Skip to main content
SpringerLink
Log in

A self-consistent approach to quantum field theory for extended particles

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

A notion of quantum space-time is introduced, physically defined as the totality of all flows of quantum test particles in free fall. In quantum space-time the classical notion of deterministic inertial frames is replaced by that of stochastic frames marked by extended particles. The same particles are used both as markers of quantum space-time points as well as natural clocks, each species of quantum test particle thus providing a standard for space-time measurements. In the considered flat-space case, the fluctuations in coordinate values with respect to stochastic frames are described by coordinate probability amplitudes related to irreducible stochastic phase space representations of the Poincaré group. Lagrangian field theory on quantum space-time is formulated. The ensuing equations of motion for interacting fields contain no singularities in their nonlinear terms, and therefore can be handled by methods borrowed from classical nonlinear analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. N. Bohr,Atomic Physics and Human Knowledge (Science, New York, 1961).

    Google Scholar 

  2. B. S. DeWitt and N. Graham, eds.,The Many-Worlds Interpretation of Quantum Mechanics (Princeton University Press, Princeton, 1973).

    Google Scholar 

  3. A. Einstein,The Principle of Relativity (Dover, New York, 1960).

    Google Scholar 

  4. H. Arzeliés,Relativistic Kinematics (Pergamon Press, Oxford, 1966).

    Google Scholar 

  5. R. F. Marzke and J. A. Wheeler, inGravitation and Relativity, R. Y. Chiu and W. F. Hoffmann, eds. (Benjamin, New York, 1964); J. Ehlers, inRelativity, Astrophysics and Cosmology, W. Israel, ed. (Reidel, Dordrecht, 1973).

    Google Scholar 

  6. E. H. Kronheimer and R. Penrose,Proc. Camb. Phil. Soc. 63, 481 (1967); N. M. J. Woodhouse,J. Math. Phys. 14, 495 (1973).

    Google Scholar 

  7. W. Heisenberg,The Physical Principles of the Quantum Theory (Dover, New York, 1960).

    Google Scholar 

  8. A. Bohr, inLectures in Theoretical Physics, vol. III, W. E. Brittin, B. W. Downs, and J. Downs, eds. (Interscience, New York, 1961).

    Google Scholar 

  9. N. N. Bogolubov, A. A. Logunov, and I. T. Todorov,Introduction to Axiomatic Quantum Field Theory (Benjamin, Reading, Massachusetts, 1975).

    Google Scholar 

  10. A. S. Wightman and S. S. Schweber,Phys. Rev. 98, 812 (1955).

    Google Scholar 

  11. N. Bohr and L. Rosenfeld,Phys. Rev. 78, 794 (1950).

    Google Scholar 

  12. B. S. DeWitt, inGravitation: An Introduction to Current Research, L. Witten, ed. (Wiley, New York, 1962).

    Google Scholar 

  13. J. Schwinger,Quantum Electrodynamics (Dover, New York, 1958), pp. XV-XVII.

    Google Scholar 

  14. L. D. Landau and I. Pomeranchuk,Bokl. Akad. Nauk SSSR 102, 489 (1955).

    Google Scholar 

  15. P. A. M. Dirac, inThe Past Decade in Particle Theory, E. C. G. Sudarshan and Y. Ne'eman, eds. (Gordon and Breach, London, 1973), pp. 753–755.

    Google Scholar 

  16. P. A. M. Dirac, inDeeper Pathways in High-Energy Physics, A. Perlmutter and L. F. Scott, eds. (Plenum, New York, 1977), p. 1.

