Abstract
The quantum mechanical description of the evolution of an unstable system defined initially as a state in a Hilbert space at a given time does not provide a semigroup (exponential) decay, law. The Wigner–Weisskopf survival amplitude, describing reversible quantum transitions, may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the evolution does riot correspond to a semigroup (experiments on the decay of the neutral K-meson system strongly support the semigroup evolution postulated, by Lee, Oehme and Yang, and Yang and Wu). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined function in the Lax–Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax–Phillips S-matrix is an inner function (the Lax–Phillips theory is essentially a theory of translation invariant subspaces). In the special case that the S-matrix is a rational inner function, we obtain the resonant state explicitly and analyze its particle (V, N, θ) content. If there is an exponential bound, the general case differs only by a so-called trivial inner factor, which does not change the complex spectrum, but may affect the wave function of the resonant state.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. D. Lax and R. S. Phillips, Scattering Theory (Academic Press, New York, 1967).
C. Flesia and C. Piron, Helv. Phys. Acta 57, 697 (1984).
L. P. Horwitz and C. Piron, Helv. Phys. Acta 66, 694 (1993).
E. Eisenberg and L. P. Horwitz, in Advances in Chemical Physics, vol. XCIX, I. Progogine and S. Rice, eds. (Wiley, New York, 1997), p. 245.
Y. Strauss, L. P. Horwitz, and E. Eisenberg, hep-th/9709036, submitted for publication.
C. Piron, Foundations of Quantum Physics (Benjamin/Cummings, Reading, 1976).
V. F. Weisskopf and E. P. Wigner, Z. Phys. 63, 54 (1930); 65, 18 (1930).
L. P. Horwitz, J. P. Marchand, and J. LaVita, J. Math. Phys. 12, 2537 (1971); D. Williams, Comm. Math. Phys. 21, 314 (1971).
L. P. Horwitz and J. P. Marchand, Helv. Phys. Acta 42, 1039 (1969).
L. P. Horwitz and J. P. Marchand, Rocky Mountain J. Math. 1, 225 (1971).
B. Winstein et al., Results from the Neutral Kaon Program at Fermilab's Meson Center Beamline, 1985–1997, FERMILAB-Pub-97/087-E, publised on behalf of the E731, E773 and E799 Collaborations, Fermi National Accelerator laboratoy, P. O. Box 500, Batavia, Illinois 60510.
T. D. Lee, R. Oehme, and C. N. Yang, Phys. Rev. 106, 340 (1957).
T. T. Wu and C. N. Yang, Phys. Rev. Lett. 13, 380 (1964).
L. P. Horwitz and L. Mizrachi, Nuovo Cimento A 21, 625 (1974); E. Cohen and L. P. Horwitz, hep-th_9808030; hep-ph_9811332, submitted for publication.
W; Baumgartel, Math. Machr. 69, 107 (1975); L. P. Horwitz and I. M. Sigal, Helv. Phys. Acta 51, 685 (1978); G. Parravicini, V. Gorini, and E. C. G. Sudarshan, J. Math. Phys. 21, 2208 (1980); A. Bohm, Quantum Mehcanics: Foundations and Applications (Springer, Berlin, 1986); A. Bohm, M. Gadella, and G. B. Mainland, Am. J. Phys. 57, 1105 (1989); T. Bailey and W. C. Schieve, Nuovo Cimento A 47, 231 (1978).
I. P. Cornfield, S. V. Formin, and Ya. G. Sinai, Ergodic Theory (Springer, Berlin, 1982).
Y. Fournès and I. E. Segal, Trans. Am. Math. Soc. 78, 385 (1955).
M. Rosenblum and J. Rovnyak, Hardya Classes and Operator Theory (Oxford University Press, New York, 1985).
L. P. Horwitz, Found Phys. 25, 39 (1995); see also D. Cocolicchio, Phys. Rev. D 57, 7251 (1998).
E. C. G. Stueckelberg, Helv. Phys. Acta 14, 322, 588 (1941); J. Schwinger, Phys. Rev. 82, 664 (1951); R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 80, 440 (1950); L. P. Horwitz and C. Piron, Helv. Phys. Acta 46, 316 (1973); R. Fanchi, Phys. Rev. D 20, 3108 (1979); A. Kyprianides, Phys. Rep. 155, 1 (1986), and references therein.
I; Antoniou, M. Gadella, I. Prigogine, and P. P. Pronko, J. Math. Phys. 39, 2995 (1998).
N. Shnerb and L. P. Horwitz, Phys. Rev. A 48, 4068 (1993); L. P. Horwitz and N. Shnerb, Found. Phys. 28, 1509 (1998).
For example, J. R. Taylor, Scattering Theory (Wiley, New York, 1972); R. J. Newton, Scattering Theory of Particles and Waves (McGraw-Hill, New York, 1976).
K. Hoffman, Banach Spaces of Analytic Functions (Prentice Hall, Englewood Cliffs, NJ, 1962).
T. D. Lee, Phys. Rev. 95, 1329 (1954); K. O. Friedrichs, Comm. Pure 6 Appl. Math. 1, 361 (1950), this result was originally conjectured by C. Piron (personal communication).
Y. Strauss and L. P. Horwitz, in preparation.
H. Lehmann, K. Symanzik, and W. Zimmerman, Nuovo Cim. 1, 1425 (1955).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Strauss, Y., Horwitz, L.P. Representation of the Resonance of a Relativistic Quantum Field Theoretical Lee–Friedrichs Model in Lax–Phillips Scattering Theory. Foundations of Physics 30, 653–694 (2000). https://doi.org/10.1023/A:1003780726075
Issue Date:
DOI: https://doi.org/10.1023/A:1003780726075