Clebsch-Gordan problem and coefficients for the three-dimensional Lorentz group in a continuous basis. I

Journal of Mathematical Physics

Volumn 15, Issue 8, 1973, Pages 1320-1331

  Mukunda, N ^a,b   Radhakrishnan, B ^a  

^a TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

^b INDIAN INSTITUTE OF SCIENCE   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 36849100206     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1666814     Document Type: Article

References (38)

1. We follow here the notation of Ref. 1.
2. In this convention, all the nonzero matrix elements of the operators [formula omitted] are real and positive.
3. This construction of [formula omitted] using oscillator operators is quite similar to that of Holman and Biedenharn, Ref. 3.
4. The point is that eigenfunetions of [formula omitted] with integral eigenvalues have [formula omitted] those with half-odd integral eigenvalues have [formula omitted]
5. The point is that if only Eq. (2.43) were known, then since [formula omitted] and [formula omitted] (similarly [formula omitted] and [formula omitted]) support one and the the same UIR [formula omitted] in principle A could have connected these two subspaces. But in fact this does not happen.
6. The UIR’s of [formula omitted] can be labeled [formula omitted] where [formula omitted] and [formula omitted] denote the constituent UIR’s of the two commuting [formula omitted] groups of which [formula omitted] is (locally) the direct product.
7. This isolation of the angular dependences of [formula omitted] in [formula omitted] is quite similar to the fact that the Hamiltonian for a nonrelativistic particle in a centrally symmetric potential [formula omitted] when expressed in spherical polar coordinates, involves partial derivatives with respect to the spherical polar angles θ, φ only via the operator of total angular momentum squared.
8. Actually, Eqs. (4.4) and (4.5) are inadequate in the sense that in the decompositions of [formula omitted] and b into UIR’s, each UIR that appears does so twice (except [formula omitted]), and this is to be kept track of.
9. We have combined Eq. (4.14) of