Elementary Particle Physics pp 57-138 | Cite as

# The Three-Body Problem

Conference paper

## Abstract

The quantum mechanical three-body problem has been studied with increasing interest in the last decade. The main progress was achieved by deriving integral equations which are not only theoretically correct, but also practically applicable. Such equations allow us in particular to investigate, besides three-body bound states, the scattering of an elementary particle from a bound two-particle system.

## Keywords

Composite Particle Projection Property Faddeev Equation Interaction Picture Resolvent Equation## Preview

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## References and Footnotes

- 1.Formal scattering theory is explained in several well known text books. Original papers, where the single-channel case is extended to multichannel problems, are: H. Ekstein, Phys. Rev. 101, 880 (1956); Nuovo Cim. 4, 1017 (1956);Google Scholar
- Many general points are discussed in the article by W. Brenig and R. Haag, Fortschritte der Physik 7, 183 (1959). For further references see T. F. Jordan, J. Math. Phys. 3, 429 (1962) and the bibliography given in Ref. 18.Google Scholar
- 2.We assume 0a=fIpcPa(p)d3p, the momentum distribution 0a(p) being a tempered test function.Google Scholar
- 3.The reduced masses are u3=m1m2/(m1+m2) and M3=m3(m1+m2)/ /(m1+m2+m3).Google Scholar
- 4.Three-body interactions are easily incorporated in all our considerations but are left out for simplicity.Google Scholar
- 5.It is rather characteristic for any multichannel situation that various different channel Hamiltonians have to be introduced which implies that there does not exista single splitting of H into a “free” Hamiltonian and the interaction.Google Scholar
- J. M. Cook, J. Math. and Phys. 36, 82 (1957); M. N. Hack, Nuovo Cim. 9, 731 (1958); 13, 231 (1959);Google Scholar
- J. M. Jauch and I. I. Zinnes, Nuovo Cim. 11, 553 (1959).MathSciNetCrossRefGoogle Scholar
- Most general proofs which also include the case of singular potentials are given by J. Kupsch and W. Sandhas, Commun. Math. Phys. 2, 147 (1966);Google Scholar
- W. Hunziker, Heiv. Phys. Acta 40, 1052 (1967).MathSciNetMATHGoogle Scholar
- 7.J. D. Dollard, J. Math. Phys. 5, 729 (1964).MathSciNetADSCrossRefGoogle Scholar
- 8.For a review see L. D. Faddeev, in Three Body Problem in Nuclear and Particle Physics, ed.: J. S. C. McKee and P. M. Rolph ( North Holland, Amsterdam, 1970 ).Google Scholar
- 10.This step is, of course, by no means trivial. Several careful investigations have been devoted to this point which, however, show that under suitable assumptions on the potential this procedure is justified. Hereby the gap is filled between time-dependent scattering theory and the usual formalism of the time-independent (stationary) theory. See, e.g., Ref. 18;Google Scholar
- J.J.H.J. M. Jauch, loc. cit. (Ref. 1);Google Scholar
- K. Hepp, Heiv. Phys. Acta 42, 425 (1969);MathSciNetGoogle Scholar
- C. Chandler and A. G. Gibson, preprint.Google Scholar
- 11.Here and in the following we assume c +o, without explicit notation.Google Scholar
- 12.L. D. Faddeev, Zh. Eksperim. i. Teor. Fiz. 39Google Scholar
- 1459.) (English transi.: Soviet Phys. - JETP 12, 1014 (1961)).Google Scholar
- 15.This difference is characteristic for all sufficiently complex situations. It becomes decisive in relativistic quantum field theory: R. Haag, Dan. Mat. Fys. Medd. 29, 12 (1955). For a review of these questions see: R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and all that ( Benjamin, New York, 1964 ).Google Scholar
