Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements

Abstract

The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to the boost of SO(n,1) is completely determined by requiring the commutation relations of SO(n,1) for the differential operators of the multiplier representation and of the parameter group of SO(n). It is shown that the bases of the space, the representation matrix elements of SO(n), are classified by the group chains of the first and the second parameter groups of SO(n), whose differential operators commute with each other, and the characteristic numbers of SO(n,1) are the same as those of the first parameter group of SO(n−1) and a complex number appearing in the multiplier. By using the scalar product defined in the space, the matrix elements for the differential operators and the computation formulas for the representation corresponding to the boost of SO(n,1) are given for all unitary representations of SO(n,1) and useful formulas containing the d matrix elements of SO(n) are obtained. By making use of these results, even for the nonunitary representation of SO(n,1) the matrix elements for the differential operators and the computation formula for the representation corresponding to the boost are obtained by defining the matrix elements with respect to the bases of the space. It is also pointed out that the unitary representations (the complementary series) corresponding to some value of the parameter, which appear in the classification using only the matrix elements of the generators, should not be included in our classification table because of divergence of the normalization integral. The continuation to SO(n+1) and the contraction to ISO(n) from the principal series are discussed.