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Czechoslovak Journal of Physics

, Volume 56, Issue 9, pp 985–997 | Cite as

A positive-definite scalar product for free Proca particle

  • Vít Jakubský
  • Jaroslav Smejkal
Article

Abstract

We implement recent results of pseudo-Hermitian quantum mechanics to the description of relativistic massive particle with spin-one. We derive a one-parameter family of Lorentz invariant positive-definite scalar products on the space of solutions of Proca equation.

Key words

free Proca particle positive-difinite metric operator 

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References

  1. [1]
    O. Klein: Z. Phys. 37 (1926) 895. V. Fock: Z. Phys. 38 (1926) 242; 39 226. W. Gordon: Z. Phys. 40 (1926) 117. P.A.M. Dirac: Proc. Roy. Soc. (London) A 117 (1928) 610. A. Proca: J. Phys. Rad. 7 (1936) 347. P.A.M. Dirac: Proc. Roy. Soc. (London) A 180 (1942) 1. W. Pauli: Rev. Mod. Phys. 15 (1943) 175.zbMATHCrossRefGoogle Scholar
  2. [2]
    E.M. Lifshitz, L.P. Pitaevskii, and V.B. Berestetskii: Quantum Electrodynamics, Oxford, 1996.Google Scholar
  3. [3]
    M. Taketani and S. Sakata: Proc. Phys.-Math. Soc. (Japan) 22 (1940) 757. I. Tamm: Compt. Rend. Acad. Sci. (U.R.S.S.) 29 (1940) 551.zbMATHGoogle Scholar
  4. [4]
    L.L. Foldy: Phys. Rev. 102 (1956) 568.zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. [5]
    H. Feshbach and F. Villars: Rev. Mod. Phys. 30 (1958) 24.zbMATHMathSciNetCrossRefADSGoogle Scholar
  6. [6]
    K.M. Case: Phys. Rev. 95 (1954) 1323.zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. [7]
    C.M. Bender and B. Boettcher: Phys. Rev. Lett. 80 (1998) 4243.MathSciNetCrossRefADSGoogle Scholar
  8. [8]
    N. Hatano and D.R. Nelson: Phys. Rev. Lett. 77 (1996) 570; Phys. Rev. B 56 (1997) 8651. C.M. Bender and K.A. Milton: Phys. Rev. D 55 (1997) R3255. C.M. Bender, S. Boettcher, and P.N. Meisinger: J. Math. Phys. 40 (1999) 2201. C.M. Bender, D.C. Brody, and H.F. Jones: Phys. Rev. Lett. 89 (2002) 0270401. G. Scolarici and L. Solombrino: J. Math. Phys. 44 (2003) 4450. C.M. Bender and H.F. Jones: Phys. Lett. A 328 (2004) 102. C.M. Bender: Czech. J. Phys. 54 (2004) 1027. H.F. Jones: Czech. J. Phys. 54 (2004) 1107. A. Mostafazadeh: Czech. J. Phys. 54 (2004) 1125.CrossRefADSGoogle Scholar
  9. [9]
    E. Caliceti, S. Graffi, and M. Maioli: Commun. Math. Phys. 75 (1980) 51. E. Caliceti and S. Gra.: J. Phys. A 37 (2004) 2239. F.M. Fernández, R. Guardiola, J. Ros, and M. Znojil: J. Phys. A: Math. Gen. 31 (1998) 10105. H. Bíla: Czech. J. Phys. 54 (2004) 1049.zbMATHMathSciNetCrossRefADSGoogle Scholar
  10. [10]
    F.G. Scholtz, H.B. Geyer, and F.J.W. Hahne: Ann. Phys. (NY) 213 (1992) 74.zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. [11]
    V. Buslaev and V. Grecchi: J. Phys. A: Math. Gen. 26 (1993) 5541. G. Alvarez: J. Phys. A: Math. Gen. 27 (1995) 4589.zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. [12]
    P. Dorey, C. Dunning, and R. Tateo: J. Phys. A: Math. Gen. 34 (2001) 5679. K.C. Shin: J. Math. Phys. 42 (2001) 2513.zbMATHMathSciNetCrossRefADSGoogle Scholar
  13. [13]
    M. Znojil: Phys. Lett. A 259 (1995) 220.MathSciNetCrossRefADSGoogle Scholar
  14. [14]
    A.A. Andrianov, F. Cannata, J.-P. Dedonder, and M.V. Ioffe: Int. J. Mod. Phys. A 14 (1999) 2675. G. Lévai and M. Znojil: J. Phys. A: Math. Gen. 33 (2000) 7165. M. Znojil: LANL arXiv math-ph/0104012, 2001; reprinted Rendiconti del Circ. Mat. di Palermo, Ser. II, Suppl. 72, 2004, p. 211. B. Bagchi, C. Quesne, and M. Znojil: Mod. Phys. Lett. A 16 (2001) 2047. B. Bagchi, C. Quesne, and R. Roychoudhury: J. Phys. A 38 (2005) L647.zbMATHMathSciNetCrossRefADSGoogle Scholar