- 16.See, in particular, H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cim. 1, 205 (1955); 1, 425 (1955); W. Zimmermann, Nuovo Cim. 10, 597 (1958). For an application to the relativistic three-body problem see D. Z. Freedman, C. Lovelace and J. M. Namyslowski, Nuovo Cim. 43, 258 (1966).Google Scholar
- 17.E. O. Alt, P. Grassberger and W. Sandhas, Nucl. Phys. B2, 167 (1967).ADSCrossRefGoogle Scholar
- 18.Compare also L. D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory, Publications of the Steklov Mathematical Institute, Vol. 69 (Leningrad, 1963) (English transi.: Israel Program for Scientific Translation, Jerusalem, 1965, distributed by Oldbourne Press, London).Google Scholar
- 19.C. Lovelace, Phys. Rev. 135, B1225 (1964).MathSciNetADSCrossRefGoogle Scholar
- 20.W. Sandhas, loc. cit. (Ref. 9).Google Scholar
- 21.L. D. Faddeev, loc. cit. (Ref. 18).Google Scholar
- 22.These considerations sketch a new derivation of the unitarity relation given in Ref. 17. For a different approach see C. Lovelace, in Strong Interactions and High Energy Physics, ed.: R. G. Moorhouse (Oliver and Boyd, London, 1964). Compare also K. L. Kowalski, Phys. Rev. 188, 2235 (1969).Google Scholar
- 23.See in particular Ref. 18.Google Scholar
- 24.The mathematics needed for the following considerations is explained, e.g., in F. Riesz and B. Sz.-Nagy, Functional Analysis (Blackie, 1956 );Google Scholar
- R. Schatten, Norm Ideals of Completely Continuous Operators ( Springer-Verlag, Berlin, 1960 );MATHCrossRefGoogle Scholar
- F. Smithies, Integral Equations ( Cambridge University Press, New York, 1958 );MATHGoogle Scholar
- R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, New York, 1953 ). For further useful publications see the bibliography given in C. Lovelace, loc. cit. (Refs. 19 and 22 ).Google Scholar
- 25.H. Rollnik, Zeitschr. f. Physik 145, 639 (1956). An extensive application of this method is given in: M. Scadron, S. Weinberg and J. Wright, Phys. Rev. 135, B202 (1964).Google Scholar
- 26.A description of these ideas has been given by the author in a seminar at the Schladming winter school 1969: Acta Physica Austriaca, Suppl. VI, 454 (1969). This may be used as a first introduction to the approach explained in the following sections.Google Scholar
- 27.K. M. Watson, Phys. Rev. 89, 575 (1953). Compare also K. M. Watson and J. Nuttall, Topics in Several Particle Dynamics (Holden-Day, San Francisco, 1967). Another early attempt is G. V. Skornjakov and K. A. TerMartirosjan, JETP 31, 775 (1956).Google Scholar
- 28.R. J. Glauber, in High Energy Physics and Nuclear Structure, ed.: G. Alexander (North Holland, Amsterdam, 1967 ). Compare also the seminar by J. M. Namyslowski, given at this Schladming winter school.Google Scholar
- 29.Y. Yamaguchi, Phys. Rev. 95, 1628 (1954). A list of further publications concerning separable potentials is given in footnote 28 of Ref. 19.Google Scholar
- 30.A. N. Mitra, Nucl. Phys. 32, 529 (1962);MATHCrossRefGoogle Scholar
- A. N. Mitra, Phys. Rev. 139, B1472 (1965);ADSCrossRefGoogle Scholar
- A. N. Mitra and V. S. Bhasin, Phys. Rev. 131, 1265 (1963);ADSCrossRefGoogle Scholar
- A. N. Mitra, in Advances in Nuclear Physics, ed.: M. Baranger and E. Vogt (Plenum Press, Inc., New York, 1969 ), Vol. I II.Google Scholar
- 31.A. G. Sitenko and V. F. Kharchenko, Nucl. Phys. 49, 15 (1963).CrossRefGoogle Scholar
- 32.R. D. Amado, Phys. Rev. 132, 485 (1963); 902 (1966). See also:MathSciNetADSMATHCrossRefGoogle Scholar
- R. Aaron, R. D. Amado, and Y. Y. Yam, Phys. Rev. 140, 81291 (1965)ADSCrossRefGoogle Scholar
