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Physica A Statistical Mechanics and its Applications

Physical role of -invariants in dual mappings over carrier spaces for Liouville multiple-quantum NMR of spin clusters within n-automorphisms: , -cubanes and some SU(6) aspects of a -met-carb

Johannes Pieter Colpa, Francis P. Temme

Published Aug 1, 1994• 7 citations
Publisher
DOI
10.1016/0378-4371(94)90054-x

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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.

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The structure and Gel’fand patterns inherent in the Heisenberg supergenerator algebra of dual Liouville-space ‖k q{K.}(k1−k n):[λ],𝒮n〉〉 tensors and the Cayley algebra of scalar invariants over a (k1−k n) field

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For many-body spin cluster problems, dual-symmetry recoupled tensors over Liouville space provide suitable bases for a generalized torque formalism using the 𝒮n-adapted density operator in which to discuss NMR and related techniques. The explicit structure of such tensors is considered in the context of the Cayley algebra of scalar invariants over a field, specified by the inner ki rank labels of the Tkq(kl−kn)s. The pertinence of both lexical combinatorial architectures over inner rank sets and SU2 propagative topologies in specifying the structure of dual recoupling tensors is considered in the context of the 𝒮n partitional aspects of spin clusters. The form of Heisenberg superoperator generators whose algebra underlies the Gel’fand pattern algebra of SU(2) and SU(2)×𝒮n tensor bases over Liouville space is presented together with both the related 𝓈-boson algebras and a description of the associated {‖2k 0〉〉} pattern sets of CF29H carrier space under the appropriate symmetry. These concepts are correlated with recent work on SU(2)×𝒮n induced symmetry hierarchies over Liouville spin space. The pertinence of this theoretical work to an understanding of multiquantum NMR in Liouville space formalisms is stressed in a discussion of the nature of pathways for intracluster J coupling, which also gives a valuable physical insight into the nature of coherence transfer in more general spin-1/2 systems.

Journal of Mathematical Physics•Jun 1, 1991•11 citations
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.
For many-body spin cluster problems, dual-symmetry recoupled tensors over Liouville space provide suitable bases for a generalized torque formalism using the 𝒮n-adapted density operator in which to discuss NMR and related techniques. The explicit structure of such tensors is considered in the context of the Cayley algebra of scalar invariants over a field, specified by the inner ki rank labels of the Tkq(kl−kn)s. The pertinence of both lexical combinatorial architectures over inner rank sets and SU2 propagative topologies in specifying the structure of dual recoupling tensors is considered in the context of the 𝒮n partitional aspects of spin clusters. The form of Heisenberg superoperator generators whose algebra underlies the Gel’fand pattern algebra of SU(2) and SU(2)×𝒮n tensor bases over Liouville space is presented together with both the related 𝓈-boson algebras and a description of the associated {‖2k 0〉〉} pattern sets of CF29H carrier space under the appropriate symmetry. These concepts are correlated with recent work on SU(2)×𝒮n induced symmetry hierarchies over Liouville spin space. The pertinence of this theoretical work to an understanding of multiquantum NMR in Liouville space formalisms is stressed in a discussion of the nature of pathways for intracluster J coupling, which also gives a valuable physical insight into the nature of coherence transfer in more general spin-1/2 systems.