Hilbert space Ψ0 and a bona fide operator TR (the renormalized T) which is related to T by a formal intertwining equation TRΔ = ΔT, where Δ maps Φ− into a space containing Ψ0. Given several generalized operators, we outline a procedure by which a subset of these can be renormalized to bona fide operators while the rest are reinterpreted as new generalized operators in the new Hilbert space. These are the rudiments of a multiplicity theory. Numerous examples illustrate the methods; in particular, the Nθ sector of the Lee model with arbitrary cutoff (including none) is treated in detail.