Summary
The many-particle scattering amplitude is projected on the matrix elements of the unitary irreducible representations of the three-dimensional Lorentz group. The usefulness of this transformation in the treatment of a certain class of integral equations satisfied by the amplitude is pointed out. A generalization taking into account a set of nonunitary representations is shown to lead to a transformation which has many of the properties of the classical Laplace transformation and can be used to obtain asymptotic expansions similar to those obtained from the Watson-Sommerfeld formula.
Riassunto
L’ampiezza di diffusione per un prooesso a più particelle viene proiettata sugli elementi di matrice délie rappresentazioni unitarie irriducibili del gruppo di Lorentz tridimensionale. Si mette in luce l’utilità di questa trasformazione nello studio di una certa classe di equazioni integrali a cui soddisfa l’ampiezza di diffusione. Si mostra che, utilizzando un certo insieme di rappresentazioni non unitarie, si ottiene una trasfor. mazione più generale che ha molte delle proprietà della trasformazione classica di Laplace e può essere usata per ottenere sviluppi asintotici simili a quelli che si ottengono dalla formula di Watson-Sommerfeld.
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References
L. Sertoeio andM. Tollbk:Nuovo Cimento,33, 413 (1964).
G. Mackey:Bull. Amer. Math. Soc,56, 385 (1950).
L. H. Loomis :Ann Introduction to Abstract Harmonie Analysis (Princeton, 1953).
M. Jacob andG. C. Wick:Ann. Phys.,7, 404 (1959);G. C. Wick:Ann. Phys.,18, 65 (1962).
E. L. Omnès:On the Three-Body Scattering Amplitude, I, II andIII, U.C.E.L. reports andPhys. Rev.,134, B 1358 (1964).
J. B. Hartle:Phys. Bev.,134, B 610, B 620 (1964).
M. Andbews andJ. Gunson:Complex Angular Momentum in Many-Partiele States, I and II, preprint (University of Birmingham, 1963). This paper gives also the connection between the matrix elements of the local representations of the rotation group and of the representations of the three-dimensional Lorentz group.
E. G. Beltrametti andG. Luzzatto:Nuovo Cimento,29, 1003 (1963).
S. Mandelstam:Ann. Phys.,19, 254 (1962).
E. P. Wignee:Ann. Math.,40, 149 (1939).
Iu. M. Shikokov:Sov. Phys., JETP.,6, 919 (1958).
H. Joos:Forts, d. Phys.,10, 65 (1962).
H. P. Stapp:Phys. Rev.,125, 2139 (1962).
V. Bargmann:Ann. Math.,48, 568 (1947).
E. P. Wignee:Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Chapt. IX, and X (New York, 1959).
M. A. Naimark:Normed Rings, Sect.28 (Groningen, 1964).
The possibility of writing down an integral equation for the relativistic threeparticle amplitudes has been investigated byA. Tucciarone:Thesis (Roma, 1964).
G. Mackey:Proc. Nat. Acad. Sci. USA,34, 156 (1948).
G. Cosenza, L. Sertorio andM. Toller:Nuovo Cimento,35, 913 (1935).
G. Tiktopoulos:Phys. Rev.,133, B 1231 (1964).
For a discussion of the concept of equivalence for nonunitary infinitedimensional representations, see :M. A. Naimark :Linear Eepresentations of the Lorentz Group, translations AMS (1957), p. 379;G. Mackey:Bull. Am. Math. Soc,69, 628 (1963).
G. Doetsch:Theorie und Anwendung der Laplace-Transformation, III Teil (New York, 1943).
A.Eedelti, W. Magnus, F. Obebhettingeb andF. Gr. Tkicomi:Higher Transcendental Functions, vol.1, formula 2.1.3.13.
Loc. cit., formula 2.1.4.17.
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Toller, M. Three-dimensional Lorentz group and harmonic analysis of the scattering amplitude. Nuovo Cim 37, 631–657 (1965). https://doi.org/10.1007/BF02749860
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DOI: https://doi.org/10.1007/BF02749860