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Cited by 76Quantum Mechanics and Non-Hermitian PhysicsQuantum Mechanics and ApplicationsSpectral Theory in Mathematical Physics

Consistency of PT-symmetric quantum mechanics

Open access
Brody, Dorje C·2016·Journal of Physics A: Mathematical and Theoretical
DOI 10.1088/1751-8113/49/10/10lt03
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Abstract

In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric—the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss.

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  • 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 &lt; 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 [46]

    The paper’s discussion of the biorthogonal framework (indexed as Abstract; no page supplied) starts with N linearly independent vectors, but does not address an eigenvalue with gα < pα. Searches for the spectral theorem, multiplicities, and pα independent eigenvectors found no passage making the attributed assertion. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

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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
2016-01-29
Volume
49
Issue
10
Pages
10LT03
ISSN
  • 1751-8113 (ISSN-L)
  • 1751-8121
DOI
10.1088/1751-8113/49/10/10lt03
OpenAlex
W1895364685
Cited by
76
References
42
Citations by year
2026: 3 · 2025: 6 · 2024: 7 · 2023: 4 · 2022: 12 · 2021: 9 · 2020: 5 · 2019: 6 · 2018: 6 · 2017: 12 · 2016: 6
FWCI
4.73
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Physics A Mathematical and Theoretical (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Surrey Research Insight Open Access (The University of Surrey) (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 Non-Hermitian Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
Keywords
PT symmetry, causality, metric operator, non-Hermitian Hamiltonian, closed quantum systems, finite-dimensional Hilbert space, open quantum systems, balanced gain and loss

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