    Google Scholar 

  17. P. A. M. Dirac, inDirections in Physics, H. Hora and J. R. Shepanski, eds. (Wiley, New York, 1978), pp. 36–37.

    Google Scholar 

  18. P. A. M. Dirac, inMathematical Foundations of Quantum Mechanics, A. R. Marlow, ed. (Academic Press, New York, 1978).

    Google Scholar 

  19. L. D. Landau, inNiels Bohr and the Development of Physics, W. Pauli, ed. (Pergamon Press, New York, 1955).

    Google Scholar 

  20. W. Pauli,Helv. Phys. Acta Suppl. 4, 68 (1956).

    Google Scholar 

  21. S. Deser,Rev. Mod. Phys. 29, 417 (1957).

    Google Scholar 

  22. D. I. Blokhintsevet al., Proceedings of the International Conference on Nonlocal Quantum Field Theory (Dubna JINR 1968); D. A. Kirzhnits,Sov. Phys.—Uspekhi 9, 692 (1967).

  23. H. S. Snyder,Phys. Rev. 71, 38 (1947); C. N. Yang,Phys. Rev. 72, 874 (1947); H. T. Flint,Phys. Rev. 74, 209 (1948); E. L. Hill,Phys. Rev. 100, 1780 (1950); A. Das,Nuovo Cimento 18, 482 (1960).

    Google Scholar 

  24. E. Prugovečki,J. Math. Phys. 19, 2260 (1978).

    Google Scholar 

  25. E. Prugovečki,Phys. Rev. D 18, 3655 (1978).

    Google Scholar 

  26. S. T. Ali,J. Math. Phys. 20, 1385 (1979);21, 818 (1980).

    Google Scholar 

  27. E. Prugovečki,J. Math. Phys. 17, 517, 1673 (1976).

    Google Scholar 

  28. S. T. Ali and E. Prugovečki,J. Math. Phys. 18, 219 (1977).

    Google Scholar 

  29. E. Prugovečki,Ann. Phys. (N.Y.)110, 102 (1978).

    Google Scholar 

  30. S. T. Ali and E. Prugovečki,Physica 89A, 501 (1977); E. Prugovečki,Physica 91A, 202 (1978).

    Google Scholar 

  31. S. T. Ali, R. Gagnon and E. Prugovečki, Conserved quantum probability currents on stochastic phase space,Can. J. Phys. 59 (1981) (in press).

  32. E. Prugovečki,Found. Phys. 9, 575 (1979).

    Google Scholar 

  33. J.-M. Lévy-Leblond, inGroup Theory and its Applications, Vol. II, E. M. Loeble ed. (Academic Press, New York, 1971).

    Google Scholar 

  34. E. Prugovečki,Quantum Mechanics in Hilbert Space, 2nd ed. (Academic Press, New York, 1981).

    Google Scholar 

  35. C. F. Dietrich,Uncertainty, Calibration and Probability (Wiley, New York, 1973).

    Google Scholar 

  36. E. Prugovečki,Int. J. Theor. Phys. 16, 321 (1977).

    Google Scholar 

  37. S. T. Ali and E. Prugovečki,Int. J. Theor. Phys. 16, 689 (1977).

    Google Scholar 

  38. R. Hofstadter,Ann. Rev. Nucl. Sci. 7, 231 (1957).

    Google Scholar 

  39. E. Prugovečki,Physica 91A, 229 (1978).

    Google Scholar 

  40. A. S. Wightman, inAspects of Quantum Theory, A. Salam and E. P. Wigner, eds. (Cambridge University Press, Cambridge, 1972).

    Google Scholar 

  41. J. Ehlers, inGeneral Relativity and Cosmology, B. K. Sachs, ed. (Academic Press, New York, 1971).

    Google Scholar 

  42. E. Prugovečki,J. Math. Phys. 19, 2271 (1978).

    Google Scholar 

  43. E. Prugovečki,Rep. Math. Phys. 18(1) (1980) (in press).

  44. S. T. Ali and E. Prugovečki, Self-consistent relativistic model of spin-1/2 particles in external electromagnetic fields,Nuovo Cimento A (1981) (to appear).

  45. E. Prugovečki, Quantum action principle and functional integration over paths in stochastic phase space,Nuovo Cimento A (1980–1981) (in press).

