Equivalence of Basis-Dependent and Basis-Independent Approach to Canonical Field Operators

Abstract

The basis‐independent approach to canonical commutation relations (CCR's), which allows arbitrary test‐function spaces for smearing the field operators, is a generalization of the basis‐dependent approach, in which the fields are smeared with an orthonormal system (or finite linear combinations thereof) to obtain an infinite set of q_k and p_k. Using recent results on continuity properties of representations of the CCR's, we show that every representation of the basis‐independent type in a separable Hilbert space can be obtained by continuous extension of a suitable representation of the basis‐dependent type where properties like irreducibility, cyclicity, etc., remain unaffected. In this sense, both approaches are equivalent, and the classification problem for CCR's is reduced from the simultaneous consideration of all representations for all possible test‐function spaces to those for a single one (up to isomorphism).