Prog. Theor. Phys. Supplement No.37 & 38 (1966) pp. 368-382
Relativistic Wave Equations for Particles with Internal Structure and Mass Spectrum
The Enrico Fermi Institute for Nuclear Studies and
the Department of Physics, The University of Chicago, Illinois, U.S.A.
(Received August 5, 1966)
We consider a class of wave equations which couple an infinite number
of tensors or spinors of all ranks. Such a system of equations
naturally possesses an infinite number of mass levels, and each
eigenfunction implicitly contains a built-in form factor. Two simple
examples of first order differential equations are examined. One is
based on the set of all finite representations
D((n/2), (n/2)) or D((n/2) + s, (n/2)) + D((n/2), (n/2) + s),
n = 0, 1, 2, …, of the Lorentz group, and gives a mass spectrum resembling the hydrogen atom, but probability densities and form factors are unphysical. The other model is based on a unitary representation of the group 0(4, 2), and has an inverted hydrogen-like spectrum. The later corresponds to a generalization of a model first proposed by Majorana.
DOI : 10.1143/PTPS.37.368
- E. P. Wigner, Ann. Math. (Princeton) 40 (1939), 149; Group Theoretical Concepts and Methods in Elementary Particles, edited by F. Gursey (Gordon and Breach, New York, 1963), p. 37.
- V. Bargmann and E. P. Wigner, Proc. Nat. Acad. Sci. 34 (1948), 211.
A. Salam, R. Delbourgo and J. Strathdee, Proc. R. Soc. A 284 (1965), 146.
B. Sakita and K. C. Wali, Phys. Rev. 139 (1965), B1355[APS].
- E. Majorana, Nuovo Cim. 9 (1932), 335.
See also an interesting review article by D. M. Fradkin, Am. J. Phys. 34 (1966), 314.
The author thanks Prof. F. Gursey and Mr. J. Cronin for first calling his attention to these papers. For related papers see references 4) and 6).
- General wave equations are discussed, for example, in E. M. Corson, Introduction to Tensors, Spinors, and Relativistic Wave Equations (Blackie and Son Ltd., London, 1953).
I. M. Gel'fand, R. A. Minlos and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Group and Their Applications, English translation (Pergamon Press, Oxford, 1963).
M. A. Naimark, Linear Representations of the Lorentz Group, English translation (Pergamon Press, Oxford, 1964). A. J. Le Couteur, Proc. R. Soc. A 202 (1950), 284; ibid. 202 (1950), 394.
- Similar operator methods were proposed by
B. Kursunoglu, Phys. Rev. 135 (1964), B761[APS].
Y. Dothan, M. Gell-Mann and Y. Ne'eman, Phys. Lett. 17 (1965), 148[CrossRef].
- Majorana's equation was rediscovered later: I. M. Gel'fand and A. M. Yaglom, Zh. Exsp. Teor. Fiz. 18 (1948), 703.
- P. A. M. Dirac, Proc. R. Soc. A 183 (1944), 284.
E. P. Wigner, Z. Phys. 124 (1947), 665.
E. M. Corson, reference 4).
H. Yukawa, Phys. Rev. 77 (1953), 219[APS];
ibid. 91 (1953), 415[APS];
ibid. 91 (1953), 416[APS].
V. L. Ginzburg and I. Ye. Tamm, Zh. Eksp. Teor. Fiz. 17 (1947), 227.
V. L. Ginzburg, Acta Phys. Polonica 15 (1956), 163, etc.
- For form factors and mass spectra, see C. Fronsdal, Symmetry Principles at High Energy, Proceedings of the Third Coral Gables Conference (W. H. Freeman and Co., New York, 1966); and other preprints from Trieste.
- We have in the meantime determined the (scalar) form factor. It is the same as Eq. (38), but has now the correct analytic structure since κ20 < κ2.
- It has come to our attention that a number of papers have been published by T. Takabayashi who considers internal structure of particles in terms of Bose operators corresponding to a relativistic oscillator, regarded as a model for SU(3). He also discusses wave equations and mass spectra using these operators. The author thanks Drs. Fronsdal and Takabayashi for private communication about their respective works in relation to the present paper.
T. Takabayashi, Prog. Theor. Phys. 36 (1966), 185[PTP].
H. Kase and T. Takabayashi, Prog. Theor. Phys. 36 (1966), 187[PTP]; and references therein.
- Working in the product space of the unitary representation of O(4, 2) and a finite (Dirac) representation of the Lorentz group, we have now obtained an equation which properly simulates the hydrogen atom regarding mass spectrum, magnetic moment and form factors.
- In this paper we considered only discrete spectra corresponding to normalizable internal wave functions. Whether we should also include continuum (ionized) states remains in open question.