{"arxiv_id":"math-ph/0107001","incoming":{"papers":73,"sentences":149,"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":4,"sentences":6,"quotes":[{"arxiv_id":"1204.6544","title":"Investigation of PT-symmetric Hamiltonian systems from an alternative point of view","url":"/paper/1204.6544","section":"Introduction","sentence":"Actually, such a method, called the algebraic method, has already been proposed [ref] for the pseudo-Hermitian Hamiltonian systems [ref] .","intent_p":0.85,"contests_p":null},{"arxiv_id":"2307.16805","title":"Poincar\\'{e} symmetries and representations in pseudo-Hermitian quantum field theory","url":"/paper/2307.16805","section":"Irreducible representations of $$","sentence":"As we assumed $ρ(J_3)$ to be diagonalizable, we can use its eigenvectors to construct a biorthonormal basis [ref] , from which we obtain a Hermitian matrix $π:_j↦_j$ such that $j(J_3)^†=πJ3jπ^-1$ is $π$-pseudo-Hermitian.","intent_p":0.95,"contests_p":null},{"arxiv_id":"2603.19006","title":"Non-Hermitian Structure and Exceptional Points in Yang-Mills Theory from Analytic Continuation of Nc","url":"/paper/2603.19006","section":"Construction of the $𝒞$ operator","sentence":"We define the $η̂$-matrix as [ref] : [equation] where the first sum runs over eigenstates with real eigenvalues, and the second sum runs over pairs of states $(ϕ_n_+, ϕ_n_-)$ with complex-conjugate eigenvalues.","intent_p":0.83,"contests_p":null},{"arxiv_id":"2405.14237","title":"Probing de Sitter Space Using CFT States","url":"/paper/2405.14237","section":"Conclusion and discussion","sentence":"We conducted our analysis by leveraging the correspondence between the dS dual and Euclidean CFT as a non-Hermitian system [ref] .","intent_p":0.8,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites it as support","papers":1,"sentences":1,"quotes":[{"arxiv_id":"0710.4432","title":"On the Role of the Normalization Factors $\\kappa_n$ and of the Pseudo-Metric ${\\cal P} \\neq {\\cal P}^\\dagger$ in Crypto-Hermitian Quantum Models","url":"/paper/0710.4432","section":"Pseudo-Hermitian Hamiltonians","sentence":"It is clear that the pseudo-Hermiticity itself is much less relevant because once we get through the difficult proof of the necessary reality of the spectrum [ref] , the pseudo-Hermiticity of a given $H$ becomes in fact fully equivalent to its quasi-Hermiticity [ref] .","intent_p":0.97,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":37,"sentences":53,"quotes":[{"arxiv_id":"2311.06466","title":"Quantization of Second Order Fermions","url":"/paper/2311.06466","section":"Introduction","sentence":"A pseudo-Hermitian Lagrangian satisfies [equation] This generalization of Hermiticity was proposed in [ref] extending the results of $PT$-symmetric quantum mechanics introduced in [ref] , and has two important features; 1) the energy spectrum of the theory is real and 2) time evolution is unitary upon the definition of a suitable internal product of states with the aid of the $η$ operator.","intent_p":0.83,"contests_p":null},{"arxiv_id":"2209.07308","title":"Constructible reality condition of pseudo entropy via pseudo-Hermiticity","url":"/paper/2209.07308","section":"Introduction","sentence":"The necessary and sufficient conditions for the pseudo-Hermiticity of $M$ are given by the following theorem [ref] .","intent_p":0.81,"contests_p":null},{"arxiv_id":"0908.1755","title":"Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty","url":"/paper/0908.1755","section":"$η$ pseudo-Hermiticity","sentence":"We recall that a Hamiltonian $H$ is called $η$ pseudo-Hermitian if it satisfies the condition [ref] [equation] where $η$ is a Hermitian operator.","intent_p":0.96,"contests_p":null},{"arxiv_id":"0710.4432","title":"On the Role of the Normalization Factors $\\kappa_n$ and of the Pseudo-Metric ${\\cal P} \\neq {\\cal P}^\\dagger$ in Crypto-Hermitian Quantum Models","url":"/paper/0710.4432","section":"Introduction","sentence":"Recently [ref] , the family of the productive symmetry-related mathematical tools has been enriched by the so called $PT$-symmetry where the vanishing commutator ([ref]) contains an antilinear operator $S=PT$ [ref] .","intent_p":0.99,"contests_p":null},{"arxiv_id":"2006.11449","title":"Bi-orthogonal harmonics for the decomposition of gravitational radiation I: angular modes, completeness, and the introduction of adjoint-spheroidal harmonics","url":"/paper/2006.11449","section":"Spheroidal Harmonic Decomposition","sentence":"Of principle relevance here are dual or what we