Foundations of Physics

, Volume 21, Issue 11, pp 1305–1314 | Cite as

Phase-space path integration of the relativistic particle equations



Hamilton-Jacobi theory is applied to find appropriate canonical transformations for the calculation of the phase-space path integrals of the relativistic particle equations. Hence, canonical transformations and Hamilton-Jacobi theory are also introduced into relativistic quantum mechanics. Moreover, from the classical physics viewpoint, it is very interesting to find and to solve the Hamilton-Jacobi equations for the relativistic particle equations.


Quantum Mechanic Relativistic Quantum Path Integral Canonical Transformation Classical Physics 
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. 1.
    R. P. Feynman, “An operator calculus having applications in quantum electrodynamics,”Phys. Rev. 84, 108 (1951), Appendix B.Google Scholar
  2. 2.
    L. van Hove, “Sur le problème des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques de la mécanique classique,”Mem. Acad. R. Belg. 26, 610 (1951).Google Scholar
  3. 3.
    W. Tobocman, “Transition amplitudes as sums over histories,”Nuovo Cimento 3, 1213 (1956).Google Scholar
  4. 4.
    S. S. Schweber, “On Feynman quantization,”J. Math. Phys. 3, 831 (1962).Google Scholar
  5. 5.
    H. Davies, “Hamiltonian approach to the method of summation over Feynman histories,”Proc. Cambridge Philos. Soc. 59, 147 (1963).Google Scholar
  6. 6.
    M. C. Gutzwiller, “Phase-integral approximation in momentum space and the bound states of an atom,”J. Math. Phys. 8, 1979 (1967).Google Scholar
  7. 7.
    P. Pearle, “Finite-dimensional path summation formulation for quantum mechanics,”Phys. Rev. D 8, 2503 (1973).Google Scholar
  8. 8.
    R. Fanelli, “Canonical transformations and phase space path integrals,”J. Math. Phys. 17, 490 (1976).Google Scholar
  9. 9.
    L. S. Schulman,Techniques and Applications of Path Integration (Wiley, New York, 1981), Secs. 27 and 31.Google Scholar
  10. 10.
    P. A. M. Dirac,The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford, 1958), Sec. 32; also published inQuantum Electrodynamics, J. Schwinger, ed. (Dover, New York, 1958).Google Scholar
  11. 11.
    A. O. Barut and I. H. Duru, “Path integration via Hamilton-Jacobi coordinates and applications to potential barriers,”Phys. Rev. A 38, 5906 (1988).Google Scholar
  12. 12.
    R. P. Feynman, “Mathematical formulation of the quantum theory of electromagnetic interaction,”Phys. Rev. 80, 440 (1950), Appendix A.Google Scholar
  13. 13.
    See Ref. 1, Appendix D.Google Scholar
  14. 14.
    G. Riazanov,Zh. Eksp. Teor. Fiz. 33, 1308 (1958); also published inSov. Phys. JETP 6, 1107 (1958).Google Scholar
  15. 15.
    R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 34–36.Google Scholar
  16. 16.
    L. S. Schulman, “Relativistic spin: Tops and wave equations,”Nucl. Phys. B 18, 595 (1970).Google Scholar
  17. 17.
    G. Rosen, “Feynman path summation for the Dirac equation: An underlying one-dimensional aspects of relativistic particle motion,”Phys. Rev. A 28, 1139 (1983).Google Scholar
  18. 18.
    T. Ichinose and H. Tamura, “Propagation of a Dirac particle. A path integral approach,”J. Math. Phys. 25, 1810 (1984).Google Scholar
  19. 19.
    T. Jacobson and L. S. Schulman, “Quantum stochastics: The passage from a relativistic to a nonrelativistic path integral,”J. Phys. A 17, 375 (1984).Google Scholar
  20. 20.
    T. Jacobson, “Spinor chain path integral for the Dirac equation,”J. Phys. A 17, 2433 (1984).Google Scholar
  21. 21.
    A. O. Barut and I. H. Duru, “Path integral derivation of the Dirac propagator,”Phys. Rev. Lett. 53, 2355 (1984).Google Scholar
  22. 22.
    A. O. Barut and I. H. Duru, “Path integral formulation of quantum electrodynamics from classical particle trajectories,”Phys. Rep. 172, 1 (1989).Google Scholar
  23. 23.
    A. O. Barut and N. Zanghi, “Classical Model of the Dirac Electron,”Phys. Rev. Lett. 52, 2009 (1984).Google Scholar
  24. 24.
    J. D. Bjorken and S. D. Drell,Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), Secs. 1.2 and 9.2; Secs. 1.3, 3.2, and 6.4.Google Scholar
  25. 25.
    H. Goldstein,Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, Massachusetts, 1980), Sec. 9.1; Sec. 10.1.Google Scholar
  26. 26.
    J. R. Klauder, “Continuous representations and path integrals revisited,” inNato Advanced Study Institute Series, Vol. 34: Path Integrals, G. J. Papadopoulos and J. T. Devreese, eds. (Plenum, New York, 1978), pp. 5–38.Google Scholar
  27. 27.
    K. Gottfried,Quantum Mechanics, Vol. I: Fundamentals (Benjamin, New York, 1966), Sec. 28.2.Google Scholar
  28. 28.
    A. Messiah,Mécanique Quantique (Dunod, Paris, 1959), Sec. VIII-6.Google Scholar
  29. 29.
    M. C. Gutzwiller, “Huygens' principle and the path integral,” inPath Summation: Achievements and Goals, S. Lundquist, A. Ranfagni, V. Sa-yakanil, and L. S. Schulman, eds. (World Scientific, Singapore, 1988), pp. 47–73.Google Scholar
  30. 30.
    G. J. Papadopoulos and J. T. Devreese, “Path-integral solutions of the Dirac equation,”Phys. Rev. D 13, 2227 (1976).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • H. Gür
    • 1
  1. 1.Department of Engineering Physics, Faculty of ScienceUniversity of AnkaraTandoganTurkey

Personalised recommendations