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Majorana Representations of the Lorentz Group and Infinite‐Component Fields
1.E. Majorana, Nuovo Cimento 9, 335 (1932).
2.D. M. Fradkin, Am. J. Phys. 34, 314 (1966).
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4.(in Russian: Fizmatgiz, Moscow, 1958).
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16.See, e.g., H. Kleinert, Ph.D. thesis, University of Colorado, 1967, where a complete account of recent work of A. O. Barut and the author is given. Nonquantized infinite‐component fields are considered in A. Böhm, Syracuse University Preprint SU‐1206‐125, 1967.
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21.P. Budini (private communication). See also P. Budini, ICTP Trieste Preprints IC/67/18 and IC/67/80 (1967).
22.One could be tempted to say that a field is transforming under a self‐coupled representation of if a first‐order invariant equation can be written down for it. This definition, however, is not equivalent to the above because one can write invariant first‐order equations for a lot of irreducible representations (for instance, for the four‐vector representation [0, 2], one can write the equation ), while an invariant Lagrangian of the type exists only for the two Majorana representations. Such a Lagrangian implies actually the existence of a complete irreducible set of invariant first‐order equations.
23.L. Castell, Nuovo Cimento 50, 945 (1967).
24.The technique of creation and annihilation operators was first applied for the realization of the representations of by P. Jordan, Z. Physik 94, 531 (1935)
24.(see a modern exposition in L. C. Biedenharn, ICTP Trieste Preprint IC/67/52, 1967).
24.For the description of unitary representations of the Lorentz group such a technique was used in P. A. M. Dirac, Proc. Roy. Soc. (London) A180, 1 (1942);
24.P. A. M. Dirac, 183, 284 (1945). , Proc. R. Soc. London
24.It has been applied in B. Kursunoglu [Modern Quantum Theory (W. H. Freeman and Co., San Francisco, 1962), p. 257] for the ladder representations of (see Appendix C).
24.In the same form as here [for the description of the Majorana representations of SL (2, C)] they have been used in P. A. M. Dirac, J. Math. Phys. 4, 901 (1963)
24.and in F. Gürsey, Relativity, Groups and Topology, C. De Witt and B. De Witt, Ed. (Gordon and Breach Science Publ., Inc., New York, 1964),
24.and more recently in A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967)
24.and C. Itzykson, Commun. Math. Phys. 4, 92 (1967). The operators should not be confused with particle creation and annihilation operators. They do not depend on coordinates (or momentum) and act only in the space of indices, which in our case is infinite‐dimensional. To avoid confusion we use different notations for the vectors in the auxiliary space X and in the Fock space H of physical states [e.g., ].
25.C. Fronsdal and R. White, Phys. Rev. 163, 1835 (1967).
26.The expressions for are simpler in a basis in which in (A7) is replaced by in such a basis the factors in (2.39) do not appear. (The basis used in the paper has the advantage to be defined also for the finite‐dimensional representations.).
27.The formulation of the theory of particles with arbitrary spin given in S. Weinberg [Phys. Rev. 133, B1318 (1964)], though convenient in a number of cases seems not to be completely satisfactory, e.g., when a minimal electromagnetic interaction for such particles has to be considered.
28.See for instance, R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That (W. A. Benjamin Inc., New York, 1964)
28.or R. Jost, The General Theory of Quantized Fields (American Mathematical Society, Providence, R.I., 1965).
29.N. N. Bogoliubov and V. S. Vladimirov, Nauchn. Dokl. Vyschei Shkoly No. 3, 26 (1958);
29.No. 2, 179 (1959).
29.A generalization of this theorem to functions of several vectors is given in J. Bros, H. Epstein, and V. Glaser, Commun. Math. Phys. 6, 77 (1967).
30.H. Epstein, J. Math. Phys. 8, 750 (1967).
31.S. Ström, Lectures in Theoretical Physics (The Univ. of Colorado Press, Boulder, Colorado, 1964), Vol. VIIA, p. 70–78.
32.V. D. Dao and V. H. Nguyen, Ann. Inst. H. Poincaré 6, 17 (1967).
33.H. Joos, Fortschr. Physik 10, 65 (1962).
34.Formula (3.38) coincides with the result of Ref. 25, see also A. O. Barut and H. Kleinert [Phys. Rev. Letters 18, 754 (1967)], where the scalar product ( ) is calculated for all unitary representations of of the type We mention that our formulas are valid for for any and for arbitrary (not necessarily collinear) 3‐momenta p and q. We remark that the matrix multiplying the invariant form factor coincides with the scalar product of two positive‐energy Dirac spinors
35.W. Rühl, Commun. Math. Phys. 6, 312 (1967).
36.The Bateman Manuscript Project: Higher Transcendental Functions Vol. I, A. Erdélyi, Ed. (McGraw‐Hill Book Co., Inc., New York, 1953), Sec. 6.13.2, Eq. (15).
37.M. E. Arons and E. C. G. Sudarshan, Syracuse University preprint, 1967 (unpublished).
38.G. Feinberg, Phys. Rev. 159, 1089 (1967).
39.It has been proved that if the Fourier transform of a local field vanishes in a domain of spacelike vectors in momentum space, then the field is a generalized free field. See G. F. Dell’Antonio, J. Math. Phys. 2, 759 (1961);
39.O. W. Greenberg, J. Math. Phys. 3, 859 (1962).
40.G. Cocho, C. Fronsdal, I. T. Grodsky, and R. White, Phys. Rev. 162, 1662 (1967);
40.R. Delbourgo, M. A. Rashid, Abdus Salam, and J. Strathdee, Phys. Letters 25B, 475 (1967);
40.C. Fronsdal, ICTP Trieste Preprint IC/67/70 (1967).
41.V. De Alfaro, S. Fubini, G. Furlan, and C. Rossetti, Torino University preprint 1966.
42.A complication arises for See I. M. Gel’fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Integral Geometry and Representation Theory (Academic Press Inc., New York, 1966), Vol. 5, Chap. III, Sec. 3. To avoid it we define in these cases to be the set of homogeneous polynomials of z and z̄ of degree of homogeneity
43.I. T. Todorov ICTP Trieste Preprint IC/66/71, 1966.
44.Analogous formulas, using a higher number of complex variables, have been introduced in R. L. Anderson, J. Fisher, and R. Raczka, ICTP Trieste Preprint IC/66/102, 1966.
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