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

The DOI matches its Crossref record; 184 of the 184 references it cites were found in a registry or an index.

Making sense of non-Hermitian Hamiltonians

Open access
Bender, Carl M·2007·Reports on Progress in Physics
DOI 10.1088/0034-4885/70/6/r03
Open this paper in CoChat — free accountOpen PDF

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.

Overview

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.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1088/0034-4885/70/6/r03

Cites

184 of 184 references verified0 citation errors · 0 not checkable

Statistics

Verification

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.

A few free checks a day · papers already verified open instantly · results are public

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?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Research works printed but not linked: 15

  1. Barton G, 1963, Introduction to Advanced Field Theory
  2. Streater R F, 1964, PCT, Spin and Statistics, and All That
  3. Fring A, 2007, J. Phys. A: Math. Theor.
  4. Bender C M, 1978, Advanced Mathematical Methods for Scientists and Engineers
  5. Bender C M, 2007, J. Math. Phys.
  6. Federov Y N, 2007, Physica
  7. Bleuler K, 1950, Helv. Phys. Acta., 23, 567
  8. Faria C F de M, 2006, Laser Phys.
  9. Ushveridze A G, 1993, Quasi-Exactly Solvable Models in Quantum Mechanics
  10. Turbiner A V, 1988, Sov. Phys.-JETP, 67, 230
  11. Curtright T, 2006, J. Math. Phys.
  12. Henkel M, 2003, Classical and Quantum Nonlinear Integrable Systems: Theory and Applications
  13. Källén G, 1955, Mat.-Fys. Medd., 30
  14. Schweber S S, 1961, An Introduction to Relativistic Quantum Field Theory
  15. Symanzik K, 1973, Nuovo Cim., 6, 77

Cited by

Sorted by citations

See all 2,984

Related

See all

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
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

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