- R. D. Amado, Annual Review of Nuclear Science 19, 635 (1969).Google Scholar
- 33.An example is the Zachariasen-Thirring model: F. Zachariasen, Phys. Rev. 121, 1851 (1961);Google Scholar
- W. Thirring, Nuovo Cim. 23, 1064 (1962).MathSciNetCrossRefGoogle Scholar
- 34.For a review see W. Sandhas, lecture given at the Symposium on the Nuclear Three-Body Problem and Related Topics, Budapest 1971 (preprint, to be published in the proceedings of the conference). Compare also:Google Scholar
- E. O. Alt, seminar given at this Schladming winter school.Google Scholar
- 35.
- K. Meetz, J. Math. Phys. 3, 690 (1962).MathSciNetADSMATHCrossRefGoogle Scholar
- 36.In practice the ideal choice can be determined by the method of J. Wright and M. Scadron, Nuovo Cim. 34, 1571 (1964).Google Scholar
- 37.The only difference is that it is now the “free Green’s function”, not the “potential” which is split into separable terms and a non-separable rest (see Sec. 9). The algebra, however, is nearly the same.Google Scholar
- 38.Our derivation of effective two-particle equations followed Ref. 17. For other approaches see:Google Scholar
- J. H. Sloan, Phys. Rev. 165, 1587 (1968);ADSCrossRefGoogle Scholar
- M. G. Fuda, Phys. Rev. 166, 1064 (1968);ADSCrossRefGoogle Scholar
- F. Riordan, Nuovo Cim. 54A, 552 (1968);ADSCrossRefGoogle Scholar
- L. Rosenberg, Phys. Rev. 168, 1756 (1968);ADSCrossRefGoogle Scholar
- R. Yaes, Phys. Rev. 170, 1236 (1968).ADSCrossRefGoogle Scholar
- Weinberg himself proposed a generalization of the quasi particle concept to the three-body (and even to the n-body) case: Phys. Rev. 133, B232 (1964).Google Scholar
- Being applied, however, to the whole kernel instead of the subsystem operators only, this method is completely different from the one developed in Ref. 17 and described in this lecture. In practice Weinberg’s approach seems to be too complicated.Google Scholar
- 39.E. O. Alt, P. Grassberger and W. Sandhas, Nucl. Phys. A139, 209 (1969);CrossRefGoogle Scholar
- V. Vanzani, Nuovo Cim. Lett. 2, 706 (1969).CrossRefGoogle Scholar
- 40.P. Grassberger and W. Sandhas, Zeitschr. f. Physik 220, 29 (1969).ADSMATHCrossRefGoogle Scholar
- 41.E. O. Alt, P. Grassberger and W. Sandhas, Phys. Rev. D1, 2581 (1970), Appendix C.Google Scholar
- 42.W. Sandhas, loc. cit. (Ref. 9);Google Scholar
- P. Grassberger and W. Sandhas,Zeitschr. f. Physik 217, 9 (1968).CrossRefGoogle Scholar
- 43.Compare in particular R. Haag, Phys. Rev. 112, 669 (1958). Nonrelativistic composite particle fields are studied in strict analogy to this approach inGoogle Scholar
- W. Sandhas, Commun. Math. Phys. 3, 358 (1966).MathSciNetADSMATHCrossRefGoogle Scholar
- 44.This is discussed in detail by W. Sandhas, loc. cit. (Ref. 9). Here also the relation to the “weak convergence method” of axiomatic field theory (see Ref. 43) is established.Google Scholar
- 45.In this way also a treatment of the four-(and even of the n-) body problem becomes possible, without starting from four-body integral equations with connected kernels. Incorporating step by step the dominant two-and three-body subsystem properties, by means of separable terms, we directly arrive at effective two-particle equations also in the four-body case: P. Grassberger and W. Sandhas, Nucl. Phys. B2, 181 (1967); Zeitschr. f. Physik 217, 9 (1968). Practical applications of this approach are given in E. O. Alt, P. Grassberger and W. Sandhas, Phys. Rev. Cl, 85 (1970). For a short description see Ref. 26.Google Scholar

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© Springer-Verlag Wien 1972