Unitary representations of the homogeneous Lorentz group in an O(2, 1) basis

Mukunda, N. (1968) Unitary representations of the homogeneous Lorentz group in an O(2, 1) basis Journal of Mathematical Physics, 9 (1). pp. 50-61. ISSN 0022-2488

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Official URL: http://link.aip.org/link/jmapaq/v9/i1/p50/s1

Related URL: http://dx.doi.org/10.1063/1.1664476


Unitary irreducible representations of the homogeneous Lorentz group, SO(3, 1), belonging to the principal series and containing integral angular momenta, are reduced with respect to the subgroup SO(2, 1). It is found that the representation {j0, ρ} of SO(3, 1), where j0 is a nonnegative integer and ρ a real number, contains each representation of SO(2, 1) of the continuous class (nonexceptional and integral) twice, and each of the discrete representations Dk(±) of SO(2, 1) once, for k = 1, 2, ..., j0. The latter representations are absent for j0 = 0. It is shown that the basis states of the representation Dk(±) (for k ≥ 2) lie in the domain of those generators of SO(3, 1) that are outside the SO(2, 1) subalgebra, while the states of the representations D1(±) do not lie in this domain. It is further shown that from the point of view of the nature of this domain, the representations D1(±) of SO(2, 1) are very intimately connected to the continuous class representations of SO(2, 1), and that these two discrete representations act as a bridge between the remaining discrete representations on the one hand, and the continuous class representations on the other.

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