Biorthogonal quantum mechanics
Open accessAbstract
The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
If a given eigenvalue λ_α has g_α < p_α, with p_α ≥ 2, the spectral theorem asserts it is possible to find p_α independent eigenvectors {|ζ_α^(m)〉〉}, m=1,...,p_α, all with eigenvalue λ_α.
“If a given eigenvalue λ α has g α < p α , with p α ≥ 2, the spectral theorem [24, 31, 46, 47] of linear algebra asserts that it is possible to find p α independent eigenvectors, {| ζ α ( m ) 〉 〉} , m = 1 , 2 , . . . , p α , all with eigenvalue λ α .”
From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [47]
Section 2 establishes linear independence only for nondegenerate eigenstates. In the discussion of exceptional points (page unavailable), Brody says that eigenvalues and their corresponding eigenstates can coalesce at a degeneracy. That is contrary to the claim that an eigenvalue with fewer independent eigenvectors than its multiplicity must nevertheless have that many independent eigenvectors. (LLM, conf=0.87)
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
The citation lists haven’t been fetched yet
References 55 · Cited by 573
Research works printed but not linked: 3
- Müller M P and Masanes L 2013 Three-dimensionality of space and the quantum bit: an information-theoretic approach New J. Phys. 15 053040 · DOI
- Brody D C and Graefe E M 2010 Coherent states and rational surfaces J. Phys. A: Math. Theor. 43 255205 · DOI
- Bender C M and Kuzhel S 2012 Unbounded C symmetries and their nonuniqueness J. Phys. A: Math. Theor. 43 444005 · DOI
- Authors
- Brody, D. C.
- Affiliations
- Brunel University of London (GB)
- Corresponding author
- Brody, Dorje C
- ORCID
- Brody, Dorje C
- Venue
- Journal of Physics A: Mathematical and Theoretical
- Publisher
- IOP Publishing
- Published
- 2013-12-24
- Volume
- 47
- Issue
- 3
- Pages
- 035305
- ISSN
- 1751-8113 (ISSN-L)
- 1751-8121
- OpenAlex
- W2011317146
- Cited by
- 573
- References
- 55
- Citations by year
- 2026: 71 · 2025: 88 · 2024: 65 · 2023: 78 · 2022: 79 · 2021: 51 · 2020: 40 · 2019: 32 · 2018: 24 · 2017: 19 · 2016: 12 · 2015: 9 · 2014: 5
- FWCI
- 4.46
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Journal of Physics A Mathematical and Theoretical (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Brunel University Research Archive (BURA) (Brunel University London) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Keywords
- non-Hermitian operators, real eigenvalues, quantum observables, spinning particles, quantum measurement, entanglement, perturbation theory, finite-dimensional Hilbert space, distributed parameter systems
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
