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Canadian Journal of Physics

Role of democratic SU2 x n dual-group carrier space and group structure in the superboson quantum-Liouville physics of identical spin ensembles: Maximal n tensor product reduction, via symbolic algebraic combinatorics

Francis P. Temme

Published Sep 1, 2001• 0 citations
Publisher

Abstract

Structural aspects of superboson mappings and their dual group-based carrier spaces inherent in quantum-Liouville NMR formalisms are presented for their conceptual role in understanding the transformational properties and the spin dynamics of identical [A]n([Formula: see text]n) spin ensembles or [AX]n systems, given in terms of {Tkq (υ)} º {|kqυ > >} tensorial bases. Interest in the explicit democratic labelling of the Liouvillian carrier subspaces of [A]n spin ensembles, prompts an examination of the inner products (ITPs) that define the projective carrier space associated with superbosons. A weak-branching (WB) limit (for (bipartite) partitions λ [Formula: see text] n, for n-indexed (SU2 ×)[Formula: see text]n) ITPs gives rise to: (i) maximal coefficient sets for [µ] [Formula: see text] [µ']([Formula: see text]n) irrep products (in Butler's notation), under a sufficiently high (n [Formula: see text] 4µ, 4µ') indexed [Formula: see text]n group, or (ii) to identical numerical {c[Formula: see text], λ' } sequence-ordered sets of reduction coefficients over similar, displaced-sequence fields — on augmenting ITP to [µ] [Formula: see text] [µ''], where µ'' > µ' > µ, — or else (iii) to some (sequence-displaced) subset of the latter. The origins of the decompositional WB limit as part of group structure may be traced to an algorithmic similarity between the Littlewood–Richardson and Young(III) combinatorial rules. Alternative approaches to the bipartite product decompositional mappings are possible, using both (subspatial-restricted) Schur-function techniques of Wybourne and [Formula: see text]16[Formula: see text]n[Formula: see text]28 symbolic algorithmic enumerations based on the SYMMETRICA discrete-maths. package of Kerber et al. (J. Symbolic Comput. 14, 195 (1993)). The nature of maximal sets and scaling in ITP decompositions is established and recognition given to the role of combinatorics, [Formula: see text]n algorithms and the superboson algebras of the SU(2) × [Formula: see text]n group in (multiple) quantized spin physics. PACS Nos.: 02.10, 03.65Ca, 33.20Vq, 33.25+k

DOI
10.1139/p01-026

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Role of democratic SU2 x n dual-group carrier space and group structure in the superboson quantum-Liouville physics of identical spin ensembles: Maximal n tensor product reduction, via symbolic algebraic combinatorics

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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.

