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An Axiomatic Formulation of the Theory of Coinciding Simple Poles and Multiple Poles
1.W. Heisenberg, Nucl. Phys. 4, 532 (1957).
2.References on multipole ghosts are found in Nagy’s review articles: K. Nagy, Nuovo Cimento Suppl. 17, 92 (1960);
2.State Vector Spaces with Indefinite Metric in Quantum Field Theory (P. Noordhoff, Groningen, 1966).
2.See also J. Lukierski, Acta Phys. Polon. 32, 551 (1967);
2.J. Lukierski, 32, 771 (1967); , Acta Phys. Pol.
2.K. Yokoyama and R. Kubo, Progr. Theoret. Phys. (Kyoto) 41, 542 (1969).
3.N. Nakanishi, Progr. Theoret. Phys. (Kyoto) 35, 1111 (1966);
3.N. Nakanishi, 38, 881 (1967).
3.See also, K. Yokoyama, Progr. Theoret. Phys. (Kyoto) 40, 160 (1968);
3.K. Yokoyama, 40, 421 (1968);
3.K. Yokoyama, 41, 1384 (1969).
4.N. Nakanishi, Phys. Rev. 140, B947 (1965).
4.See also N. Nakanishi, Phys. Rev. 147, 1153 (1966); , Phys. Rev.
4.N. Nakanishi, Progr. Theoret. Phys. (Kyoto) 39, 1585 (1968);
4.J. Arafune, Progr. Theoret. Phys. (Kyoto) 40, 620 (1968);
4.S. Naito, Progr. Theoret. Phys. (Kyoto) 41, 500 (1969).
5.N. Nakanishi, Progr. Theoret. Phys. (Kyoto) 41, 233 (1969).
5.See also N. Nakanishi, Progr. Theoret. Phys. (Kyoto) 41, 780 (1969);
5.M. Minami, Progr. Theoret. Phys. (Kyoto) 41, 1328 (1969).
6.D. Z. Freedman and J.‐M. Wang, Phys. Rev. 153, 1596 (1967). There are a number of papers on the unequal‐mass conspiracy.
7.N. Nakanishi, Phys. Rev. 136, B1830 (1964);
7.D. Z. Freedman, C. E. Jones, and J.‐M. Wang, Phys. Rev. 155, 1645 (1967); , Phys. Rev.
7.N. Nakanishi, Progr. Theoret. Phys. (Kyoto) 41, 516 (1969);
7.N. Nakanishi and N. Seto, Progr. Theoret. Phys. (Kyoto) 41, 1094 (1969).
8.M. Ida, Progr. Theoret. Phys. (Kyoto) 43, 808 (1970).
9.For details, see N. Nakanishi, Progr. Theoret. Phys. (Kyoto) Suppl. 43, 1 (1969).
10.Though it is natural to consider the poles on the S plane from the physical point of view, it is more convenient mathematically to work on the λ plane. Both approaches are equivalent if See also Sec. 7.
11.R. G. Cooke, Linear Operators (Macmillan, London, 1953), p. 350. Extension to a complex functional was made by Bohnenblust and Sobczyk.
12.Here may be singular near and but this causes no difficulty because is continuous in z.
13.If we consider the trivially reducible case, S is of course reducible.
14.We do not use the property of S in this section.
15.A similar construction of possible basis vectors of was made by Ida (Ref. 8), but unfortunately he did not prove their linear independence. Indeed, in his notation, given linear independent vectors belonging to but not to for are not necessarily linearly independent.
16.In this case, basis vectors were obtained also by Ida (Ref. 8) in a different, more complicated way.
17.For example, the self‐conjugateness of S and H yields only the following results in the two cases considered in Sec. 6: In Sec. 6A, all coefficients are real if and in Sec. 6B, and Re if
18.Note that does not depend on
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