Abstract
First the “frame problem” is sketched: The motion of an isolated particle obeys a simple law in Galilean frames, but how does the Galilean character of the frame manifest itself at the place of the particle? A description of vacuum as a system of virtual particles will help to answer this question. For future application to such a description, the notion of global particle is defined and studied. To this end, a systematic use of the Fourier transformation on the Poincaré group is needed. The state of a system of n free particles is represented by a statistical operator W, which defines an operator-valued measure on ^Pn (^P is the dual of the Poincaré group). The inverse Fourier–Stieltjes transform of that measure is called the characteristic function of the system; it is a function on Pn. The main notion is that of global characteristic function: it is the restriction of the characteristic function to the diagonal subgroup of Pn; it represents the state of the system, considered as a single particle. The main properties of characteristic functions, and particularly of global characteristic functions, are studied. A mathematical Appendix defines two functional spaces involved.
Similar content being viewed by others
REFERENCES
J.-Y. Grandpeix, and F. Lurc¸at, “Particle description of zero energy vacuum II: Basic Vacuum System,” following article.
S. Weinberg, The Quantum Theory of Fields, Vol. I (Cambridge University Press, Cambridge, 1995).
J. Frenkel, Doklady Akademii Nauk SSSR 64, 507 (1949).
W. E. Thirring, Principles of Quantum Electrodynamics (Academic, New York, 1958).
L. Rosenfeld, Nucl. Phys. 10, 508 (1959).
S. S. Schweber, Rev. Mod. Phys. 58, 449–509 (1986).
S. S. Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonoga (Princeton University Press, Princeton, 1994).
F. Lurc¸at, Phys. Rev. 173, 1461 (1968).
F. Lurc¸at, Ann. Phys. 106, 342 (1977).
N. Wiener, The Fourier Integral and Certain of its Applications (Cambridge University Press, Cambridge, 1933; republished by Dover, New York).
E. Hewitt, and K. A. Ross, Abstract Harmonic Analysis, Vol. I (Springer, Berlin, 1963) and Vol. II (Springer, Berlin, 1970).
A. A. Kirillov, Elements of the Theory of Representations, translated from Russian by E. Hewitt (Springer, Berlin, 1976).
J. Dixmier, C*-Algebras (North-Holland, Amsterdam, 1977).
P. Bonnet, J. Funct. Anal. 55, 220–246 (1984).
Nghiäm Xuan Hai, Commun. Math. Phys. 12, 331–350 (1969).
Nghiäm Xuan Hai, Commun. Math. Phys. 22, 301–320 (1971).
A. M. Yaglom, “second-order homogeneous Random Fields,” Proceedings IVth Berkeley Symposium Math. Statistics and Probability, Vol. 2 (University of California Press, Berkeley, 1961).
U. Fano, Rev. Mod. Phys. 29, 74–93 (1957).
T. D. Newton, and E. P. Wigner, Rev. Mod. Phys. 21, 400–406 (1949).
J. P. Baton, and G. Laurens, Phys. Rev. 176, 1574–1586 (1968).
H. Joos, Fortschr. Physik 10, 65–146 (1962).
W. H. Klink, and G. J. Smith, Commun. Math. Phys. 10, 231–244 (1968).
W. H. Klink, “Induced representation theory of the Poincaré group,” in Mathematical Methods in Theoretical Physics (Boulder, Colorado, 1968), K. T. Mahantappa and W. E.
Brittin, eds. (Lectures in Theoretical Physics, Vol. 11D) (Gordon & Breach, New York, 1969).
P. Eymard, Bull. Soc. Math. France 92, 181–236 (1964).
P. Eymard, “A Survey of Fourier Algebras,” in Applications of Hypergroups and Related Measure Algebras (Seattle, Washington, 1993), W. C. Connett, M.-O. Gebuhrer, and A. L. Schwartz, eds. (Contemporary Mathematics, Vol. 183) (American Mathematical Society, Providence, Rhode Island, 1995), pp. 111-128.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grandpeix, JY., Lurçat, F. Particle Description of Zero-Energy Vacuum I: Virtual Particles. Foundations of Physics 32, 109–131 (2002). https://doi.org/10.1023/A:1013852931455
Issue Date:
DOI: https://doi.org/10.1023/A:1013852931455