It was shown in I [J. Math. Phys. 30, 66 (1989)] that the eigenfunctions for the reduced motion of the quantum relativistic bound state with O(3,1) symmetric potential have support in an O(2,1) invariant subregion of the full spacelike region. They form irreducible representations of SU(1,1) [in the double covering of O(2,1)] parametrized by the unit spacelike vector nμ, taken in I as the direction of the z axis (the spectrum is independent of this choice), for which this O(2,1) is the stabilizer. Lorentz transformations move these representations on an orbit whose range is the single‐sheeted hyperboloid covered by this spacelike vector, providing a set of induced representations of SL(2,C). From linear combinations of functions from the irreducible representations of SU(1,1), the representations of the SU(2) subgroup of SL(2,C) on the orbit are extracted and the differential equations that are the eigenvalue equations for the Casimir operators of SL(2,C) are solved. It is found that these SU(2) representations form a basis for the principal series in the canonical representations of Gel’fand. There is a natural scalar product, obtained from group integration on SL(2,C), for which the canonical basis forms an orthogonal set, and the representation is unitary. Since the scalar product [over the O(2,1) invariant measure space] of SU(1,1) irreducible representations is invariant under the action of the little group, the remaining group measure [on the coset space SL(2,C)/SU(1,1)] is the volume on the hyperboloidal spacelike hypersurface dμn=d4n δ(n2−1). The family of Hilbert spaces (ℋn) that carries the representations of O(3,1) is therefore embedded in a larger Hilbert space ℋ with measure d4ydμn, where the { y} are the space‐time coordinates of the restricted region associated with nμ. The representations with nonrelativistic limit coinciding with the known Schrödinger solutions for corresponding spherically symmetric potential problems are in the double covering (half‐integer values for the lowest L level) of O(3,1).

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R.
Arshansky
and
L. P.
Horwitz
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(
1989
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I. M. Gel’fand, R. A. Minlos, and P. Shapiro, Representations of the Rotation and Lorentz Groups and Their Application (Pergamon, New York, 1963).
5.
R. Arshansky and L. P. Horwitz, to be published in J. Math. Phys.
6.
N. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, RI, 1986), p. 106.
7.
We thank H. Bacry for a discussion of this point.
8.
M. A. Naimark, Linear Representations of the Lorentz Group (Pergamon, New York, 1964).
9.
As we have pointed out in I, in addition to n+⩾0 there is a set of solutions with n+⩽0; since the action of the Lorentz group affects these indices only through the generators of the SU(1,1) little group, the two sequences are disconnected [the corresponding analysis based on χn+k−n* gives rise to unitarily inequivalent representations of SL(2,C)].
10.
See Ref. 6, p. 144.
11.
See Ref. 6, p. 150.
12.
See Ref. 6, p. 129.
13.
See Ref. 6, p. 133.
14.
This condition replaces the requirements of square integrability, since we do not consider these functions as elements of a Hilbert space as yet. We shall carry out such an embedding in a later section.
15.
We remark that for the solutions with n+⩽0 corresponding to χu+k−n*,Mk→−Mk in Eq. (6.7).
16.
See Ref. 6, p. 143.
17.
See Ref. 4, p. 194;
see Ref. 8, p. 104.
18.
See Ref. 4, p. 206.
19.
Higher Transcendental Functions, Vol. 2, Bateman Manuscript Project, California Institute of Technology, edited by A. Erdelyi (McGraw‐Hill, New York, 1953).
20.
See Ref. 6, p. 158.
21.
See Ref. 8, p. 144.
22.
See Ref. 6, p. 168.
23.
R. Arshansky, Ph.D. thesis, Tel Aviv University, 1987, p. 160.
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