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Partial Group‐Theoretical Treatment for the Relativistic Hydrogen Atom
1.V. Bargmann, Ann. Math. 48, 568 (1947).
2.L. Pukanszky, Math. Ann. 156, 96 (1964).
3.S. S. Sannikov, Zh. Eksperim. i Teor. Fiz. 49, 1913 (1965)[
3.English transl.: S. S. Sannikov, Soviet Phys.‐JETP 22, 1306 (1966].
4.A. O. Barut and C. Fronsdal, Proc. Roy. Soc. (London) A287, 532 (1965).
5.The authors are grateful to J.‐J. Loeffel for his help in the understanding of this question. The property which is emphasized in the text is essentially due to the fact that one has for two sets of square‐integrable solutions of Eq. (1) instead of one when
6.The dimensionality of the space of homogeneous polynomials of degree l in the q‐dimensional space is The Laplace operator transforms this space into the space of homogeneous polynomials of degree The kernel of this transformation is the space of spherical harmonics of degree l, the dimensionality of which is then The reduction of the “triangular” representations of SU(q) with respect to SO(q) corresponds to the following relations , .
7.Clearly, we must use in our approach reducible representations of the group We used the theory of spherical harmonics of the q‐dimensional Euclidean space in order to obtain the reduction of the representation. In that sense, our method, although complete, is only partially group theoretical.
8.The notation is that of L. J. Schiff, Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1955) p. 338.
9.Z is supposed to be small
10.L. C. Biedenharn, Phys. Rev. 126, 845 (1962).
11.P. C. Martin and R. J. Glauber, Phys. Rev. 109, 1307 (1958).
12.The q‐dimensional nonrelativistic Kepler problem has, as a noninvariance group, the group V. Fock, Z. Physik 98, 145 (1935);
12.V. Bargmann, Z. Physik 99, 576 (1936); , Z. Phys.
12.S. P. Alliluyev, Zh. Eksperim. i Teor. Fiz. 33, 200 (1957)
12.[English transl.: S. P. Alliluyev, Soviet Phys.‐JETP 6, 260 (1957)].
12.The Lie algebra of this group can be realized with Poisson brackets. [For , see H. Bacry, Nuovo Cimento 41, 222 (1966)].
12.A theorem in symplectic manifolds proves that the same Lie‐Poisson algebra can be built for all q‐degrees‐of‐freedom problems [see H. Bacry, H. Ruegg, and J.‐M. Souriau, Commun. Math. Phys. 3, 323 (1966);
12.H. Bacry, “Dynamical Groups and Relativity,” lectures given in Boulder Colorado Summer Institute for Theoretical Physics, 1966 (to be published).
12.See also direct proofs in N. Mukunda, “Dynamical Symmetries and Classical Mechanics, Realizations of Lie Algebras in Classical Mechanics,” preprints, Syracuse University, Syracuse, New York, 1966;
12.J. Rosen, “On Realizations of Lie Algebras and Symmetries in Classical and Quantum Mechanics,” preprint;
12.J.‐M. Souriau, Compt. Rend. Acad. Sci. 263, 1191 (1966).
12.If we use the definition of phase space given in H. Bacry, Commun. Math. Phys. 5, 97 (1967), a particle with nonzero spin corresponds to a four‐degrees‐of‐freedom problem. Consequently, one can hope to build explicitly the hydrogen‐atom levels with the aid of a unitary representation of SO(5, 1).
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