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Making sense of non-Hermitian Hamiltonians
Open accessAbstract
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive! Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
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“In the so-called PT -symmetric formulation [39, 40, 41, 43], for example, an operator qualifies as an observable if it is simultaneously invariant with respect to both parity ( P ) and time-reversal ( T ) operations.”
From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [43]
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“In non-Hermitian quantum mechanics [39, 40, 41, 43], the formal definition of an operator representing a physical observable posits on entirely different criteria and are more relaxed with respect to the Hermiticity requirement in Hermitian quantum mechanics.”
From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [43]
In the citing paper
“This inner product, however, results in indefinite norms for the non-Hermitian, but PT -symmetric case [238].This observation will become crucial below, since this means that for the geometric approach the angle (67) will have to be re-defined.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [238]
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Metadata
- Authors
- Bender, C. M.
- Affiliations
- Los Alamos National Laboratory (US)
- Corresponding author
- Bender, Carl M
- ORCID
- Bender, Carl M
- Venue
- Reports on Progress in Physics
- Publisher
- IOP Publishing
- Published
- 2007-05-30
- Volume
- 70
- Issue
- 6
- Pages
- 947-1018
- ISSN
- 0034-4885 (ISSN-L)
- 1361-6633
- OpenAlex
- W1997323544
- Cited by
- 2,984
- References
- 162
- Citations by year
- 2026: 167 · 2025: 204 · 2024: 237 · 2023: 233 · 2022: 260 · 2021: 199 · 2020: 211 · 2019: 198 · 2018: 183 · 2017: 172 · 2016: 170 · 2015: 137 · 2014: 128 · 2013: 111 · 2012: 113
- FWCI
- 38.76
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Reports on Progress in Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy
- Los Alamos National Laboratory
- 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
- non-Hermitian Hamiltonian, PT symmetry, real energy spectrum, unitary evolution, Lee model, ghost states, renormalization
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 ›
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