Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 001, 19 pages      arXiv:0901.0700      http://dx.doi.org/10.3842/SIGMA.2009.001
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

Three-Hilbert-Space Formulation of Quantum Mechanics

Miloslav Znojil
Nuclear Physics Institute ASCR, 250 68 Rez, Czech Republic

Received October 29, 2008, in final form December 31, 2008; Published online January 06, 2009

Abstract
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H(auxiliary) and H(standard)) we spot a weak point of the 2HS formalism which lies in the double role played by H(auxiliary). As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator H(gen) of the time-evolution of the wave functions may differ from their Hamiltonian H.

Key words: formulation of Quantum Mechanics; cryptohermitian operators of observables; triplet of the representations of the Hilbert space of states; the covariant picture of time evolution.

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References

  1. Styer D.F. et al., Nine formulations of quantum mechanics, Amer. J. Phys. 70 (2002), 288-297.
  2. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
    Bender C.M., Boettcher S., Meisinger P.M., PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201-2229, quant-ph/9809072.
  3. Messiah A., Quantum mechanics, North Holland, Amsterdam, 1961, Vols. 1, 2.
  4. http://gemma.ujf.cas.cz/~znojil/conf/index.html
  5. Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics" (Prague, June 2003), Editor M. Znojil, Czech. J. Phys. 54 (2004), no. 1.
    Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics II" (Prague, June 2004), Editor M. Znojil, Czech. J. Phys. 54 (2004), no. 10.
    Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics III" (Istanbul, June 2005), Editor M. Znojil, Czech. J. Phys. 55 (2005), no. 9.
    Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics IV" (Stellenbosch, November 2005), Editors H. Geyer, D. Heiss and M. Znojil, J. Phys. A: Math. Gen. 39 (2006), no. 32.
    Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics V" (Bologna, July 2006), Editor M. Znojil, Czech. J. Phys. 56 (2006), no. 9.
    Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics VI" (London, July 2007), Editors by A. Fring, H. Jones and M. Znojil, J. Phys. A: Math. Theor. 41 (2008), no. 24.
    Proceedings of the Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics VII" (Benasque, July 2008), Editors A. Andrianov et al., SIGMA 5 (2009).
  6. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  7. Scholtz F.G., Geyer H.B., Hahne F.J.W., Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Physics 213 (1992), 74-101.
  8. Dieudonne J., Quasi-Hermitian operators, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Pergamon, Oxford, 1961, 115-122.
    Williams J.P., Operators similar to their adjoints, Proc. Amer. Math. Soc. 20 (1969), 121-123.
  9. Znojil M., Time-dependent version of cryptohermitian quantum theory, Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874.
  10. Bender C.M., Turbiner A., Analytic continuation of eigenvalue problems, Phys. Lett. A 173 (1993), 442-446.
    Günther U., Stefani F., Znojil M., MHD α2-dynamo, squire equation and PT-symmetric interpolation between square well and harmonic oscillator, J. Math. Phys. 46 (2005), 063504, 22 pages, math-ph/0501069.
    Jakubský V., Thermodynamics of pseudo-Hermitian systems in equilibrium, Modern Phys. Lett. A 22 (2007), 1075-1084, quant-ph/0703092.
    Mostafazadeh A., Loran F., Propagation of electromagnetic waves in linear media and pseudo-Hermiticity, Europhys. Lett. EPL 81 (2008), 10007, 6 pages, physics/0703080.
  11. Mostafazadeh A., Hilbert space structures on the solution space of Klein-Gordon type evolution equations, Classical Quantum Gravity 20 (2003), 155-172, math-ph/0209014.
  12. Feshbach H., Villars F., Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles, Rev. Mod. Phys. 30 (1958), 24-45.
  13. Jakubský V., Smejkal J., A positive-definite scalar product for free proca particle, Czech. J. Phys. 56 (2006), 985-997, hep-th/0610290.
    Smejkal J., Jakubský V., Znojil M., Relativistic vector bosons and PT-symmetry, J. Phys. Stud. 11 (2007), 45-54, hep-th/0611287.
    Zamani F., Mostafazadeh A., Quantum mechanics of Proca fields, arXiv:0805.1651.
  14. Bender C.M., Brody D.C., Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002), 270402, 4 pages, Erratum, Phys. Rev. Lett. 92 (2004), 119902, quant-ph/0208076.
  15. Mostafazadeh A., Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205-214, math-ph/0107001.
    Mostafazadeh A., Pseudo-hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. 43 (2002), 2814-2816, math-ph/0110016.
    Mostafazadeh A., Pseudo-Hermitian quantum mechanics, arXiv:0810.5643.
  16. Znojil M., Quantum knots, Phys. Lett. A 372 (2008), 3591-3596, arXiv:0802.1318.
    Znojil M., Identification of observables in quantum toboggans, J. Phys. A: Math. Theor. 41 (2008), 215304, 14 pages, arXiv:0803.0403.
  17. Znojil M., On the role of the normalization factors κn and of the pseudo-metric PP in crypto-Hermitian quantum models, SIGMA 4 (2008), 001, 9 pages, arXiv:0710.4432.
  18. Cannata F., Dedonder J.-P., Ventura A., Scattering in PT-symmetric quantum mechanics, Ann. Physics 322 (2007), 397-433, quant-ph/0606129.
  19. Jones H.F., Scattering from localized non-Hermitian potentials, Phys. Rev. D 76 (2007), 125003, 5 pages, arXiv:0707.3031.
    Znojil M., Scattering theory with localized non-Hermiticites, Phys. Rev. D 78 (2008), 025026, arXiv:0805.2800.
    Znojil M., Discrete PT-symmetric models of scattering, J. Phys. A: Math. Theor. 41 (2008), 292002, 9 pages, arXiv:0806.2019.
  20. Mostafazadeh A., Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric, Phys. Lett. B 650 (2007), 208-212, arXiv:0706.1872.
  21. Znojil M., Which operator generates time evolution in quantum mechanics?, arXiv:0711.0535.
  22. Figueira de Morisson Faria C., Fring A., Time evolution of non-Hermitian Hamiltonian systems, J. Phys. A: Math. Gen. 39 (2006), 9269-9289, quant-ph/0604014.
  23. Bender C.M., Wu T.T., Anharmonic oscillator, Phys. Rev. 184 (1969), 1231-1260.
  24. Sibuya Y., Global theory of second order linear differential equation with polynomial coefficient, North Holland, Amsterdam, 1975.
    Dorey P., Dunning C., Tateo R., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 34 (2001), 5679-5704, hep-th/0103051.
    Dorey P., Dunning C., Tateo R., Supersymmetry and the spontaneous breakdown of PT symmetry, J. Phys. A: Math. Gen. 34 (2001), L391-L400, hep-th/0104119.
    Dorey P., Dunning C., Tateo R., The ODE/IM correspondence, J. Phys. A: Math. Theor. 40 (2007), R205-R283, hep-th/0703066.
    Davies E.B., Linear operators and their spectra, Cambridge, Cambridge University Press, 2007.
  25. Caliceti E., Graffi S., Maioli M., Perturbation theory of odd anharmonic oscillators, Comm. Math. Phys. 75 (1980), 51-66.
    Buslaev V., Grecchi V., Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A: Math. Gen. 26 (1993), 5541-5549.
    Fernández F.M., Guardiola R., Ros J., Znojil M., Strong-coupling expansions for the PT-symmetric oscillators V(r) = aix + b(ix)2 + c(ix)3, J. Phys. A: Math. Gen. 31 (1998), 10105-10112.

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