The basic resolvents of position and momentum operators form a total set in the resolvent algebra

Abstract

Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents (aQ+bP-ir)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(aQ + bP - i r)^{-1}$$\\end{document} for real constants a,b,r≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a,b,r \ e 0$$\\end{document}. This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {R}$$\\end{document} generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results are obtained for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.