Takayoshi Maekawa

Note on the computation formula of the boost matrices of SO(n−1,1) and continuation to the d matrices of SO(n)

The general formula of computing the boost matrices of SO(n−1,1), which is valid for the single‐ and double‐valued representations and is similar to that of Vilenkin and Wolf, is given. It is noted that a phase factor of a unit magnitude in the boost matrices must be taken into account in analytic continuation to the d matrices of SO(n), and then the formula of computing the d matrices of SO(n) is given. It is remarked that the d matrices of SO(n) are expressed in terms of those of SO(n−1).

Formula for the computation of the representation matrix elements of the group SO (n)

The formulas of expressing the infinitesimal operators of the parameter groups of SO (n) in terms of the Euler angles are given. By using these, recursion relations which are useful for calculations of the representation matrix elements of SO (n) are obtained. Expressions for the d functions with the highest and some weights are obtained explicitly, and it is shown that the d functions of SO (2j+1) or SO (2j) with special weights agree with those of SO (3) or SO (4) respectively. By using the results, a formula of computing the D−matrix elements of SO (n) are given in terms of lowering operators corresponding to those of Pang and Hecht.

Explicit form of the Haar measure of U(n) and differential operators

The Haar measure of the group U(n) is explicitly introduced using the Euler‐like parameters, and the differential operators of the first‐ and second‐parameter groups are given. The D‐matrix elements are defined through the Gel’fand and Tsetlin basis and the orthogonality and the completeness relations for the d‐matrix elements are given explicitly.

Infinite‐dimensional representations of the graded Lie algebra (Sp(4):4). Representation of the para‐Bose operators with real order of quantization

Infinite‐dimensional representations for the system of the two para‐Bose operators, which generate the graded Lie algebra (GLA) (Sp(4):4), are constructed by using the irreducible representations (discrete series) of the Lie algebra Sp(4)≊SO(3,2). It is shown that there are four kinds of the irreducible representations of the GLA’s (Sp(4):4), i.e., cases (I), (II), (III), and (IV). Case (I) is described by the three irreducible representations of Sp(4) and corresponds to the para‐Bose quantization with real order of the quantization greater than 1 and case (IV) corresponding to the ordinary Bose quantization by the two irreducible representations of Sp(4). Cases (II) and (III), which are described by the four and two irreducible representations of Sp(4), respectively, and cannot be obtained by the method of Fock space, express the representations of the graded Lie algebra (Sp(4):4).

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

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.

On the computation formulas of the SO(n−1,1) representation matrix elements

The formulas for computing the boost matrix elements are found for all classes of the unitary irreducible representations of SO(n−1,1) by defining the invariant scalar product in the space consisting of the D functions of SO(n−1) and assuming that functions with some property exist for the complementary series. The normalization constants of the bases are completely determined by requiring that the boost matrix elements in the finite transformations agree with those obtained by the method of the infinitesimal operators in the infinitesimal transformations.

On the unitary irreducible representations of SO(n,1) and U(n,1) in the scalar product with the nonintertwining operator

The irreducible representations of the noncompact groups SO(n,1) and U(n,1) are discussed in the space of functions on the compact groups SO(n) and U(n), and then it is shown that the Hermitian form scalar product with a nonintertwining operator can be introduced for each of the complementary and the discrete series of the unitary irreducible representations (UIR) of the groups SO(n,1) and U(n,1). The positive definite scalar product can be introduced for all classes except for a discrete series in the case of U(n,1).

On the Wigner and the Racah coefficients of suq(2) and suq(1,1)

An explicit expression for the Wigner coefficients coupling two positive (or negative) discrete series of the representations is obtained and the Racah coefficients for these cases are found with a single sum.

Note on the linear representations of any dimensional Lorentz group and their matrix elements

The linear representations of the group SO(n,1) are studied in the space of functions given on the maximal compact subgroup SO(n) without making use of a special parametrization for elements of the group. The scalar products, which are invariant under the transformation by any element of SO(n,1), are introduced into the space together with the action of any representation operator of SO(n,1) on the function. The computation formula for the representation matrix elements is obtained.

Note on the Explicit Form of Invariant Operators for O(n)

A complete set of invariants of O(n) is constructed explicitly and a method of deriving the corresponding invariants of O(p, q) is briefly remarked.