Ghost busting: Making sense of non-Hermitian Hamiltonians

  • Carl M. Bender


The Lee model is an elementary quantum field theory in which mass, wave-function, and charge renormalization can be performed exactly. Early studies of this model in the 1950’s found that there is a critical value of g 2, the square of the renormalized coupling constant, above which g 0 2 , the square of the unrenormalized coupling constant, is negative. For g 2 larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. In this non-Hermitian regime a new state appears whose norm is negative. This state is called a ghost. It has always been thought that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess \( \mathcal{P}\mathcal{T} \) symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having \( \mathcal{P}\mathcal{T} \) symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a time-independent operator called \( \mathcal{C} \). In terms of \( \mathcal{C} \) one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown that time evolution in such a theory is unitary. In this talk the \( \mathcal{C} \) operator for the Lee model in the ghost regime is constructed in the V/Nθ sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of g 2.


Energy Eigenvalue Bare Coupling Symmetric Quantum Versus Particle Charge Renormalization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ben05]
    Bender, C.M.: Quantum Mechanics Based on Non-Hermitian Hamiltonians. In: Takei, Y. (ed) Proceedings of the International Conference “Recent Trends in Exponential Asymptotics,” Kyoto, July, 2004. RIMS Kokyuroku Symposium Series 1424, 64–77 (2005). This paper gives a background of \( \mathcal{P}\mathcal{T} \)-symmetric quantum mechanics and gives some references.Google Scholar
  2. [BenBroJon02]
    Bender, C.M., Brody, D.C., Jones, H.F.: Complex Extension of Quantum Mechanics, Phys. Rev. Lett. 89, 270401–4 (2002)CrossRefMathSciNetGoogle Scholar
  3. [BenMeiWan03]
    Bender, C.M., Meisinger, P.N., and Wang, Q.: Perturbative Calculation of the Hidden Symmetry Operator in \( \mathcal{P}\mathcal{T} \)-Symmetric Quantum Mechanics, J. Phys. A: Math. Gen. 36, 1973–83 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. [BenJon04]
    Bender, C.M. and Jones, H.F.: Semiclassical Calculation of the \( \mathcal{C} \) Operator in \( \mathcal{P}\mathcal{T} \)-Symmetric Quantum Mechanics, Phys. Lett. A 328, 102–9 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. [BenBroRefReu04]
    Bender, C.M., Brod, J., Refig, A.T., Reuter, M.E.: The \( \mathcal{C} \) Operator in \( \mathcal{P}\mathcal{T} \)-Symmetric Quantum Theories, J. Phys. A: Math. Gen. 37, 10139–10165 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. [BenBroJon04-1]
    Bender, C.M., Brody, D.C., Jones, H.F.: Scalar Quantum Field Theory with Cubic Interaction, Phys. Rev. Lett. 93, 251601–4 (2004)CrossRefMathSciNetGoogle Scholar
  7. [BenBroJon04-2]
    Bender, C.M., Brody, D.C., Jones, H.F.: Extension of \( \mathcal{P}\mathcal{T} \)-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction, Phys. Rev. D 70, 025001 (2004) (19 pages)CrossRefGoogle Scholar
  8. [BenCavMilSha05]
    Bender, C.M., Cavero-Pelaez, I., Milton, K.A., Shajesh, K.V.: \( \mathcal{P}\mathcal{T} \)-Symmetric Quantum Electrodynamics, Phys. Lett. B 613, 97–104 (2005)CrossRefMathSciNetGoogle Scholar
  9. [BenMeiWan05]
    C. M. Bender, P. N. Meisinger, and Q. Wang: Wilson Polynomials and the Lorentz Transformation Properties of the Parity Operator, J. Math. Phys. 46, 052302 (2005) (13 pages)CrossRefMathSciNetGoogle Scholar
  10. [BenBraCheWan05-1]
    Bender, C.M., Brandt, S.F., Chen, J., Wang, Q.: The \( \mathcal{C} \) Operator in \( \mathcal{P}\mathcal{T} \)-Symmetric Quantum Field Theory Transforms as a Lorentz Scalar, Phys. Rev. D 71, 065010 (2005) (7 pages)CrossRefGoogle Scholar
  11. [Lee54]
    Lee, T.D.: Some Special Examples in Renormalizable Field Theory, Phys. Rev. 95, 1329–34 (1954)MATHCrossRefGoogle Scholar
  12. [KälPau55]
    Källén, G., Pauli, W.: On the Mathematical Structure of T. D. Lee’s Model of a Renormalizable Field Theory, Mat.-Fys. Medd. 30, No. 7 (1955)Google Scholar
  13. [Sch61]
    Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Co., Evanston (1961), Chap. 12Google Scholar
  14. [Bar63]
    G. Barton: Introduction to Advanced Field Theory. John Wiley & Sons, New York (1963), Chap. 12MATHGoogle Scholar
  15. [Kle04]
    Kleefeld, K.: Non-Hermitian Quantum Theory and its Holomorphic Representations: Introduction and Applications, hep-th/0408028 and hepth/0408097Google Scholar
  16. [AbaBroSugWhi75]
    Abarbanel, H.D.I., Bronzan, J.D., Sugar, R.L., White, A.R.: Reggeon Field Theory: Formulation and Use, Phys. Rept. 21, 119–182 (1975); Brower, R., Furman, M., Moshe, M.: Critical Exponents for the Reggeon Quantum Spin Model, Phys. Lett. B 76, 213-9 (1978); Harms, B., Jones, S., Tan, C.-I.: Complex Energy Spectra in Reggeon Quantum Mechanics with Quartic Interactions, Nucl. Phys. B 171 392–412 (1980) and New STructure in the Energy Spectrum of Reggeon Quantum Mechanics with Quartic Couplings, Phys. Lett. B 91B, 291-5 (1980)CrossRefGoogle Scholar
  17. [Fis78]
    Fisher, M.E.: Yang-Lee Edge Singularity and φ 3 Field Theory, Phys. Rev. Lett. 40, 1610–13 (1978); Cardy, J.L.: Conformal Invariance and the Yang-Lee Singularity in Two Dimensions, ibid. 54, 1354-6 (1985); Cardy, J.L. and Mussardo, G.: S Matrix of the Yang-Lee Edge Singularity in Two Dimensions, Phys. Lett. B 225, 275-8 (1989); Zamolodchikov, A.B.: Two-Point Correlation Function in Scaling Lee-Yang Model, Nucl. Phys. B 348, 619-41 (1991)CrossRefGoogle Scholar
  18. [BenBraCheWan05-2]
    Bender, C.M., Brandt, S.F., Chen, J., Wang, Q.: Ghost Busting: \( \mathcal{P}\mathcal{T} \)-Symmetric Interpretation of the Lee Model, Phys. Rev. D 71, 025014 (2005) (11 pages)CrossRefGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Carl M. Bender
    • 1
  1. 1.Physics DepartmentWashington UniversitySt. LouisUSA

Personalised recommendations