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Cited by 2,984Quantum Mechanics and Non-Hermitian PhysicsQuantum Mechanics and ApplicationsSpectral Theory in Mathematical Physics

Making sense of non-Hermitian Hamiltonians

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Bender, Carl M·2007·Reports on Progress in Physics
DOI 10.1088/0034-4885/70/6/r03
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Abstract

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 PT-symmetric formulation, an operator qualifies as an observable if it is simultaneously invariant with respect to both parity and time-reversal operations.

    “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]

    Section 4.4, “Observables,” gives a different criterion: an observable must satisfy Aᵀ = CPT A CPT. It does not require an observable to be invariant under parity and time reversal separately; the Introduction defines PT symmetry as invariance under their combined operation. (LLM, conf=0.90)

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  • In non-Hermitian quantum mechanics, the formal definition of an operator representing a physical observable posits on entirely different criteria and is more relaxed with respect to the Hermiticity requirement.

    “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]

  • The standard Hermitian inner product yields indefinite norms for non-Hermitian PT-symmetric systems, so the geometric angle must be redefined.

    “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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Research works printed but not linked: 15

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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
DOI
10.1088/0034-4885/70/6/r03
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
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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