- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Foldy-Wouthuysen transformations in an indefinite-metric space. I. Necessary and sufficient conditions for existence

R. A. Krajcik* and Michael Martin Nieto

  • Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545

  • *Present address: Institute of Geophysics and Planetary Physics, Scripps Institute of Oceanography, University of California, San Diego, La Jolla, Calif. 92093.
PDF

Phys. Rev. D 13, 2245 – Published 15 April, 1976

DOI: https://doi.org/10.1103/PhysRevD.13.2245

Abstract

We prove that the necessary and sufficient conditions that a pseudounitary (Foldy-Wouthuysen) transformation exists which will diagonalize a nondiagonal pseudo-Hermitian matrix θ on a (nonsingular) indefinite-metric space are that all the eigenvalues of θ be real and all the eigenvectors of θ have nonzero norm. Physical applications are discussed. For example, the 2 × 2 case is discussed in general and for the Sakata-Taketani spin-0 field and the Lee model. This theorem also allows one to show that one can transform all the Bhabha Poincaré generators to a form which decouples the different mass (and normed) states.

View figure in article

See Also

Foldy-Wouthuysen transformations in an indefinite-metric space. II. Theorems for practical calculations

R. A. Krajcik and Michael Martin Nieto
Phys. Rev. D 13, 2250 (1976)

Foldy-Wouthuysen transformations in an indefinite-metric space. III. Relation to Lorentz transformations for first-order wave equations and the Poincaré generators

R. A. Krajcik and Michael Martin Nieto
Phys. Rev. D 15, 416 (1977)

Foldy-Wouthuysen transformations in an indefinite-metric space. IV. Exact, closed-form expressions for first-order wave equations

R. A. Krajcik and Michael Martin Nieto
Phys. Rev. D 15, 426 (1977)

References (27)

  1. W. Pauli and V. Weisskopf, Helv. Phys. Acta 7, 709 (1934)
  2. R. A. Krajcik and M. M. Nieto, following paper, Phys. Rev. D 13, 2250 (1976)
  3. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950) S. Tani, Prog. Theor. Phys. 6, 267 (1951)
  4. M. H. L. Pryce, Proc. R. Soc. A150, 166 (1935) R. Becker, Nachr. Akad. Wiss. Göttingen, Math. Physik Kl. II, No. 1, 39 (1945), [3]
  5. H. C. Lee, Proc. Lond. Math. Soc. 50, 230 (1949)
  6. R. Nevanlinna, Ann. Ac. Sci. Fenn. (Suomalaisen Tiedeakatemian Toimituksia) Ser. A I. Math.-Phys. No. 108 (1952) ibid.No. 113 (1952) ibid.No. 115 (1952) ibid.No. 163 (1954) ibid.No. 222 (1956)
  7. L. K. Pandit, Nuovo Cimento Suppl. 11, 157 (1959)
  8. K. L. Nagy, Nuovo Cimento Suppl. 17, 92 (1960)
  9. Yu. P. Ginzburg and I. S. Iokhvidov, Usp. Math. Nauk 17, No. 4, 3 (1962) [Russ. Math. Surv. 17, No. 4, 1 (1962)]
  10. K. L. Nagy, State Vector Spaces with Indefinite Metric in Quantum Field Theory (Noordhoff, Groningen, 1966)
  11. Pandit, in Sec. 8 of [7] J. W. Helton [Bull. Am. Math. Soc. 81, 1028 (1975)] J. Bognár, Indefinite Inner Product Spaces (Springer, New York, 1974) Ju. L. Daleckiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Trans. Math. Mono. 43, (Amer. Math. Soc., Providence, 1974)
  12. E. Fischbach, M. M. Nieto, and C. K. Scott, Prog. Theor. Phys. 48, 574 (1972)
  13. S. Sakata and M. Taketani, Sci. Pap. Inst. Phys. Chem. Res. Tokyo 38, 1 (1940) M. Taketani and S. Sakata, Proc. Phys. Math. Soc. Jpn. 22, 737 (1940)
  14. M. TaketaniS. Sakata[13] Prog. Theor. Phys. Suppl. No. 1, 84 (1955) ibid.No. 1, 98 (1955)
  15. W. Heitler, Proc. R. Irish Acad. 49, 1 (1943)
  16. R. J. Duffin, Phys. Rev. 54, 1114 (1938)
  17. N. Kemmer, Proc. R. Soc. London A173, 91 (1939)
  18. G. Petiau, Acad. R. Belg. Cl. Sci. Mem. Collect. 8 16, No. 2 (1936)
  19. Ig. Tamm, Compt. Rend. (Doklady) Acad. Sci. URSS 29, 551 (1940)
  20. H. Feshbach and F. Villars, Rev. Mod. Phys. 30, 24 (1958)
  21. K. M. Case, Phys. Rev. 95, 1323 (1954)
  22. F. Strocchi, Nuovo Cimento 26, 955 (1962)
  23. T. D. Lee, Phys. Rev. 95, 1329 (1954)
  24. G. Källén and W. Pauli, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 30, No. 7 (1955)
  25. W. Heisenberg, Nucl. Phys. 4, 532 (1957)
  26. R. A. Krajcik and M. M. Nieto, Phys. Rev. D 10, 4049 (1974) ibid.11, 1442 (1975) ibid.11, 1459 (1975) ibid.13, 924 (1976)
  27. R. A. Krajcik and M. M. Nieto (in preparation), paper V of the series of [26]

Outline

Information

Phys. Rev. D 13, 2245– Published 15 April, 1976
  • Received 15 September 1975

© 1976 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation