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Making sense of non-Hermitian Hamiltonians
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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
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