{"paper":{"title":"Central Characters of $G_{\\mathrm{NC}}$, Darboux Normalization, and the Kinematical Inequivalence of NCQM and QM","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Generic noncommutative quantum mechanics sectors are not unitarily equivalent to ordinary quantum mechanics as G_NC representations","cross_cats":["math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"S. Hasibul Hassan Chowdhury","submitted_at":"2026-02-28T07:49:53Z","abstract_excerpt":"We analyze generalized Bopp shifts and Darboux normalization in two-dimensional noncommutative quantum mechanics (NCQM) from the viewpoint of the unitary representation theory of the kinematical symmetry group \\(G_{\\mathrm{NC}}\\). This group is a step-two nilpotent Lie group with three-dimensional center, and the regular part of its unitary dual \\(\\widehat{G_{\\mathrm{NC}}}\\) is labelled by central characters \\((\\hbar,\\vartheta,B_{\\mathrm{in}})\\). Ordinary two-dimensional quantum mechanics (QM) appears inside \\(\\widehat{G_{\\mathrm{NC}}}\\) as the family of Weyl-Heisenberg representations inflate"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that a generic nondegenerate NCQM sector (ℏ₀,ϑ₀,B₀), with ℏ₀,ϑ₀,B₀≠0 and ℏ₀−B₀ϑ₀≠0, is not unitarily equivalent to the ordinary QM sector (ℏ₀,0,0) as a G_NC-representation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the kinematical symmetry group is correctly identified as the step-two nilpotent Lie group G_NC with three-dimensional center whose regular unitary dual is labelled by central characters (ℏ,ϑ,B_in), and that ordinary QM corresponds exactly to the inflated Weyl-Heisenberg representations with central character (ℏ,0,0).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generic nondegenerate NCQM sectors with nonzero central character parameters are not unitarily equivalent to ordinary QM as representations of G_NC.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generic noncommutative quantum mechanics sectors are not unitarily equivalent to ordinary quantum mechanics as G_NC representations","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0679025efdc3bce0d00b46ade51555bb197953c9e4de5c5ec35c17bd49668fad"},"source":{"id":"2603.00524","kind":"arxiv","version":2},"verdict":{"id":"31ae7cb6-edcb-4d92-ab09-4248e519357c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T18:56:01.113544Z","strongest_claim":"We prove that a generic nondegenerate NCQM sector (ℏ₀,ϑ₀,B₀), with ℏ₀,ϑ₀,B₀≠0 and ℏ₀−B₀ϑ₀≠0, is not unitarily equivalent to the ordinary QM sector (ℏ₀,0,0) as a G_NC-representation.","one_line_summary":"Generic nondegenerate NCQM sectors with nonzero central character parameters are not unitarily equivalent to ordinary QM as representations of G_NC.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the kinematical symmetry group is correctly identified as the step-two nilpotent Lie group G_NC with three-dimensional center whose regular unitary dual is labelled by central characters (ℏ,ϑ,B_in), and that ordinary QM corresponds exactly to the inflated Weyl-Heisenberg representations with central character (ℏ,0,0).","pith_extraction_headline":"Generic noncommutative quantum mechanics sectors are not unitarily equivalent to ordinary quantum mechanics as G_NC representations"},"references":{"count":14,"sample":[{"doi":"","year":1978,"title":"F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,Deformation theory and quantization. I. Deformations of symplectic structures, Annals Phys.111(1978) 61–110","work_id":"cec0774e-49eb-42e3-afb6-4067fff6dfb2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1978,"title":"F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,Deformation theory and quantization. II. Physical applications, Annals Phys.111(1978) 111–151","work_id":"6ac1765c-3e0c-483e-b85a-8505cbc5574a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"On Quantum Mechanics on Noncommutative Quantum Phase Space","work_id":"07f5e632-b24f-4e0c-b757-f6220069cebb","ref_index":3,"cited_arxiv_id":"hep-th/0309006","is_internal_anchor":true},{"doi":"","year":2004,"title":"Noncommutative Quantum Mechanics and Seiberg-Witten Map","work_id":"3718214d-f1b6-482e-b28c-8184539d13a0","ref_index":4,"cited_arxiv_id":"hep-th/0401180","is_internal_anchor":true},{"doi":"","year":2016,"title":"A comparative review of four formulations of noncommutative quantum mechanics","work_id":"54c34589-b4e4-41b3-8595-048178b619fa","ref_index":5,"cited_arxiv_id":"1603.07176","is_internal_anchor":true}],"resolved_work":14,"snapshot_sha256":"83cd2071bec73fe6e77daa73a72c4030499569da51e2dca0d125c1818fd3cb45","internal_anchors":7},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}