AIP conference proceedings•Jan 1, 2007•0 citations
Structural aspects of superboson mappings and their dual group-based carrier spaces inherent in quantum-Liouville NMR formalisms are presented for their conceptual role in understanding the transformational properties and the spin dynamics of identical [A]n([Formula: see text]n) spin ensembles or [AX]n systems, given in terms of {Tkq (υ)} º {|kqυ > >} tensorial bases. Interest in the explicit democratic labelling of the Liouvillian carrier subspaces of [A]n spin ensembles, prompts an examination of the inner products (ITPs) that define the projective carrier space associated with superbosons. A weak-branching (WB) limit (for (bipartite) partitions λ [Formula: see text] n, for n-indexed (SU2 ×)[Formula: see text]n) ITPs gives rise to: (i) maximal coefficient sets for [µ] [Formula: see text] [µ']([Formula: see text]n) irrep products (in Butler's notation), under a sufficiently high (n [Formula: see text] 4µ, 4µ') indexed [Formula: see text]n group, or (ii) to identical numerical {c[Formula: see text], λ' } sequence-ordered sets of reduction coefficients over similar, displaced-sequence fields — on augmenting ITP to [µ] [Formula: see text] [µ''], where µ'' > µ' > µ, — or else (iii) to some (sequence-displaced) subset of the latter. The origins of the decompositional WB limit as part of group structure may be traced to an algorithmic similarity between the Littlewood–Richardson and Young(III) combinatorial rules. Alternative approaches to the bipartite product decompositional mappings are possible, using both (subspatial-restricted) Schur-function techniques of Wybourne and [Formula: see text]16[Formula: see text]n[Formula: see text]28 symbolic algorithmic enumerations based on the SYMMETRICA discrete-maths. package of Kerber et al. (J. Symbolic Comput. 14, 195 (1993)). The nature of maximal sets and scaling in ITP decompositions is established and recognition given to the role of combinatorics, [Formula: see text]n algorithms and the superboson algebras of the SU(2) × [Formula: see text]n group in (multiple) quantized spin physics. PACS Nos.: 02.10, 03.65Ca, 33.20Vq, 33.25+k
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 S n –based, uniform auxiliary dual tensorial (UDT) sets, explicitly with even maximal all–alike j i ( k i ) over v ≡ ( j 1 ⋯ j 2 n ), or ( k 1 ⋯ k 2 n ), sub–rank auxiliary labelling, the specific time–reversal invariance–dependent dual group invariants and their independent cardinalities (| SI | (2 n ) ) are fundamental to understanding simply reducible (SR) dynamical spin system structure over (Liouvillian) carrier space, a quantum physics property implicit in dual quasi–particle (QP) formalisms associated with, for example, uniform identical multiple spin NMR systems . Such UDT sets stand in total contrast to the distinct j i ( k i ) of ( j 1 ⋯ j″ n )(( k 1 ⋯ k″ n )) sub–rank orthogonal tensorial sets with their Z aa′ graph invariants which for reasons discussed in the text are of restricted validity. For UDTs as operator bases related to the (spin) interaction network–based tensors of high S n permutational NMR Hamiltonians (Liouvillians), the use of QP boson (superboson) formalisms and their dual projective maps (F. P. Temme 1993 Physica A 198 , 245–261) prove invaluable. Once the independent cardinality of the multiple invariants of these UDTs (alias the SU (2) × S 2 n group invariants) (here denoted | SI | (2 n ) ) are known in terms of democratic re–coupling (DR) and polyhedral combinatorial (PC) sampling applied to the time–reversal invariance (TRI) properties of the uniform multiple spin (sub–)system, so the SR dual carrier spaces , ℍ˜, ℍ˜ v follow directly as well–defined entities. For UDTs of general (2 n > 6)–fold ensembles, the augmented models of TRI discussed here (i.e. beyond linear re–coupled Weyl forms) are essential, due to the presence of non–Abelian degeneracy. A regular geometric automorphic limit is established for sequential (2 n )–based | SI | (2 n ) total series, specific to (higher) uniform ( j 1 ⋯ j i ), ( k 1 ⋯ k i )–based dual tensors. By treating the | SI | (2 n ) obtained from the DR–based TRIs as group measures and invoking an automorphic geodesic augmentation, certain otherwise inaccessible, additional | SI | (2 n′ ) (and hence UDTs) may be established; these are based on sporadic higher (regular) (2 n ≫ 12)–fold ensemble invariants and are obtained via finite group duality properties implicit in the concept of group measures. The PC DR–based views of general (2 n ) interacting spins of UDTs expressed here give specific insight into dynamic structure over (explicit TRI invariant–defined) Liouvillian carrier spaces. This lattice point (Erdösian) view of DR shows that UDT invariants are distinct and more contracted than predicted by the earlier linear–re–coupling–based | D 0 ( U )|(⨷ SU (2)) approach. The presence of degeneracy in general high (2 n ) UDTs as non–Abelian entities ensures that the homomorphic–based, | D 0 ( U )| enumeration (with its implied linear graphical re–coupling) is invalid for high (2 n ) uniform (NMR) multi–spin systems whose TRI properties are governed by DR re–coupling. The prime focus of the report concerns the positive conceptual value of PC lattice–point–based DR, as applied to the S 2 n –invariants which define UDTs (of NMR) and their completeness. Brief contrasts with the earlier, more restricted O (2 k + 1) (overall) invariants of ( O ( n ) ⊃ ⋯ ⊃ G )–based democratic re–coupled quantum systems are given for completeness.
Quasi-Particle Dual Mappings on H~v\widetilde{\mathbb{H}}_vHv​ Carrier Spaces for Dynamical Structures of S 2n -Invariant-Based, Integer-Rank Tensorial Sets: A Democratic-Recoupling S 2n -Compatible Overview of v-Auxiliary Labelling in Superboson Algebras of Uniform Multispin NMR Ensembles
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.