The resolvent algebra: A new approach to canonical quantum systems

Detlev Buchholz, Hendrik Grundling
2008 Journal of Functional Analysis  
The standard C * -algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian. Here a novel C * -algebra of the canonical commutation relations is presented which does not suffer from such problems. It is based on the resolvents of the
more » ... al operators and their algebraic relations. The resulting C * -algebra, the resolvent algebra, is shown to have many desirable analytic properties and the regularity structure of its representations is surprisingly simple. Moreover, the resolvent algebra is a convenient framework for applications to interacting and to constrained quantum systems, as we demonstrate by several examples. linear bosonic constraints in the context of the resolvent algebra, and find that it is considerably simpler than in the Weyl algebra. All proofs for our results are collected in Section 10. Mollifiers and resolvents There are several concepts of when a selfadjoint operator A on a Hilbert space H is affiliated with a concretely represented C * -algebra A ⊂ B(H). One is that the resolvent (iλ1 − A) −1 ∈ A for some λ ∈ R \ 0 (hence for all λ ∈ R \ 0). This notion is used by Damak and Georgescu [6] (and is weaker than the one used by Woronowicz [29] ) and it implies the usual one, i.e. that A commutes with all unitaries commuting with A (but not conversely). Observe that then Thus the resolvent (iλ1 − A) −1 = M acts as a "mollifier" for A, i.e. MA and AM are bounded and in A, and M is invertible such that M −1 MA = A = AMM −1 . This suggests that as AM and MA in A carries the information of A in bounded form, we can "forget" the original representation, and study the affiliated A abstractly through these elements. In the literature, the bounded operators A λ := iλA(iλ1 − A) −1 are called "Yosida approximations" of A [22, p. 9]. We want to apply this idea to a bosonic field as above, i.e. for a fixed Hilbert space H we assume that there is a common dense invariant core D ⊂ H for the selfadjoint operators φ(f ), f ∈ X on which the φ(f ) satisfy the canonical commutation relations. One may be tempted to find mollifiers for the operators φ(f ) in the (concretely represented) Weyl algebra Δ(X, σ ) = C * exp iφ(f ) f ∈ X ⊂ B(H), but unfortunately this is not possible because of [5]: Proposition 2.
doi:10.1016/j.jfa.2008.02.011 fatcat:jkj5jkmkcvb4nhcoolduujbi4u