Geodesic and re–coupling–induced limits toS2ngroup invariants of uniform (k1⋯k2n) dual tensorial sets in spin physics, or NMR: SR dynamical structure on {ℍv} via polyhedral re–coupling and time–reversal invariance

Abstract

ForSn–based,uniform auxiliarydual tensorial (UDT) sets, explicitly with even maximal all–alikeji(ki) overv≡ (j1⋯j2n), or (k1⋯k2n), sub–rank auxiliary labelling, the specific time–reversal invariance–dependentdual group invariantsand their independent cardinalities (|SI|(2n)) 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 thedistinctji(ki) of (j1⋯j″n)((k1⋯k″n)) sub–rank orthogonal tensorial sets with theirZaa′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 highSnpermutational NMR Hamiltonians (Liouvillians), the use of QP boson (superboson) formalisms and their dual projective maps (F. P. Temme 1993PhysicaA198, 245–261) prove invaluable. Once the independent cardinality of the multiple invariants of these UDTs (alias theSU(2) ×S2ngroup invariants) (here denoted |SI|(2n)) 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 theSR dual carrier spaces, ℍ˜, ℍ˜vfollow directly as well–defined entities. For UDTs of general (2n> 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 (2n)–based |SI|(2n)totalseries, specific to (higher) uniform (j1⋯ji), (k1⋯ki)–based dual tensors. By treating the |SI|(2n)obtained from the DR–based TRIs as group measures and invoking an automorphic geodesic augmentation, certain otherwise inaccessible, additional |SI|(2n′)(and hence UDTs) may be established; these are based on sporadic higher (regular) (2n≫ 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 (2n) 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 |D0(U)|(⨷SU(2)) approach. The presence of degeneracy in general high (2n) UDTs as non–Abelian entities ensures that the homomorphic–based, |D0(U)| enumeration (with its implied linear graphical re–coupling) is invalid for high (2n) 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 theS2n–invariants which define UDTs (of NMR) and their completeness. Brief contrasts with the earlier, more restrictedO(2k+ 1) (overall) invariants of (O(n) ⊃ ⋯ ⊃G)–based democratic re–coupled quantum systems are given for completeness.