Systems containing more than two relativistic particles are analyzed from the point of view of irreducible representations of the Poincaré group. Corresponding Clebsch—Gordan coefficients are calculated for three‐particle systems, and recoupling functions (which are the analog of the Racah coefficients) are defined.

## REFERENCES

1.

2.

The decomposition of the direct product associated with many relativistic particles is described in the lectures of A. S. Wightman, “Invariance in Relativistic Quantum Mechanics,”

*Dispersion Relations and Elementary particles*, edited by C. deWitt and R. Omnes (John Wiley & Sons, Inc., New York, 1961), Sec. IV, p. 159.The present paper follows a different line of reasoning. For references dealing with two relativistic particles see Ref. 1 of

A. J.

Macfarlane

, J. Math. Phys.

4

, 490

(1963

).3.

A. J.

Macfarlane

, Rev. Mod. Phys.

34

, 41

(1962

). We shall generally refer to this paper as M, often making use of its results.4.

U. Fano and G. Racah,

*Irreducible Tensorial Sets*(Academic Press Inc., New York, 1959), Chap. 11.5.

We write $(a\alpha b\beta |c\gamma )$ instead of $(a\alpha b\beta |abc\gamma )$ as

*a b*on the right are obvious.6.

The reason for the upper limit of φ to be π and not $2\pi $ is given in Appendix B.

7.

It is shown in Appendix B that the weight sin φ is attached to the segment $(0,\pi ).$ The functions $(N+12)12PN(cos\phi )$ were chosen for the sake of orthonormality.

8.

See M (3.17), M (3.26)‐M (3.29). We discussed here the form of the function

*P*only; the origin of the other factors on right‐hand side of (19) is made clear in Ref. (3). In particular, $2w12[\lambda (w2,k\u20322,k\u20332)]\u221212$ is a normalization factor assuring the orthonormality of the right‐hand side of (19). Note that (21) differs from M (3.17) in that it involves $R\u22121(ki,L\u22121(r))$ rather $R(ki,L(r));$ these rotations do not equal each other.9.

Since $R3$ in this context is the first parametric group, a rotation

*R*transforms $DlmL$ into $\Sigma n\u2009Dln(R)DnmL.$ Accordingly, there are $(2L+1$ independent linear bases for the representation*L*of $R3$ in the space of functions over $R3;$ and we choose $m,\u2212L\u2a7dm\u2a7dL$ as a label for these bases.10.

Relations (30) and (31) follow from the unitarity of the where $(NN\u2032K000)$ is conventional 3‐

*D*matrices, the orthogonality of the CGc’s of $R3,$ and the relation$PN(cos\u2009\phi )PN\u2032(cos\u2009\phi )\u2009=\u2009\Sigma K(2K+1)(NN\u2032K000)2PK(cos\u2009\phi )$

*j*symbol.11.

Greek letters denote masses or invariant energies of corresponding 4‐momenta; e.g., $\gamma 2\u2009=\u2009c2,xgr;2\u2009=\u2009x2,$ etc.

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© 1966 The American Institute of Physics.

1966

The American Institute of Physics

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