Abstract
Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.
Similar content being viewed by others
References
Antoine, J. P. (1969). Journal of Mathematical Physics 10, 53.
Bargmann, V. and Wigner, E. P. (1948). Proceedings of the National Academy of Sciences of the United States of America 34, 211.
Bernicha, A., López Castro, G., and Pestieu, J. (1994). Physical Review D 50, 4454.
Bernicha, A., López Castro, G., and Pestieu, J. (1996). Nuclear Physics A 597, 623.
Bohm, A. (1966). Boulder Lectures in Mathematical Physics 9A, 255.
Bohm, A. (1978a). The Rigged Hilbert Space and Quantum Mechanics (Lecture Notes in Physics, Vol. 78), A. Bohm and J. D. Dollard, eds., Springer-Verlag, Berlin.
Bohm, A. (1978b). Letters of Mathematical Physics 3, 455.
Bohm, A. (1981). Journal of Mathematical Physics 22, 2813.
Bohm, A. (1999). Physical Review A 60, 861.
Bohm, A., Maxson, S., Loewe, M., and Gadella, M. (1997). Physica A 236, 485.
Gel'fand, I. M. and Vilenkin, N. Ya. (1994). Generalized Functions, Vol. 4, Academic Press, New York, p. 103.
Hagiwara, K. et al., (2002). Physical Review D 66, 010001.
Maurin, K. (1968). General Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers (PWN), Warszawa.
Roberts, J. E. (1966). Journal of Mathematical Physics 7, 1097.
Sirlin, A. (1991a). Physical Review Letters 67, 2127.
Sirlin, A. (1991b). Physics Letters B 267, 240.
Stuart, R. G. (1991). Physics Letters B 262, 113.
Stuart, R. G. (1997). Physical Review D 56, 1515.
Veltman, M. (1964). Physica 29, 186.
Wigner, E. P. (1939). Annals of Mathematics 40, 149.
Willenbrock, S. and Valencia, G. (1991). Physics Letters B 259, 373.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kielanowski, P. Relativistic Gamow Vectors. International Journal of Theoretical Physics 42, 2339–2355 (2003). https://doi.org/10.1023/B:IJTP.0000005961.79955.4a
Issue Date:
DOI: https://doi.org/10.1023/B:IJTP.0000005961.79955.4a