  46. J. Scherk,Rev. Mod. Phys. 47, 123 (1975).

    Google Scholar 

  47. R. P. Feynman, M. Kislinger, and F. Ravndal,Phys. Rev. D 3, 2706 (1971).

    Google Scholar 

  48. Y. S. Kim and M. E. Noz,Phys. Rev. D 10, 3521 (1973);12, 129 (1975).

    Google Scholar 

  49. Y. S. Kim and M. E. Noz,Phys. Rev. D 15, 335 (1977).

    Google Scholar 

  50. M. Fierz,Helv. Phys. Acta 23, 731 (1950).

    Google Scholar 

  51. A. Ferretti,Nuovo Cimento 27, 1503 (1963).

    Google Scholar 

  52. E. Prugovečki, General aspects of stochastic quantum field theory for extended particles,Found. Phys. 11 (1981) (to appear).

  53. H. Snyder,Phys. Rev. 71, 38 (1947).

    Google Scholar 

  54. Yu. A. Gol'fand,Sov. Phys.—JETP 10, 356 (1960);16, 184 (1963); V. G. Kadyshevskii,Sov. Phys.—Doklady 7, 1031 (1963).

    Google Scholar 

  55. J. G. Taylor,Phys. Rev. D 19, 2336 (1979).

    Google Scholar 

  56. R. Penrose, inBattelle Rencontres, C. M. DeWitt and J. A. Wheeler, eds. (Benjamin, New York, 1968).

    Google Scholar 

  57. S. A. Fulling,Phys. Rev. D 7, 2850 (1973).

    Google Scholar 

  58. J. A. Wheeler,Geometrodynamics (Academic Press, New York, 1962).

    Google Scholar 

  59. C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973).

    Google Scholar 

  60. R. Penrose, inQuantum Gravity, C. J. Isham, R. Penrose, and D. W. Sciama, eds. (Clarendon Press, Oxford, 1975).

    Google Scholar 

  61. G. Rosen,Int. J. Theor. Phys. 17, 1 (1978).

    Google Scholar 

  62. G. B. Whitham,Linear and Nonlinear Waves (Wiley, New York, 1974).

    Google Scholar 

  63. R. Rajaraman,Phys. Reports 21C, 227 (1975).

    Google Scholar 

  64. R. Jackiw,Rev. Mod. Phys. 49, 681 (1977).

    Google Scholar 

  65. L. D. Faddeev and V. E. Korepin,Phys. Reports 42C, 1 (1978).

    Google Scholar 

  66. N. N. Bogoliubov and D. V. Shirkov,Introduction to the Theory of Quantized Fields, (Interscience, New York, 1959).

    Google Scholar 

  67. H. Lehman, K. Symanzik, and W. Zimmermann,Nuovo Cimento 1, 205 (1955);6, 319 (1957).

    Google Scholar 

  68. C. S. Moravetz and W. A. Strauss,Comm. Pure Appl. Math. 25, 1 (1972).

    Google Scholar 

  69. R. Raczka,J. Math. Phys. 16, 173 (1975).

    Google Scholar 

  70. J. Goldstone and R. Jackiw,Phys. Rev. D 11, 1486 (1975).

    Google Scholar 

  71. E. Wigner,Helv. Phys. Acta Suppl. 4, 210 (1956).

    Google Scholar 

  72. O. Klein,Helv. Phys. Acta Suppl. 4, 58 (1956).

    Google Scholar 

  73. E. Prugovečki, Stochastic quantization of geometrodynamic curved spacetime,Nuovo Cimento B (1981) (in press).

  74. E. Prugovečki, Quantum spacetime operationally based on propagators for extended test particles,Hadronic J. 4 (1981) (in press).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Canada

    Eduard Prugovečki

Authors
  1. Eduard Prugovečki
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by an NSERC grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prugovečki, E. A self-consistent approach to quantum field theory for extended particles. Found Phys 11, 355–382 (1981). https://doi.org/10.1007/BF00727074

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00727074

Keywords

Search

Navigation