  15. [15]
    W.D. Heiss and H.L. Harney: Eur. Phys. J. D 17 (2001) 149. W.D. Heiss: Czech. J. Phys. 55 (2005) 1107. U. Gunther and F. Stefani: Czech. J. Phys. 55 (2005) 1099.CrossRefADSGoogle Scholar
  16. [16]
    H. Langer and C. Tretter: Czech. J. Phys. 54 (2004) 1113.MathSciNetCrossRefADSGoogle Scholar
  17. [17]
    M. Znojil: Phys. Lett. A 285 (2001) 7. V. Jakubský and M. Znojil: Czech. J. Phys. 54 (2004) 1101.zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. [18]
    A. Ramirez and B. Mielnik: Rev. Mex. Fis. 49S2 (2003) 130.MathSciNetGoogle Scholar
  19. [19]
    S. Albeverio, S.-M. Fei, and P. Kurasov: Lett. Math. Phys. 59 (2002) 227. M. Znojil and V. Jakubský: J. Phys. A: Math. Gen. 38 (2005) 5041. V. Jakubský and M. Znojil; J. Phys. 55 (2005) 1113. S.-M. Fei: Czech. J. Phys. 55 (2005) 1085.zbMATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Mostafazadeh: J. Math. Phys. 43 (2002) 205; 2814; 3944; 6343.zbMATHMathSciNetCrossRefADSGoogle Scholar
  21. [21]
    B. Bagchi, S. Mallik, and C. Quesne: Mod. Phys. Lett. A 17 (2002) 1651. A. Mostafazadeh: LANL arXiv quant-ph/0310164, 2003. F. Kleefeld: AIP Conf. Proc. 660 (2003) 325. M. Znojil: arXiv:hep-th/0408081, 2004. F. Kleefeld: arXiv:hep-th/0408028, 2004.zbMATHMathSciNetCrossRefADSGoogle Scholar
  22. [22]
    A. Mostafazadeh and A. Batal: J. Phys. A: Math. Gen. 37 (2004) 11645.Google Scholar
  23. [23]
    G.S. Japaridze: J.Phys. A: Math. Gen. 35 (2002) 1709. H.B. Geyer, F.G. Scholtz, and I. Snyman: Czech. J. Phys. 54 (2004) 1069. H.B. Geyer and I. Snyman: Czech. J. Phys. 55 (2005) 1091. W.D. Heiss: Czech. J. Phys. 55 (2005) 1107. R. Kretschmer and L. Szymanowski: Phys. Lett. A 325 (2004) 112.zbMATHMathSciNetCrossRefADSGoogle Scholar
  24. [24]
    F.G. Scholz and H.B. Geyer: Phys. Lett. B, in print.Google Scholar
  25. [25]
    A. Mostafazadeh: J. Phys. A: Math. Gen. 38 (2005) 6557.zbMATHMathSciNetCrossRefADSGoogle Scholar
  26. [26]
    A. Mostafazadeh: Class. Quant. Grav. 20 (2003) 155; arXiv quant-ph/0307059. A. Mostafazadeh and F. Zamani: arXiv quant-ph/0312078, 2003. A. Mostafazadeh: Ann. Phys. (NY) 309 (2004) 1; Czech. J. Phys. 54 (2004) 93; J. Math. Phys. 44 (2003) 974.zbMATHMathSciNetCrossRefADSGoogle Scholar
  27. [27]
    M. Znojil: Czech. J. Phys. 54 (2004) 151; 55 (2005) 1187.MathSciNetCrossRefADSGoogle Scholar
  28. [28]
    M. Znojil: J. Phys. A: Math. Gen. 39 (2006) 441.zbMATHMathSciNetCrossRefADSGoogle Scholar
  29. [29]
    E.P. Wigner: Ann. Math. 40 (1939) 149.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    W.I. Fushchich and A.G. Nikitin: Symmetries of Equations of Quantum Mechanics, Allerton Press, New York, 1994.zbMATHGoogle Scholar
  31. [31]
    Ch. Fronsdal: Phys. Rev. 113 (1958) 1367. D.I.Weaver, C.L. Hammer, and R.H. Good, Jr.: Phys. Rev. B 135 (1964) 241.MathSciNetCrossRefADSGoogle Scholar
  32. [32]
    P.M. Mathews: Phys. Rev. 143 (1966) 985.zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Institute of Physics, Academy of Sciences of Czech Republic 2006

Authors and Affiliations

  • Vít Jakubský
    • 1
  • Jaroslav Smejkal
    • 2
  1. 1.Nuclear Physics InstituteAcad. Sci. CRŘežCzech Republic
  2. 2.Institute of Technical and Experimental PhysicsCzech Technical University PraguePraha 2Czech Republic

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