will refer to as adjoint functions, and their role in bi-orthogonal decomposition [ref] .","intent_p":0.89,"contests_p":null}]}],"unlabelled_sentences":89},"outgoing":{"papers":0,"sentences":0,"groups":[],"unlabelled_sentences":0},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":{"arxiv_id":"math-ph/0107001","kind":"model","paper_version":3,"prompt_version":"held-up-v3.5","generated_at":"2026-10-27T13:59:14+00:00","loaded_at":"2026-10-28T01:01:59.449708+00:00","evidence":"substantial","account":[],"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":3,"entries":[{"p":0.99,"quote":null,"citing":[{"self":false,"year":2021,"title":"Conserved quantities, exceptional points, and antilinear symmetries in non-Hermitian systems","arxiv_id":"2104.11265"}],"verified":true,"paraphrase":"Proceeded by direct construction using spectral decomposition techniques from this paper to show the existence of conserved operators."},{"p":0.92,"quote":null,"citing":[{"self":false,"year":2020,"title":"Non-reciprocal phase transitions","arxiv_id":"2003.13176"}],"verified":true,"paraphrase":"Following this paper, expressed the matrix and the operators in terms of Pauli matrices."},{"p":0.83,"quote":null,"citing":[{"self":false,"year":2026,"title":"Non-Hermitian Structure and Exceptional Points in Yang-Mills Theory from Analytic Continuation of Nc","arxiv_id":"2603.19006"}],"verified":true,"paraphrase":"Defined its eta-matrix as this paper did."}],"papers_all":10,"self_papers":0,"self_entries":[]},{"kind":"supports_this_paper","label":"Cites it as support","papers":2,"entries":[{"p":0.96,"quote":null,"citing":[{"self":false,"year":2023,"title":"Observation of higher-order exceptional points in pseudo-Hermitian radio-frequency circuits","arxiv_id":"2305.00900"}],"verified":true,"paraphrase":"Demonstrated that its proposed system's eigenfrequency characteristics are consistent with those of a pseudo-Hermitian system as in this paper."},{"p":0.95,"quote":null,"citing":[{"self":false,"year":2024,"title":"Three perspectives on entropy dynamics in a non-Hermitian two-state system","arxiv_id":"2404.03492"}],"verified":true,"paraphrase":"Said that this realizes a general underlying result of this paper, that a theory with orthogonal inner product and non-real eigenvalues is not pseudo-Hermitian."}],"papers_all":4,"self_papers":0,"self_entries":[]}],"counts":{"by_kind":{"extends":1,"comparison":4,"method_or_tool":10,"supports_this_paper":4},"dropped":{"relied_on":0,"unknown_ids":0,"account_lines":0,"contest_vetoed":0,"below_threshold":0,"contest_drawback":0,"contest_without_basis":0,"contest_not_one_sentence":0,"contest_on_group_citation":0},"papers_read":37,"self_papers":0,"citing_papers":253,"sentences_read":38,"citing_sentences":420,"unambiguous_read":38,"typed_independent":18,"independent_papers":253,"by_kind_independent":{"extends":1,"comparison":4,"method_or_tool":10,"supports_this_paper":4}},"input_sha256":"ade1ae393d5be239b07b4f0f0333cec327025188e076b10caec8c5875299edd5","method":"For a paper with five or more citing arXiv papers, Pith selected up to 40 citing sentences its citation classifier typed, read those whose citation markers point at this paper alone, and wrote one paraphrase for each later paper in each group, each resting on one sentence. A later paper is listed under a label only when the classifier gives one of its sentences that label at intent probability 0.8 or more (0.95 for support), Pith's reading of the sentence finds that the citing paper itself did it, and a second check of the paraphrase against the sentence passes. The section lists uses only: building on the paper, using its method, tool, data or values, comparing with it and citing it as support. It does not list criticism or disagreement, so their absence says nothing about how the paper has been received. Code counted the papers, wrote the summary from the listed entries, set the evidence level and linked each entry to its citing paper. Quotes are copied from the citing sentence and shown only when the citing paper's licence is CC BY, CC BY-SA or CC0; otherwise the paraphrase is shown. Papers that share an author with this one are listed apart. A paper with fewer citing papers gets a line counted by code, which names uses only: a later paper whose only label is a contest is counted as not described.","line":null,"summary":"5 later papers by other authors are listed below: 3 used its method or tool and 2 cited it as support."},"_links":{"html":"https://pith.science/paper/math-ph/0107001"}}