Construction of a complete set of states in relativistic scattering theory

Journal of Mathematical Physics

Volumn 38, Issue 1, 1997, Pages 115-138

  Pelc, Oskar ^a   Horwitz, L P ^b,c,d  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^b INSTITUTE FOR ADVANCED STUDY   (United States)

^c TEL AVIV UNIVERSITY   (Israel)

^d BAR ILAN UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031282770     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531845     Document Type: Article

References (32)

1. D. Iagolnitzer, The S Matrix (North-Holland, Amsterdam, 1978).
2. P. A. M. Dirac, The Principles of Quantum Mechanics, 1st ed. (Clarendon, Oxford, 1930), 3rd ed. (1947).
3. J. E. Roberts, J. Math. Phys. 7, 1097 (1966); Commun. Math. Phys. 3, 98 (1966); A. Böhm, The Rigged Hilbert Space in Quantum Physics, in Boulder Lectures in Theoretical Physics, edited by A. O. Barut, Vol. 9A (Interscience, New York, 1996).
4. J. E. Roberts, J. Math. Phys. 7, 1097 (1966); Commun. Math. Phys. 3, 98 (1966); A. Böhm, The Rigged Hilbert Space in Quantum Physics, in Boulder Lectures in Theoretical Physics, edited by A. O. Barut, Vol. 9A (Interscience, New York, 1996).
5. J. E. Roberts, J. Math. Phys. 7, 1097 (1966); Commun. Math. Phys. 3, 98 (1966); A. Böhm, The Rigged Hilbert Space in Quantum Physics, in Boulder Lectures in Theoretical Physics, edited by A. O. Barut, Vol. 9A (Interscience, New York, 1996).
6. J.-P. Antoine, J. Math. Phys. 10, 53 (1969).
7. J.-P. Antoine, J. Math. Phys. 10, 2276 (1969).
8. See also: S. J. L. van Eijndhoven and J. de Graaf, A Mathematical Introduction to Dirac's Formalism (North Holand, Amsterdam, 1986); A. Böhm and M. Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lecture Notes in Physics (Springer, Berlin, 1989). This approach has become standard in quantum field theory; see for instance N. N. Bogolubov, A. A. Logunov, and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory (Benjamin, New York, 1975).
9. See also: S. J. L. van Eijndhoven and J. de Graaf, A Mathematical Introduction to Dirac's Formalism (North Holand, Amsterdam, 1986); A. Böhm and M. Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lecture Notes in Physics (Springer, Berlin, 1989). This approach has become standard in quantum field theory; see for instance N. N. Bogolubov, A. A. Logunov, and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory (Benjamin, New York, 1975).
10. See also: S. J. L. van Eijndhoven and J. de Graaf, A Mathematical Introduction to Dirac's Formalism (North Holand, Amsterdam, 1986); A. Böhm and M. Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lecture Notes in Physics (Springer, Berlin, 1989). This approach has become standard in quantum field theory; see for instance N. N. Bogolubov, A. A. Logunov, and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory (Benjamin, New York, 1975).
11. L. P. Horwitz and O. Pelc, J. Math. Phys. (in press).
12. A. O. Barut and R. Ra̧czka, Theory of Group Representations and Applications (PWN, Warszawa, 1977).
13. 0 (and not only isomorphic to it).
14. E. P. Wigner, Ann. Math. 40, 149 (1939); Group Theory (Academic, New York, 1959).
15. E. P. Wigner, Ann. Math. 40, 149 (1939); Group Theory (Academic, New York, 1959).
16. V. Bargmann, J. Math. Phys. 5, 862 (1964).
17. Bargmann showed [V. Bargmann, Ann. Math. 59, 1 (1954)] that for r + s≥3, all the ray representations of ℘ are (equivalent to) true representations, so in this case assumption 2.2 is not a restriction].
18. (1) is expressible in terms of irreducible representations and since these are identified with particle types (see the discussion in the introduction of Ref. 7), this requirement is actually part of the physical interpretation.
19. ℘.
20. K. Maurin, General Eigenfunction Expansions and Unitary Representations of Topological Groups (PWN, Warszawa, 1968).
21. n).
22. F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967).
23. l ℒ(Ω) are considered.
24. † in Appendix A 4).
25. I. M. Gel'fand, G. E. Shilov, and N. Ya. Vilenkin, Obobshchennye Funktsii i Deistviya Nad Nimi (Gosudarstvennoe Izdatel'stvo Fiziko-Matematicheskoi Literatury, Moskow, 1958-1960), Vols. I-V. English translation: Generalized Functions (Academic, New York, 1964).
26. I. M. Gel'fand, G. E. Shilov, and N. Ya. Vilenkin, Obobshchennye Funktsii i Deistviya Nad Nimi (Gosudarstvennoe Izdatel'stvo Fiziko-Matematicheskoi Literatury, Moskow, 1958-1960), Vols. I-V. English translation: Generalized Functions (Academic, New York, 1964).
27. J. M. Jauch and J.-P. Marchand [Helv. Phys. Act. 39, 325 (1966)] seem to be the first to use the notation "(·|·)" but this was only for φ(x) = 〈x|φ) and they did not consider (x| as a distinct entity.
28. M. Reed and B. Simon, in Ref. 19, p. 147.
29. L. Schwartz, Théorie des distributions (Hermann & Cie., Paris, 1957-1959), Vols. I, II.
30. The notion of a "complete set of commuting observables" may be formulated also in a pure algebraic manner, see J.-P. Antoine, G. Epifanio, and C. Trapani, Helv. Phys. Act. 56, 1175 (1983).
31. This is only in the sense of the Parseval equality, which is, in a Hubert space, characteristic to a complete orthonormal system. This is not completeness in the usual sense since it is only in Φ while the (x|'s are not, in general, in Φ. Orthogonality is not even defined since there is no inner product in Φ′.
32. -).