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, ${\mathrm{T:\Phi }}_{-}{\mathrm{\to \Phi }}_{+}{\mathrm{\supset \Phi }}_0.$ 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 R_λ of generalized operators called the resolvent of T. Formally, R_λ = (λ − T)⁻¹, an equation to which we give more precise interpretations. The ambiguities in determining R_λ are associated with the arbitrary adjustments that are characteristic of renormalization programs. When appropriate conditions are met, we can construct from R_λ a new Hilbert space Ψ₀ and a bona fide operator T_R (the renormalized T) which is related to T by a formal intertwining equation T_RΔ = 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.