Abstract
Starting from a unitary, Lorentz invariant two-particle scattering amplitude, we show how to use an identification and replacement process to construct a unique, unitary particle–antiparticle amplitude. This process differs from conventional on-shell Mandelstam s, t, u crossing in that the input and constructed amplitudes can be off-diagonal and off-energy shell. Further, amplitudes are constructed using the invariant parameters which are appropriate to use as driving terms in the multi-particle, multichannel non-perturbative, cluster decomposable, relativistic scattering equations of the Faddeev-type integral equations recently presented by Alfred, Kwizera, Lindesay and Noyes. It is therefore anticipated that when so employed, the resulting multi-channel solutions will also be unitary. The process preserves the usual particle–antiparticle symmetries. To illustrate this process, we construct a J=0 scattering length model chosen for simplicity. We also exhibit a class of physical models which contain a finite quantum mass parameter and are Lorentz invariant. These are constructed to reduce in the appropriate limits, and with the proper choice of value and sign of the interaction parameter, to the asymptotic solution of the non-relativistic Coulomb problem, including the forward scattering singularity, the essential singularity in the phase, and the Bohr bound-state spectrum.
Similar content being viewed by others
References
M. Alfred P. Kwizera J. V. Lindesay H. P. Noyes (2004) ArticleTitle“A non-perturbative, finite particle number approach to relativistic scattering theory,” SLAC-PUB-8821 Found. Phys. 34 IssueID4 581–616 Occurrence Handle10.1023/B:FOOP.0000019627.19038.50
J. V. Lindesay, PhD thesis, available as SLAC Report No. SLAC-243 (1981).
J. V. Lindesay A. J. Markevich H. P. Noyes G. Pastrana (1986) Phys. Rev. D 33 2339 Occurrence Handle10.1103/PhysRevD.33.2339 Occurrence Handle1:CAS:528:DyaL28XhslGqsL8%3D
A. J. Markevich, “Relativistic three-particle scattering theory,” PhD Thesis, Stanford University (1985).
A. J. Markevich (1986) Phys. Rev. D 33 2350 Occurrence Handle10.1103/PhysRevD.33.2350 Occurrence Handle1:CAS:528:DyaL28XhvFCmsLw%3D
G. F. Chew S Matrix Theory of Strong Interactions: A Lecture Note and Reprint Volume (Benjamin, 1961).
G. F. Chew, The Analytic S Matrix: A Basis for Nuclear Democracy (Benjamin, 1966).
L. D. Faddeev, Zh. Eksp. Teor. Fiz. 39, 1459 (1960); Sov. Phys. -JETP 12, 1014 (1961); see also L. D. Faddeev, Mathematical Aspects of the Three-Body Problem in Quantum Scattering Theory (Davey, New York, 1965). For the extension to the four-body problem, see e.g. O. A. Yakubovsky, Yad. Fiz. 5, (1967); Sov. J. Nucl. Phys. 5, 937 (1967).
B. Lippman (1956) Phys. Rev. 102 264 Occurrence Handle10.1103/PhysRev.102.264
M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, 1956), pp. 197–209.
H. P. Stapp T. J. Ypsilantis N. Metropolis (1957) Phys. Rev. 105 302 Occurrence Handle10.1103/PhysRev.105.302 Occurrence Handle1:CAS:528:DyaG2sXkt1Cjuw%3D%3D
N. F. Mott H. S. W. Massey (1949) The Theory of Atomic Collisions EditionNumber2 Oxford University Press New York
L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1955), Sec. 20.
Author information
Authors and Affiliations
Corresponding author
Additional information
Work supported in part by Department of Energy contract DE-AC03-76SF00515
Rights and permissions
About this article
Cite this article
Lindesay, J., Noyes, H.P. Construction of Non-Perturbative, Unitary Particle–Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms. Found Phys 35, 699–741 (2005). https://doi.org/10.1007/s10701-005-4563-8
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10701-005-4563-8