Beyond simple point sets to the indistinguishable point-set tensorial limits of R–W algebras: multiple G-invariants and the carrier-map-based quantal-basis completeness of uniform NMR spin systems
Francis P. Temme
Augmented quasiparticle (QP) mappings, as applied to indistinguishable point sets of (Liouvillian) democratic-recoupled (DR) tensors, provide for a 1:1 invariant labelling of the underlying (disjoint) dual projective map carrier subspaces, where the Liouville pattern basis set is defined via superboson unit-tensor actions on a null space, |∅⟩⟩. The co-operative-action Liouville algebras described here imply parallel limitations to Jucys graph recoupling and its related Racah–Wigner (R–W) algebras once DR indistinguishable point tensorial sets are involved, as in non-SR dominant (NMR) spin symmetry. The importance of G-invariants, as labels for disjoint carrier subspaces in such automorphic spin symmetries, arises from their essential role in defining the quantal-completeness of indistinguishable point sets. From the established properties of augmented-QPs as super-bosons (Temme 2002 Int. J. Quantum Chem. 89 429) (i.e., beyond the earlier Hilbert-space-based Louck and Biedenharn boson pattern views), insight into Atiyah and Sutcliffe's (A–S) assertions (Atiyah and Sutcliffe 2002 Proc. R. Soc. A 458 1089) on the limitations of graph recoupling theory to distinct point sets is obtained. This clarifies the wider analytic intractible of automorphic DR spin systems—beyond the Lévi–Civitá cyclic-commutation (R–W) approach (Lévy-Leblond and Lévy-Nahas 1965 J. Math. Phys. 6 1372) which holds for a mono-invariant problem. For (rotating-frame) density matrix approaches to [ A ] n , [ A ] n ( X ) and (dual) NMR systems, the focus is necessarily on the specialized nature of indistinguishable point sets within multiple invariant-theoretic-based, dynamical spin physics. Here, the GI(s) (GI-cardinality) constitute an important part of the dual irrep set, , with combinatorics, as a central facet of invariant theory, playing a crucial role in the concept of 'quantal completeness' and the impact of A–S's assertion on the existing NMR tensorial physics. Clearly, the role of Liouvillian Yamanouchi projection, now as disjoint subspatial-based transformational properties, defines such DR bases and their unit tensors. A brief outline is given of the structure of augmented general-indexed Lévi–Civitá superoperators with their multiple GI-labelled carrier subspaces.
Journal of Physics A Mathematical and Theoretical•Dec 12, 2007•0 citations