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Journal of Mathematical Chemistry

Indistinguishability in DR n-fold point-sets & their $${\mathcal{S}}_n$$ -invariant dual projective mappings: limitations imposed on Racah–Wigner algebras for Liouville spin dynamics of [A] n X multi-invariant NMR systems

B. C. Sanctuary, Francis P. Temme

Published Jul 7, 2007• 1 citation
Publisher
DOI
10.1007/s10910-007-9236-8

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Indistinguishability in DR n-fold point-sets & their Sn{\mathcal{S}}_nSn​ -invariant dual projective mappings: limitations imposed on Racah–Wigner algebras for Liouville spin dynamics of [A] n X multi-invariant NMR systems
Augmented quasiparticle (QP) mapping techniques for indistinguishable point sets, based on (Liouvillian tensorial) democratic recoupling (DR), are shown to induce a full 1:1 invariant‐labelling of the underlying (disjoint) projective mapping subspaces. This theoretical simple reducibility SR‐driven view of dual group algebras underlies certain parallel limitations of Jucys graph recoupling and (in consequence) standard Racah‐Wigner (R‐W) algebras for tensorial point sets once it is applied to indistinguishable point sets. The explicit invariants (for labelling of all disjoint carrier subspaces) and their cardinality are central to automorphic dual (symmetry) tensorial formalisms, because they provide the essential quantal‐completeness condition for DR Liouvillian tensorial sets. By drawing on augmented‐QP properties [2002, Int. J. Quantum Chem. 89, 429; 1993, Physica A 198 245]—beyond the classic Hilbert boson‐pattern overview of Louck & Biedenharn [1979, Permutation group in physics ..,Springer] of simple carrier space projective mapping—a definitive theoretical basis is derived for Atiyah and Sutcliffe's recent assertions [2002, Proc. Roy Soc., Lond. A 458, 1089], concerning the specific limitations of graph recoupling theory to distinct point sets. Direct insight is obtained into the central question of why general index ‘n’ matrix formulations of multi‐invariant DR tensorial set‐based, (automorphic) spin problems are not amenable to analytic treatment—even utilising indirect Lévi‐Civitá‐based cyclic‐commutation (R‐W) approaches, e.g. of [Lévy‐Leblond, Lévy‐Nahas, 1965, J. Math. Phys. 6, 1372], unless the tensorial system is constrained to a mono‐invariant form. The importance of this (DR) tensorial‐set work comes from its focus on the specialised nature of indistinguishable point sets in (multi)invariant theoretic views of quantum dynamical spin physics.
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.